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Bài giảng Vật lý điện tử__em-outreach | Môn Vật lý điện tử | Trường Đại học Bách Khoa Hà Nội. Tài liệu gồm 20 trang giúp bạn tham khảo ôn tập đạt kết quả cao trong kỳ thi sắp tới. Mời bạn đọc đón xem.

I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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I.THEMICROSCOPE
I.A.PRINCIPLESOFTHETRANSMISSIONELECTRONMICROSCOPE(TEM)
I.A.1.OriginoftheTransmissionElectronMicroscope
I.A.2.ComparisonofLight(LM)andElectronMicroscopes(Fig.I.1)
a.Similarities(Arrangementandfunctionofcomponentsaresimilar)
1)Illuminationsystem:producesrequiredradiationanddirectsitontothespecimen.Consistsofasource,whichemitstheradiation,anda
condenserlens,whichfocusestheilluminatingbeam(allowingvariationsofintensitytobemade)onthespecimen.
2)Specimenstage:situatedbetweentheilluminationandimagingsystems.
3)Imagingsystem:Lenseswhichtogetherproducethefinalmagnifiedimageofthespecimen.Consistsofi)anobjectivelenswhichfocusesthe
beamafteritpassesthroughthespecimenandformsanintermediateimageofthespecimenandii)theprojectorlens(es)whichmagnifiesa
portionoftheintermediateimagetoformthefinalimage.
4)Imagerecordingsystem:Convertstheradiationintoapermanentimage(typicallyonaphotographicemulsion)thatcanbeviewed.
b.Differences
1)Opticallensesaregenerallymadeofglasswithfixedfocallengthswhereasmagneticlensesareconstructedwithferromagneticmaterialsand
windingsofcopperwireproducingafocallengthwhichcanbechangedbyvaryingthecurrentthroughthecoil.
2)MagnificationintheLMisgenerallychangedbyswitchingbetweendifferentpowerobjectivelensesmountedonarotatingturretabovethe
specimen.Itcanalsobechangedifoculars(eyepieces)ofdifferentpowerareused.IntheTEMthemagnification(focallength)oftheobjective
remainsfixedwhilethefocallengthoftheprojectorlensischangedtovarymagnification.
3)TheLMhasasmalldepthoffield,thusdifferentfocallevelscanbeseeninthespecimen.Thelarge(relative)depthoffieldintheTEMmeans
thattheentire(thin)specimenisinfocussimultaneously.
4)Mechanismsofimageformationvary(phaseandamplitudecontrast).
5)TEMsaregenerallyconstructedwiththeradiationsourceatthetopoftheinstrument:thesourceisgenerallysituatedatthebottomofLMs.
6)TEMisoperatedathighvacuum(sincethemeanfreepathofelectronsinairisverysmall)somostspecimens(biological)mustbedehydrated
(i.e.dead!!).
7)TEMspecimens(biological)arerapidlydamagedbytheelectronbeam.
8)TEMscanachievehighermagnificationandbetterresolutionthanLMs.
9)Pricetag!!!(100xmorethanLM)
I.A.3.Photons/Electrons
a.Dualconceptofwaveandparticle(Fig.I.4)
Lighthaspropertiesbothofaparticleandawave.Thisdualnatureisrequiredtosatisfactorilyexplaintheresultsofvariousphysicalexperiments.
Thediffractionoflight(bendingaroundcorners)illustratesthewavenature.TheWAVETHEORYisbasedonthestatisticalnatureofeventsand
haslittlemeaningwithrespecttothebehaviorofsingleparticles.ThewavetheorywasdevelopedandexpoundedbyHuygens(16291695)and
Hooke(16381703).TheCORPUSCULARTHEORYwasproposedbyNewton(16421727)andbecamethemoreacceptedtheoryevenafter
demonstrationofdiffractionbyYoung(17731829)andinterferencebyFresnel(17881827).Atabout1850,thewavetheorycamebackinto
favoruntilabout1900whenmoreevidenceforthecorpusculartheorywasdiscovered.
QUANTUMTHEORY(PlanckandEinstein)providesthebasisforexplainingthephenomenaofINTERFERENCE,DIFFRACTIONandthe
PHOTOELECTRICeffect(lightfallingoncertainmetalscausethemtoemitelectrons)andthusformsacommonbasisforexplainingthenatureof
actionoflight.Thetransferofenergybetweenlightandmatteroccursonlyindiscretequantitiesproportionaltothefrequencyofthe
lightwave.
E=hJ
where
E=energyofphoton(joules)
h=Planck'sconstant(6.624
10
34
joulesec)
J=frequency(cycles/sec)
b.Electronvelocityandwavelength
Therelationbetweenthewavelength(l )ofaparticleofmass,m,movingatavelocity,v,isgivenbytheDeBrogliewaveequation:
(1)
Anelectronofchargee(1.6
10
19
coulomb),andmassm(9.11
10
28
gm),whenpassingthroughapotentialdifferenceofVvolts(expressedin
joules/coulomb),hasakineticenergy:
(2)
Solvingforvelocity:
(3)
substitutingintotheDeBroglieequation(1):
(4)
Since1joule=10
7
dyne
cm=10
7
cm
2
gm/sec
2
(5)
Thus,forexample,ifV=60,000volts,l =0.005nm.
Fromequation(3)theelectronvelocitycanbecalculatedifVisknown:
(6)
Thefollowingtableillustratesthat,athighvoltage,electronvelocityiscomparabletothespeedoflightinavacuum(c=3
10
10
cm/sec).
Theequationbreaksdownwhenthevelocityoftheelectronapproachesthespeedoflightbecausearelativisticcorrectionmustbemadeforthe
valueofthemasswhere:
(7)
Therelationbetweenl andVismorecorrectlygivenby:(seeHall(1966)pp.3334)
(8)
Thefollowingtableisobtainedwhenrelativityeffectsareincluded:
c.Interference/diffraction/coherence
Anideallenssystemobtainsanexactimageoftheobject(eachpointfaithfullyreproduced).Thephenomenaofdiffractionmakesthis
unattainable(Figs.I.5,I.6).
Diffractionphenomenainvolvesthebendingofthepathofradiationpassingclosetoanobstacle(Fig.I.6).Thisresultsinaspreadingofthe
radiationintotheregionbehindtheobstructionthatthewavespassed.Thediffractionatedgescontributestothecontrastatwhichanedgecan
beobserved.Diffractionalsolimitstheresolvingpowerofthemicroscopesincetheimagepointproducedbyalensisadiffractionimageofthe
openingofthelensortheaperturerestrictingtheeffectiveopeningofthelens(Fig.I.5).
Ifthelightsourceandtheplaneatwhichthediffractionpatternisobservedareatfinitedistancesfromtheedge,thephenomenoniscalled
Fresneldiffraction.Thepatternisdescribedasduetointerferencebetweenthenondiffractedlightandawaveoflightdiffractedattheedge.
Theresultingsuperpositiongivesrisetoaseriesofdiffractionfringesorientedparalleltotheedgeandrepresentingperiodicallyvaryingbrightness,
maximaandminima.(Figs.I.7,1.8,I.9).
Coherence:Aprerequisiteforinterferenceisasuperpositionofwavesystemswhosephasedifferenceremainsconstantintime.Twobeams
arecoherentif,whencombined,theyproduceaninterferencepattern.Thebeamsareincoherentwhentheyareincapableofproducingan
interferencepattern.Twobeamsoflightfromselfluminoussourcesareincoherent.Iflightfromthetwosourcesfallsonascreen,theresultant
intensityissimplythesumofthetwointensitieswhichwouldoccurfromeachsourceseparately(Figs.I.10,I.11).Inpracticeanemittingsource
hasfiniteextentandeachpointofthesourcecanbeconsideredtogeneratelight.EachsourcegivesrisetoasystemofFresnelfringesatthe
edge.Thesuperpositionofthesefringesystemsisfairlygoodforthefirstmaximaandminimabutfartherawayfromtheedgeshadowtheoverlap
ofthefringepatternsbecomessufficientlyrandomtomakethefringesdisappear.Thenatureofwaves,phase,amplitude,andinterferenceare
illustratedinFigs.I.12andI.13.
d.Resolution
1)Definitions:
RESOLUTION:abilitytodistinguishcloselyspacedpointsasseparatepoints.
RESOLUTIONLIMIT:smallestseparationofpointswhichcanberecognizedasdistinct.
RESOLVINGPOWER:resolutionachievedbyaparticularinstrumentunderoptimumviewingconditions.
2)Distinctionbetweenresolutionandresolvingpower:
Notethedistinctionbetweenresolutionandresolvingpower.Resolvingpowerisapropertyoftheinstrumentandisaquantitythatmaybe
estimatedontheoreticalgrounds.Resolutionisequaltoorpoorerthantheresolvingpowerandisthequantityobservedunderanygivenset
ofexperimentalconditions.IntheTEM,especiallywithbiologicalsamples,theresolutionachievedmaybeconsiderablyinferiortothetheoretical
resolvingpoweroftheinstrument.
Microscopyisthescienceofseeingtheverysmall.Underidealconditions,theeyeresolvesabout1minuteofarc(=1/60degree=2.9x10
4
radian;recallthereare2p radiansin360°)andsinceitcanfocusdowntoabout250mm,thesmallestobjectwecanresolveisabout0.07mm
(70mm)(Fig.I.14).Thislimitisrelatedtothesizeofthereceptorsintheretinaoftheeye.Thefunctionofamicroscopeistomagnifytheimage
fallingontheretina(Fig.I.15).Theadvantageoflightandelectronmicroscopesisthattheyeffectivelygettheobjectclosertotheeyesoa
magnifiedimageisobtainedandmoredetailcanbediscerned.
TENNISBALLANALOGY:Eyecanresolve3cmat100meters,thusatennisballisclearlyvisible.Butifthetennisballisheldupagainstawhite
background,thevisibilitydecreases(becauseofthedecreaseincontrast).
3)Abbecriteriaofresolution:
Thefundamentalnatureoflightposeslimitsonthedetailthatcanberesolved(Fig.I.16).Abbe(1893)showedthatthesmallestresolvable
distanceisabout1/2thewavelengthoflightused.Thus,1/2thewavelengthoftheradiationusedistheultimateresolvingpowerofany
instrument.Thislimitstheusablemagnificationofopticalmicroscopesto<1000X.Atfirstitwasthoughtthatxraymicroscopeswouldbeuseful
buttherefractiveindexofsubstancesforxraysisnearly=1,thusrefractinglensescannotbemadeforxrays,andconsequently,xrayscannot
beeasilyfocusedtoformimages.
4)Magnificationlimits:
Themaximummagnificationofaninstrumentislimitedaccordingtothefollowingrelationship:
Thus,fortheLM,witharesolvingpowerofapproximately0.25µm,themaximum(useful)magnificationisabout250µm/0.25µm=1000X.Thevalue
usedfortheresolvingpoweroftheeyeinthisexample(250µm)representsamorerealisticviewingcondition.Anymagnificationabovethevalue
givenbytheaboveformularepresentsemptymagnification,sincesuchmagnificationleadstonomoreusefulinformationbutratheramagnified
blur.
AccordingtotheAbbecriteria,at60,000volts,theTEMshouldhavearesolvingpowerofabout0.0025nm.Thisshouldallowamaximumuseful
magnificationofabout100milliontimes!!!Inpractice,themaximumusefulmagnificationoftheTEMoperatedat60kVislimitedtomuchlessthan
1,000,000X.Thus,althoughtheLMnearlyobeystheAbbecriteria,theTEMfallsshortbyaconsiderableamount.Themainlimitingfactorinthe
TEM,withrespecttoachievingthetheoreticalresolvingpoweroftheinstrument,concernsthenatureoftheimaginglensesandtheprocessof
imageformation.
5)Rayleighcriteria(practicalbutarbitrary):(Fig.I.17).
Itistheabilityofmicroscopestomakeobjectpointswhichareclosetogetherappearintheimageasseparatepoints.Anideallenstakeseach
objectpointandrepresentsitexactlyasapointintheimage.Areallenstakeseachobjectpointandspreadsitoutintoacirculardisk(Airy
disk)intheimageplanewhosediameterdependsontheangularapertureofthelens.
Theshortestdistancebetween2disksatwhichthetwodisksappearpartiallyseparatedcorrespondstoabout1/2thewidthofthedisks.The
distance,d,inobjectspaceisgivenby:
d=0.61l /n
sina
where
l =wavelenghtoftheradiation
n=refractiveindexofthemedia
a=semiangularapertureofthelens
NOTE:n
sina=thenumericalaperture(N.A.)ofthelens.
Thus,tomaximizeresolvingpower,l mustbedecreased,nincreased,oraincreased.Recallthatweareconcernedatthemomentwithan
aberrationfreeopticalsystem.Thevaluefortheconstant(0.61)iscontroversialbecauseitdependsonthecoherenceoftheradiationanda
criteriaofvisibility.
FortheLM,usingoilimmersionoptics(n=1.5),sina=0.87,andvioletlight(l =400nm),d=0.2µm.Theonlywaytoimprovetheresolutionisto
uselightofshorterwavelengthsinceN.A.cannotbeincreasedbeyond~1.5.FortheTEM,n=1(vacuum),sina=10
2
andl =0.005nmfor60kV
electrons,thusd=0.3nm.
I.A.4.Optics(LensTheory)
a.Basiclawsofclassicalgeometricaloptics
1)Rectilinearpropagationoflightwhenn(refractiveindex)isconstant.
2)Lawofreflection:
3)Lawofrefraction(Snell'sLaw):
4)Independenceofrays.Theassumptionismadethatlightraystravelindependentlythroughspace.
Theselawsholdforelectrons,except#4,ifthecurrentdensityistoohighwhennegativechargedelectronscaninterfere.
b.Classicalvs.electronoptics
1)Classicaloptics:Therefractiveindexchangesabruptlyatasurfaceandisconstantbetweenthesurfaces.Therefractionoflightatsurfaces
separatingmediaofdifferentrefractiveindicesmakesitpossibletoconstructimaginglenses.Glasssurfacescanbeshaped.
2)Electronoptics:Here,changesintherefractiveindexaregradualsoraysarecontinuouscurvesratherthanbrokenstraightlines.Refraction
ofelectronsmustbeaccomplishedbyfieldsinspacearoundchargedelectrodesorsolenoids,andthesefieldscanassumeonlycertaindistributions
consistentwithfieldtheory.
c.Geometricalandphysicaloptics
Thefundamentalprinciplesofopticsgovernthedesignandoperationofboththelightandelectronmicroscopes.Thebasicopticalprinciples
involvingtheuseofrefractileelementsorlensesinordertoformmagnifiedimagesareidenticalinboththeLMandTEM.TheTEMdiffersfromthe
LMonlyintheradiationitusesandinthewayinwhichtheradiationisbentorrefracted.
Geometricalopticsdealswiththestudyofthepathsfollowedby'rays'oflightorelectronsthroughlensesandapertures,andthegeometrical
constructionsusedtofindtherelativepositionsandsizesofobjectsandtheirimages.Arayoflightorelectronsisdefinedasaninfinitelythin
pencilorbeam.Physicalopticsshowsthatthisanabstractionandcannotphysicallyexistbecauseof'diffraction'whichdealswiththewave
natureoflightandelectrons.Allresultsobtainedingeometricalopticscanbederivedfromtheprinciplesofphysicaloptics,alongwithother
phenomenasuchasinterferenceanddiffractionwhicharenotexplicableinsimplegeometricalterms.
d.Idealversesreallenses:
Lensesareusedtobendraysoflightorelectronssotheyaredeflectedinapredictablewayfromtheiroriginalpaths.Thepropertiesofanideal
lens,possessinganaxisofrotationalsymmetryare:
1)Eachrayofthebundleofrayswhichpassesfromanobjectpointwillberefractedbytheideallenstomeetinoneimagepoint.
2)Raysoriginatingfrompointswhichlieonaplaneperpendiculartotheaxis,mustbeimagedinaplanewhichisalsoperpendiculartotheaxis.
3)Theimageappearsliketheobjectirrespectiveofthemagnification,sotherelativelineardimensionsoftheobjectarepreservedintheimage.
Inpractice,theimagingbyanyreallensdoesnotcorrespondtothatoftheideallensowingtothefactthatanobjectpointisrepresentedbya
diffractionimage(Airydisc)ofthelensopeningortheapertureusedforrestrictingtheeffectiveopeningofthelens.Thisisaresultofthewave
likepropertiesoflight.Lensaberrationsalsocontributetomoreorlesspronounceddeviationsfromthepropertiesoftheideallens.
Thesinglerefractingsurfaceofsphericalcurvatureisthefundamentalunitoffocusingactionbyglasslenses.Sphericalrefractingsurfacesactas
lensesforparaxialrayswhicharethoseraysthatpassclosetotheprincipalaxisofthelens.RayswithlargeangleswillNOTobeyideallens
action.
Afundamentaldifferencebetweenlightandmagneticlensesisthattheelectronbeamdoesnotchangeinforwardvelocityasitpassesthrough
themagneticfield(lightraysslowdownwhenpassingintoamediumofhigherrefractiveindex).Refractioniscontinuouswithelectronswhenthey
areinthemagneticfield:lightisrefractedonlyattheinterfacebetweenmediaofdifferingrefractiveindex.Theelectronsalsofollowspiral
trajectoriesthroughthemagneticfield(seealsoSec.I.A.5.c(Magneticfieldsandmagneticlenses)
e.Raydiagrams:(Figs.I.20I.28)
Themethodofconstructionofraydiagramsisbasedonthreesimpleprinciples:
1)Allraysenteringthelensparalleltotheaxisarebroughttoacommonpointontheaxis,thefocalpoint.
2)Allrayspassingthroughthegeometricalcenterofthelensareundeviatedandpassstraighton,nomatterfromwhichdirectiontheycome.
3)Principleofreversibility:ifthedirectionofarayisreversedinanysystemtherayexactlyretracesitspaththroughthesystem.Thisapplies
onlytothelocationoflightpathsandnottotheintensityofthelight.
Theaboveprinciplesarebasedontheassumptionsthatwearedealingwithathinlensandconcernedwiththepathsofparaxialrays.The
standardconventionistodrawdiagramswithraysthattravelfromlefttoright.Theobjectistotheleft(infront)ofthelensandtheimageisto
theright(behind)ofthelens.
f.Definitions:
Realimage:oneatwhichlightraysphysicallyreunite,sothataphotographicplateplacedatthepositionofarealimageisexposed.
VirtualImage:onefromwhichlightraysappeartodiverge;raysarenotinfactconcentratedatthepositionofavirtualimage,sothata
photographicplateplacedatthepositionoftheimageisnotexposed(byfocusedrays).Placinganopticalsystemsuchastheeyebehindthe
lens,willenablethedivergentraystobefocusedtoformarealimage.TheintermediatelensofanTEMissometimesusedthiswayinorderto
reducethefinalsizeoftherealimageformedbytheprojectorlens(es).
Converging(positive)lens:bendsraystowardtheaxis.Ithasapositivefocallength.Formsarealinvertedimageofanobjectplacedtothe
leftofthefirstfocalpointandanerectvirtualimageofanobjectplacedbetweenthefirstfocalpointandthelens.
Diverging(negative)lens:bendsthelightraysawayfromtheaxis.Ithasanegativefocallength.Anobjectplacedanywheretotheleftofa
diverginglensresultsinanerectvirtualimage.Itisnotpossibletoconstructanegativemagneticlensalthoughnegativeelectrostaticlensescan
bemade.
g.Lensformula(thinlensequation):
where
f=focallengthofthethinlens(sameradiusofcurvatureforboth
sphericalsurfaces)
o=distanceofobjectfromlens(positivetotheleft)
i=distanceofimagefromthelens(positivetotheright)
NOTE:Foravirtualimage,ihasanegativevalue.
h.Magnification:
Foraconverginglens,iftheobjectismorethantwicethefocallengthfromthelens,thentheimageformedisreal,inverted,andsmallerinsize
thantheobject(M<1).Whentheobjectisatadistance=2f,theimageandobjectarethesamesize(M=1);whenitisbetweenfand2f,
theimageislargerthantheobject(M>1),andwhenitis<f,theimageisvirtual,erect,andlargerthantheobject(M>1).
i.Angularapertureofthelens(2a)(Fig.I.29)
Theaperturedeterminesthetotalamountofradiationarrivingfromtheobjectwhichcanbefocusedto
formanimage.Theaperturethuscontrolstheabilityofthelenstogatherinformationabouttheobject.
Thisdependsontheangleoftheconeofraysitisabletoacceptfromtheobject.Bringinganobject
closertotheeyeincreasestheangularaperture,butthereisalimittotheclosenessthattheobjectcan
bebroughttotheeye(~25cmcorrespondingtoanangleofabout0.9°fora4mmexitpupildiameterof
theeyelens;atypicalLMwithanoilimmersionobjectivelenshas2aof~175°).
j.Simplevs.compoundmicroscope(Figs.I.30I.32)
Inprinciple,arealimageofanydesiredmagnificationcanbeobtainedfromasinglepositivelens,butin
practicethisiscumbersomebecauseofthelonglensimagedistance.Oneormorelensescanbeusedto
magnifytheimageinstages(totalmagnificationequalingtheproductofthemagnificationsofeachlens).
Theimageformedbyonelensconstitutestheobjectforthesubsequentlens,whetherornotareal
intermediateimageisformed.
Comparisonofoneversestwostagemagnification:
Thefollowingdescriptionillustrateshowdifferentpathlengthsarerequiredtoachieveamagnificationof10,000Xusingeitheroneortwolenses
withf=2.0cm.
OneStageSystem:
TwoStageSystem:
Assumebothlenseshavef=2.0cmandarearrangedsoeachgives100Xmagnification.
Thus,M=100=x
i
/x
o
foreachstage.
FirstStage:
x
o
=2.02cm
x
i
=202.00cm
Therealimageformedinthefirststagebecomestheobjectforthesecondlens.
SecondStage:x
o
,x
i
sameasinthefirststage.
Totallengthofsystem
=lengthoffirststage+lengthofsecondstage
=(x
o
+x
i
)+(x
o
+x
i
)
=2(x
o
+x
i
)
=2(2.02+202.00)=408.04cm(4.08meters)
k.Problemset
Thefollowingproblemsareforyourownedificationtotesthowwellyouunderstandthebasicsoflensoptics.
1.Foreachsituationbelow,drawanaccurate(usegraphpaper)raydiagramshowingthepathof(atleasttwo)raysfromtheobjecttoimage.
Indicatewhethertheimageisrealorvirtualandspecifythedistanceoftheimagefromthelensandthemagnificationoftheimagewithrespect
totheobject.Ineachexample,thelensisconverging,withidenticalfrontandbackfocalpoints.Distancesareinarbitraryunits.Letf=lens
focallength,x
o
=distanceofobjectinfrontofthelens,andx
i
=distanceofimagebehindthelens.Theobjectmaybeanysize.
2.Inthefollowingexamples,therearetwoconverginglensesinsuccession.Drawraydiagramsshowingtheformationofboththeintermediate
andfinalimagesandgiveappropriateinformationaboutthenature(real/virtual),positions,andsizesoftheintermediateandfinalimages.What
isthemagnificationateachstageofimageformationandwhatisthemagnificationofthefinalimage?Thedistancefromthecenteroflens1(L1)
tothecenteroflens2(L2)equalsD.
I.A.5.ElectronOptics/ElectronLenses
a.Electronemission
Innershellelectronsinmetalatomsshieldtheelectricfieldofthenucleusthroughtherepulsiveforcestheyexertontheoutershell(valence)
electrons.Thustheattractionbetweenthevalenceelectronsandthenucleusisreduced.Metalatomsarecharacterizedashavingtwoloosely
boundvalenceelectronswhichmigratefreely(thisiswhymetalsaregoodelectricalconductors)andcanescapefromthemetalcompletelyif
sufficientadditionalenergyisimpartedtothem.Asthetemperatureofametalisincreased,thekineticenergyoftheelectronsincreasesbecause
ofincreasedthermalvibrationsofthemetalions,whichcollidemorefrequentlywiththeelectrons.Thermionicemissionisthetermusedto
describetheprocessbywhichthermalenergyissuppliedtolooselyboundelectronsinordertoformasourceofelectrons.
Atroomtemperatureelectronsareeffectivelypreventedfromescapingthesurfaceofthemetalowingtotheattractiveforceofthepositively
chargedions.Asthetemperatureisincreasedsomeelectronsacquiresufficientenergytoovercometheattractionandleavethemetal
temporarily.Metal,shapedasathinwire,caneasilybeheatedbypassinganelectriccurrentthroughit.Sincethemetalsurfacebecomes
positivelycharged,acertainlevelofenergy(workfunction)mustbesuppliedtoallowelectronstoescapefromthesurface.Eachmetalhasa
characteristicworkfunction.Tungsten,withalowworkfunction,emitsmoreelectronsthanmetalswithhigherworkfunctions(seealsopp.2930
andFig.I.59).
Ifastrongelectrostaticfieldisappliedinavacuumbetweenthewire(givenanegative,cathode,potential)andananode,theelectricfieldwill
causeelectronstoaccelerateawayfromthewiretowardstheanodesurface(Fig.I.33).Thespeedoftheelectronsdependsonthestrengthof
theelectrostaticfield(voltage)betweenthecathodeandanode(equation(3),Sec.I.A.3.b)Thenumberofelectronswhichleavethewire
dependsonthetemperaturetowhichthewireisheated,whichdependsonhowmuchfilamentcurrentpassesthroughthewire.
A"V"shapedwirewillhavethehighesttemperatureatthetip.Electronswithdrawnfromthefilamenttipcarryelectricchargestotheanode.This
electriccurrent,whichflowsbetweenthefilamentandtheanode,iscalledthebeamcurrent.
1)Electricfield/Equipotentials
Anelectricallychargedobjecthasassociatedwithitanelectricfield.Thus,anelectricallychargedparticle,whenbroughtnearachargedobject,
isinfluencedbyanelectricalforceinthevicinityoftheobject.Theforceisdirectedtowardthechargedobjectifthechargesareofopposite
signsandawayfromtheobjectiftheyareofsimilarsign.DEFINITION:Thedirectionofanelectricfieldisdefinedasthedirectionoftheforce
actingonapositivecharge.(Figs.I.3435)
Alongthelinesofforceconnectingthetwocharges,theelectricpotentialwillchangegraduallybetweentheextremevaluesrepresentedbythe
twocharges.DEFINITION:Equipotentiallinesdefinethepointsalongthelinesofforcewithidenticalelectricalpotential.Theseequipotential
surfacesarealwaysorientedperpendiculartothelinesofforce.Thechangesintheelectricpotentialaregradualinspace.
Electronswhichenterafieldbetweentwoparallelplatesinadirectionparalleltotheplatesareaffectedbytheforcedirectedperpendicularto
theplates(FigI.36).Theelectronswillbeattractedtowardthepositiveplate.Thepathchangesinaseriesofgradualstepsattheequipotential
surfaces.
FigureI.37showshowtheelectronpathis"refracted"attheequipotentialsurface.TheresultisfundamentallythesameasthatgivenbySnell's
Lawofrefractioninlightoptics.Aconsequenceofthisisthatasphericallycurvedequipotentialsurfaceexhibitsthepropertiesofalens.
FiguresI.38andI.39showhowbothpositive(converging)andnegative(diverging)electrostaticlensescanbeformed.Thisfeatureof
electrostaticlensesdiffersfromelectromagneticlenseswhichcanonlyactasconverginglenses.
2)Advantages/Disadvantagesofelectronlenses:
Thefactthattherefractiveindexdoesnotchangeabruptlyinelectronlenseshasoneadvantageinthattherearenotroublesomereflectionsat
equipotentialsasatglassinterfaces.Thereisaseriousdisadvantageinthatequipotentialscannotbeshapedandcombinedinarbitraryfashionto
correctforchromaticaberrationandothererrors.
Althoughelectronmicroscopeswhichemployelectrostaticlenseshavebeenmade,mostmicroscopesuseelectromagneticlenses.Amajorreason
isthatelectrostaticlensesaremoresensitivetothequalityofthevacuumandcleanlinessofthecomponentsthanareelectromagneticlenses.
Somelensaberrationsaremoresevereforelectrostaticlensescomparedtoelectromagneticlenses.Electrostaticlensesrequireverypowerful
electrostaticfieldswhichcanleadtoelectricalbreakdownor"arcingover"insidethecolumn,especiallyunderpoorvacuumconditions.Forthis
reason,electrostaticlensescannotbemadewithfocallengthsasshortasmagneticlenses.
b.Electrostaticlens
Abasicunderstandingofelectrostaticlensesisimportantfortwomainreasons:i)theelectrongunuseselectrostaticlensactiontoformthe
primarybeamsource,andii)itisquitecommonforachargetodeveloponthenonconductingcontaminationwhichmayaccumulateonphysical
apertures(suchastheobjectiveaperture)andtransformthemintoweakelectrostaticlenseswhichcandistorttheelectronimage.
3)Propertiesofelectrostaticlenses:
a)Anyaxiallysymmetricalelectrostaticfieldhasthepropertiesofalensforraysconfinedtotheparaxialregion.Alltheideallensformulasapply
toelectrostaticlenses.
b)Forelectronlenses,replaceÃ(f)forrefractiveindexinthelensequations(f=valueofthepotentialontheaxis).Intheimageformingsystem
ofmostTEMs,fisthesameonbothsidesofthelens(SeeSec.I.A.5.c).
c)Ifboundedbyregionswherefisconstant,anelectrostaticlensisalwaysconvergent.
c.Magneticfieldsandmagneticlenses
1)Magneticfield
Anelectriccurrentpassingthroughaconductorgivesrisetoamagneticfield.TheconventionisthatNisthedirectioninwhichthelinesofthe
magneticfieldpoint(Fig.I.40).Themagneticfluxisthetotalnumberoflinesandthefluxdensityisthenumberoflinesperunitareaofasurface.
UsetheRIGHThandruletodeterminethedirectionofthefield(thumbpointstowardcurrentdirectionandfingerscurlindirectionoffield.NOTE:
Byconvention,thedirectionofelectronflowisoppositetothatofcurrentflow.
Fluxdensitydependsonthepropertiesofthematerialsurroundingtheconductor.Ironinducesahigherfluxdensitythanairoravacuum.The
propertyofthematerialwhichaffectsthefluxdensityiscalledthepermeability,m,ofthematerial.Forairandvacuum,m=1.0.For
ferromagneticmaterialsitcanbeaslargeasseveralhundredthousand.
Iftheconductorhastheshapeofacircularloop,thelinesofforceformcirclesaroundtheloop.Thefluxdensityisgreatestatthecenterofthe
loop.Themagnetinthecenteroftheloopisorientedperpendiculartotheplaneoftheloopifthecurrentthroughtheloopissufficientlystrongto
eliminatetheinfluenceoftheearth'sfield.ThesideoftheloopatwhichthelinesofforceleavetheloopistheNorth(N)poleofthemagnet(Figs.
I.41,I.42).
Ifthewireiswoundinseveralturnsaroundacylindricalsurface(solenoid),eachturnwillcontributetotheinducedmagneticfield.Theflux
densityinthecenterofthecoilisgivenby:
Inthecaseofairandnonmagneticmaterials,µ=1.0andB=H.
Thepermeabilityofirondependsonthefieldstrength,H,anddecreasestounityathighfieldstrengthorwhenthefluxdensity,B=H(Fig.I.43).
Thehighpermeabilityofironisduetotheinducedmagneticfieldorientingmicroscopiccrystalregionsactingastinymagnetsintheiron.Allthese
tinymagnetsaddtheirmagneticfieldstotheinducedfield(Fig.I.44).Whenallmicromagnetsarenearlyoriented,theironwillaffecttheflux
densitytoadecreasingamountsincethereorientationofmicromagnetsisnearingcompletion.Thus,thepermeabilityofironathighfieldstrengths
approachesthatofemptyspace.Atthispointtheironreachesmagneticsaturation.
2)Hysteresis
Thestrengthofthelensdependstosomeextentonthepreviousmagnetichistoryof
thelens.Whenthecurrentinalensisreduced,thedecreaseinmagnetizationdoes
notretracethesamepathobtainedwhenthecurrentwasincreased(Fig.I.45).
Inductionofmagnetizationinvolvesaphysicalmovementwithinthemagnetized
material,requiringtheovercomeofacertaindegreeofinertia.Asaresult,
magnetizationtendstolagbehindthemagnetizingforceapplied.Inducedmagnetic
fluxcanonlybereturnedtozerobyapplicationofacurrentintheoppositedirection.
Aconsequenceofhysteresisisthatthelevelofcurrentusedtoenergizeamagnetic
lensDOESNOTpreciselyspecifythelensstrength(i.e.focallength).
NormalizationofTEMlensesisaccomplishedbyreducingthelenscurrenttozero
somepredeterminednumberoftimes.Hysteresismayalsobeminimizedbytakinga
lenstosaturationandthenreturningittotheworkingcurrentwithoutovershooting.
Whenthefieldstrengthisreducedtozero,somemagnetizationstillremainsintheiron
(residualmagnetizationorremanence).Anadvantageofsoftironisthefactthat,
whenusedinanelectromagnet,hysteresisislow.
Introductionofpiecesofironinamagneticfielddrasticallyaffectsthefluxdensity.
Magneticmaterialhasashieldingeffect,theeffectbeinggreaterthegreaterthe
permeability.Permalloy(mmetal)hasamaximumpermeabilityof80,000140,000
comparedtoironusedintransformers(~7000).mmetalpermeabilityislimitedtolower
fieldstrengthsthanthatatwhichironstillretainshighpermeability.Thismetalis
thereforeusefulforshieldingtheTEMfromexternalmagneticfields.
d.Theelectromagneticlens
1)Lensdesigndevelopment(Figs.I.46I.48)
Theefficiencyofthemagneticfieldproducedbyashortsolenoidwasfirstimproved
byencasingtheenergizingcoilinasheathofsoftironwhichhasthepropertyofconcentratingthelinesofforceinamagneticfieldandthus
becomingmagnetizedbyinduction.Inthiswayamuchmorepowerfulaxialmagneticfieldisobtainedforthesameamountofcurrentflowing
throughthesolenoid.Furtherdevelopmentinvolvedencasingtheentirecoilwithsoftironexceptatanarrowannulargapintheinsideofthecoil.
Thisproducesagreaterconcentrationofthemagneticfieldalongashortaxialdistance.Toachieveshorterfocallengthlenses(andobtaingreater
magnifications)asoftironpolepiecewithanopenaxialborewasintroducedatthepositionoftheannulargap.
MagneticlensesusedinTEMsarealwaysconstructedwithanironcircuittoproduceahighfieldstrengthacrossashortgap.Themagneticfields
forTEMlensesareintherangeof10,00020,000gauss.
2)Forcesactingonacurrentinamagneticfield
Theforceonanelectroninamagneticfieldisalwaysatrightanglestothevelocityandthedirectionofthefield(Figs.I.49,I.50,andI.52).The
fieldonlyactsonthevelocitycomponentwhichisdirectedperpendiculartothelinesofforce.Usethelefthandrule(Fig.I.51:firstfingerfor
fielddirection,middlefingerforcurrentdirection,andthumbfordirectionofforce).Rayspassingthroughthelensareturnedthroughanangle
whichdoesNOTdependonthedistanceoftheraysfromtheaxis.Allelectronscontainedinagivenmeridionalplanebeforeenteringthefieldare
containedinarotatingmeridionalplaneastheypassthroughthelens,andthentheyleavethelenscoplanar.
Whenelectronsenterthelenstheyencounterasidewaysforcewhichcausestheelectrontorotate
asitcontinuesthroughthelens(Figs.I.53I.57).Sincetheradialcomponentofthemagneticfield
reversesafterthecenterofthelens,therotationalvelocitysetupinthefirsthalfofthelensis
countered.Theelectronenteredthelenswithoutangularmomentumabouttheaxisandleaves
withoutangularmomentum.Theneteffectisadeflectiontowardtheaxis,whichitmustcrossatthe
focalpointf
2
.Theanglebetweentheobjectvectorandtheimagevectoris180°+q,whereas,for
glasslensesandelectrostaticlenses,theanglebetweenarealobjectandtheimageisexactly180°.
Sincetheradialforceisdirectedtowardtheaxis,thelensisconvergentnomatterwhatthedirection
ofthefield.
3)Propertiesofamagneticlens:
Anyaxiallysymmetricmagneticfieldhasthepropertiesofanideallens.Alltheformulasfortheideallensmaybeapplied.
Magneticlensesarealwaysconvergent.Theconventional,axiallysymmetriclensisalwaysboundedbyregionswhicharefieldfree,the
consequencebeingthatthenetactionofelectronlensesisinevitablyconvergent.Limitedregionsmaybedivergentbutnotthelensasawhole.
Theseriousconsequenceofthisisthatneithersphericalorchromaticaberrationscanbecorrectedasisdoneinlightopticsbytheusedoublets
ofpositiveandnegativelenses.
Intheabsenceofelectrostaticfields,therefractiveindexisthesameinobjectandimagespace,thereforef
1
=f
2
.
Electronstravelingthroughaxiallysymmetricfieldsexperienceaspiraltrajectoryofdiminishingradius.Theimagevectorisatanangle180°+qto
theobjectvector.
Thedeflectionoftheelectrontowardstheaxismeansthatanelectronenteringthelensparalleltoitsaxiswillcrosstheaxisafterhavingpassed
thelens.Thedeflectionwillincreasewiththedistancefromtheaxis.Thus,abeamofelectronsinparallelpathsparalleltotheaxisofthelenswill
befocusedtoanimagepointontheaxiswhichrepresentsthesecond(back)focalpointofthelens(f
2
).Notethatmagneticlensesarehighly
inefficientinthatonlyaminorportionofthetotalfieldstrengthisactuallyeffectiveinfocusingtheelectron.
4)Magneticlensfocallength
Inamagneticelectronlensthefocallengthisdeterminedbythefieldstrengthinthelensgapandbythespeedoftheelectrons(determinedby
theacceleratingvoltage).
Formagneticlenses,focusingisachievedbyvaryingthecurrentwhichpassesthroughtheelectromagnet.Thisinturnchangesthestrengthof
themagneticfieldandtherebyaltersthefocallengthofthelensandisequivalenttoacombinedchangeinboththe"refractiveindex"and
"curvatureofsurface".Forabeamofmoreenergeticelectrons,thelenscurrenthastobeincreasedinordertokeepthefocallengthconstant.
FocallengthandcurrentareNOTlinearlyrelated:strengthincreasesinasigmoidfashion(Fig.I.45)ascurrentincreasesuntilapointisreached
wherethelensissaturatedandnofurtherincreaseinlensstrengthcanbeachieved.
Sincethefocallengthofthelensisdirectlyproportionaltotheacceleratingvoltage,avariationinthevelocityoftheelectronsintheimaging
beamaffectsimagequalitybyeliminatingperfectfocus(chromaticaberration).
5)Magneticlensdesign:(Figs.I.46,I.47).
Condenserlensesusuallyhavearelativelylargeboreandspacingwhichresultsinalongfieldandlongfocallength.
Typicalconstructionoftheobjectivelensproducesastrongfieldofshortaxialextent(i.e.shortfocallengthbetween(1.53mm)necessaryfor
formationofimagesathighmagnification.Thespecimenisplacedwithinthemagneticfieldoftheobjectivelens.Thus,anyfieldintroducedby
contaminantsinthespecimencandistortthefieldofthelens.Notethatthisalsomeansthatpartofthelensfield("prefield")isonthefrontside
oftheobjectandaffectstheelectronbeambeforeitpassesthroughtheobject.
Mostofatypicalmagneticlensliesoutsidethevacuumofthemicroscope.Onlythoseregionsthroughwhichtheelectronbeampassesareinhigh
vacuum.Magneticlensesmustbewatercooledtodissipatethelargeamountsofheatproducedbythecurrentsintheelectromagnetcoils.
DATE NAME EVENT
1897 J.J.Thompson Discoverstheelectron
1924 LouisdeBroglie Identifiesawavelengthtomovingelectrons
l =h/mv
where
l =wavelength
h=Planck'sconstant
m=mass
v=velocity
(Foranelectronat60kVl=0.005nm)
1926 H.Busch Magneticorelectricfieldsactaslensesforelectrons
1929 E.Ruska Ph.Dthesisonmagneticlenses
1931 Knoll&Ruska Firstelectronmicroscopebuilt
1931 Davisson&Calbrick Propertiesofelectrostaticlenses
1934 Driest&Muller SurpassresolutionoftheLM
1938 vonBorries&Ruska FirstpracticalEM(Siemens)10nmresolution
1940 RCA CommercialEMwith2.4nmresolution
1945  1.0nmresolution
Fig.I.1Comparisonoflightandelectronmicroscopes.Ineach
instrument,illuminationfromthesource(lamp,filamentinthe
electrongun)isfocusedbythecondenserlensontothespecimen.
Afirstmagnifiedimageisformedbytheobjectivelens.Thisimage
isfurthermagnifiedbytheprojectorlensontoagroundglass
screen(light)orfluorescentscreen(electrons).(FromAgar,p.8)
Fig.I.2CrosssectionalviewofthePhilipsEM300.(FromAgar,p.40)
Toviewtheanimatedmicroscope,clickhere.
Note:QuickTimePlugInrequired
(Thiscanbedownloadedathttp://apple.com/quicktime/download)
Fig.I.3Sectionthroughacomplex
doublecondenser6lensPhilips
EM200electronmicroscope.(From
Meek,p.99)
Fig.I.4Thestatisticalnatureofdiffraction
patterns.(FromHall,p.13)
"Theprecisenatureof'electronwaves'or'matterwaves'isverydifficulttounderstandordescribeinmaterialterms.Electronwavesarenot
electromagneticradiationofthekindtowhichlight,Xraysandradiowavesbelong.Theyconstituteasortofquantumor'packet'ofradiationwhich
accompanieseachindividualelectron,followingitspathandnotradiatingoutwardsfromit."(Meek,1976,pp.4849).
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.198
50,000 0.0055 1.326 0.442
100,000 0.0039 1.875 0.625
1,000,000 0.0012 5.930 1.977!
V l (nm)
v(
10
10
cm/sec)
v/c
10,000 0.0123 0.593 0.195
50,000 0.0055 1.326 0.414
100,000 0.0039 1.875 0.548
1,000,000 0.0012 5.930 0.941
FigI.5Aperfectpointsourcecannotbe
imagedbyalensasaperfectpointimagedue
tothepresenceoftheapertureAB(which
maybetheedgeofthelens).Diffractionat
thisaperturegivesrisetoaseriesoffringes,
whichsurroundtheimageformedofthepoint
source.Thepatternproducediscalledan'Airy'
disk.(FromMeek1sted.,p.35)
Fig.I.6Wheneverawavefrontstrikesa
barrier,itcanbendaroundthecorner'by
givingrisetoasecondarywavefrontatthe
edge,sinceeachpointonthewavefrontcan
giverisetoanewsourceofwaves.This
phenomenoniscalled'diffraction'.(FromMeek
1sted.,p.22)
Fig.I.7Diffractionpattern(right)formedbyanirregularlyshapedaperture
(left).(FromYoung,p.95)
Fig.I.8(a)
Photographof
theedgeofa
razorblade
illuminatedby
monochromatic
(blue)light
rendered
coherentby
passing
througha
narrowslit.(b)
Microdensitometer
tracingofthe
negativefrom
whichthe
photograph
wasmade.
FourFresnel
fringescanbe
distinguished.
(FromMeek1st
ed.,p.27)
(a)
(b)
(a) (c)
(b)
Fig.I.9(a)Fresnelfringesformed
byelectrons.Thesefringesare
formedoutsidetheedgeofahole
(white)inacarbonfilm(black).(b)
Amicrodensitometertracingofthe
fringesystem;thepatternis
identicalwiththeFresnelfringe
systemformedbyvisiblelight(see
Fig.I.8).(c)Anunderfocusedimage
ofaholeinafilm,showinga
completesystemofabout40
Fresnelfringesinsidethehole.
(FromMeek1sted.,pp.2930)
Fig.I.10Diffractionimagesoftwo
easilyresolvedpoints.(From
Slayter,p.243)
Fig.I.11Imagesoftwo
incoherentlyilluminatedpointsat
thelimitofresolution.(a)individual
intensities;(b)summedintensities.
(FromSlayter,p.244)
Fig.I.12Examplesofwhatoccurswhentwo
wavesofthesamewavelengthandequal
amplitudeadd.Ineachexample,thetwo
separatewavesareshownontheleftand
theirsumorresultantwaveontheright.The
differentexamplesarecharacterizedbyvarying
phasedifferences.Thephaseofawave
(usuallyexpressedasafractionofthe
wavelengthorindegrees)isthepositionofa
crestrelativetosomearbitrarypoint.Although
thephaseofagivenwavevarieswithtimeas
thewavetravels,thedifferenceinphaseofthe
twowavesofthesamewavelengthwiththe
samevelocity,isindependentoftime.Such
wavescaninterferewithoneanother.The
resultantwavehasthesamewavelength,l .
(a)Phasedifferencezero.Thewavestotally
reinforceandaresaidtobe"inphase"orto
showconstructiveinterference.
(b)Phasedifferencel /4.Partialreinforcement,
givingawaveofamplitude1.4(intensity2.0).
(c)Phasedifferencel /2.Thewavesare"outof
phase"andthereistotaldestructive
interferencetogivenoresultantwaveora
wavewithamplitude0(intensity0).(From
GluskerandTrueblood,p.19)
Fig.I.13Photographofan
interferencepatterninwater
wavesformedbythe
superpositionofwavesfromtwo
sourcesoscillatinginphasewith
thesamefrequency.(FromYoung,
p.22)
Fig.I.14Theangularapertureofthe
eyeisrelativelysmall.(FromMeek,
1sted.,p.13)
Fig.I.15Increasingtheangular
apertureoftheeyewithalens.The
lensallowstheobjecttobeheld
closertotheeye,whichisthereby
enabledtogathermoreinformation.
(FromMeek1sted.,p.14)
Fig.I.16Theinteractionofwaves
withanobstacle.Theboatridesthe
longwavelengthoceanwave,but
reflectsthesmallwavelength
surfaceripple.Anobserverwho
wishestodetectthepresenceof
theboatcandosoonlyby
observingwaveswhichhave
wavelengthssmallerthan,or
comparableto,thelengthofthe
boat.(FromSherwood,p.19)
Maximummagnification=
resolvingpoweroftheeye
resolvingpowerofthemicroscope
(a)
Fig.I.17(a)TwoAiry
disksrepresenting
twoimagepoints
shownatincreasing
separationfromleft
toright.Inthe
pictureatthe
extremerightthe
twodiskscanbe
distinguishedas
separatedisks.
(b)Theintensity
distributionatanAiry
disk.R,halfwidthof
thecentralmaximum
representedbya
bellshapedcurve.
(c)TheRayleigh
criteriaforresolution.
(FromSjostrand,
p.115)
i=r
Fig.I.18Reflection.(FromSlayter,
p.4)
Fig.I.19Refraction.(FromSlayter,
p.6)
Fig.I.20Principalraydiagramsshowing
imageformationbyaconvexlens.
(a)Whentheobjectdistanceisgreater
thanthefocallength,areal,inverted
imageisformed.
(b)Whentheobjectdistanceisless
thanthefocallength,avirtual,erect
imageisformed;itspositionis
obtainedbyprojectingtheprincipal
raysbackward.Theraysappearto
comefrompointQ.(FromYoung,
p.127)
Fig.I.21Definitionof
principalfocus,F,inimage
spaceofalens.(From
Sjostrand,p.20)
Fig.I.22Focusingeffectoflenson
raysoriginatingfrompointson
principalaxislocatedatdifferent
distancesfromthelens.(From
Sjostrand,p.21)
Fig.I.23Definitionofprincipal
focusinobjectspaceF1ofa
lens.(FromSjostrand,p.21)
Fig.I.24VirtualimageQof
objectpointlocatedbetween
principalfocusinobjectspace
andthelens.(FromSjostrand,
p.21)
Fig.I.25Constructionoftheimage
ofanobjectbymeansofray
tracing.(FromSjostrand,p.22)
Fig.I.26Magnifyingeffectof
apositivelens.(From
Sjostrand,p.22)
Fig.I.27Virtualmagnified
imageofobjectlocated
betweenprincipalfocusin
objectspaceandthelens.
(FromSjostrand,p.22)
Fig.I.28Amagnifyingglassformsan
enlarged,erectvirtualimage.The
angularsizeofthisimageis
approximately4"/15"or4/15.The
angularsizeoftheobjectatthe
closestdistanceforcomfortable
viewingis1"/10"or1/10.The
magnificationinthissituationis(4/15)/
(1/10),or22/3.(FromYoung,p.130)


Fig.I.29Theangularapertureofa
lens.Theangle2a isthe
acceptanceangleofthelens,and
thelargeritcanbemade,themore
informationcanthelenstransmit.A
largelensofhighaperturecan
thereforetellusmoreaboutan
objectthanasmalllensoflow
power.(FromMeek1sted.,p.12)
Fig.I.30Raydiagramforhighmagnification
modeofoperation.Notethateachlens
formsarealimage,withimageinversion.
I0istheimageformedbytheobjective
lensO,I1isformedbythefirstprojector
lensP1andI2bythesecondprojectorP2,
onthescreen.(FromAgar,p.30)
Fig.I.31Left:raydiagramofthe3realimagemediumandhighmagnificationimagingsystemofa3lens
microscope.Center:2realimageverylowor'scan'magnificationrangeofa3lensmicroscope.Right:2real
imagelowmagnificationsystemofa3lensmicroscope.(FromMeek1sted.,pp.118,120121)
Fig.I.32Raydiagramforacompleteelectron
microscope.FilamentF,condenser1lensC1,
condenser2lensC2,condenserapertureCA,
specimenS,objectivelensO,objectiveaperture
OA(inbackfocalplane).1stintermediateimage
andselectorapertureSA.IntermediatelensP1,
secondintermediateimageI2,projectorlensP2
andfinalimageonthefluorescentscreenFC.
(FromAgar,p.35)
a)f=2.0
x
o
=5.0

e)f=4.0
x
o
=3.0
b)f=10.5
x
o
=21.0
f)f=13.3
x
o
=13.3
c)f=3.5
x
o
=3.0
g)f=3.142
x
o
=0.0
d)f=5.0
x
o
=2.5
 
a)D=8.0

L1:f=2.0

x
o
=4.0
L2:f=3.0

b)D=8.0
L1:f=2.0
x
o
=6.0
L2:f=3.0

c)D=4.0
L1:f=2.0
x
o
=3.0
L2:f=2.0
FigI.33Accelerationofelectronin
electricfield.(FromSjostrand,p.26)
Fig.I.34Linesofforceatapositively
chargedsphericalbody.(FromSjostrand,
p.32)
Fig.I.35Linesofforceandequipotential
surfaces(stippledlines)associatedwith
twoequalchargesofoppositesign.
(FromSjostrand,p.32)
Fig.I.36Equipotentialsurfacesattwoparallel
platesofoppositechargeswiththepathofan
electronindicatedwithinthehomogeneous
partofthefield.(FromSjostrand,p.33)
Fig.I.37Refractionofelectronatan
equipotentialsurface.(FromSjostrand,p.33)
Fig.I.38.Negativelensactionof
electrostaticfieldatanaperturewhen
V2V1>V3V2.(FromSjostrand,
p.34)
Fig.I.39.Positivelensactionatan
aperturewhenV2V1<V3V2.
(FromSjostrand,p.34)
Fig.I.40Magneticfieldinducedbycurrent
passingthroughaconductor.(From
Sjostrand,p.35)
Fig.I.41Magneticfieldinducedbycurrent
passingthroughasolenoid.(From
Sjostrand,p.37)
Fig.I.42Solenoidwithironcore.(From
Sjostrand,p.40)
Fig.I.43a.Dependenceofpermeabilityonfluxdensity,B.
b.RelationshipbetweenfluxdensityBandfieldstrength.
(FromSjostrand,p.38).
Fig.I.44Magnetization.(FromSlayter,
p.361)
Fig.I.45Curves
showinghowthe
magnetizationof
softiron(lens
strength)is
relatedtothe
magnetizing
force(lens
current).An
unmagnetized
lensstartsfrom
thepointOand
followsthepath
OAaslens
current
increases.Atthe
pointA,further
increaseinlens
currentproduces
nofurther
increaseinlens
strength;thelensissaidtobe'saturated'.Whenlenscurrent
isreduced,thepathOAisnotretraced;adifferentpathABis
followed.Thisdisplacementiscalled'hysteresis'.Atzero
current(pointB),someresiduallensstrengthremains;thisis
called'remanence'.Tobringthelensbacktozerostrength,a
reversecurrentOCmustbeapplied.Lensstrengththen
increaseswithincreasingreversecurrent,followingthepath
CD.Thepolarityofthelenschanges,butthisdoesnotaffect
itsfocusingpower;onlythespiralelectronpathisreversed.
Becauseofhysteresis,itisnotpossibletocalibratealens
currentmeteraccuratelyintermsoflensstrengthor
magnification.
Fig.I.46Evolutionofmagneticelectron
lenses.(A)Shortsolenoidusedasa
magneticlens.(B)Softironcasing
enclosingoutersurfaceofthesolenoid,
thusconcentratingthefield.(C)Soft
ironencasingthesolenoidexceptata
narrowannulargaptherebyreducing
themagneticfieldtoaveryshortregion
alongthelensaxis.(D)Modern
objectivelensconsistingofasoftiron
encasedsolenoidandsoftironpole
piecessoastohaveanenormously
concentratedfieldatthelevelofthe
annulargap.(FromWischnizter2nded.,
p.33)
Fig.I.47(a)Amagneticlensconsisting
ofatightlywoundcoilandasoftiron
shroudsurroundingthecoilexceptfor
asmallgap.Thefieldisconcentratedin
thatgap.(b)Shortfocallength
electromagneticlenswithpolepieces.
(FromSjostrand,p.50)
Fig.I.48Fieldstrengthdistribution
curves.ThecurvesADcorrespondto
therespectivelensesillustratedinFig.
I.46.Eachrepresentsthefieldstrength
alongthelongaxisofthelens.The
changesintheshapeofthecurves
representtheshorteningor
concentrationofthefieldovera
shorteraxialdistance.Hz=
longitudinalmagneticfield.Z=
distancealongtheaxisofsymmetry.
(FromWischnizter2nded.,p.33)
Fig.I.49TheforceFactingon
astraightconductorina
homogeneousmagneticfield
offluxdensityBwhencurrent
Iispassedthroughthe
conductor.(FromSjostrand,
p.43)
Fig.I.50Pathofanelectronina
magneticfield.(A)Inalarge,uniform
magneticfield.(B)Inasmall,uniform
magneticfield.(FromWischnitzer2nd
ed.,p.25)
Fig.I.51Lefthandthumbrule.
(FromSjostrand,p.43)
Fig.I.52Thompson'sexperiment.A
streamofelectronsoriginatingfroma
sourceandpassing,invacuo,through
amagneticfieldproducedbyapairof
magnetswillbedeflected.The
directionofdeflectiondemonstrates
thatelectronsarenegativelycharged
particlesofmatter.(FromWischnitzer
2nded.,p.25)
Fig.I.53Actionofasolenoidonan
electronbeam.Anelectriccurrent
passingthroughthecoilproducesan
axialmagneticfield.Thisisthe
refractingmediumfortheelectrons.An
electronstartingatapointontheaxisA
andatanangletoitfollowsaspiral
path,returningtotheaxisatthepoint
B.Theactionisbasicallysimilartothat
oftheconverginglightlensshownin
Fig.I.20.(FromMeek1sted.,p.8)
Fig.I.54Actionofthemagnetic
lens.(a)Inperspective.(b)
Electrontrajectoryin
projection,alongdirectionof
propagation.(c)Electron
trajectoryinprojection,side
view.(FromSlayter,p.358)
Fig.I.55Componentsofthe
vectorHneartheaxisoffields
withaxialsymmetry.His
representedbytwo
components,Hz,the
componentinthez(axial)
direction,andHr,the
compnentinther(radial)
direction.(FromHall,p.85)
Fig.I.56Theycomponentof
themagneticfieldina
magneticlensisoriented
perpendiculartothedirection
ofanelectronenteringthe
lensalongapathparallelto
thelensaxis.Thisy
componentwillaffectthe
electron,deflectingitinthex
directionasindicatedbythe
arrowmarkedvx.(From
Sjostrand,p.48)
Fig.I.57Thezcomponentofthemagneticfield
andthexvelocitycomponentoftheelectron
inamagneticlensinteract,deflectingthe
electronintheydirectiontowardthelens
axis.(FromSjostrand,p.49)
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