Electron diffraction in TEM

1. Analysis of diffraction patterns generated through TEM through concept of reciprocal
space with an insight into EELS
Report
by
Mainak Saha
Roll No: 14/MM/65
Submitted to
Prof. Durbadal Mandal
Department of Metallurgical and Materials Engineering
National Institute of Technology(NIT) Durgapur
Durgapur-713209,West Bengal
14 February,2018

2. Abstract
Transmissionelectronmicroscope (TEM) canoperate infourmainmodes:brightfield(BF),darkfield
(DF),electrondiffraction(ED) andenergy-dispersive analysisof X-rays(EDX).A TEMworkinginall
modes(BF,DF,ED and EDX) is usuallycalledthe analytical electronmicroscope.Itisa powerful tool
for studyof nanocrystals,whichare invisible inlightmicroscopesanddonot diffractX-rays
sufficiently.TEM/BFincombinationwithimage analysisyieldsaquantitative descriptionof
nanocrystal shapes.TEM/EDXgiveselemental compositionof nanoparticles.TEM/EDincombination
withcrystallographicdatabasesidentifiesknowncrystal structures. TEM/DFmaydifferentiate
monocrystalsfromtwins.
Introduction
Diffraction, crystals, electronwaves,and the transmissionelectronmicroscope
-Diffraction oflight by an optical grating
If monochromaticandspatiallycoherent(laser) lighttransmitsa transparentdiffractiongratingwith
perpendiculardiffractionslitsbeingseparatedbythe distance dalongthe horizontal x-axisa
diffractionpatternwithdiscrete diffractionordersisformed.Constructive interference of light
emittedbyeachof the slits inthe diffractiongratingshowninFig.1 leadstohorizontallyseparated
diffractionspotsatanglesθm forwhichholds
sin(θm)d=mλ
Experimentalsetupof diffractionof laserlightbyanoptical gratingand itsprojectionontoa detector
by an optical setupconsistingof anobjectivelens,andobjectiveaperture,anda projectorlens.
Differentdiffractionorders(butnotdifferentwavelengths!)are indicatedbycolor.
For the more complex discussionof electrondiffractionfromathree-dimensional lattice itisuseful
to describe thisphenomenoninmomentumspace.The momentumvectorof the incidentradiation
isthengivenby
~ k0 = (kx,ky,kz)=(0,0,2π/λ)
the diffractedradiationbehindthe diffractiongratingisnow splitupintopartial waveswith
momenta
~ kl = (2π sinθm/λ,0,2πcosθm/λ)=(2πm/d,0,2πcosθm/λ)

3. The discrete kx-componentsof the scatteredwave vectorsmayare obviouslyindependentof the
wavelengthandonlyafunctionof the scatteringobject.Theyare thuscalledreciprocal lattice points
alongthe horizontal momentumaxisatdistances
gm =2πm/d
away fromthe momentumvectorof the unscatteredpartial wave whichisequal tothatof the
incidentlight~k0.A three-dimensional latticemayhave separate periodicitiesalongthree different
noncolineardirections~a,~ b,and ~c. The reciprocal lattice vectorsare thenindexednotbyasingle
index m,butby the three (Miller) indicesh,k,andl.The correspondinglatticevectoristhusgivenby
~g(h,k,l) =(h~a∗,k~ b∗,l~c∗)
If we nowplace a recordingscreenat some cameralengthL awayfrom the diffractiongratingwe
expectthe spotsonthe screentoappearat distances
The factor Lλ isalsocalledthe calibrationfactor.
Electronwaves
Louisde Broglie inhis1924 doctoral thesisproposedthe wave nature of mass-carryingparticles,
givingthemthe wavelength
(hbeingPlanck’sconstant,m0the electronrest mass,andEkin = |eV0|the kineticenergyof the
electron,withe beingitscharge andV0 the acceleratingvoltage) isveryshort,almost2ordersof
magnitude shorterthantypical X-raywavelengths.Forelectronbeamenergiesachievableina
standardTEM, i.e.E ≥ 20 kV,the wavelengthcanbe approximatedbythe followingexpression:

4. For example,inastandardmediumvoltage TEM(U = 200 kV) the electronwavelengthisλ= 0.0251 ˚
A. Only3 yearsafterthe revolutionarydiscoveryof the wave nature of electrons,electron
diffraction,aphenomenonbasedonthe wave nature of these small particles,wasable toprovide
diffractionpatternsof crystallinematerialsverysimilartothose obtainedbyX-rays.Today,electron
diffractionhasdevelopedintoavery powerful tool in structural crystallographyandmaterials
science andhas againsplitintoseveral subdisciplines.Inlightof the large bodyof literature available
on electronscatteringanddiffractionitseemsimportanttoemphasizethatwithinthischapterthe
term’diffraction’istreatedsynonymouslywith’elasticscattering’,i.e.aprocesswhichpreservesthe
energyof the incidentelectron.If the acceleratingvoltage of the electronisinthe range of 20 - 200
V (sometimesupto600 V) one speaksof low-energyelectrondiffraction(LEED).Electronsof this
energyhave a wavelengthof about1 ˚A(2.7 ˚A at 20 V and 0.5 ˚A at 600 V),i.e.comparable tothe
distance betweenatoms,andcanonlypenetrate the firstatomiclayers,or 5-10 ˚ A of the sample.
LEED istherefore exclusivelyusedtostudythe atomicstructure of surfaces,mainlyunderultra-high
vacuumconditions.Athigherbeamenergies,asinmediumenergyelectrondiffraction(MEED,
acceleratingvoltage between1kV and 5 kV or reflectionhighenergyelectrondiffraction(RHEED,
electronenergyrange between40keV and100 keV) the scatteringanglesbecometoosmall towork
witha normallyincidentelectronbeam,butinstead,amore (RHEED) or less(MEED) grazing
incidence setuphastobe chosen.Thisdeviationof the angle of incidence fromthe surface normal
makesRHEED muchmore sensitive tosurface roughnessthanLEED,withMEED beingaboutin
betweenbothworlds.Atsufficientlyhighenergy(above 20keV) electronsare fastenoughto
penetrate severalnanometersof material withouttoomuchabsorptionorcharging.Although
transmissionthroughverythinsamplesandevenatomicresolutionimagingisalsopossible atlower
energy,transmissionelectrondiffractionintransmissionelectronmicroscopes (TEMs) ismost
commonlydone atacceleratingvoltagesabove 20kV and insome casesup to 1 or even3 MV.
Because of the versatilityof TEMs andtheiravailabilityinalarge numberof laboratoriesaroundthe
globe thischapterwill focusontransmission highenergyelectrondiffraction(HEED) only.Electron
Diffractionpatternsof crystalline materialscontainawealthof informationandhave helpedto
reconstructthe atomic positionsof complex crystal structures,largelybyapplyingthe verysame
techniquesdevelopedinX-raycrystallography.However,beinglimitedinspace,the final sectionof
thischapterwill onlybe able toprovide anoverview of aselectionof electrondiffractiontechniques
whichshowcase the unique capabilitiesof electrondiffraction comparedtoother(e.g.X-rayor
neutron) diffractiontechniques.The electriccharge of the electronsnotonlyallowsthemtobe
acceleratedbyelectrostaticfields,butalsodeflectedandfocusedbyboth,magneticaswell as
electricfieldsaccordingtothe Lorentzforce.
While roundmagneticlenseshave aninherentpositive spherical aberration,multipoleelements
may be usedto compensate thisspherical (andalsohigherorder) aberrations,producingelectron

5. probesas small as0.5 ˚A indiameter.In additiontothe correctionof lensaberrationsinthe probe-
forminglenssystemthe formationof suchsmall probesalsorequiresahighspatial coherence.A
critical parameterdescribingthe amountof currentavailableata certaindegree of spatial
coherence isthe brightnessβ, whichisthe amountof currentemittedintoacertainsolidangle Ω
froman area A. For rotationallysymmetricsystemsone canalsowrite
where α isthe angularradiusof the cone of emittedelectrons,andrthe radiusof the source.
Because of the increasedmomentumof electronsathigheracceleratingvoltages,the effectivecone
radiusα isproportional tothe electronwavelength,andisthusinverse proportional tothe square
root of the electron’skineticenergy.Forthisreasonthe reducedbrightness,isoftenusedinstead.
The electronmicroscope usedforthispracticumhasa fieldemissiongun(FEG).A FEG, todaythe
standardelectronsource inhigh-performance TEMs,providesasource of nanometerdimensions
and sub-eV energyspread,withabrightnessperunitbandwidthgreaterthancurrent-generation
synchrotrons.When alsotakingintoaccountthe muchstrongerinteractionof electronswith
matter,the amountof coherentlyscatteredsignal thatcanpotentiallybe collected fromacertain
volume of material exceedseventhatof free-electronlasers.FEGshave a much higherbrightness
than thermionicelectronguns,suchasa LaB6 source is used.
The needto understandatomicprocessesinsolidshasledtoan increasingdemand fornew imaging,
diffractionandspectroscopymethodswithhighspatial resolution. Thisneedhasbeenreinforcedby
the growinginterestinnanoscience andnanotechnology. Particularly,transmissionelectron
microscopy(TEM) isof unrivalledvalue,becauseitcanprovide structural informationwithexcellent
spatial resolution(downtoatomicdimensions) viahighresolutionTEMimagingandelectron
diffraction. These structural datacanbe supplementedbychemical informationfromthe same
specimenregion,obtainedusinganalytical techniques,suchasenergy-dispersiveX-rayspectroscopy
(EDS) and electronenergy-lossspectroscopy(EELS).Due tothe broad range of inelasticinteractions
of the highenergyelectronswiththe specimenatoms,rangingfromphononinteractionsto
ionisationprocesses,electronenergy-lossspectroscopyoffersunique possibilitiesforadvanced
materialsanalysis. Itcan be usedto map the elementalcompositionof aspecimen,butalsofor
studyingthe physical andchemical propertiesof awide range of materialsandbiologicalmatter.
Beginningwiththe increasinguse of EEL spectrometersduringthe eightiesof the lastcentury[1],
the energy-filteringtechnique wasamajorstepforwardfor two-dimensionalmappingof the
spectral featuresvisibleinanEEL spectrum. The wide distributionof energy-filteringTEMs (EFTEM)
ledto numerouspractical applicationsbothinthe materialsandbiological sciences[4]. Inthe
meanwhile,EFTEMisa widelyusedmethodforbothoverview and nanoscale characterisationof
thinsamples,applicable tomostchemical elementsandespeciallysensitive tolightelements.With
the introductionof highbrightnesselectronsourcesandspherical aberrationcorrectors,the image
resolutionof the TEMreachesnowthe 50 picometre level. Especiallyscanningtransmission
electronmicroscopy(STEM) equippedwithaCs-probe correctorisextremelyusefulforhigh

6. resolutionimaging[6] andthe parallel chemical analysisbyEELSand EDS because of the much
increasedprobe currentinthe correctedbeam.
Electron energylossSpectroscopy
A beamelectronina (S)TEMmay be inelasticallyscatteredwhenitinteractswiththe atomic
electronsinthe specimen. The electronbeamlosesenergyandisbentthroughasmall angle (5 -
100 milliradians). The energydistributionof all the inelasticallyscatteredelectronsprovides
informationaboutthe local environmentof the atomicelectronswhichinturnrelatestothe
physical andchemical propertiesof the specimen. Thisisthe basisof electronenergy-loss
spectroscopy(EELS). Much of the informationobtainablefromEELSis similartothatof X-ray
absorptionspectrometry(XAS) inthe synchrotron.
The firstpeak,the most intense foraverythinspecimen,occursat0 eV and istherefore calledthe
zero-losspeak. Itrepresentselectronswhichhave notbeenscatteredinthe specimen(transmitted
electrons) andwhichhave beenelasticallyscatteredviainteractionwiththe atomicnuclei.
The low-lossorvalence region of anEEL spectrum(< 50 eV) providessimilarinformationtothat
providedbyoptical spectroscopy,containingvaluable informationaboutthe bandstructure andin
particularaboutthe dielectricpropertiesof amaterial (e.g.,bandgap,surface plasmons). The most
prominentpeak,centredat24 eV,comesfroma plasmaresonance of the valence atoms. Signal
intensitiesinthe low-lossregionare largerthaninthe high-lossregionof the spectrum.Athigher
energylosses(>50 eV),where the numberof inelasticallyscatteredelectronsismuchlower,the
spectrumshowscharacteristicfeaturescalled“ionisationedges”(due totheirtypical shape,arapid
rise followedbyamore gradual fall). These edgesare the exactequivalentof anabsorptionedge in
XASand arise fromthe same process. The edgesare formedwhenaninner-shellelectronabsorbs
enoughenergyfromabeamelectrontobe excitedtoa state above the Fermi level. Notall
ionisationedgesare saw-toothedlikethe carbonK-edge infigure1,but exhibitmore complex
edgeshapessuchasthe L2,3-edge of titaniuminfigure 1. This L2,3-edge includessharpexcitations
at the onsetof the ionisationedge,calledwhite-lines,whichare typical forelementsinthe firstrow
of transitionelementsand forthe rare earthelements. The ionisationedgescanbe usedfor the
analysisof almostall chemical elementsinparticularforthe lighterelements,the edge onsetgives
the ionisationenergyandallowsthe qualitativeanalysisandincase of verythin samplesthe edge
intensitiesare proportional tothe concentrationof the correspondingelements.
RecentadvancesinEELS methodologyTEM-EELSinstrumentationisbasedona magneticprism, in
whicha uniformmagneticfieldisgeneratedbyanelectromagnet withspeciallydesignedpole
pieces. The prismbendsthe inelasticallyscatteredelectronsbyabout90o, dispersesthemaccording
to theirdifferentkineticenergiesandalsohasa focussingaction. The spectraare recordedwitha
charge coupleddevice(CCDcamera). Due tothe enormousdynamicrange of anEEL spectrum,
spanningmanyordersof magnitude,anddue torestrictionsindynamicrange andsensitivityof CCD
basedspectrometers,acomplete EELspectrumnormallyhastobe recordedinseveral segmentsby
changingilluminationconditionsand/oracquisitiontimes.Energyfiltersare usedtoformimages
fromelectronsthathave sufferedaspecificenergyloss[energyfilteredTEM(EFTEM) or electron
specificimaging(ESI)]. Inparticular,twotypesof energy-filteringinstrumentsare mostoftenused:

7. firstlypost-columnenergyfilters,withasingle prismgeometryandmultipole lenses,canbe
attachedand retrofittedtopracticallyanyTEMor STEM instrument. The post-columnfilterisnot
onlyan efficientimagingsystem, butalsoaversatile instrumentforacquiringEELspectra inhigh
quality. Secondlythe in-columnfilter,whosespectrometerconsistsof fourmagneticprismsthatare
arrangedsymmetricallyaccordingtothe shape of a GreekOmega,islocatedbelow the projector
lenssystemof the TEM. Thistype of filtercan be alsousedfor spectroscopy,butthe main
applicationfieldishighqualityimagingsuchaslarge areaelemental mappingandenergy-filtered
electrondiffractionstudies.Anotherimportantdevelopmentwasthe introductionof
monochromatorsforthe electronsource,whichpavedthe wayforacquiringEEL spectraat high
energyresolution,typicallyinthe 100 - 200 meV range (e.g.,the Wienfilterapproach). The
improvedenergyresolutionopensnewpossibilitiesforstudyingdetailedelectronicstructure and
bondingeffectsevaluatedfromnearedge fine structuresof the ionisationedges,butalsoaccurate
bandgap and dielectricfunctionmeasurementsviathe low-losspartof the spectrum.Several years
ago, the problemof the highdynamicrange of the EEL spectrumcouldbe addressedbyacquiring
the elasticandinelasticpartsat nearlycoincidenttimesatidentical experimental conditions,
offeringthe advantage of usingthe low-lossregime asareference forquantitative dataanalysis.
Thissystemworkswithan additional electrostaticvertical deflectorandisnow successfullyusedby
several groupsworldwide.AdvancesinX-raydetectionwiththe adventof large areaor four-
quadrantsolidstate silicondriftdetectors andthe rapidprogressinEDS data processing,pushed
the ideasforhighlyefficientelemental mappingincombinationwithEELS. The collaboration
betweenGatan,Brukerandthe TU Graz enabledthe fastrecording of multimodal datasothat
STEM-images,low-lossandhigh-lossEELSandEDS data can now be acquiredwitha speedof 1,000
to 1,500 spectrapersecond.While the conventional methodof EELSmappingcombinesTEM/STEM
imageswiththe local concentrationof the elements,the developmentof more powerfulcomputers
and data reductionproceduresnowadaysallowsa“holistic”approach:the whole spectral
informationisgatheredforeachpointof animage,generatingathree-dimensional datasetwhichis
oftencalledaspectrumimage. The spectrum-imagingtechniqueoffersvariousadvantages:the
wealthof data allowsacertainamountof “postexperimentmicroscopy”andbecause of the
completenessof data,interpretationmistakescanbe avoided. Additionally,adata-evaluation
software canautomaticallyidentifyandhighlightthe mostprominentfeatures,e.g.,chemical
phases. Therefore,spectrum-imagingtechniquesare the essential basisforsuccessful EELSmapping
at the atomicscale.
Elemental mappingathighspatial resolutionOne of the mostcommonlyusedapplicationsof STEM-
EELS or EFTEM isto derive compositional informationbyrecordingenergy-filteredimagesusingthe
elementcharacteristicionisationedges.
Althoughadvancedenergy-filtersorspectrometersnow enablethe fastacquisitionof elemental
maps almostona routine basis,care must be takendue to several experimental limitationsof EELS:
generally,the signal-to-noise ratioof the elementalsignal isverylow whichismainlycausedbythe
highuncharacteristicbackgroundbelow the ionisationedgesandthe low ionisationcross-sections
for heavierelementsandforelementsoccurringatlow concentrations.Anotherimportantlimitation
comesfrommultiple scatteringof the inelasticsignal inthickerspecimens,restrictingelemental
mappingtospecimenthicknesseswellbelow the meanfree pathlengthsof the inelastically
scatteredelectrons(specimenthickness<70 nmfor mostmaterials).

8. Alternatively,EELSspectrafrom thickerspecimensmaybe deconvolvedbyusingthe low-losspartof
the EELS spectrum,recoveringasingle scatteringdistributionEELSspectrum. Typical procedures
involve Richardson-LucyorFourierlogmethods,whichhoweverdecreasethe signal-to-noise ratio
of the spectrum. Whencrystalline specimensare studied,afrequentproblemisthe preservationof
diffractioncontrastininelasticimaging,whichcanbe reducedbyemployingadvancedillumination
conditions(hollow-coneorrocking-beamillumination).
ImagesshowingMgO/Ni core shell nanoparticlesinvestigatedwithaCS-correctedSTEMandEELS
spectrumimagingrevealingthe Ni core andthe MgO shell,the insetshowsthe Z-contrastimage of
the Ni core.
Chemical bondinginformation
Edge fine structuresarouse considerableinterestinthe applicationof EELS,especiallyinthe fieldof
materialsscience,because theycanbe usedtoextractinformationregardinglocal charge
distributions,coordinationnumbersandbondingcharacteristics. The fine structuresare dividedinto
the near edge-fine structures(ELNES) within50eV of the edge onsetandthe extendedfinestructure
(EXELFS) aboutsome 100 eV above the edge onset. Mostof the ionisationedgescontainamore
complex structure thancan be explainedinsimple atomicterms. The ionisationedgesare often
modifiedbythe solidstate environmentof the atomundergoingthe inner-shellexcitationandthis
informationcanbe usedas a “fingerprint”fromelementsinsimilarchemical environments[26,27].
In the meanwhilethe calculationof ELNESstructureshas reacheda highdegree of sophistication
[28]. One importantdevelopmentforELNESstudieswasthe introductionof the monochromated
(S)TEMs whichhelpedtoimprove the instrumental energyresolutiontovaluesaslow as100 meV.
In combinationwithadvancedenergy-filtersand/orspectrometersitwasnow possible tofully
exploitinformational the spectral detailscontainedinthe EEL spectrum. It wasnot at all obvious,
however, atthe time of the developmentduringthe late ninetiesthatthese toolswouldgainsuch
importance. The reducedenergyspreadof the TEMimmediatelyopenedpossibilitiesfordetailed
studiesof the near-edge fine structuresatK- andL2,3-ionisationedgesof oxides,perovskitesand

9. similarmaterials[29]. The knowledgeaboutthese ELNESstructuresathighenergyresolutionisan
importantbasisforSTEM-EELS imagingof chemical bondinginmaterials. Figure 4showsthe ELNES
structure of the V L2,3 white linesof V2O5recordedat differentenergyresolutions. The L3 white
line revealsthe symmetryof the vanadiumsite beingsurroundedbysix oxygenatomsforminga
stronglydistortedoctahedronunit.
Comparisonof the V L2,3- and O-ionisationedgesof V2O5whichhave beenrecordedwithdifferent
energyresolution;
a) 200 kV TEM withLaB6 cathode (0.7 eV);
b) 200 kV TEM withSchottkyemitter(0.6eV);
c) 200 kV TEM witha monochromatorand a HR-energyfilter(0.3eV);
d) X-rayabsorptionspectrumrecordedwithanenergyresolutionof 0.08 eV.
Physical propertymappingThe low-lossspectrumcontainsenergylossestovalence orconduction
electronswhichproviderichinformationaboutthe physical propertiesof aspecimene.g.,inter-or
intrabandtransitions,bandgapsandthe dielectricproperties. One importantapplicationof low-loss
EELS liesinthe studyof optical propertiesof metallicnanostructureswhichmaydrasticallychange at
the nanometre scale asa functionof size,morphologyandenvironment. Inparticular,collective
oscillationsof quasi free electronsonametallic/dielectricinterface,i.e.,surface plasmons,are
increasinglystudiedinthe (S)TEM,takingadvantage of itsunbeatenhighspatial resolution
(comparedtothe scanningoptical near-fieldmicroscope). Here the introductionof
monochromators,providinganenergyresolutionof 100 meV or evenless[33],wasmainly
responsible forthe bigsuccessof the technique.Since the earlyseventiesof the lastcenturyEELS
has beenusedforstudyingsurface plasmons ,butit wasonlyin 2007 whentwoindependentgroups
showedhowfastelectronscanbe usedto map localizedsurface plasmonsof single noble metal
nanoparticles[36,37]. In these pioneeringworksSTEM-EELSwasused,butlaterit wasalso shown

10. that a monochromatedEFTEMcan yieldcomparable results[38,39]. It is now widelyacceptedthat
EELS or EFTEM are the mostadvancedmethodstoprobe plasmonicmodesandtheyare increasingly
usedto studynanostructuresof increasingcomplexity. Forexample STEM-EELSwasusedfor
studyingdarkplasmonicbreathingmodesinsilvernanodisks[40],formorphingaplasmonic
nanodiskintoananotriangle ,forrevealinguniversaldispersionsof surface plasmonsinflat
nanostructures , for studyingthe 3D distributionof surface plasmonsaroundametal nanoparticle.
a) High-angle annulardarkfield(HAADF) image of a 30 nm thick Au-nanostar on a 15 nm thin silicon
nitride membrane (left) andcorrespondingelectronenergyloss (EEL) maps at 0.8 eV (A), 1.35 eV (B)
and 1.70 eV (C) integrated over an energy width of 150 meV. The EELS maps were generated using
the STEM-EELS approach with a monochromated 200 keV electron beam with 150 meV energy
resolution (FWHM). The sample was prepared by electron beam lithography, raw data are
presented. B) The regions 1-3 in the spectrum image (left) mark the areas from which EEL spectra
were extracted. The peaks labelled by A, B and C corresponds to the energies of the EELS maps
shown in (a).
Diffraction of electronwavesby a three-dimensional lattice
Due to fact that the electronsinan atomare much more delocalizedthanitsprotons,every(neutral)
atom producesa sharplypeakedpositiveelectrostaticpotential withthe centerof thispeakatthe
positionof the core of thisatom.Accordingto equation, electronsare deflectedbyelectrostaticand
magneticfields.Neglectinganymagneticcontributionstothe scatteringthe electronsare thus
deflectedbythe electrostaticpotentialof atoms.Since incrystalline specimenthe atomsare located
on a regularlattice inthree-dimensional space we candescribe the scatteringpropertyof the whole
crystal by only describingthe electrostaticpotentialof itsunitcell inreciprocal space.The Fourier
coefficientsof the electrostaticpotentialof the unitcell are calledstructure factorsandare givenby

11. where γ = 1/q1−v2/c2 isthe Lorentzfactor forthe fast electronatvelocityv,Vcell isthe unitcell
volume,fj
el(s) isthe atomicscatteringfactorforscatteringparameters= |~g|/2, ~g isa reciprocal
lattice vector,and~rj isthe positionof the jthatomwithinthe unitcell.The atomicscatteringfactor
isthe radial part of Fouriertransformof the electrostaticpotential of the atomj alone.Since the
potential of anisolatedatomis isotropicinangle andnotinfinitelysharplypeaked,thisscattering
factor fallsof withscatteringparameters.Itcan be well approximatedbythe sumof a 4 Gaussians[.
The electronstructure factorsare onlynon-zeroonpointsinreciprocal space whichlie onthe
reciprocal lattice.However,thisreciprocal lattice isconvolutedbythe shape transformof the
crystal,i.e. the Fouriertransformof itsshape inreal space.Since TEM specimenmustbe verythin
for the electronbeamtobe able to passthrough them, butmay be verywide laterally,the reciprocal
lattice pointshave some finite extentalongthe directionparallel tothe wave vectorof the incident
electronbeam,butare verysharplydefinedinthe lateral directions.Fornanocrystals,whichare also
laterallyconfined,the reciprocal lattice pointsextendalsolaterally. Elasticallyscatteredelectrons
maintaintheirmomentumandmaytherefore scatteronlytoreciprocal lattice vectorswhichlie ona
sphere (Ewaldsphere).Reflectionswhichare notintersectedbythe Ewaldsphere maystill be
excited,but,because of the nonvanishingexcitationerrorsg(reciprocal space distance betweenthe
Ewaldsphere andthe reciprocal lattice pointforthe infinitelyextendedcrystal),inmostcaseswitha
reducedintensity.Kinematical scatteringtheory,whichneglectsthe possibilitythatanelectron
scattersmore than once on its paththroughthe sample predictsthatthe intensityinthe diffraction
patternIIh,k,l ∝|U(~q)|2,i.e.thatitisproportional tothe amplitude squaredof the structure factor
U(~gh,k,l). Forelasticscatteringof the incidentelectronwavethe kineticenergyof the electronis
preserved,andwiththatalsothe absolute value of itsmomentumvector.Thismeansthatin
reciprocal space the wave vectorsof all scatteredelectronsmustlie onthe surface of a sphere of the
sphere.
Optical systemsbuiltof lenses
For settingupthe optical diffractioncamera,andfor understandingthe electrondiffraction
experimentitisimportantthatyoureview the followingprinciplesof geometrical optics:
The lensequation.
The magnificationof animage:M = himage/hobject = dimage/dobject.
The fact that inthe back focal plane a lensproducesthe Fouriertransformof the lightfieldinthe
objectplane.
Experimental setupsof an optical diffraction camera and an electronmicroscope
The transmissionelectronmicroscope (TEM)

12. Basedon the Lorentzforce givenbyequation ithas beendiscoveredbyBusch that magneticcoils
will focusanelectronbeam.Thishassoonafterledto the developmentof
Diagram illustratingthe Ewaldsphere constructionandthe physical meaningof the excitationerror
sg. The fast beamelectroncanonlyscatterelasticallytoreciprocal space vectorswhichlie onthe
Ewaldsphere (dashedcurve).The distancebetweenagivenpointinreciprocal space andthe Ewald
sphere inthe directionof the specimen’ssurface normal (assumedtobe inthe z-directioninthis
illustration) iscalledexcitationerrorsg.
the first transmissionelectronmicroscope(TEM) byKnoll andRuska an achievementwhichhas
beenhonouredwiththe Nobelprice in1986. ModernTEMs are equippedwithatleast4
electromagneticlenses(gunlensandcondenserlenssystemandobjective pre-fieldlens) to
collimate orfocusthe beamon the sample andat least4 lensesbelow the sample(objective lens
and a projectorlenssystem) tofocuseitheramagnifiedimage orthe magnifieddiffractionpattern
on a detectorwhichcan be a CCD camera, photographicfilm, digital imagingplates,ora
spectrometer.Inthispracticumwe will use aCCD camerafor detectingimagesanddiffraction
patterns.Figure showsthatby simplychangingthe currentrunningthroughlensesinthe projector
lenssystemthe TEMcan switchbetweenimage anddiffractionmode.Likewise,the condenserlens
systemcan be usedto continuouslyvarythe illuminationconvergence angleandwiththatthe size of
the electronprobe onthe sample,changingfromparallel toconvergentillumination,facilitating
parallel orconvergentbeamelectrondiffraction(CBED).The areaof the sample contributingtothe
diffractionpatterninparallel-beamillumination maybe selectedin2ways:a) by an aperture inthe
condenserlenssystemorb) bya selectedareaaperture inthe positionof the firstintermediate
image (indicatedbyanarrowin Figure 3b).Inthispracticum we will use selected areaelectron
diffraction(SAED),i.e.applythe secondoptionof selectingthe contributingarea.The reasonforthis
isthe experimental simplicity.Withoutchanginganylenscurrentsoralignment,orthe illumination
conditionsonthe sample we mayquicklyswitchbetweenaverylarge fieldof view forrecordingan
overviewimage andasmall fieldof view for selectingthe areafromwhichthe diffractiondatashall
stem,simplybyinsertingandpositioningasmall aperture.There are afew problemsof this

13. approach:a) Because of the fixedmagnificationof the objective lensthe smallestareathatcan be
selectedbyaverysmall (e.g.5 µm aperture) hastypicallyadiameternotmuchsmallerthan80 nm.
b) Due to aberrationsof the objective lenshigh-angle diffractioninformationmaystemfroma
slightlylargerareathanthe one selectedfromthe (bright-field) image.
Two differentmodesof operationof aTEM: a) High-resolutionimagingmode:Anincidentplane
wave scatterselasticallyaccordingtothe differentlattice planesandthe diffractedbeamsinterfere
witheachother.Thisinterference patternmayinsome casesbe interpretedasdirectlyrepresenting
the atomic structure.b) For parallel illuminationthe diffractionmode isa(conventional)spot
pattern.
Resolutionof modernTEMmicroscopesiswell below1nm, whichmakesthemsuitabletoolsforstudy
of nanoparticles and nanostructuresin bulk materials.Both nanoparticles and nanostructures cover
broadrange of applicationsinmaterialsscience (inorganic nanoparticles,nanolayers,defectsinmetals
andalloys),polymerscience (syntheticpolymernanocomposites,blockcopolymers,polymermicelles)
andbiology(morphologyof cellsandviruses).Thiscontributionisfocusedonanalysisof nanoparticles,
namely nanocrystals.
A standardmoderntransmissionelectronmicroscope(TEM) operatesinthreemodes:brightfield(BF),
dark field (DF), and electron diffraction (ED, SAED). In TEM/BF we detect transmitted electronsand
receive a"standard"TEMmicrograph:a directimage,inwhichthedarkareasappearduetodiffraction
and/or absorption contrast. In TEM/DF we detect (a selected portion of) scattered/diffracted
electrons and obtain a direct image with "inverted" contrast. In TEM/ED we detect
scattered/diffracted electrons in a different way and obtain electron diffraction pattern, which is
analogous to X-ray and neutron diffraction patterns.
Asan extraoption,TEMmicroscope canbe equippedwithaEDXdetectorandworkinmode of energy
dispersive analysis of X-rays (EDX, EDS). In such a case we can detect characteristic X-rays and, as a
result, perform elemental analysis in nanoscale. A TEM microscope equipped with the four basic
modes (BF, DF, ED, EDX) is usually called the analytical electron microscope.

14. It isa matterof course thatthere are also more advancedand/orspecial modesof TEM,such as high-
resolutiontransmissionelectronmicroscopy(HRTEM),convergentbeamelectrondiffraction(CBED),
scanningtransmissionelectronmicroscopy(STEM),energy-filteredtransmissionelectronmicroscopy
(EFTEM), cryo transmission electron microscopy (cryoTEM) - just to name a few. Nevertheless, this
contribution concentrates on the explanation of four basic modes (BF, DF, ED, EDX) and their
application on study of nanocrystal shapes, elemental composition and crystal structure.
Imaging and diffraction mode in TEM.
Single crystal diffraction in XRD and ED.
For single crystal X-ray diffraction,we needtouse four-circle diffractometersandcollectdiffractions
step-by-step,usingeitherpointdetectorsor2D-detectors,suchasCCDcameras,imageplatesof films.

15. In a TEM microscope, we can see the whole plane of reciprocal lattice in one image. The basic
explanation is again relatively straightforward, employing just Ewald's construction, calculation of
electron wavelength and one simple formula from Fourier theory.
CeO2 nanocubes in TEM: (a) bright field, (b) dark field
(c)electron diffraction, (d) EDX spectrum.
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