1

•• InIn thethe diffractiondiffraction patternpattern consideredconsidered asas anan

exampleexample forfor indexingindexing thethe spots,spots, itit isis noticednoticed thatthat

thethe ll indexindex forfor allall diffractiondiffraction spotsspots == 00

•• ThisThis meansmeans thatthat thethe planesplanes (which(which diffractdiffract thethe

beam)beam) dodo notnot intersectintersect thethe zz--axisaxis (or(or allall thethe

planesplanes areare parallelparallel toto thethe zz--axis)axis)

•• TheThe centrecentre spotspot isis formedformed byby thosethose electronselectrons

whichwhich areare notnot scatteredscattered andand passpass straightstraight

throughthrough thethe crystalcrystal..

•• MeaningMeaning thatthat thethe diffractiondiffraction patternpattern isis orientedoriented

withwith thethe electronelectron beambeam parallelparallel toto thethe zz--axisaxis..

•• TheThe orientationorientation ofof thethe specimenspecimen isis defineddefined byby

statingstating thatthat thethe zonezone axisaxis (ZA)(ZA) ofof thethe diffractiondiffraction

patternpattern isis [[001001]]

DETERMINATION OF THE ZONE AXIS (BEAM DIRECTION)DETERMINATION OF THE ZONE AXIS (BEAM DIRECTION)

h1k1l1

h2k2l2

beam

Zone of reflecting planes

ZA: is a zone axis

ZA

h3k3l3

•• The zone axis [UVW] is determined by using the zone axis equation:The zone axis [UVW] is determined by using the zone axis equation:

U = kU = k11ll22 -- kk22ll11

V = lV = l11hh22 -- ll22hh11

W = hW = h11kk22 -- hh22kk11

•• Where hWhere h11kk11ll11 and hand h22kk22ll22 are the Miller indices of any two spots in the DP. are the Miller indices of any two spots in the DP.

•• In a simple way we can use:In a simple way we can use:

DETERMINATION OF THE ZONE AXIS (BEAM DETERMINATION OF THE ZONE AXIS (BEAM

DIRECTION)DIRECTION)

hh1 1 kk1 1 ll1 1 hh1 1 kk1 1 ll1 1

x x xx x x

hh2 2 kk2 2 ll2 2 hh2 2 kk2 2 ll22

2

INDEXING A PATTERN WITH MANY DIFFRACTION SPOTSINDEXING A PATTERN WITH MANY DIFFRACTION SPOTS

•• ForbiddenForbidden spotsspots areare thethe expectedexpected reflectionsreflections whichwhich maymay notnot occuroccur inin thethediffractiondiffraction patternspatterns fromfrom certaincertain crystalscrystals..

•• InIn aa simplesimple cubiccubic crystalcrystal,, allall reflectionsreflections occuroccur andand therethere isis nono forbiddenforbidden spotsspots..

•• Consider,Consider, forfor example,example, thethe diffractiondiffraction patternpattern formedformed fromfrom NiNi (FCC(FCC crystal)crystal) asasshownshown inin FigureFigure 44..88.. (the(the beambeam directiondirection isis [[001001])])..

•• TheThe latticelattice parameterparameter ofof NiNi isis aaNiNi == 00..3535 nm,nm, λλ == 00..004004 nmnm andand LL == 800800 mmmm.. AAfterftermeasuringmeasuring thethe RR values,values, wewe havehave::

FORBIDDEN ELECTRON PATTERN SPOTSFORBIDDEN ELECTRON PATTERN SPOTS

3

Which gives:

( )21

21

21

176.0111

lkh

ad lkh

++

==

( )22

22

22

124.0222

lkh

ad lkh

++

==

8

4

22

22

22

21

21

21

=++

=++

lkh

lkh

•• So the only possible Miller indices are:So the only possible Miller indices are:

•• of {200}of {200}

•• of {220}of {220}

•• The diffraction pattern is indexed as shown in Figure 4.9The diffraction pattern is indexed as shown in Figure 4.9

•• Note the absence of alternate diffraction spotsNote the absence of alternate diffraction spots

111 lkh ++

222 lkh ++

4

�� To explain why some expected spots are being absent (or forbidden), To explain why some expected spots are being absent (or forbidden),

let us consider the FCC lattice structure let us consider the FCC lattice structure

�� InIn thisthis lattice,lattice, itit isis possiblepossible toto drawdraw anan extraextra planeplane PQRS,PQRS, containingcontaining

thethe faceface--centeringcentering atoms,atoms, inin whichwhich thethe numbernumber andand arrangementarrangement ofof

thethe atomsatoms isis exactlyexactly thethe samesame asas inin thethe originaloriginal ((100100)) planesplanes OAGFOAGF

andand CBHECBHE..

�� ThisThis planeplane (PQRS)(PQRS) isis indexedindexed ((200200))..

�� TheThe electronelectron beambeam cannotcannot distinguishdistinguish betweenbetween planesplanes mademade upup ofof

thethe cornercorner atomsatoms ofof thethe unitunit cellcell andand thosethose mademade upup ofof thethe faceface--

centeringcentering atomsatoms..

5

•• ThereThere areare twicetwice thethe numbernumber ofof planesplanes indexedindexed ((200200)) atat a/a/22 latticelattice

spacingspacing andand ((100100)) planesplanes dodo notnot existexist anyany moremore..

•• Therefore,Therefore, diffractiondiffraction spotsspots fromfrom thesethese planesplanes ((100100)) willwill notnot appearappear inin

thethe diffractiondiffraction patternpattern

•• TheThe selectionselection rulesrules forfor thethe cubiccubic systemsystem allowallow reflectionsreflections toto appearappear inin

thethe diffractiondiffraction patternpattern onlyonly whenwhen::

FCC crystal h, k, l are all even

BCC crystal h + k + l = even

•• TheThe reciprocalreciprocal latticelattice isis directlydirectly relatedrelated toto thethe realreal latticelattice andand isis

commonlycommonly usedused inin diffractiondiffraction..

•• ToTo constructconstruct thethe reciprocalreciprocal lattice,lattice, aa normalnormal toto eacheach setset ofof planesplanes inin thethe

realreal crystalcrystal latticelattice isis drawndrawn..

•• Then,Then, makemake offoff thethe pointspoints alongalong thesethese normalsnormals atat distancesdistances 11/d/d fromfrom thethe

originorigin..

•• ForFor example,example, thethe ((200200)) planesplanes ofof thethe FCCFCC latticelattice givegive riserise toto aa pointpoint

••200200,, inin thethe reciprocalreciprocal latticelattice atat aa distancedistance 11/d/d200200,, fromfrom thethe originorigin •• 000000

inin aa directiondirection normalnormal toto thethe ((200200)) planesplanes asas shownshown inin FigureFigure 44..1515..

•• InIn aa similarsimilar way,way, thethe ((020020),), ((002002)) andand ((111111)) planesplanes asas shownshown..

RECIPROCAL LATTICERECIPROCAL LATTICE

6

�� WhenWhen thesethese pointspoints areare combinedcombined theythey givegive thethe basicbasic arrayarray ofof thethe FCCFCC

reciprocalreciprocal lattice,lattice, whichwhich cancan bebe extendedextended byby thethe additionaddition ofof pointspoints forfor allall thethe

otherother planesplanes..

�� FurtherFurther pointspoints areare alsoalso addedadded toto representrepresent thethe negativenegative sideside ofof thethe planesplanes asas

shownshown inin FigureFigure..

�� TheThe fullfull reciprocalreciprocal latticelattice forfor FCCFCC isis asas shownshown inin FigureFigure 44..1616

�� YouYou mustmust notenote thatthat thethe FCCFCC reciprocalreciprocal latticelattice isis actuallyactually thethe BCCBCC latticelattice..

�� TheThe BCCBCC reciprocalreciprocal latticelattice isis thethe FCCFCC latticelattice..

7

EwaldEwald

circlecircle

00

incidentincidentbeambeam

diffracteddiffractedbeambeam

22θθ

AA

1/1/λλBB

θθ

TheThe relationshiprelationship betweenbetween thethe

diffractiondiffraction patternpattern andand reciprocalreciprocal

latticelattice cancan bebe demonstrateddemonstrated byby

thethe EwaldEwald spheresphere constructionconstruction

TheThe EwaldEwald spheresphere passespasses

throughthrough aa reciprocalreciprocal latticelattice point,point,

whichwhich isis aa distancedistance 11/d/d fromfrom thethe

originorigin..

THE EWALD SPHERETHE EWALD SPHERE

1/d

8

From geometry, we find that:From geometry, we find that:

oror

That is: That is: Bragg’sBragg’s Law is satisfied. Law is satisfied.

d

d

2sin

1

1 λθ

λ

==

θλ sin2d=

�� IndexingIndexing aa diffractiondiffraction patternpattern fromfrom anan unknownunknown specimenspecimen isis notnot anan

easyeasy tasktask..

�� AA trialtrial andand errorerror approachapproach isis used,used, togethertogether withwith somesome intuitionintuition andand

knowledgeknowledge aboutabout thethe specimenspecimen..

�� TheThe correctcorrect formulaformula ofof thethe dd--spacingsspacings andand interplanarinterplanar anglesangles mustmust bebe

foundfound..

�� TheThe symmetrysymmetry ofof thethe diffractiondiffraction patternpattern cancan alsoalso helphelp inin determiningdetermining

thethe typetype ofof crystalcrystal fromfrom whichwhich thethe diffractiondiffraction patternpattern waswas obtainedobtained..

INDEXING A PPATERN FROM AN UNKNOWN SPECIMENINDEXING A PPATERN FROM AN UNKNOWN SPECIMEN

9

h2+k

2+l

2 Primitive cubic Face-centred cubic Body-centred cubic

1 100 - -

2 110 - 110

3 111 111 -

4 200 200 200

5 210 - -

6 211 - 211

7 - - -

8 220 220 220

9 221/300 - -

10 310 - 310

11 311 311 -

12 222 222 222

13 320 - -

14 321 - 321

15 - - -

16 400 400 400