Effect of Symmetry on the Orientation...

EffectofSymmetryontheOrientationDistribution

27-750Texture,Microstructure&Anisotropy

A.D.Rolle<

Lastrevised:20thJan.‘16

2

Objectives•  HowtoconvertEuleranglestoanorientaDonmatrix,andback.•  HowtoconvertanorientaDonmatrixtoMillerindices,andback.•  Reviewofsymmetryforbothcrystalsandmaterialsprocessing.•  Explainsymmetryoperators,theirmatrixrepresentaDon,andhowtousethemto

findallthesymmetricallyequivalentdescripDonsofagiventexturecomponent.•  ToillustratetheeffectofcrystalandsamplesymmetryontheEulerspacerequired

foruniquerepresentaDonoforientaDons.•  TopointoutthatsamplesymmetryisstaDsDcalratherthanphysical.Alsothat

orientaDonsrelatedbysamplesymmetryare,ingeneral,physicallydisDnct.•  ToexplainwhyEulerspaceisgenerallyrepresentedwitheachangleintherange

0-90°,insteadofthemostgeneralcaseof0-360°forφ1andφ2,and0-180°forΦ.•  Topointoutthespecialcircumstanceofcubiccrystalsymmetry,combinedwith

orthorhombicsamplesymmetry,andthepresenceof3equivalentpointsinthe90x90x90°“box”or“reducedspace”.

•  Toexplaintheconceptof“fundamentalzone”.•  Part1ofthelecturedealswithorientaDonmatrices,EuleranglesandMillerindices.•  Part2providesaqualitaDvedescripDonofsymmetryanditseffects.•  Part3providesthequanDtaDvedescripDonofhowtoexpresssymmetryoperaDons

asrotaDonmatricesandapplythemtoorientaDons.

InClassQuestions:11.  Whatisafundamentalzone?2.  Whatisthedifferencebetween“crystal

symmetry”and“samplesymmetry”?3.  Whichkindofsymmetryisphysicalinnature,

andwhichoneisstaDsDcal?4.  Whichpointgroupisapplicabletocubicmetals?

Hexagonalmetals?TetragonalDn?5.  Listthecommontypesofsamplesymmetry.

3

InClassQuestions:21.  WhatlimitsontheEuleranglesareappropriatetocubic-

monoclinicsymmetry?2.  Giveanexampleofasymmetryoperatorinthe

hexagonalgroup622.3.  ExplainhowtostartwithasetofEulerangles,apply

symmetryandobtainanewsetofEuleranglesforthesymmetricallyequivalentorientaDon.

4.  Ifweapplythesymmetryofagivenpointgroup,bywhatfactoristhevolumeoftheresulDngsub-spacedecreased,comparedtotheoriginalorientaDonvolume?

5.  BasedonamatrixrepresentaDonoforientaDons,dowele_-mulDplyorright-mulDplytoapplyasamplesymmetryoperator?

4

InClassQuestions:31.  Howcanwetestourschemetocalculatesymmetry-

relatedorientaDonstomakesurethatweapplycrystalandsamplesymmetrycorrectly?

2.  Ifwechangethesamplesymmetrytomonoclinic,howlargeaspaceshouldweuseinEuleranglespace?

3.  Ifwechangethecrystalsymmetrytohexagonal,howdoesthischangetherangeofEuleranglesrequired?Canyouexplainwhythe3rdEulerangleonlyneedstohavetherange0-60°?

4.  DrawastereographicprojecDonandaddovalstorepresentdiads,trianglestorepresenttriadsandsquarestorepresenttetradsunDlyouhaveadiagramoftheO(432)pointgroup.

5

Part1:Millerindices↔ ︎Matrix↔︎Eulerangles

•  InthissecDon,wefindouthowtoconvertfromonerepresentaDonoforientaDon(texturecomponent)toanother.

6

7

FormmatrixfromMillerIndices

ˆ b = (u, v, w)u2 + v2 + w2ˆ n = (h, k, l)

h2 + k2 + l2

ˆ t =ˆ n × ˆ b ˆ n × ˆ b aij =Crystal

Sample

b1 t1 n1b2 t2 n2b3 t3 n3

!

"

####

$

%

&&&&

Obj/notationAxisTransformationMatrixEulerAnglesComponents

Basicidea:wecanconstructthecompleterotaDonmatrixfromtwoknown,easytodeterminecolumnsofthematrix.KnowingthatwehavecolumnsratherthanrowsisaconsequenceofthesenseofrotaDon,whichisequivalenttothedirecDoninwhichtheaxistransformaDoniscarriedout.

8

BungeEuleranglestoMatrix

Rotation1(φ1):rotateaxes(anDclockwise)aboutthe(sample)3[ND]axis;Z1.


Rotation2(Φ):rotateaxes(anDclockwise)aboutthe(rotated)1axis[100]axis;X.

Rotation3(φ2):rotateaxes(anDclockwise)aboutthe(crystal)3[001]axis;Z2.

Basicidea:constructthecompleteorientaDonmatrixfromindividual,easytounderstandrotaDonsthatarebasedonthethreedifferentEulerangles.DemonstratetheequivalencebetweentherotaDonmatrixconstructedfromtheserotaDons,andthematrixderivedfromdirecDoncosines.“RotaDon”inthiscontextmeans“transformaDonofaxes”.

9

BungeEuleranglestoMatrix,contd.

€

Z1 =

cosφ1 sinφ1 0−sinφ1 cosφ1 00 0 1

$

%

& & &

'

(

) ) ) ,

A=Z2XZ1Obj/notationAxisTransformationMatrixEulerAnglesComponents

€

X =

1 0 00 cosΦ sinΦ0 −sinΦ cosΦ

$

%

& & &

'

(

) ) )

€

Z2 =

cosφ2 sinφ2 0−sinφ2 cosφ2 00 0 1

$

%

& & &

'

(

) ) )

Combinethe3rotaDonsviamatrixmulDplicaDon;1stontheright,lastonthele_:

EachindividualrotaDonisa2-DrotaDonaboutXorZ

10

MatrixwithBungeAngles

cosϕ1 cosϕ2− sinϕ1sinϕ2 cosΦ

sinϕ1 cosϕ2+cosϕ1sinϕ2 cosΦ

sinϕ2 sinΦ

−cosϕ1sinϕ2− sinϕ1cosϕ2 cosΦ

− sinϕ1sinϕ2+cosϕ1cosϕ2 cosΦ

cosϕ2 sinΦ

sinϕ1 sinΦ −cosϕ1sinΦ cosΦ

$

%

& & & & & & & &

'

(

) ) ) ) ) ) ) )

A=Z2XZ1= (hkl)[uvw]


WhenyoumulDplyeverythingout…

11

CompareMatrices

aij = Crystal

Sampleb1 t1 n1b2 t2 n2b3 t3 n3

!

"

# #

$

%

& &

cosϕ1 cosϕ2− sinϕ1sinϕ2 cosΦ

sinϕ1 cosϕ2+cosϕ1sinϕ2 cosΦ

sinϕ2 sinΦ

−cosϕ1sinϕ2− sinϕ1cosϕ2 cosΦ

− sinϕ1sinϕ2+cosϕ1cosϕ2 cosΦ

cosϕ2 sinΦ

sinϕ1 sinΦ −cosϕ1sinΦ cosΦ

$

%

& & & & & & & &

'

(

) ) ) ) ) ) ) )

[uvw] [uvw] (hkl)(hkl)


Basicidea:thecompleteorientaDonmatrixthatdescribesanorientaDonmustbenumericallythesame,coefficientbycoefficient,regardlessofwhetheritisconstructedfromtheEulerangles,orfromtheMillerindices.ThereforewecanequatethetwomatrixdescripDons,entrybyentry.

≡

12

MillerindicesfromEuleranglematrix

ComparethematrixformedfromtheMillerindiceswiththeEuleranglematrix.Extractindicesfromthe1stand3rdcolumns

h = nsinΦsinϕ2k = nsinΦcosϕ2

l = ncosΦu = # n cosϕ1cosϕ2 − sinϕ1sinϕ2 cosΦ( )

v = # n −cosϕ1sinϕ2 − sinϕ1 cosϕ2 cosΦ( )w = # n sinΦ sinϕ1

n,n’=arbitraryfactorstomakeintegersfromrealnumbersObj/notationAxisTransformationMatrixEulerAnglesComponents

13

EuleranglesfromOrientationMatrix

Also,ifthesecondEulerangleistooclosetozero,or180°,thenthestandardformulaefailbecausesine(Φ)approacheszero(seenextslide).Theformulaetotherightdealwiththisspecialcase,wherethe1stand3rdanglesarelinearlydependent;distribuDngtherotaDonbetweenthemisarbitrary.

if a33 ≈1, Φ = 0, ϕ1 = ATAN2 a12, a11( ) / 2, and ϕ2 =ϕ1

Corrected-a32informulaforφ118thFeb.05;correcteda33=1case13thJan08;addeda33~-113iv14

€

ϕ1 = tan−1 a31 sinΦ−a32 sinΦ%

& '

(

) * = ATAN2 a31 sinΦ,−a32 sinΦ( )

Φ = cos−1 a33( )

ϕ2 = tan−1 a13 sinΦa23 sinΦ%

& '

(

) * = ATAN2 a13 sinΦ,a23 sinΦ( )

Notes:therangeofinversecosine(ACOS)is0-π,whichissufficientforΦ;fromthis,sin(Φ)canbeobtained.Therangeofinversetangentis0-2π,sonumericallyonemustusetheATAN2(y,x)funcDon)tocalculateφ1andφ2.CauDon:inExcel,onehasATAN2(x,y),whichisthereverseorderofargumentscomparedtotheusualATAN2(y,x)inFortran,C(use‘doubleatan2(doubley,doublex);‘)etc.!

if a33 ≈ −1, Φ = 0, ϕ1 = ATAN2 a12, a11( ) / 2, and ϕ2 = −ϕ1

14

SpecialCase:Φ=0

cosϕ1 cosϕ2−sinϕ1 sinϕ2

sinϕ1 cosϕ2+cosϕ1 sinϕ2

0

−cosϕ1 sinϕ2−sinϕ1 cosϕ2

−sinϕ1 sinϕ2+cosϕ1 cosϕ2

0

0 0 1

"

#

$$$$$$$$$$

%

&

''''''''''

=

cos2ϕ1 sin2ϕ1 0−sin2ϕ1 −cos2ϕ1 00 0 1

"

#

$$$$

%

&

''''

A=Z2IZ1=


Setφ1 = φ2

IistheIdentitymatrix

15

SpecialCase:Φ=180°,cosΦ=-1

cosϕ1 cosϕ2+sinϕ1 sinϕ2

sinϕ1 cosϕ2−cosϕ1 sinϕ2

0

−cosϕ1 sinϕ2+sinϕ1 cosϕ2

−sinϕ1 sinϕ2−cosϕ1 cosϕ2

0

0 0 1

"

#

$$$$$$$$$$

%

&

''''''''''

=

cos2ϕ1 sin2ϕ1 0sin2ϕ1 −cos2ϕ1 00 0 1

"

#

$$$$

%

&

''''

A=Z2XZ1=


Setφ1 = -φ2

X =1 0 00 −1 00 0 −1

"

#

$$$

%

&

'''

Part2:CrystalandSampleSymmetry16

This section provides a qualitative description of symmetry and its effects

17

Crystalvs.SampleSymmetry•  AnunderstandingoftheroleofsymmetryisessenDalin

texture.•  TwoseparateanddisDnctformsofsymmetryare

relevant:–  CRYSTALsymmetry–  SAMPLEsymmetry

•  Crystalsymmetryisalwayspresent(evenif,inprinciple,heavilydefectedcrystalsmayexhibitlowersymmetry)andmustalwaysbeaccountedfor.SamplesymmetryisonlypresentonastaDsDcalbasis,i.e.onlyapolycrystalcanexhibitsamplesymmetryforthetextureasawhole.

•  TypicalusageliststhecombinaDoncrystal-samplesymmetryinthatorder,e.g.cubic-orthorhombic.

18

Take-HomeMessageTheessenDalpointsofthislectureare:1.  Crystalsymmetrymeansthatacrystalcanberotatedinvariouswaysthat

leaveitunchanged(physicallyindisDnguishable).InorientaDonspace,theequivalentresultisthatanygivenorientaDonisrelatedtoexactlyasmanyotherorientaDonsastherearesymmetryoperators(understandableintermsofgrouptheory).

2.  BecausetherearemulDpleequivalentpoints,weusuallydivideupthefullorientaDonspace(e.g.360°x180°x360°inEulerspace)anduseonlyasmallporDonofit(e.g.90°x90°x90°).

3.  Samplesymmetryhasthesamesortofeffectascrystalsymmetry,eventhoughitisastaDsDcalsymmetry(andorientaDonsrelatedbyasamplesymmetryelementarephysicallydisDnguishableanddohaveamisorientaDonbetweenthem.Samplesymmetryisevidentinpolefigureswhereascrystalsymmetryisevidentininversepolefigures.

4.  Unfortunately,Eulerspace(sphericalangles)meansthatthedividingplaneshaveoddshapesandsoforthecommoncubic-orthorhombiccombinaDon,wehave3copiesofthefundamentalzoneintherange0-90°foreachangle.Rodriguesspaceisfarsimplerinthisrespectbecausesymmetryoperatorsalwaysleadtoflatdividingplanes.

Deformationvs.SampleSymmetry•  Notethatthetextureofamaterialisaconsequenceof

itsthermomechanicalhistory(andthereforeofferscluesaboutthathistory).

•  The(staDsDcal)samplesymmetryisdirectlyrelatedtothesymmetryoftheprecedingdeformaDon.

•  Ingeneral,thesamplesymmetryreflectsthelowestsymmetrydeformaDonthatwasimposedonthematerial.

•  RollingisaplanestraindeformaDon,withorthorhombicsymmetry.Wecallit“planestrain”because(intheidealcase)allthestrainoccursinoneplane,theRD-NDplanewithzerostrainintheTD(paralleltotheaxesoftherolls).Torsionisasimpleshear,withmonoclinicsymmetry.Wiredrawingisanaxi-symmetricdeformaDonwithcylindricalsymmetry.UpseCngoruniaxialcompressionisalsoanaxi-symmetricdeformaDonwithcylindricalsymmetry.

19

Rolling:anvilWire.com

torsion:ejsong.com

20

PoleFigureforWireTexture•  (111)polefigureshowinga

maximumintensityataspecificanglefromaparDculardirecDoninthesample,andshowinganinfiniterotaDonalsymmetry(C∞).

•  F.A.=FiberAxis(paralleltothelengthofthewire)

•  AparDcularcrystaldirecDoninallcrystalsisalignedwiththisfiberaxis(i.e.aparDcularsampledirecDon).

•  Foracubicmaterial,thiscombinaDonwouldbeclassedascubic-cylindrical.

Inthiscase,<100>//F.A.

{111}forth-armoury.com

21

EffectofSymmetry

TheessenDalpointaboutapplyingasymmetryoperaDonisthat,onceithasbeendone,youcannottellthatanythinghaschanged.Inotherwords,therotatedorreflectedobjectisphysicallyindisDnguishablefromwhatyoustartedwith.

•  IllustraDonof3-fold,4-fold,6-foldrotaDonalsymmetry

NotethesymbolsusedtodenotethedifferentrotaDonalsymmetryelements.

EffectofSymmetry:Example

•  Notehowonecanre-labeltheaxesbutleavethephysicalobject(crystal)unchanged.

•  Whichsymmetryoperatorwasapplied(fromO(432))?

22

[100]

[100][010] [010][001]

[001]

23

StereographicprojecDonsofsymmetryelementsandgeneralpolesinthecubicpointgroupswithHermann-MauguinandSchoenfliesdesignaDons.Notethepresenceoffourtriadsymmetryelementsinallthesegroupson<111>.Cubicmetalsmostlyfallunder. Groups:

mathemaDcalconcept,veryusefulforsymmetry

m3m

24

SampleSymmetry

Torsion,shear:Monoclinic,2.

Rolling,planestraincompression,mmm.

Axisymmetric:C∞Otherwise,triclinic.

25

FundamentalZone•  Thefundamentalzone(orasymmetricunit)istheporDonorsubsetof

orientaDonspacewithinwhicheachorientaDon(ormisorientaDon,whenwelaterdiscussgrainboundaries)isdescribedbyasingle,uniquepoint.

•  ThefundamentalzoneistheminimumamountoforientaDonspacerequiredtodescribeallorientaDons.

•  Example:thestandardstereographictriangle(SST)fordirecDonsincubiccrystals.

•  Thesizeofthefundamentalzonedependsontheamountofsymmetrypresentinbothcrystalandsamplespace.Moresymmetry⇒smallerfundamentalzone.

•  NotethatinEulerspace,the90x90x90°regiontypicallyusedforcubic[crystal]+orthorhombic[sample]symmetryisnotafundamentalzonebecauseitcontains3copiesoftheactualzone!

26

SymmetryIssues•  Crystalsymmetryoperatesinaframea<achedtothecrystalaxes.

•  BasedonthedefiniDonofEulerangles,crystalsymmetryelementsproducerelaDonsbetweenthesecond&thirdangles.

•  Samplesymmetryoperatesinaframea<achedtothesampleaxes.

•  SamplesymmetryproducesrelaDonsbetweenthefirst&secondangles.

•  ThecombinaDonofcrystalandsamplesymmetryiswri<enascrystal-sample,e.g.cubic-orthorhombic,orhexagonal-triclinic.

27

ChoiceofSectionSize•  Quad,Diadsymmetryelementsareeasytoincorporate,butTriadsarehighlyinconvenient.

•  Four-foldrotaDonelements(andmirrorsintheorthorhombicgroup)areusedtolimitthethird,φ2,(first,φ1)anglerangeto0-90°.

•  Secondangle,Φ,hasrange0-90°(diffracDonaddsacenterofsymmetry).

28 SectionSizesCrystal-Sample

•  Cubic-Orthorhombic:0≤φ1≤90°,0≤Φ≤90°,0≤φ2≤90°

•  Cubic-Monoclinic:0≤φ1≤180°,0≤Φ≤90°,0≤φ2≤90°

•  Cubic-Triclinic:0≤φ1≤360°,0≤Φ≤90°,0≤φ2≤90°

•  But,theselimitsdonotdelineateafundamentalzone.

29 φ2

Φ

φ1

Takeapoint,e.g.“B”;operateonitwiththe3-foldrotaDonaxis(bluetriad);thesetofpointsrelatedbythetriadareB,B’,B’’,withB’’’beingthesamepointasB.Thepointis,thatasyouoperateonapoint(orientaDon)withasymmetryoperaDon,ingeneralalltheEulerangleschange.

Pointsrelatedbytriadsymmetryelementon<111>(triclinic

samplesymmetry)

[Bunge]

30

Effectof3-foldaxis

sectioninφ1cutsthroughmorethanonesubspace

[Bunge]

RegionsI,IIandIIIarerelatedbythetriadcrystalsymmetryelement,i.e.120°about<111>,combinedwithsamplesymmetry.

Anglesmeasuredinradians

31

Exampleof3-foldsymmetryThe“S”component,{123}<634>hasangles{59,37,63}also{27,58,18},{53,74,34}andoccursinthreerelatedlocaDonsinEulerspace.10°sca<ershownaboutcomponent;shapeisnotsphericalbecausethevolumeelementsizevariesassin(Φ).RegionsI,IIandIIIarerelatedbythetriadcrystalsymmetryelement,i.e.120°about<111>,combinedwithsamplesymmetry.

[Randle&Engler,Xig.5.7]

32

Scomponent

inφ2sections

RegionsI,IIandIIIarerelatedbythetriadsymmetryelement,i.e.120°about<111>.10°sca<ershownabouttheScomponent,sothateachofthe3equivalentposiDonsiscross-secDonedinmulDplesecDons.

[Randle&Engler,Xig.5.7]

33

OrthorhombicSampleSymmetry(mmm)RelationshipsinEuler

Spaceφ1=0° 180° 270° 360°90°

Φ

diad

‘mirror’‘mirror’

2-foldscrewaxischangesφ2byπΦ=180°

Note:thisslideillustrateshowthesetof3diads(+idenDty)insamplespaceoperateonagivenpoint.TherelaDonshiplabeledas‘mirror’isreallyadiadthatactslikea2-foldscrewaxisinEulerspace.

34

Samplesymmetry,detail

TablesforTextureAnalysisofCubicCrystals,SpringerVerlag,1978

35 CrystalSymmetryRelationships(432)inEulerSpace

φ2=0° 180° 270° 360°90°

Φ

4-foldaxis

3-foldaxis

Note:pointsrelatedbytriad(3-fold)havedifferentφ1values

Other4-fold,2-foldaxis:actonφ1also

Φ=180°

36

Crystalsymmetry:detail


37

Howmanyequivalentpoints?•  Forcubic-orthorhombiccrystal+samplesymmetry,weusea

range90°x90°x90°forthethreeangles,givingavolumeof90°2(orπ2/4inradians).

•  Inthe(reduced)spacethereare3equivalentpointsforeachorientaSon(texturecomponent).Bothsampleandcrystalsymmetriesmustbecombinedtogethertofindthesesets.

•  IfwelookwithinthefullorientaSonspacethenthereareasmanyequivalentpointsastherearesymmetryelements.Thusforcubiccrystalsymmetrythereare24equivalentpoints;forhexagonal12etc.

•  Fewer(e.g.Copper)ormore(e.g.cube)equivalentpointsforeachcomponentarefoundinthereducedspaceifthethecomponentcoincideswithoneofthesymmetryelements.

38

VolumeofOrientationSpace•  O(432)has24operators(i.e.order=24);O(222)has4

operators(i.e.order=4):whynotdividethevolumeofEulerspace(8π2,or,360°x180°x360°)by24x4=96togetπ2/12(or,90°x30°x90°)?

•  Answer:weleaveoutatriadaxis(becauseoftheawkwardshapesthatitwouldgiveus),sowedivideby8x4=32togetπ2/4(90°x90°x90°).

•  ThisisanillustraDonofgrouptheoryinacDon.EachsetofsymmetryoperatorsdividesuptheorientaDonspacebyafactorequaltothenumberofelementsinthegroup.

•  Thereasonis,essenDally,aquesDonofcounDng:eachpointinonezonehasexactlyonepointinanotherzonerelatedbyagivenoperator.

39

Grouptheoryapproach•  Again,thisisanillustraDonofgrouptheoryinacDon.Each

setofsymmetryoperatorsdividesuptheorientaDonspacebyafactorequaltothenumberofelementsinthegroup.

•  Crystalsymmetry:acombinaDonof4-and2-foldcrystalaxes(2x4=8elements)reducetherangeofΦfromπtoπ/2,andφ2from2πtoπ/2.

•  Samplesymmetry:the2-foldsampleaxes(4elementsinthegroup)reducetherangeofφ1from2πtoπ/2.

•  Volumeof{0≤φ1,Φ,φ2≤π/2}isπ2/4.•  Thereasonis,essenDally,aquesDonofcounDng:eachpoint

inonezonehasexactlyonepointinanotherzonerelatedbyagivenoperator.

40

SpecialPoints,PartialFibers

Copper: !2!Brass: !3!S: ! !3!Goss: !3!Cube: !8!Dillamore:2!

Humphreys&Hatherly

No.ofpointsinthe90x90x90°space

WhythisvariaDon?Pointsthatfallontheedgemayappearmore,orfewerDmesthanthestandardcountof3.

41

SampleSymmetryRelationshipsinEulerSpace:specialpoints

φ1=0° 180° 270° 360°90°

Φ

diad

diaddiad

Cubeliesonthecorners

Copper,Brass,Gosslieonanedge,sothattheycoincidewithasymmetryelement

Φ=90°

42

3DViewsa)Brassb)Copperc)Sd)Gosse)Cubef)combinedtexture1:{35,45,90},Brass,2:{55,90,45},Brass3:{90,35,45},Copper,4:{39,66,27},Copper5:{59,37,63},S,6:{27,58,18},S,7:{53,75,34},S8:{90,90,45},Goss9:{0,0,0},cube*10:{45,0,0},rotatedcube*Notethatthecubeexistsasalinebetween(0,0,90)and(90,0,0)becauseofthelineardependenceofthe1stand3rdangleswhenthe2ndangle=0.

FigurecourtesyofJae-hyungCho

43

SpecialPoints:Explanations•  Pointscoincidentwithsymmetryaxesmayalsohave

equivalentpoints,o_enontheedge.Cube:shouldbeasinglepoint,buteachcornerisequivalentandvisible.

•  Goss,Brass:asinglepointbecomes3becauseitisontheφ2=0plane.

•  Copper:2pointsbecauseonepointremainsintheinteriorbutanotheroccursonaface;alsotheDillamoreorientaDon.

•  NoteforlateraboutvolumefracDons:althoughthevolumesenclosedbythe12°capture(misorientaDon)distanceshowninthefigurevarywiththe2ndeulerangle,tofirstorderitisthesymmetryplanesthatcontrolthevolumefracDons.SoS,forexample,has3complete“blobs”whereasGosshasonly3x¼andsowillcontainonly~1/4ofthevolumefracDonofS,basedonarandom(uniform)texture.

44 Table 4.F.2. fcc Rolling Texture Components: Euler Angles and Indices

Name Indices Bunge(ϕ1,Φ,ϕ2)RD= 1

Kocks(ψ,Θ,φ)RD= 1

Bunge(ϕ1,Φ,ϕ2)RD= 2

Kocks(ψ,Θ,φ)RD= 2

copper/1st var.

{112}〈111̄〉 40, 65, 26 50, 65, 26 50, 65, 64 39, 66, 63

copper/2nd var.

{112}〈111̄〉 90, 35, 45 0, 35, 45 0, 35, 45 90, 35, 45

S3* {123}〈634̄〉 59, 37, 27 31, 37, 27 31, 37, 63 59, 37, 63S/ 1st var. (312)<0 2 1> 32, 58, 18 58, 58, 18 26, 37, 27 64, 37, 27S/ 2nd var. (312)<0 2 1> 48, 75, 34 42, 75,34 42, 75, 56 48, 75, 56S/ 3rd var. (312)<0 2 1> 64, 37, 63 26, 37, 63 58, 58, 72 32, 58, 72brass/1st var.

{110}〈1̄12〉 35, 45, 0 55, 45, 0 55, 45, 0 35, 45, 0

brass/2nd var.

{110}〈1̄12〉 55, 90, 45 35, 90, 45 35, 90, 45 55, 90, 45

brass/3rd var.

{110}〈1̄12〉 35, 45, 90 55, 45, 90 55, 45, 90 35, 45, 90

Taylor {4 4 11}〈11 11 8̄〉 42, 71, 20 48, 71, 20 48, 71, 70 42, 71, 70Taylor/2nd var.

{4 4 11}〈11 11 8̄〉 90, 27, 45 0, 27, 45 0, 27, 45 90, 27, 45

Goss/1st var.

{110}〈001〉 0, 45, 0 90, 45, 0 90, 45, 0 0, 45, 0

Goss/2nd var.

{110}〈001〉 90, 90, 45 0, 90, 45 0, 90, 45 90, 90, 45

Goss/3rd var.

{110}〈001〉 0, 45, 90 90, 45, 90 90, 45, 90 0, 45, 90

[Kocks,Tomé,Wenk]

45

Meaningof“Variants”•  Theexistenceofvariantsofagiventexturecomponentisa

consequenceof(staDsDcal)samplesymmetry.•  IfonepermutestheMillerindicesforagivencomponent

(forcubics,onecanchangethesignandorder,butnotthesetofdigits),thendifferentvaluesoftheEuleranglesarefoundforeachpermutaDon.

•  Ifapolefigureisplo<edofallthevariants,oneobservesanumberofphysicallydisDnctorientaDons,whicharerelatedtoeachotherbysymmetryoperators(diads,typically)fixedinthesampleframeofreference.

•  EachphysicallydisDnctorientaDonisa“variant”.ThenumberofvariantslisteddependsonthechoiceofsizeofEulerspace(typically90x90x90°)andthealignmentofthecomponentwithrespecttothesamplesymmetry.

46

Part3:QuantitativeTreatmentofSymmetry

•  InsecDon,wedescribethemethodsfordescribingsymmetryoperaDonsasrotaDonmatricesandhowtoapplythemtotexturecomponents,alsoexpressedasmatrices.

•  NotaDon:orientaDon:g(crystal)symmetryoperator:OC(sample)symmetryoperator:OS

47

Rotations:deWinitions•  RotaDonalsymmetryelementsexistwheneveryoucan

rotateaphysicalobjectandresultisindisDnguishablefromwhatyoustartedoutwith.

•  RotaDonscanbeexpressedinasimplemathemaDcalformasunimodularmatrices(or,orthogonalmatrices),o_enwithelementsthatareusuallyeitheroneorzero(butnotalways!).Eachrowandeachcolumnhasunitlength.

•  Whenrepresentedasmatrices,theinverseofarotaDonisequaltothetransposeofthematrix.

•  RotaDonsaretransformaDonsofthefirstkind;determinant=+1.

•  ReflecDons(notneededhere)aretransformaDonsofthesecondkind;determinant=-1.

•  CrystalsymmetryalsocaninvolvetranslaDonsbutthesearenotincludedinthistreatment.

48

RotationMatrixfromAxis-AnglePair

€

gij = δij cosθ + rirj 1− cosθ( )+ εijkrk sinθ

k=1,3∑

=

cosθ + u2 1− cosθ( ) uv 1− cosθ( ) + w sinθ uw 1− cosθ( ) − v sinθuv 1− cosθ( ) − w sinθ cosθ + v 2 1− cosθ( ) vw 1− cosθ( ) + usinθuw 1− cosθ( ) + v sinθ vw 1− cosθ( ) − usinθ cosθ + w2 1− cosθ( )

'

(

) ) )

*

+

, , ,

Wri<enoutasacomplete3x3matrix:

49

SymmetryOperatorexamples•  Diadonz:[uvw]=[001],θ=180°-subsDtutethevaluesofuvwandangleintotheformula

•  4-foldonx:[uvw]=[100]θ=90°

€

gij =

cos180 + 02 1− cos180( ) 0*0 1− cos180( ) +1*sin180 0*1 1− cos180( ) − 0sin1800*0 1− cos180( ) − w sin180 cos180 + 02 1− cos180( ) 0*1 1− cos180( ) + 0sin1800*1 1− cos180( ) + 0sin180 0*1 1− cos180( ) − 0sin180 cos180 +12 1− cos180( )

#

$

% % %

&

'

( ( (

=

−1 0 00 −1 00 0 1

#

$

% % %

&

'

( ( (

€

gij =

cos90 +12 1− cos90( ) 1*0 1− cos90( ) + w sin90 0*1 1− cos90( ) − 0sin900*1 1− cos90( ) − 0sin90 cos90 + 02 1− cos90( ) 1*0 1− cos90( ) +1sin900*1 1− cos90( ) + 0sin90 0*0 1− cos90( ) −1sin90 cos90 + 02 1− cos90( )

#

$

% % %

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'

( ( (

=

1 0 00 0 10 −1 0

#

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% % %

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'

( ( (

50

-

Matrixrepresentationoftherotationpointgroups

Whatisagroup?Agroupisasetofobjectsthatformaclosedset:ifyoucombineanytwoofthemtogether,theresultissimplyadifferentmemberofthatsamegroupofobjects.RotaDonsinagivenpointgroupformclosedsets-tryitforyourself!

Note:the3rdmatrixinthe1stcolumn(x-diad)ismissinga“-”onthe33element;thisiscorrectedinthisslide.Also,inthe2ndfromthebo<om,lastcolumn:the33elementshouldbe+1,not-1.Insomeversionsofthebook,inthelastmatrix(bo<omrightcorner)the33elementisincorrectlygivenas-1;herethe+1iscorrect.

Kocks,Tomé,Wenk:Ch.1TableII

The 4 operators enclosed in orange boxes are also the 222 point group, appropriate to orthorhombic symmetry

51 Nomenclatureforrotationelements

•  Ingeneral,thenotaDonidenDfiestheaxisaboutwhichtherotaDonisperformed.

•  Thusa2-foldaxis(180°rotaDon)aboutthez-axisisknownasaz-diad,orC2z,orL0012

•  Triad(120°rotaDon)about[111]asa111-triad,or,120°-<111>,or,L1113etc.

•  Apointgroupiswri<enasO(432),forexample,wheretheentryintheparenthesesindicatesthesymmetry.

52

Howtouseasymmetryoperator?1.  ConvertMillerindicesthatrepresentanorientaDonortexture

componenttoa(orientaDon)matrix.2.  PerformmatrixmulDplicaDonwiththesymmetryoperator,Oc,

andtheorientaDonmatrix,g,toobtaintheneworientaDon(matrix),g’.Becarefultogettheordercorrect!Forthestandardcoordinate/axistransformaDonusedhere,thecrystalsymmetryoperatoralwaysle_-mulDplies(pre-mulDplies)theorientaDon.SincetheorientaDonmatrixconvertsfromsampletocrystalaxes,youcanthinkofapplyingcrystalsymmetryaYertransformingaquanDtyintothecrystalframe.

g’=OCg3.  ConvertthematrixbacktoMillerindices.4.  Thetwosetsofindicesrepresent(forcrystalsymmetry)

indisDnguishableobjects.

53

Example

Goss:(011)[100]:=

1 0 00 1

212

0 − 12

12

"

#

$$$$$$

%

&

''''''

−1 0 00 −1 00 0 1

"

#

$$$

%

&

'''

−1 0 00 − 1

2− 1

20 − 1

212

"

#

$$$$$$

%

&

''''''

Samplesymmetry:rightmulDply(post-mulDply)thematrix.

g

g’

gOC

g’=OCg

whichis(0-11)[-100]

Pre-mulDplybyz-diad: 1 0 00 1

212

0 − 12

12

"

#

$$$$$$

%

&

''''''

54

CrystalsymmetryO(432)

actingonthe(231)[3-46]Scomponent

0

40

80

120

160

0 50 100 150 200 250 300 350

S_xtal_symm

52

232 232

52

121

152

52

301

332

52

232 232

301

121

152

332

152

301

121

152

301

332332

121

Φ

φ2

φ1 values noted

Notethatall24variantsarepresent

Homeworkexercise:youcanmakethissameplotbyusing,e.g.,MatlabtocomputethematrixofeachsymmetricallyequivalentorientaDonandthenconverDngeachnewmatrixtoEulerangles;thenuse“sca<er3”toplotin3D.

Eulerangles:(52.9,74.5,33.7)

55

OrderofMatrices•  AssumethatweareusingthestandardaxistransformaSon(passiverotaSon)definiDonoforientaDon(asfound,e.g.inBunge’sbook).

•  Orderdependsonwhethercrystalorsamplesymmetryelementsareapplied.

•  Foranoperatorinthecrystalsystem,Oxtal,theoperatorpre-mulDpliestheorientaDonmatrix.

•  Thinkofthesequenceasfirsttransformintocrystalcoordinates,thenapplycrystalsymmetryonceyouareincrystalcoordinates.

•  Forsampleoperator,Osample,post-mulDplytheorientaDonmatrix.

56

SymmetryRelationships•  Notethattheresultofapplyinganyavailableoperatorisequivalentto

(physicallyindisDnguishableinthecaseofcrystalsymmetry)fromthestarDngconfiguraDon(notmathemaDcallyequalto!).

•  Also,ifyouapplyasamplesymmetryoperator,theresultisgenerallyphysicallydifferentfromthestarDngposiDon.Why?!BecausethesamplesymmetryisonlyastaSsScalsymmetry,notanexact,physicalsymmetry.TwoorientaDonsrelatedviaasamplesymmetryarephysicallydisDnct(bycontrasttocrystalsymmetry).

•  DonotconfusethiswiththestandardexpressionforthetransformaDonofasecondranktensor:T’=OTOT

€

" g = OcrystalgOsample=← → % g

NB:ifonewritesanorientaDonasanacSverotaSon(asinconDnuummechanics),thentheorderofapplicaDonofsymmetryoperatorsisreversed:premulDplybysample,andpostmulDplybycrystalsymmetry!

57

Symmetry:How-to•  Howtofindallthesymmetricallyequivalentpoints?•  ConvertthecomponentofinteresttomatrixnotaDon.•  Makealistofallthesymmetryoperatorsinmatrixform(24forcubiccrystal

symmetryor12forhexagonalcrystalsymmetry,and4fororthorhombicsamplesymmetry).Thisisthesamelistofoperatorsshownas3x3matricesintheTablequotedfromtheKocksbook,slide35inthisset.

•  Crystalsymmetry:idenDty(1),plus90/180/270°about<100>(9),plus180°about<110>(6),plus120/240°about<111>(8).

•  Samplesymmetry:idenDty,plus180°aboutRD/TD/ND(4).•  Loopthrougheachsymmetryoperatorinturn,withseparateloopsfor

sampleandcrystalsymmetry.•  Foreachresult,convertthematrixtoEulerangles.•  Howcanyoubesurethatyouhaveappliedtheoperatorscorrectly?Answer:

makeapolefigureofthesetofsymmetricallyrelatedorientaDons.Crystalsymmetryrelatedpointsmustplotontopofoneanotherwhereassamplesymmetryrelatedpointsgiveriseto(ingeneral)mulDplesetsofpoints,relatedbythesamplesymmetrythatshouldbeevidentinthepolefigure.

CrystalSymmetryCorrectlyApplied•  IfyoustartwiththeGossorientaDonandapplycrystal

symmetrybyle_-mulDplying,youshouldobtainpolefigureslikethis.TheapplicaDonofcrystalsymmetrydidnotproduceanyneworientaDonsasfarasthepolefigureisconcerned.Notethatx(//RD)pointsrightinthisexample.

58

PlotscourtesyofVahidTari,2011

CrystalSymmetryIncorrectlyApplied•  IfyoustartwiththeGossorientaDonandapplycrystal

symmetrybyright-mulDplying,youshouldobtainpolefigureslikethis.TheapplicaDonofcrystalsymmetrydoesproduceneworientaDonsinthepolefiguresbutthisincorrect.TheresultofapplyingcrystalsymmetrymustbeindisDnguishablefromtheiniDalstate.Notethatxpointsrightinthisexample.

59

PlotscourtesyofVahidTari,2011

TextureComponentVariants•  VariantsofagiventexturecomponentshareMillerindicesthatarethesame

exceptforwhatevervariaDonsareallowedbycrystalsymmetry.ThisresultsinorientaDonsthatareusually,butnotinvariably,differenti.e.havefinitemisorientaDonsbetweenthem.

•  Toobtainasetofvariants,oneappliestheappropriatesetofsamplesymmetryoperatorstoasingleorientaDonthatdefinesthecomponent,e.g.fromapairofMillerindicesforhkl//ND,uvw//RD.Fororthorhombicsamplesymmetry,forexample,{Osample}=O(222)whichhasfouroperators,sothereareamaximumoffourvariants.

•  Asaconsequenceofthewaythateachcomponentalignswiththesamplesymmetryoperators,thecubeandGosscomponentshaveonlyonevariant,thecopperandbrasshavetwo,whereastheScomponenthasallfourvariants.

60

g{ }variants = g Osample{ }

61

Summary•  SymmetryoperatorshavebeenexplainedintermsofrotaDonmatrices,withexamplesofhowtoconstructthemfromtheaxis-angledescripDons.

•  TheeffectofsymmetryontherangeofEuleranglesneeded,andtheshapeoftheplo�ngregion.

•  TheparDculareffectofsymmetryoncertainnamedtexturecomponentsfoundinrolledfccmetalshasbeendescribed.

•  Inlaterlectures,wewillseehowtoperformthesameoperaDonsbutwithoronRodriguesvectorsandquaternions.

62

SupplementalSlides•  Thefollowingslidesprovide:•  DetailsoftherangeofEulerangles,andtheshapeoftheplo�ngspacerequiredforCODs(crystalliteorientaDondistribuDons)orSODs(sampleorientaDondistribuDons)asafuncDonofthecrystalsymmetry;

•  AddiDonalinformaDonaboutthedetailsofhowsymmetryelementsrelatedifferentlocaDonsinEulerspace.

63

•SincecrystalsymmetryoperatorsarecloselylinkedtolowindexdirecDons,itishelpfultoconvertaxis-angledescripDonstomatricesandviceversa.•TherotaDoncanbeconvertedtoamatrix,g,(passiverotaDon)bythefollowingexpression,whereδistheKroneckerdelta,θistherotaDonangle,ristherotaDonaxis,andεisthepermutaDontensor.

€

gij = δij cosθ + rirj 1− cosθ( )+ εijkrk sinθ

k=1,3∑

AxisTransformationfromAxis-AnglePair

Comparewithactiverotationmatrix!24

A rotation is commonly written as ( ,θ) or as (n,ω). The figure illustrates the effect of a rotation about an arbitrary axis, OQ (equivalent to and n) through an angle α (equivalent to θ and ω).

ˆ r

ˆ r

gij = δ ij cosθ − eijknk sinθ

+ (1− cosθ)ninj(This is an active rotation: apassive rotation ≡ axistransformation)

Rotations (Active): Axis- Angle Pair

64

Geometryof{hkl}<uvw>

e1//[uvw]^

e’1^

e2//t^

e’2^

e3//(hkl)^e’3

^

^

[001][010]

[100]

MillerindexnotaDonoftexturecomponentspecifiesdirecDoncosinesofcrystaldirecDons//tosampleaxes.Formthesecondaxisfromthecross-productofthe3rdand1staxes.

SampletoCrystal(primed)

t=hklxuvwObj/notationAxisTransformationMatrixEulerAnglesComponents

65

Matrix,MillerIndices

•  ThegeneralRotaDonMatrix,a,canberepresentedasinthefollowing:

•  HeretheRowsarethedirecDoncosinesforthe3crystalaxes,[100],[010],and[001]expressedinthesamplecoordinatesystem(polefigure).

[100]xtaldirection

[010]xtaldirection

[001]xtaldirection

€

a11 a12 a13a21 a22 a23a31 a32 a33

"

#

$ $ $

%

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' ' '


66

Matrix,MillerIndices

•  Thecolumnsrepresentcomponentsofthreeotherunitvectors:

[uvw]≡RD TD ND≡(hkl)

€

a11 a12 a13a21 a22 a23a31 a32 a33

"

#

$ $ $

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' ' '


• HeretheColumnsarethedirecDoncosines(i.e.hkloruvw)forthesampleaxes,RD,TDandNormaldirecDonsexpressedinthecrystalcoordinatesystem.Comparetoinversepolefigures.

67

CrystalSymmetryElemente.g.rotationon[001](associatedwithφ2)

SampleSymmetryElemente.g.diadonND

(associatedwithφ1)

[Bunge]

68

Table of Effect of Symmetry Elements on Euler Angles

Symmetry Element Bunge Symm-etric

(Kocks)

2-fold axis on Sample X π-φ1 π-Φ φ2±π −Ψ π-Θ φ±π2-fold axis on Sample Y -φ1 π-Φ φ2±π π-Ψ π-Θ φ±π2-fold axis on Sample Z φ1-π Φ φ2 Ψ±π Θ φ

2-fold axis on Crystal x φ1±π π-Φ π-φ2 Ψ±π π-Θ -φ2-fold axis on Crystal y φ1±π π-Φ −φ2 Ψ±π π-Θ π-φ2-fold axis on Crystal z φ1 Φ φ2±π Ψ Θ φ±π3-fold axis on Crystal z φ1 Φ φ2±2π/3 Ψ Θ φ±2π/34-fold axis on Crystal z φ1 Φ φ2±π/2 Ψ Θ φ±π/26-fold axis on Crystal z φ1 Φ φ2±π/3 Ψ Θ φ±π/3

[Kocks]

Thesearesimplecases:seedetailedchartsattheendofthissetofslides

69

Othersymmetryoperators•  Symmetryoperatorsofthesecondkind:theseoperatorsincludetheinversioncenterandmirrors;determinant=-1.

•  Theinversion(=centerofsymmetry)simplyreversesanyvectorsothat(x,y,z)->(-x,-y,-z).

•  Mirrorsoperatethroughamirroraxis.Thusanx-mirrorisamirrorintheplanex=0andhastheeffect(x,y,z)->(-x,y,z).

70 Examplesofsymmetryoperators

•  Diadonz:(1stkind)

• Mirroronx:(2ndkind)

−1 0 00 1 00 0 1

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#

$ $ $

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' ' '

−1 0 00 −1 00 0 1

"

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' ' '

−1 0 00 −1 00 0 −1

"

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InversionCenter:(2ndkind)

71

Howmanyequivalentpoints?•  EachsymmetryoperatorrelatesapairofpointsinorientaDon(Euler)space.

•  ThereforeeachoperatordividestheavailablespacebyafactoroftheorderoftherotaDonaxis.Infact,theorderofgroupissignificant.Iftherearefoursymmetryoperatorsinthegroup,thenthesizeoforientaDonspaceisdecreasedbyfour.

•  ThissuggeststhattheorientaDonspaceissmallerthanthegeneralspacebyafactorequaltothenumberofgeneralpoles.

72

CrystalliteOrientationDistribution

SectionsatconstantvaluesofthethirdangleKocks:Ch.2Xig.36

73

SampleOrientationDistribution

SectionsatconstantvaluesoftheXirstangleKocks:Ch.2Xig.37

74


75


76

Determinantofamatrix•  MulDplyeachsetofthreecoefficientstakenalonga

diagonal:tople_tobo<omrightareposiDve,bo<omle_totoprightnegaDve.

•  |a|=a11a22a33+a12a23a31+a13a21a32-a13a22a31-a12a21a33-a11a32a23=εi1i2…inai11ai22…aiNN

a11 a12 a13a21 a22 a23a31 a32 a33

!

"

# # #

$

%

& & &

+ -

77

SymmetryandProperties•  Forlater:whenyouuseamaterialproperty(ofasingle

crystal,forexample)toconnecttwophysicalquanDDes,thenapplyingsymmetrymeansthattheresultisunchanged.Inthiscasethereisanequality.Thisequalityallowsustodecreasethenumberofindependentcoefficientsrequiredtodescribeananisotropicproperty(Nye).

78

Anisotropy•  GivenanorientaDondistribuDon,f(g),onecanwritethe

followingforanytensorpropertyorquanDty,t,wheretherangeofintegraDonisoverthefundamentalzoneofphysicallydisDnguishableorientaDons,SO(3)/G.

•  “SO(3)”meansallpossibleproperrotaDonsin3Dspace(butnotreflecDons);“G”meanstheset(group)ofsymmetryoperators,e.g.(432)fortheproperrotaDonsinthecubicsystem;SO(3)/GmeansthespaceofrotaDonsdividedbythesymmetrygroup.

t = t g( ) f g( )dgSO 3( ) / G∫

79

Homework•  Thisdescribesthepartofthehomework(Hwk3)thatdealswithlearninghowtoapply

symmetryoperatorstocomponentsandfindallthesymmetricallyrelatedposiDonsinEulerspace.–  A.Writethesymmetryoperatorsforthecubiccrystalsymmetry(pointgroup432)asmatricesintoafile(orasanarrayina

Matlabscript).Itissensibletoputthreenumbersonaline,sothattheappearanceofthenumbersissimilartothewayinwhicha3x3matrixiswri<eninabook.Youcansimplycopywhatwasgivenintheslides(takenfromtheKocksbook).Matlabexamples,withtheIDoftheoperatorwri<enasthe3rdindex:e(:,:,1)=[100;010;001];e(:,:,2)=[100;0-10;00-1];etc.

–  [AlternaDvely,youcanworkoutwhateachmatrixisbasedontheactualsymmetryoperator.Thisismoreworkbutwillshowyoumoreofwhatisbehindthem.]

–  B.Writethesymmetryoperatorsfortheorthorhombicsamplesymmetry(pointgroup222)asmatricesintoaseparatefile(orasanarrayinaMatlabscript).

–  C.Writeacomputercodethatreadsinthetwosetsofsymmetryoperators(cubiccrystal,asksforanorientaDonspecifiedas(six)Millerindices,(h,k,l)[u,v,w],andcalculateseachneworientaDon,whichshouldbewri<enoutasEulerangles(meaning,converttheresult,whichisamatrix,backtoEulerangles).NotethattheidenDtyoperatorisalwaysincludeasthefirstsymmetryoperator.So,evenifyouapplynosymmetry,intermsofloopsinyourprogram,yougoatleastoncethrougheachloopwherethefirstDmethroughisapplyingtheidenDtyoperator(onesonthediagonal,zeroselsewhere).

–  D.Listalltheequivalentpointsfor{123}<63-4>fortriclinic(meaning,nosamplesymmetry).IneachlisDng,idenDfythepointsthatfallintothe90x90x90regiontypicallyusedforplo�ng.

–  E.Listalltheequivalentpointsfor{123}<63-4>formonoclinic(useonlytheND-diadoperator,i.e.180°aboutthesamplez-axis).IneachlisDng,idenDfythepointsthatfallintothe90x90x90regiontypicallyusedforplo�ng.

–  F.Listalltheequivalentpointsfor{123}<63-4>fororthorhombicsamplesymmetries(useall3diadsinaddiDontotheidenDty).IneachlisDng,idenDfythepointsthatfallintothe90x90x90regiontypicallyusedforplo�ng.

–  G.Repeat3fabovefortheCoppercomponent,(112)[11-1].–  H.Howmanydifferentpointsdoyoufindforeachofthethreesamplesymmetries?–  I.Howmanypointsfallwithinthe90x90x90regionthatwetypicallyuseforplo�ngorientaDondistribuDons?

•  Studentsmaycodetheprobleminanyconvenientlanguage(Excel,C++,Pascal….)butarestronglyencouragedtodotheexercisesinMATLAB:beverycarefuloftheorderinwhichyouapplythesymmetryoperators!

Effect of Symmetry on the Orientation...

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