Geometry of Crystals Polycrystals and Phase Transformations

245 Pages·2018·55.605 MB·english

by  Harshad K. D. H. Bhadeshia

Geometry Crystals
of ,
Polycrystals Phase
, and
Transformations
Harshad K. D. H. Bhadeshia
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2018 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
Version Date: 20170802
International Standard Book Number-13: 978-1-138-07078-3 (Hardback)
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Contents
PrefaceandAcknowledgments xi
Author xiii
Acronyms xv
I BasicCrystallography 1
1 IntroductionandPointGroups 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Thelattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
PrimitiverepresentationofCubic-F . . . . . . . . . . . . . . . 7
1.3 Bravaislattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Numberofequivalentindices . . . . . . . . . . . . . . . . . . . 12
1.5 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Weisszonelaw . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Symmetryoperations . . . . . . . . . . . . . . . . . . . . . . . . 15
Five-foldrotation . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.9 Crystalstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Structureofgraphene . . . . . . . . . . . . . . . . . . . . . . . 20
1.10 Pointgroupsymmetry . . . . . . . . . . . . . . . . . . . . . . . . 21
Pointsymmetryofchesspieces . . . . . . . . . . . . . . . . . . 23
Octahedralintersticesiniron . . . . . . . . . . . . . . . . . . . 23
1.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 StereographicProjections 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Projectionofsmallcircle . . . . . . . . . . . . . . . . . . . . . 29
2.2 Utilityofstereographicprojections . . . . . . . . . . . . . . . . . 31
2.3 Stereographicprojection:constructionandcharacteristics . . . . . 32
RadiusoftraceofgreatcircleonWulffnet . . . . . . . . . . . . 35
Tracesofplates . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Stereographicrepresentationofpointgroups . . . . . . . . . . . . 38
Mirrorplaneequivalentto2 . . . . . . . . . . . . . . . . . . . 38
Thepointgroups3mandm3 . . . . . . . . . . . . . . . . . . . 39
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 StereogramsforLowSymmetrySystems 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Hexagonalsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Anglesinthehexagonalsystem . . . . . . . . . . . . . . . . . . 48
Growthdirectionofcementitelaths . . . . . . . . . . . . . . . . 50
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 SpaceGroups 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Screwaxesandglideplanes . . . . . . . . . . . . . . . . . . . . 55
4.3 Cuprite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Locationofatomsincupritecell . . . . . . . . . . . . . . . . . . 58
SpacegroupofFe-Si-Ucompound . . . . . . . . . . . . . . . . 60
Cementite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Diamondandzincsulfide . . . . . . . . . . . . . . . . . . . . . 62
4.5 Shapeofprecipitates . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 TheReciprocalLatticeandDiffraction 67
5.1 Thereciprocalbasis . . . . . . . . . . . . . . . . . . . . . . . . . 67
Weisszonelaw . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Crystallographyofdiffraction . . . . . . . . . . . . . . . . . . . 70
5.3 Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Solutionofelectrondiffractionpattern . . . . . . . . . . . . . . 71
Anotherdiffractionpatternsolution . . . . . . . . . . . . . . . 73
Disorderedandorderedcrystals . . . . . . . . . . . . . . . . . 74
5.4 Diffractionfromthincrystals . . . . . . . . . . . . . . . . . . . . 76
5.5 Neutrondiffraction . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 DeformationandTexture 81
6.1 Slipinasingle-crystal . . . . . . . . . . . . . . . . . . . . . . . 81
Elongationduringsingle-crystaldeformation . . . . . . . . . . 83
εmartensite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Orientationdistributionfunctions . . . . . . . . . . . . . . . . . . 88
Euleranglesrelatingtwoframes . . . . . . . . . . . . . . . . . 89
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 Interfaces,OrientationRelationships 93
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Symmetricaltiltboundary . . . . . . . . . . . . . . . . . . . . . 94
7.3 Coincidencesitelattices . . . . . . . . . . . . . . . . . . . . . . 96
7.4 Representationoforientationrelationships . . . . . . . . . . . . . 96
Coordinatetransformation . . . . . . . . . . . . . . . . . . . . 97
7.5 MathematicalmethodfordeterminingΣ . . . . . . . . . . . . . . 100
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8 CrystallographyofMartensiticTransformations 103
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Shapedeformation . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3 Bainstrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
II A FewAdvanced Methods 111
9 OrientationRelations 113
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.2 Cementiteinsteels . . . . . . . . . . . . . . . . . . . . . . . . . 114
Bagaryatskiorientationrelationship . . . . . . . . . . . . . . . 114
9.3 Relationsbetweenfccandbcccrystals . . . . . . . . . . . . . . . 118
Kurdjumov–Sachsorientationrelationship . . . . . . . . . . . . 119
9.4 Relationshipsbetweengrainsofidenticalstructure . . . . . . . . 122
Axis-anglepairsandrotationmatrices . . . . . . . . . . . . . . 122
Doubletwinning . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.5 Themetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Planenormalsanddirectionsinanorthorhombicstructure . . . 126
9.6 Moreaboutthevectorcrossproduct . . . . . . . . . . . . . . . . 127
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10 Homogeneousdeformations 133
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
10.2 Homogeneousdeformations . . . . . . . . . . . . . . . . . . . . 134
Bainstrain:undistortedvectors . . . . . . . . . . . . . . . . . 136
10.3 Eigenvectorsandeigenvalues . . . . . . . . . . . . . . . . . . . . 138
Eigenvectorsandeigenvalues . . . . . . . . . . . . . . . . . . . 139
10.4 Stretchandrotation . . . . . . . . . . . . . . . . . . . . . . . . . 140
10.5 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Deformationsandinterfaces . . . . . . . . . . . . . . . . . . . 143
10.6 Topologyofgraindeformation . . . . . . . . . . . . . . . . . . . 144
10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
11 Invariant-planestrains 153
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Tensiletestsonsingle-crystals . . . . . . . . . . . . . . . . . . 157
Transitionfromeasyglidetoduplexslip . . . . . . . . . . . . . 160
11.2 Deformationtwins . . . . . . . . . . . . . . . . . . . . . . . . . 161
Twinsinfcccrystals . . . . . . . . . . . . . . . . . . . . . . . . 161
11.3 Correspondencematrix . . . . . . . . . . . . . . . . . . . . . . . 164
11.4 AnalternativetotheBainstrain . . . . . . . . . . . . . . . . . . 165
11.5 Steppedinterfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Interactionofdislocationswithinterfaces . . . . . . . . . . . . 168
fcctohcptransformationrevisited . . . . . . . . . . . . . . . . 172
11.6 Conjugateofaninvariant-planestrain . . . . . . . . . . . . . . . 178
Combinedeffectoftwoinvariant-planestrains . . . . . . . . . . 179
11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
12 Martensite 185
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.2 Shapedeformation . . . . . . . . . . . . . . . . . . . . . . . . . 185
12.3 Interfacialstructureofmartensite . . . . . . . . . . . . . . . . . . 188
12.4 Phenomenologicaltheoryofmartensitecrystallography . . . . . . 190
12.5 Stage1:Calculationoflatticetransformationstrain . . . . . . . . 193
Determinationoflatticetransformationstrain . . . . . . . . . . 194
12.6 Stage2:Determinationoftheorientationrelationship . . . . . . . 197
Martensite-austeniteorientationrelationship . . . . . . . . . . 197
12.7 Stage3:Natureoftheshapedeformation . . . . . . . . . . . . . . 198
Habitplaneandtheshapedeformation . . . . . . . . . . . . . 199
12.8 Stage4:Natureofthelattice-invariantshear . . . . . . . . . . . . 201
Lattice–invariantshear . . . . . . . . . . . . . . . . . . . . . . 202
12.9 Textureduetodisplacivetransformations . . . . . . . . . . . . . 204
12.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
13 Interfaces 211
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
13.2 Misfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Symmetricaltiltboundary . . . . . . . . . . . . . . . . . . . . 214
Interfacebetweenalphaandbetabrass . . . . . . . . . . . . . 217
13.3 Coincidencesitelattices . . . . . . . . . . . . . . . . . . . . . . 219
Coincidencesitelattices . . . . . . . . . . . . . . . . . . . . . 219
Symmetryandtheaxis-anglerepresentationsofCSL’s . . . . . . 222
13.4 TheO-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
alpha/betabrassinterfaceusingO-latticetheory . . . . . . . . 227
13.5 Secondarydislocations . . . . . . . . . . . . . . . . . . . . . . . 228
Intrinsicsecondarydislocations . . . . . . . . . . . . . . . . . 229
13.6 TheDSClattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
13.7 Somedifficultiesassociatedwithinterfacetheory . . . . . . . . . 233
13.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Appendices 239
A Matrixmethods 241
A.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
A.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
B Generalrotationmatrix 247
Index 249
Preface
Tostatetheobvious,crystalscontainorder.Evenwhenthatisdisturbedlocally,the
perturbations themselves may sometimes form regular patterns. This is evident in
the structure of interfaces where two or more crystals meet in an apparently hap-
hazardmanner.Disordercansometimesbeignoredwithoutcompromisingsomeof
the consequences of long-range order. Crystalline solid solutions in which atoms
aredispersedatrandomwouldfailthestrictdefinitionoflong-rangeorder,butthey
neverthelessshowthecharacteristicsofcrystalswhenprobedbyX-rays.Weshallat-
temptinthisbooktounderstandnotonlytheeleganceofindividualcrystals,butalso
ofclustersofspace-fillingcrystalsandtransformationsbetweencrystallinephases.
Crystallographyishardlyanew subjectso therearenumerousbooksavailable,
manyofwhicharebeaconsofscholarship.Sowhyanothertextonthismuchmooted
topic?First,asBuckminsterFullerpointedout,therehasbeenamassiveexpansionin
humanknowledge,withtheprocesscontinuingatanunabatingpace.Newsubjects
spring up with notorious regularity and some of these have come to be regarded
as essential first steps in the higher education curriculum. Modern undergraduate
studentsarethereforefacedwithamuchgreaterpaletteofdistinctcoursesthanhas
beenthecaseinthepast.
This book is partitioned into two, the first part of which is meant to be self-
containedanddealswithwhatIfeelisessentiallearningforanystudentinthemate-
rialsciences,physics,chemistry,earthsciences,andthenaturalsciencesingeneral.
Thispart,intentionallyconcise,isbasedonasetofninelecturesthatIgiveannually
to undergraduatestudentsof the Natural Sciences Triposin CambridgeUniversity.
It is of generic value and has just sufficient material to deliver concepts. It covers
crystals, polycrystals, interfaces, and transformationsthat occur by the disciplined
motion of atoms. I feel that most books on crystallography are too detailed to ac-
commodatewithinthescheduleofacontemporaryundergraduate.
The second part has depth which would be appreciated most in the context of
research.ItisadevelopmentfromabookthatItaughtandpublishedin1987onthe
GeometryofCrystals.Thetreatmentislimitedtophenomenadominatedbycrystal-
lography.
Thebookcontainsworkedexamplesthroughoutbecausecrystallographyisasub-
ject that thrives on practice. Video lectures and other electronic materials that can
enhancethecontentofthisbookareavailablefreelyon
http://www.phase-trans.msm.cam.ac.uk/teaching.html
Crystallographyhasrules,establishedbyconvention,butmanyofthesehaveex-
ceptions and there sometimes are multiple conventions.Donnay in his 1943 paper
proposed rules for defining the crystallographic orientation of a crystal [American
Mineral28(1943)313–328],punctuatedbyexceptions,specialcases, anddifficul-
ties. Inthe BungeconventionforEulerangles,thesample frameis rotatedintothe
crystalframe,butthereisanotherconventionwherethereverseisthecase.Myview
isthatintheapplicationofcrystallography,onlyalimitednumberofconventionsare
important. For example, the handednessof cell axes should be consistent through-
out. Similarly, although only three indices are necessary to define vectors in three
dimensions, the four index system for the hexagonalclass permeates the literature
andhenceneedstobetaught.Theemphasisofthebookistoteachconceptsrather
thanrules,whichnecessarilyrequirerotelearning– I amuncertainas towhetherI
havedoneenoughtominimizetheuseofconventions.
Myinterestincrystallographystemsprimarilyfromresearchonsolid-statephase
transformations. I have benefitted enormously from the writings of the late J. W.
ChristianandC.M.Wayman.Andfromincisivequestionsposedbyundergraduates.
IhavebeenabletotaproutinelyintotheknowledgeandwisdomofJohnLeakeand
Kevin Knowles whenever I felt confused about the fine detail of crystallography.
To all of these characters, I shall remain perpetuallygratefulfor the educationand
camaraderie.
H.K.D.H.Bhadeshia
Cambridge
Acronyms
bcc Body-centeredcubic ILS Invariant-linestrain
bct Body-centeredtetragonal P Primitive
fcc Face-centeredcubic I Body-centered
hcp Hexagonalclose-packed F Face-centered
IPS Invariant-planestrain R Trigonal
Part I
Basic Crystallography