Continuum Models for Phase Transitions and Twinning in Crystals (Applied Mathematics)

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Author: Mario Pitteri | G. Zanzotto

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Continuum Models for Phase Transitions and Twinning in Crystals

APPLIED MATHEMATICS Editor: R.J. Knops This series presents texts and monographs at graduate and research levels covering a wide variety of topics of current research interest in modern and traditional applied mathematics, numerical analysis, and computation. 1 Introduction to the Thermodynamics of Solids J.L. Ericksen (1991) 2 Order Stars A. Iserles and S.P. Nørsett (1991) 3 Material Inhomogeneities in Elasticity G. Maugin (1993) 4 Bivectors and Waves in Mechanics and Optics Ph. Boulanger and M. Hayes (1993) 5 Mathematical Modelling of Inelastic Deformation J.F. Besseling and E van der Geissen (1993) 6 Vortex Structures in a Stratified Fluid: Order from Chaos Sergey I. Voropayev and Yakov D. Afanasyev (1994) 7 Numerical Hamiltonian Problems J.M. Sanz-Serna and M.P. Calvo (1994) 8 Variational Theories for Liquid Crystals E.G. Virga (1994) 9 Asymptotic Treatment of Differential Equations A. Georgescu (1995) 10 Plasma Physics Theory A. Sitenko and V. Malnev (1995) 11 Wavelets and Multiscale Signal Processing A. Cohen and R.D. Ryan (1995) 12 Numerical Solution of Convection-Diffusion Problems K.W. Morton (1996) 13 Weak and Measure-valued Solutions to Evolutionary PDEs J. Málek, J. Necas, M. Rokyta and M. Ruzicka (1996) 14 Nonlinear Ill-Posed Problems A.N. Tikhonov, A.S. Leonov and A.G. Yagola (1998) 15 Mathematical Models in Boundary Layer Theory O.A. Oleinik and V.M. Samokhin (1999) 16 Robust Computational Techniques for Boundary Layers P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan and G. I. Shishkin (2000) 17 Continuous Stochastic Calculus with Applications to Finance M. Meyer (2001) 18 Spectral Computations for Bounded Operators Mario Ahues, Alain Largillier and Balmohan V. Limaye (2001) 19 Continuum Models for Twinning in Crystals M. Pitteri and G. Zanzotto (2001) (Full details concerning this series, and more information on titles in preparation are available from the publisher.)

Continuum Models for Phase Transitions and Twinning in Crystals MARIO PITTERI G. ZANZOTTO

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0849303273 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Dedication to J.L. Ericksen scientist, teacher and friend

Contents

List of ﬁgures

9

List of tables

15

Foreword

19

1 Introduction 1.1 Outline of chapter contents 1.2 Some experimental observations

21 25 28

2 Preliminaries 2.1 Basic notation 2.2 Some notions of elementary group theory 2.2.1 Basic deﬁnitions 2.2.2 Conjugacy 2.2.3 Group actions and symmetry 2.3 Linear and orthogonal transformations 2.3.1 Tensors with period two 2.3.2 Simple shears 2.3.3 Finite groups of tensors or matrices 2.4 Aﬃne transformations 2.5 Continuum mechanics 2.5.1 Deformation 2.5.2 Thermodynamic potentials and their invariance 2.5.3 Stability of equilibrium

41 41 42 42 42 43 45 48 48 50 52 53 54 56 58

3 Simple lattices 3.1 Deﬁnitions and global symmetry 3.2 Geometric symmetry and crystal systems 3.2.1 Crystallographic point groups and holohedries 3.2.2 Crystal classes and crystal systems 3.2.3 Laue groups 3.3 Arithmetic symmetry and Bravais lattice types 3.3.1 Lattice groups

61 62 66 66 68 70 72 72

3

4

CONTENTS

3.4 3.5 3.6

3.7

3.8

3.3.2 Conjugacy in O (crystal systems) and in GL(3, Z) (Bravais lattice types) 3.3.3 Centerings The fourteen Bravais lattices Fixed sets of lattice groups 3.5.1 An example Symmetry-preserving stretches for simple lattices 3.6.1 Commutation relations 3.6.2 Structure of the ﬁxed sets 3.6.3 The Bain stretch in the centered cubic lattices Lattice subspaces, packings and indices 3.7.1 Lattice rows and lattice planes 3.7.2 Close-packed structures 3.7.3 Miller indices and crystallographic equivalence 3.7.4 Miller-Bravais indices for hexagonal lattices Lattice groups and ﬁxed sets for planar lattices

4 Weak-transformation neighborhoods and variants 4.1 Reconciliation of global and local symmetries 4.2 Symmetry-breaking stretches for simple lattices 4.3 Small deformations and weak phase transformations 4.3.1 Small symmetry-preserving stretches 4.3.2 Small symmetry-breaking stretches 4.4 Constructing the small symmetry-breaking stretches 4.5 Variant structures (local orbits) in the wt-nbhds 4.5.1 An example 4.5.2 General deﬁnitions 4.5.3 Variants and cosets 4.5.4 Variant structures and conjugacy classes

73 76 78 87 90 91 91 94 96 96 96 97 98 101 102 107 108 111 113 114 115 118 120 120 122 123 124

5 Explicit variant structures 127 5.1 Variant structures in cubic wt-nbhds 128 5.1.1 Tetragonal conjugacy class and variant structure 130 5.1.2 Rhombohedral conjugacy class and variant structure 131 5.1.3 Orthorhombic conjugacy classes and variant structures 132 5.1.3.1 Orthorhombic ‘cubic edges’ variants 133 5.1.3.2 Orthorhombic ‘mixed axes’ variants 134 5.1.4 Monoclinic conjugacy classes 135 5.1.4.1 Monoclinic ‘cubic edges’ variants 135 5.1.4.2 Monoclinic ‘face-diagonals’ variants 135 5.1.5 Triclinic conjugacy class and variant structure 138 5.2 Variant structures in hexagonal wt-nbhds 138 5.2.1 Orthorhombic conjugacy class and variant structure 141 5.2.2 Monoclinic conjugacy classes and variant structures 142

CONTENTS

5.2.2.1 Monoclinic ‘basal diagonals’ variants 5.2.2.2 Monoclinic ‘basal side-axes’ variants 5.2.2.3 Monoclinic ‘optic axis’ variants 5.2.3 Triclinic conjugacy class and variant structure 5.3 Kinematics of weak phase transformations 5.4 Irreducible invariant subspaces for the holohedries 5.4.1 General properties 5.4.2 Reduced actions and reduced symmetry groups on the i.i. subspaces 5.4.3 Decompositions of Sym under the action of the holohedries 5.4.3.1 Triclinic decompositions 5.4.3.2 Monoclinic decompositions 5.4.3.3 Orthorhombic decompositions 5.4.3.4 Rhombohedral decompositions 5.4.3.5 Tetragonal decompositions 5.4.3.6 Hexagonal decompositions 5.4.3.7 Cubic decompositions

5

142 142 146 146 146 149 150 153 154 155 155 156 157 160 162 163

6 Energetics 6.1 Invariance of simple-lattice energies 6.2 The Cauchy-Born hypothesis 6.2.1 The Born rule 6.2.2 Failures of the Born rule 6.3 Thermoelastic constitutive equations for crystals 6.3.1 Invariance of the response functions of elastic crystals 6.4 Energy minimizers and their general properties 6.4.1 Multiplicity of the symmetry-related minimizers 6.4.2 Multiphase crystals: minimizers that are not symmetry-related 6.4.3 Lack of convexity and symmetry-induced instabilities 6.5 Constitutive functions for weak phase transitions 6.5.1 Weak and symmetry-breaking transformations 6.5.2 Domain restrictions for the constitutive functions 6.5.3 Energy wells in the wt-nbhds 6.6 In the vicinity of an energy well 6.6.1 Thermal expansion and compressibility of a crystal 6.6.2 The elasticity tensor 6.6.3 Temperature-dependence of the elastic moduli 6.7 Anisotropic elasticity

165 167 169 170 171 172 173 175 175

7 Bifurcation patterns 7.1 Introduction 7.1.1 The Landau theory 7.2 Isolated critical points and bifurcation points

199 199 200 204

176 178 179 180 181 182 186 187 189 193 194

6

CONTENTS

7.3 7.4

7.5 7.6

7.7 7.8

7.2.1 Neighborhoods of bifurcation points 7.2.2 Genericity Reduced bifurcation problems; order parameters Analysis of the reduced bifurcation problems 7.4.1 Reduced problem (1) 7.4.2 Reduced problem (2) 7.4.3 Reduced problem (3) 7.4.4 Reduced problem (4) 7.4.5 Reduced problem (5) 7.4.6 Reduced problem (6) 7.4.7 Comparison with the kinematic transitions of §5.3 Behavior of the moduli along the transitions Examples of energy functions for simple lattices 7.6.1 A schematic 1-dimensional example 7.6.2 Energies for cubic-to-tetragonal and for tetragonalto-monoclinic transitions 7.6.3 Orientation relationships and lattice correspondence Relation with the Landau theory General references

206 207 208 211 213 214 216 218 221 222 223 225 226 227 228 231 234 235

8 Mechanical twinning 8.1 Coherence and rank-1 connections 8.2 The twinning equation 8.3 Solutions of the twinning equation 8.3.1 Diﬀerent descriptions of the same twin and cosets 8.3.2 Crystallographically equivalent twins 8.3.3 Reciprocal twins 8.3.4 Generic twins 8.3.5 Type-1 and Type-2 (conventional) twins 8.3.6 Compound twins 8.3.7 Conventional twins and rationality conditions 8.4 Short remarks 8.4.1 Experimental data 8.4.2 Mechanical twinning and the Born rule 8.4.3 Growth twins

239 240 244 248 249 250 251 252 252 254 256 257 257 257 259

9 Transformation twins 9.1 General properties 9.1.1 Procedure to determine the transformation twins 9.2 Rk-1 connections in a cubic wt-nbhd 9.2.1 Tetragonal variant structure 9.2.2 Rhombohedral variant structure 9.2.3 Orthorhombic variant structures 9.2.3.1 Orthorhombic ‘cubic edges’ wells 9.2.3.2 Orthorhombic ‘mixed axes’ wells

263 263 268 270 270 271 271 271 272

CONTENTS

7

9.2.4 Monoclinic variant structures 9.2.4.1 Monoclinic ‘cubic edges’ wells 9.2.4.2 Monoclinic ‘face-diagonals’ wells 9.2.5 Triclinic variant structure 9.3 Rk-1 connections in a hexagonal wt-nbhd 9.3.1 Orthorhombic variant structure 9.3.2 Monoclinic variant structures 9.3.2.1 Monoclinic ‘basal diagonals’ wells 9.3.2.2 Monoclinic ‘basal side-axes’ wells 9.3.2.3 Monoclinic ‘optic axis’ wells 9.3.3 Triclinic variant structure 9.4 The Mallard law

273 273 278 280 281 281 281 281 282 283 284 285

10 Microstructures 10.1 Piecewise homogeneous equilibria 10.2 Generalized solutions 10.2.1 The minors relations 10.2.2 The N -well problem 10.3 Examples of microstructures that are not laminates 10.4 Habit planes in martensite 10.4.1 Geometrically nonlinear theory 10.4.2 Self-accommodation in shape memory alloys 10.4.3 Wedges and other microstructures

287 287 289 291 292 295 297 297 299 300

11 Kinematics of multilattices 11.1 Crystals as multilattices 11.1.1 Descriptors and conﬁguration spaces for deformable multilattices 11.1.2 Essential descriptions of multilattices 11.2 The global symmetry of multilattices 11.2.1 Indeterminateness of the descriptors (P0 ,..., Pn−1 , ea ) 11.2.2 Indeterminateness of the descriptors (P0 , εσ ) 11.2.3 Nonessential descriptors of multilattices 11.3 The aﬃne symmetry of multilattices 11.3.1 Space groups; crystal class and crystal system of a multilattice 11.4 The arithmetic symmetry of multilattices 11.4.1 Lattice groups of multilattices 11.4.2 Fixed sets of lattice groups 11.4.3 Relation between the arithmetic and the space-group symmetries 11.5 Examples 11.5.1 Three-dimensional 2-lattices and hexagonal closepacked structures 11.5.2 The structure of quartz as a 3-lattice

301 302 304 305 306 306 309 311 314 315 318 318 320 321 324 324 327

8

CONTENTS

11.6 Weak-transformation neighborhoods 11.7 The energy of a multilattice and its invariance 11.7.1 Minimizing out the internal variables of complex crystals 11.7.2 Local invariance of multilattice energies; the example of quartz 11.8 Twinning in multilattices 11.8.1 A proposal for a class of twins 11.8.2 Two examples 11.8.3 A model for stress relaxation

333 335 336 339 341 343 345 348

References

351

Index

371

List of ﬁgures 1.1

1.2 1.3

1.4

1.5

1.6

1.7

2.1 2.2

Penetration growth twins: parts of the interpenetrating dodecahedra, the iron cross schematically represented in (a)1 , are easily found in cubic crystals, for instance in pyrite as in (a)2 ; (b)1 is a schematic description of a typical twin arrangement in staurolite, two twinned crystals of which are shown in (b)2 . Contact growth twins: (c)1 is a schematic example of twinning in cassiterite (tinstone) while an actual twinned crystal is shown in (c)2 . (d) shows a typical contact twin in gypsum The Baumhauer experiment; (a) schematic description; (b) picture of an actual twinned calcite crystal X-ray diﬀraction of a hexagonal metal, along (a) the sixfold hexagonal axis; (b) one of the twofold axes in the basal plane; (c) the sixfold axis after the twinning shear sketched in (d) has occurred; notice the twofold-like spots (a) A wedge of orthorhombic martensite grown in cubic austenite of a CuAlNi alloy. (b) A blowup of an apparently homogeneous region of martensite, which actually consists of twin bands, each one made of other twin bands. (c) A cartoon of the deformation relating two individuals of a (compound) twin Patches of twins in (a) Rochelle salt and (b), (c) CuAlNi shape memory alloy. (c) shows a detail of the twin bands in one of the patches of (b) Experimental phase diagram of zirconia (ZrO2 ) showing the progressive tetragonal-orthorhombic-monoclinic symmetry reduction, and the related triple point. The holohedries are detailed in §§5.1.1, 5.1.3.1, and 5.1.4.1 (a) twinning and (b) slip in simple lattices. Compare slip along the plane represented by a dotted line in (b) with twinning of the same lattice along the same plane, drawn in Fig. 1.4(c) The ‘shear elements’ K1 , K2 , η1 , η2 Finite deformation 9

29 30

30

31

32

34

35 49 54

10

LIST OF FIGURES

Lattice bases ea and ¯ ea = mba eb , m ∈ GL(2, Z), and elementary cells for a 2-dimensional lattice 3.2 The names, International Symbols, and symmetry hierarchy of the seven crystal systems. The number of elements in each holohedry is indicated in parenthesis. Table 3.1 gives a complete list of elements in the groups representing the systems. Another widely used notation for the cubic system is m3m. See Remark 5.6 for the inclusion ¯3m → 6/mmm 3.3 The triclinic lattice, with no symmetry axes or planes 3.4 The two monoclinic lattice types, whose twofold axis is along the y axis. (a) The primitive monoclinic lattice. (b) The base-centered monoclinic lattice. The extra lattice point in this type are always at the center of two opposing rectangular faces of the standard nonunit cell 3.5 (a) The primitive orthorhombic lattice. (b) The basecentered orthorhombic lattice. The three twofold axes are the coordinate axes 3.6 (a) The face-centered orthorhombic lattice. (b) The bodycentered orthorhombic lattice. The three twofold axes are the coordinate axes 3.7 The two tetragonal lattice types, whose fourfold axis is along the z axis. (a) The primitive tetragonal lattice. (b) The body-centered tetragonal lattice 3.8 (a) The rhombohedral lattice, with basis given by (3.49). The vectors (3.50) generating the hexagonal sublattice are indicated by ˜ ea . The threefold axis is the z axis, along ˜ e3 , while the three twofold axes are along ˜ e1 , ˜ e2 and −˜ e1 − ˜ e2 , respectively. (b) The hexagonal lattice, in which the sixfold axis is the z axis. Also the shaded base-centered orthohexagonal double cell is put in evidence. (c) The basal plane in (b), showing the lattice vectors and the twofold symmetry axes 3.9 The primitive cubic lattice 3.10 The face-centered cubic (f.c.c.) lattice. (a) The rhombohedral unit cell given by (3.53). (b) The close-packed nets in the f.c.c. lattice – see §3.7.2 – and the lattice vectors (3.54) 3.11 The body-centered cubic (b.c.c.) lattice. (a) The unit cell of the lattice vectors in (3.55). (b) The rhombohedral unit cell with basis (3.56) 3.12 The inclusion relations for the 3-dimensional Bravais lattice types (or, the lattice groups up to GL(3, Z)-conjugacy). The denominations combine the notation for holohedries and centerings in Table 3.1

3.1

65

70 79

80

81

81

82

83 85

85

86

89

LIST OF FIGURES

3.13 One of the primitive orthorhombic ﬁxed sets, including (parts of) three 2-dimensional primitive tetragonal ﬁxed sets, whose intersection is a 1-dimensional primitive cubic ﬁxed set. The forms of the related metrics are also shown. The intersections of the tetragonal ﬁxed sets with the triangle intercepting each axis at 1 are fully drawn 3.14 The body-centered tetragonal cell for the f.c.c. lattice – see (3.54) – obtained from an equally oriented b.c.c. cell by the Bain stretch (3.81) 3.15 Close-packing of spheres. The stacking of two close-packed nets (gray is lower) is schematically represented in (a). By placing the third layer on top of the ﬁrst and repeating this arrangement of the two layers we obtain the h.c.p. structure. The other possibility to place the third layer, giving the f.c.c. structure when repeated, is shown in (b) 3.16 Traces of congruent but not crystallographically equivalent nets (the lattice vector e3 is orthogonal to the page) 3.17 Representation of Q> 2 modulo dilations in the C11 –C12 plane. Traces of various ﬁxed sets are shown 3.18 The four crystal systems and the ﬁve Bravais types of nets 3.19 Section C11 = C22 = C33 = 1/3 of the Wigner-Seitz domain of 3-dimensional lattices, showing the primitive cubic (P ), f.c.c. (F ) and b.c.c. (I) reduced cells; and, among the others, two rhombohedral (R2 and R3), two centered tetragonal (T I2 and T I3) and one body-centered orthorhombic (OI1) ﬁxed sets 4.1

5.1

5.2

GL(3, Z)-orbits in the spaces B and Q> 3 . A wt-nbhd of a cubic basis (and metric) and the included tetragonal variants are indicated schematically The tree of holohedral subgroups PL (ea ) when ea is a cubic basis. Vertical and oblique lines indicate inclusion, while horizontal lines indicate conjugacy within Cijk . Corresponding to the centering of the cubic basis ea indicated under Cijk , and in the same position, one ﬁnds, under the name of each holohedral subgroup in the table, the centering of the bases which belong to the wt-nbhd Nea and have that holohedry as symmetry group. For simplicity only the inclusions of the monoclinic holohedries in two of the rhombohedral holohedries are shown, the others can be easily reconstructed The tree of holohedral subgroups PL (ea ) when ea is a hexagonal basis

11

90

97

99 100 103 104

106

121

129 140

12

LIST OF FIGURES

5.3

7.1 7.2

7.3

7.4

7.5

The tree of holohedral subgroups PL (ea ) when ea is a tetragonal basis. Vertical and oblique lines indicate inclusion, while horizontal lines indicate conjugacy within Tk . Corresponding to the centering of the tetragonal basis ea indicated under Tk , and in the same position, one ﬁnds, under the name of each holohedral subgroup in the table, the centering of the bases which belong to the wt-nbhd Nea and have that holohedry as symmetry group ¯ Solid lines represent the stable Turning point for θ ≥ θ. branches ¯ (a), (b) Pitchforks for the (y = 0)-branch stable above θ. The derivatives of Φ are taken at the transition, and the stable branches are represented by solid lines. (c) Energy proﬁle at various temperatures for case (a). The analogue for case (b) can be obtained from Fig. 7.3 by restricting the attention to neighborhoods of the origin that only contain the unstable branches (dotted lines there) A 1-dimensional example based on the sextic potential in (7.27). Both energy and its derivative, which measures stress, are plotted versus the order parameter y, which is a measure of strain, for various values of θ near the transition ¯ At the shown temperature between θ¯ and temperature θ. θM the dotted lines plot energy and stress versus strain in the absence of the restabilizing sextic term (a) Phase diagram in the temperature-pressure plane; T3 is our Tk , etc. The thin lines represent stability boundaries, and the thick lines the Maxwell lines, concurring into the triple point. The darker gray area represents states in which all three phases coexist as (meta)stable equilibria. The bifurcation diagram along the straight line t-s is shown in Fig. 7.5. (b) Energy landscape illustrating the variety of wells existing in the typical point of the three-phase coexistence region highlighted in darker gray in (a); y3 and y6 are the order parameters Bifurcation pattern along the s-t line shown in Fig. 7.4. The points marked here correspond to the ones having the same name in that ﬁgure. The dotted lines indicate the unstable portions of the equilibrium branches

149

214

215

228

233

233

LIST OF FIGURES

8.1

9.1

13

Schematic description of three martensitic, symmetryrelated wells and one austenitic well, where the reference is set. The upper part shows reciprocal (as in §8.3.3) rank-1 connections between the wells OU1 and OU1 H1 when the data in the twinning equation are (F1 , H1 ) (see (8.20), (8.12)) and (U1 , H1 ) (see (8.25), (8.27)). The lower part contains the construction of a habit plane between austenite and a simple laminate of variants U1 and U3 of martensite; see the formulae in §10.4.1, which involve variant U2 instead of U3

248

The generic ((a)) and the nongeneric ((b)) twins in the monoclinic ‘cubic axes’ variant structure; the variants are indicated by dots and the rank-1 connections by lines. The full set of possible twins is the union of (a) and (b)

277

11.1 Two arithmetically inequivalent planar 3-lattices with the same space group 11.2 The hexagonal close-packed (h.c.p.) structure 11.3 Projection of the Si atoms of left-handed hexagonal β-quartz onto the basal plane, and descriptors ε0σ = (ea0 , p10 , p20 ) for the 3-lattice given by (3.50) and (11.71) 11.4 Projection as in Fig. 11.3 for the left-handed α-quartz + + + structure and descriptors ε+ σ = (ea , p1 , p2 ) for the 3lattice given in (3.50), (11.76) with λ > 0 11.5 (a) twinning in β-tin. (b) details the shuﬄe displacements: for each shift pi the vectors Sp i and ¯ pi (dashed) are drawn; their diﬀerence agrees with (11.118) 11.6 (01¯ 12) twinning mode in hcp metals. Both in (a) and (b) atomic positions in the upper half plane before deformation are indicated by gray open and ﬁlled circles

323 326

329

331

346

347

List of tables

3.1 3.2

3.3

The 7 crystal systems and the related 14 lattice types (§3.3.2)

71

The four nonholohedral crystal classes to be added to the seven crystal systems in Table 3.1 to give the complete list of the eleven Laue groups. The generators are given in terms of the same orthonormal basis i , j , k , as in Table 3.1

72

The f.c.c. and b.c.c. lattice group elements with positive determinant associated with the bases (3.53) and (3.56), respectively. See the text for the notation. Notice the common rhombohedral lattice subgroup

78

5.1

Cubic holohedry

130

5.2

Tetragonal holohedries

131

5.3

Rhombohedral holohedries

132

5.4

Orthorhombic ‘cubic edges’ holohedries

133

5.5

Orthorhombic ‘mixed axes’ holohedries

134

5.6

Monoclinic ‘cubic edges’ holohedries

136

5.7

Monoclinic ‘face diagonals’ holohedries

137

5.8

Triclinic holohedry

139

5.9

Hexagonal holohedry

139

5.10 Orthorhombic holohedries

141

5.11 Monoclinic ‘basal diagonals’ holohedries

143

5.12 Monoclinic ‘basal side-axes’ holohedries

144

5.13 Monoclinic ‘optic axis’ holohedries

145

5.14 Triclinic holohedry

146

6.1

Restrictions on the right Cauchy-Green tensor of the deformation between undistorted states of anisotropic solids 15

189

16

LIST OF TABLES

7.1

8.1

9.1 9.2 9.3

The reduced bifurcation problems, listed by the dimension of the order-parameter space and the generators of the reduced group P. The corresponding general (G) and θcontrolled (θ) – see §7.4.7 – symmetry-breaking continuous transitions in simple lattices are indicated in the second and third table, without explicit mention for 1-dimensional order-parameter spaces (problems (1) and (2)), where they coincide. Wherever necessary, the centerings of the high-symmetry phase and the corresponding centering of the low-symmetry one, in the same order, are indicated. A comparison of G- or θ-symmetry breaking with the kinematically possible continuous paths of §5.3 is sketched in §7.4.7. The overall notation is taken from §5.4.3, which contains the details about the i.i. subspaces and the related centerings. Also, ‘tp’ means turning point, a change of stability but not of symmetry – see §7.4.1

212

Some commonly reported twin data. The ρs and σs are suitable composition-dependent real numbers. ‘I’ means irrational, with approximate values in double quotes

258

Rank-1 connections in the tetragonal variant structure Rank-1 connections in the rhombohedral variant structure Rank-1 connections in the orthorhombic ‘cubic edges’ variant structure 9.4 Representatives of equivalence classes of rank-1 connections in the orthorhombic ‘mixed axes’ variant structure 9.5 Representatives of equivalence classes of compound twins in the monoclinic ‘cubic edges’ variant structure 9.6 Representatives of equivalence classes of Type 1 and 2 rank-1 connections in the monoclinic ‘cubic edges’ variant structure 9.7 Representatives of equivalence classes of conventional twins in the monoclinic ‘face diagonals’ variant structure. In the second column r, s denote for short the corresponding components of a∗ 9.8 More representatives of equivalence classes of Type-1 and Type-2 twins in the monoclinic ‘face diagonals’ variant structure 9.9 Representative of compound twins in the orthorhombic variant structure; indices in italics are standard orthorhombics 9.10 Representative of compound twins in the monoclinic ‘basal diagonals’ variant structure; indices in italics are standard monoclinics

270 271 272 273 274

274

278

278 281

282

LIST OF TABLES

9.11 Representatives of classes of Type-1 and Type-2 twins in the monoclinic ‘basal diagonals’ variant structure; indices are standard monoclinics 9.12 Representative of compound twins in the monoclinic ‘basal side-axes’ variant structure; indices in italics are standard monoclinics 9.13 Representatives of classes of Type-1 and Type-2 twins in the monoclinic ‘basal side-axes’ variant structure; indices are standard monoclinics 9.14 Representatives of classes of compound twins in the monoclinic ‘optic axis’ variant structure; indices are standard monoclinics

17

282

283

283

284

Foreword When the publisher Chapman & Hall invited us to write a book for its series in Applied Mathematics and Mathematical Computation, the best we could contribute appeared to be an outline of the work done in the last two decades on the nonlinear thermoelastic theories for twinning and phase transitions in crystalline materials. The developments of this elastic approach to crystal mechanics, rich in new theoretical aspects as well as applications, have been so extensive and ramiﬁed that we felt a smooth, wellhoned presentation of them might be diﬃcult to obtain. However, the very abundance of and interest in the results already available were well worthy of an organized survey covering the essential notions on the subject. Our eﬀort has been to fulﬁll in this way some of the needs of the ever-increasing, strongly interdisciplinary community devoted to this highly interesting research area, which stretches from deep theoretical issues in mathematics to very practical problems in materials science. We brieﬂy touched upon various subjects, such as crystallography, bifurcation theory, Landau theory, variational calculus, which are natural tools for the development of the model. Of course, we made no attempt to be comprehensive, also because a great deal of investigation is currently being carried out in many diﬀerent directions to further clarify and broaden the scope of the proposed theory. The purpose of our presentation is to provide a valuable aid for the basic training of those who are approaching the subject for the ﬁrst time, through which they can come close to the ideas of current research. Also, this book may provide a convenient reference volume for the workers already active in the ﬁeld. The reader is assumed to be familiar with the fundamentals of calculus, linear algebra, and continuum mechanics, although some basic information needed from these disciplines is presented in chapter 2. Material has been drawn from many sources, whose treatment and notation have been slightly or more substantially revised to allow the various topics to blend together more smoothly. References are given in each case, and consulting the original works is necessary for anyone undertaking research in these areas. Any omissions, inconsistencies, and mistakes are our responsibility. Key Words: Continuum Mechanics, Crystalline Materials, Crystallography, Materials Science, Mathematical Physics, Metallurgy, Microstructure, Mineralogy, Phase Transitions, Thermoelasticity, Twinning. 19

20

FOREWORD

Acknowledgements: Special thanks are due to A. DeSimone, J. L. Ericksen, G. P. Parry, and L. Truskinovsky, for reading portions of the typescript and giving helpful advice and criticism. We would also like to acknowledge the continuing interaction, over the years, with J.M. Ball, K. Bhattacharya, F. Cardin, P. Cermelli, S. Conti, G. Fadda, I. Fonseca, R.L. Fosdick, K.F. Hane, R.D. James, D. Kinderlehrer, P.H. Leo, M. Luskin, I. M¨ uller, S. M¨ uller, P. Podio Guidugli, which helped us shape our ideas. G. Fadda has been very helpful with the ﬁgures; also, C. Chu and R.D. James, and C. Brogiato, have kindly permitted the use of some of their photos. We are very thankful to R.J. Knops for watching over the progress of this work at various times. The writing of this volume was made easier and more fruitful by several arrangements. We would like to mention the encouragement of our home institution, and the warm hospitality of the Department of Aerospace Engineering & Mechanics and of the Institute for Mathematics and its Applications at the University of Minnesota on a number of occasions. We gratefully acknowledge the ﬁnancial support of the Italian Ministero per l’Universit` a e la Ricerca Scientiﬁca through the research project ‘Mathematical Models for Materials Science’, of the Italian CNR and Gruppo Nazionale per la Fisica Matematica, and of the European Union through the TMR network ‘Phase Transitions in Crystalline Solids’. Our appreciation is also directed to the publishers and authors who have granted us permission to use their material; credit is given wherever any such sources have been used. Last but not least, we are particularly grateful to our families for their patience and support throughout the progress of this work.

CHAPTER 1

Introduction Since the mid-seventies there has been a strong and remarkably successful eﬀort to use 3-dimensional nonlinear thermoelasticity theory for modelling the behavior of crystalline solids in the range of ﬁnite deformations. Extremely common examples of this behavior are encountered in mechanical twinning and solid state phase transitions with loss of symmetry. The contributions to this subject by a number of authors stem from some earlier works by Ericksen, in which he proposed a theory of material symmetry for nonlinearly elastic solids that can describe crystalline substances having many nontrivially related natural states. His approach to the material symmetry of crystals is, in a sense, more local than the by now standard one of Coleman and Noll (1964), also adopted by Truesdell and Noll (1965); according to them there is a reference conﬁguration such that the related material symmetry group is one of the 32 crystallographic point groups, augmented by the central inversion. This basically extends to nonlinearly elastic crystals the assumption of classical linear elasticity that the invariance of the energy is dictated by the point group describing the geometric symmetry of the underlying crystal lattice in a natural state, that is, an unstressed stable equilibrium state, taken as reference conﬁguration. While this view is universally adopted for determining the independent elastic moduli of crystalline solids of any given symmetry (see Love (1927)), it is too rigid when extended to nonlinear regimes, where material symmetry is not necessarily imposed by the geometric symmetry of any speciﬁc reference conﬁguration. The approach of Ericksen (1978) thus aimed at an elastic theory capable of accounting for the common phenomena of crystal mechanics mentioned above. These could hardly be described1 by means of the earlier elastic models, even nonlinear, in which invariance is based on a given point group. Ericksen (1970), (1977), (1980b), (1987) achieved his goal by relying on molecular models to construct the global material symmetry that the constitutive equations of a crystal should exhibit. This invariance group turns out to be an inﬁnite, discrete group in which there are nonorthogonal as well as simple shearing transformations. These notions allow elasticity theory to become ﬂexible enough to deal with the aforemen1

There are, though, geometrically linear proposals, like the 1-dimensional model by Lifshitz (1948) for twinning, and the theory of equilibrium for crystals undergoing martensitic transformations developed by Eshelby (1961), Khachaturyan (1961) and others – see §10.4.1 21

22

INTRODUCTION

tioned phenomena, which laid for long outside its range. Indeed, some of the nonorthogonal transformations contained in such a large symmetry group play an important role in the description of mechanical twins, in particular transformation twins, as pairwise homogeneous coherent natural states for crystals. This description includes properties of twins considered to be true by the mineralogists, Friedel (1926) and Buerger (1945) among others. On the basis of these ideas and of the earlier calculations of Ericksen (1981a), a twinning equation has been derived which describes the allowed twinning operations and the kinematics of twins. This equation has been studied by many authors and a great deal is now known, although a full explicit classiﬁcation of natural states is still missing. This view of material symmetry, based on a very large invariance of the constitutive equations, is nevertheless compatible, for ‘small but ﬁnite’ deformations, with the standard nonlinear scheme based on point groups, because when restricted to a suitable neighborhood of any given conﬁguration, the large invariance reduces to the point group symmetry dictated by that conﬁguration. Therefore the new theory is reconciled with the classical one; in particular, it includes the typical restrictions of the linear theory on the elastic moduli, which are successfully used by metallurgists and materials scientists. In many circumstances the results obtained so far describe well the great variety of twinned conﬁgurations and microstructures experimentally observed in crystals2 by metallurgists and crystallographers. Among others, typical examples are the internally twinned wedge-like formations observed in martensitic phase transitions of shape-memory alloys (SMA) – see §1.2. These materials, owing to their peculiar thermomechanical properties, are extremely interesting from both the theoretical and the experimental points of view. The shape memory eﬀect in the alloy seems to be a consequence of the following ability of the material to form a ﬂexible variant structure. The crystal undergoes a solid-state martensitic transformation from a highsymmetry austenitic phase to many variants of a low-symmetry martensitic phase. In this process self-accommodating equilibrium variant mixtures, especially involving microstructures made of periodic patterns of ﬁne twin bands, are capable of forming in a great variety of morphologies in response to imposed boundary conditions such as given loads or deformations. Very common are, for instance, the wedge- or spear-like martensitic microstructures that grow into the austenite as in Fig. 1.4(a). The wedge consists of two sets of ﬁne martensitic twins, that is, of laminates of ﬁne bands of diﬀerently oriented martensitic variants – see Fig. 1.4(b). This particular microstructure is very important because it provides an easy way to initiate 2

Some interesting materials are not properly crystalline but can often be treated as crystals for our purposes; for instance, various metallic (substitutional) alloys have the atoms of one of the alloying elements randomly interspersed within the crystalline lattice of one or the other constituents. This is the case with the common carbon steels, or with some shape memory alloys.

INTRODUCTION

23

the transformation, and is thus important for the reversibility of the transformation – see Bhattacharya (1991). It must be emphasized that equations for the interface (habit plane) between austenite and martensite have been obtained from kinematical considerations in the so-called crystallographic theory of martensite – see §10.4. However, we refer to a more fundamental and unifying energetic framework, based on the calculus of variations, which not only allows for a wider range of geometries to be examined, but also does it with a minimum of a priori assumptions regarding the position and features of the interfaces and microstructures. Our theoretical understanding of the peculiar thermomechanical behavior of the SMA is based on the analysis of the related phase transitions. Roughly, the establishment of thermoelastic equilibrium between the parent austenitic phase and variants of martensitic phases depends on the composition of the alloy, the temperature and the internal stress. Nucleation and growth of the martensite are controlled by shear strains that develop between adjacent martensite regions as the alloy is cooled or stressed. The balancing of such internal strains and any external stress causes the martensite regions to grow in an array of self-accommodating plates. The orientations of the plates with respect to the orientation of their neighbors is the one that is energetically most favorable in that particular strain ﬁeld. Whether the ﬁeld results from an applied stress or from a change in temperature, in the material as a whole the plates assume a variety of orientations and a range of sizes. The strain associated with one variant is compensated by the strain in the other variants, so that the growth of multiple self-accommodating ﬁelds of martensite plates is energetically favored over the growth of one single plate. In this way the morphology of the crystal and its macroscopic form can easily be altered with changes in the thermal and mechanical environment, and this in turn greatly inﬂuences the macroscopic mechanical properties it exhibits. The shape-memory eﬀect is then obtained by returning the crystal completely to the parent austenitic phase, which has no variants, by changing the temperature of the body. An early application of the shape memory eﬀect was roughly the following: the skeleton of a large parabolic antenna, to be part of a telecommunications satellite, was built with a shape memory alloy which was in the austenitic phase at the operating temperature. By lowering the temperature the alloy transformed into the martensitic phase, which became highly deformable due to the presence of the twins. This allowed the antenna to be folded into a small package that ﬁt inside the satellite. When the latter reached its operating position, the temperature forced the alloy to transform back to austenite, all the martensitic twins disappeared, and the skeleton unfolded to recover the shape in which it was manufactured. There has been a strong eﬀort in the last decade to understand this mechanism precisely, in the context of thermoelasticity and the calculus of variations. Indeed, while the possible approaches are various, the theoretical modelling of these phenomena is based on a nonconvex energy (or a

24

INTRODUCTION

nonmonotone stress-strain relation) whose early prototype are the equations of the van der Waals type for a liquid and its vapor. The theory that seems capable of the most detailed predictions for static problems is the 3-dimensional nonlinearly elastic version of this, complemented by the aforementioned proposal regarding the symmetry group of the energy of a crystalline material. The calculus of variations based on this approach has greatly improved our understanding of many aspects of these phenomena of crystal mechanics. For instance, the ﬁne-scale microstructures, such as the rapidly oscillating laminar twin systems and other periodic patterns in the observed arrangements of phases and variants, suggest competition between the bulk energy of the material and the loading or other environmental eﬀects, and are indeed being modelled in this way. We will only very shortly mention some results concerning microstructures in chapter 10. Of course, describing the dynamics of complex crystalline microstructures is an interesting and challenging problem, for which much needs to be done, and it will not be addressed here. In particular, we would like to better understand the hysteretic behavior of the martensitic phase changes and of the microstructural arrangements of phases that are commonly observed in connection with these transitions (see Ball et al. (1995)), and the pseudoelastic behavior with all its complexities (M¨ uller and Villaggio (1977), M¨ uller and Wilmanski (1980), M¨ uller (1989), M¨ uller and Xu (1991), Fedelich and Zanzotto (1992), Fu et al. (1992), Fu et al. (1993), Raniecki and Lexcellent (1994), Leclercq and Lexcellent (1996)). A qualitative idea is that metastability is an important factor for hysteresis: during its time evolution the system, while trying to minimize energy, can be constrained to remain near relative minima by energy barriers, and further forcing is needed to have it reach the absolute minimum. Preliminary results by Friesecke and McLeod (1996) support this interesting point of view; see also Truskinovsky and Zanzotto (1995), (1996). The above framework has also been extended to equilibrium theories for various materials and phenomena, among which micromagnetism and magnetostriction (James and Kinderlehrer (1990), (1993), Kinderlehrer (1992), DeSimone (1993), (1994a,b), (1995), DeSimone and Podio Guidugli (1995), (1996), James and Wuttig (1998), James et al. (1999), DeSimone and James (2002)), ferroelectrics and nematic elastomers (DeSimone (1999), Dav`ı (2001), Dolzmann (2001), DeSimone and Dolzmann (2002), Conti et al. (2002)), and thin ﬁlms (Roitburd et al. (1998), Bhattacharya and James (1999)). Related numerical work is detailed by Collins and Luskin (1989), (1991), Collins et al. (1991), Luskin and Ma (1992), Collins et al. (1993), Chipot et al. (1995), Luskin (1996b), Li and Luskin (1998a,b), and Dolzmann (2001). These extensions are outside the scope of this book and, together with various aspects of the nonlinear elastic approach mentioned above and of its applications, can be found in Ericksen (1980b), Ball and James (1992),

1.1 OUTLINE OF CHAPTER CONTENTS

25

(2002), M¨ uller (1998), James (1999), DeSimone et al. (2000), James and Hane (2000), Pedregal (2000), and in the many references to the original works given there. The main focus of this book is on the thermoelastic constitutive equations for crystalline solids in small but ﬁnite deformations, and on the consequences of the invariance of those equations in the modelling of twinning and phase transformations in such solids in the absence of imposed loads or boundary constraints, and for distortions that are not too large. The invariance of the constitutive functions is suggested by the lattice description of crystals. This description calls for a self-contained treatment of the global symmetry of simple lattices, which puts in perspective some classical results of crystallography. The same line of thought is applied to complex lattices in chapter 11, possibly introducing new ideas for their crystallography. 1.1 Outline of chapter contents 1. Introduction We present some main experimental facts regarding twins originated during growth, or along phase transformations, or by mechanical deformation. Some of the early geometric theories for twinning are also mentioned. 2. Preliminaries This chapter surveys some useful notions from group theory, linear algebra, and continuum mechanics. As customary in elasticity, the Lagrangian, or referential, description of deformation will be adopted. 3. Simple lattices Here the molecular model of a simple (monatomic) lattice is introduced to describe periodicity of crystalline solids. The lattice basis is used as a descriptor of the periodic structure, and its indeterminateness is detailed. This is regarded as the global invariance of the lattice, the same for all lattices. A connection with the standard crystallographic concepts of holohedry, crystal system and Bravais lattice type is then obtained, and the 14 types are described explicitly. The concepts of lattice group (or arithmetic holohedry), of its ﬁxed set, and of symmetry-preserving stretch, play an important role in the development of the theory, and are analyzed here to the extent possible, giving examples for 2-dimensional lattices (nets). 4. Weak-transformation neighborhoods and variants Here we show that each lattice conﬁguration is the center of a neighborhood (wt-nbhd) in which the global invariance of the lattice, generally described by an inﬁnite, discrete group, reduces to a ﬁnite subgroup, which is precisely the lattice group of the center. We also ﬁnd it useful to describe the wtnbhds in terms of symmetry-breaking stretches, of which the symmetrypreserving ones of the previous chapter are a special case. If a wt-nbhd is

26

INTRODUCTION

so described, the (local) invariance in it can be expressed in terms of the holohedry of the center and of its holohedral subgroups. We show how this representation allows for an explicit (local) description of the wt-nbhds, and of how the stretches organize themselves into variants; the elements of each variant are ﬁnite in number and are symmetry-related. 5. Explicit variant structures In this chapter we give the explicit description of the wt-nbhds of centers whose symmetry is maximal (cubic or hexagonal), and give hints at how this can also give information about wt-nbhds of centers of nonmaximal symmetry. For each system whose symmetry is contained in a cubic or hexagonal group, the stretches in any existing variant structure, the corresponding holohedral subgroups, and the generating elements from a given stretch in the structure are given in table form. This information is useful for addressing the question of how symmetry can change, as far as pure kinematics is concerned, along a continuous path of (small but ﬁnite) stretchings of a lattice. It is also used to classify the irreducible invariant subspaces of symmetric second-order tensors under the action of the holohedries, a kinematic preliminary for the analysis of bifurcation patterns in chapter 7. 6. Energetics We ﬁrst introduce a nonconvex energy function per unit cell of the lattice, following ideas originated by Cauchy, and analyze its global invariance. Later we also examine some of the main consequences of Cauchy’s ideas, as for instance the Cauchy relations which are not actually satisﬁed by many real materials. Elasticity theory is then brought into the picture by connecting this molecular framework to the continuum description via a hypothesis due to Cauchy and corrected by Born, the so called Born rule. In a wt-nbhd the large global invariance of the elastic energy turns out to reduce to the point-group symmetry of the center, thus recovering the standard material symmetry of nonlinear elasticity as well as the restrictions it induces on the elastic moduli, used in linear elasticity. We then use the local symmetry to get informations on the possible energy minimizers, or energy wells, in a wt-nbhd. 7. Bifurcation patterns As a ﬁrst application of the thermoelastic approach to crystal mechanics presented this far, we give some results regarding the bifurcation patterns that are possible during solid-state phase transitions involving changes of symmetry in simple lattices. This follows a scheme ﬁrst introduced by Landau, and is but a section of the general theory of bifurcations and phase transformations, which in its generality is outside the scope of this book. Finally, we give some example of lattice energies suitable for the description of certain phase transitions.

1.1 OUTLINE OF CHAPTER CONTENTS

27

8. Mechanical twinning Here we introduce the notion of mechanical twinning deformations, regarded as a particular class of stress-free stable coherent equilibria for crystals, and discuss various related results. In all of this chapter no surface energy will be associated to the boundary separating the two individuals of a twinned crystal. The theory ﬁts essentially materials whose crystalline structure is described by a simple lattice, or that behave as such; however in chapter 11 we also sketch a kinematics of twinning and shuﬄe displacements in multilattices. We derive and study the twinning equation, which is the main restriction on the orientations exhibited by mechanical twins. Particular attention is paid to the so-called Type-1 and Type-2 twins, which involve operations of period two and cover the vast majority of the observed mechanical twins. A few words are devoted to caution in the use of the Born rule and to a model for growth twins proposed by Zanzotto (1990), which has interesting applications in the Earth Sciences. 9. Transformation twinning In this chapter we consider the case that all the relevant energy wells have representatives in a wt-nbhd, and focus on these. The possible twins in this case can be explicitly and systematically analyzed, and we do so for the case of wt-nbhds of lattice conﬁgurations of maximal symmetry (cubic or hexagonal), taking advantage of the local representation of such neighborhoods given in chapter 5. We present the twin data in table form, stressing the cases in which the twins are not of the conventional Type 1 or 2. 10. Microstructures This chapter is an essential and sketchy introduction to the modelling of various complicated microstructures that are actually often found in crystals of shape memory alloys. Also a condition for the existence of habit planes between austenite and possibly twinned martensite is recalled, which is in agreement with the so-called crystallographic theory of martensite. 11. Kinematics of multilattices Within a molecular description of crystalline conﬁgurations we introduce the notion of multilattice, and introduce the relevant variables of the model. We discuss the groups of transformations leaving multilattices invariant. As for simple lattices, these have a classical description in terms of aﬃne isometries, the space groups, and a not so classical arithmetic counterpart, the lattice groups of multilattices. As an example of these symmetries we consider the symmetry-related Dauphin´e twins in quartz. Then we analyze the implications of the global and local symmetries of multilattices on the material symmetry of crystalline substances; in particular, on the invariance properties of their energy function. Based on these, we follow Ericksen’s interpretation of a procedure suggested by Born, and sketch the elimination of the internal variables for the energy of a multilattice, which yields a

28

INTRODUCTION

possibly multivalued simple-lattice energy function for crystalline solids. Finally, we mimic the construction of the conventional Type-1 and Type-2 twins in chapter 8 to propose a deﬁnition of twins in multilattices that describes a number of cases, ﬁtting also the shuﬄe displacements. The examples given are the most common twinning mode in hexagonal metals and the twinning mode of β-tin. The kinematics just sketched works well for multilattice conﬁgurations referred to any of the smallest elementary cells possible, these being always well deﬁned. Unfortunately many modes of deformation, including twins, can involve cells that are multiples, usually by small integers, of a smallest cell. A uniﬁed kinematics of smallest as well as multiple cells for multilattices remains to be formulated, and its availability, which is actively pursued, will be of utmost interest for the treatment of phase transformations and related phenomena in complex crystals. 1.2 Some experimental observations In nature crystalline ediﬁces very seldom appear in the form of well individualized single crystals. Rather, they almost invariably come in complex associations in which a number of homogeneous and congruent parts of the same compound intergrow, either in a parallel fashion or with one part penetrating into another. Such polycrystals often occur in random patterns; however, when the various individuals are reoriented in ways that are typical and characteristic for each crystalline species, the formations are called twins, with a broad interpretation of the word which refers to siamese twins. The rigid transformations taking one individual into another are called twinning operations or twin laws. Often, though not always, the cohesion between the diﬀerent individuals in a twin is as strong as the internal cohesion of the single crystal. Aggregates of this kind have been known and studied by crystallographers and mineralogists at ﬁrst, and later also by metallurgists, for at least two centuries, and have long been considered to have a particular significance in crystal mechanics. Yet, neither in the mineralogical nor in the metallurgical literature is there a common agreement on what exactly constitutes a twin, and there is no precise deﬁnition which is generally accepted. Each author takes a view that others are likely to share in the general outlines, but often not in the details. For instance, in the Introductory Remarks to the Symposium on Twinning held in Madrid, Buerger (1960) mentions that the reader of the proceedings will ﬁnd a divergence of views on certain points, no attempt having been made by the editor to force any uniformity into the treatment or results, and stresses that one point of divergence concerns the kind of symmetry which can relate the individuals of a twin pair. The disagreement appears to arise from a diﬀerent deﬁnition of a twin. Deﬁnitions and diverse remarks on the phenomenon of twinning are given by Evans (1912), Friedel (1926) pp. 421–427, 479–483, 490–493, (1933),

1.2 SOME EXPERIMENTAL OBSERVATIONS

29

(a)1

(b)1

(c) 1

(a)2

(b)2

(c) 2

(d)

Figure 1.1 Penetration growth twins: parts of the interpenetrating dodecahedra, the iron cross schematically represented in (a)1 , are easily found in cubic crystals, for instance in pyrite as in (a)2 ; (b)1 is a schematic description of a typical twin arrangement in staurolite, two twinned crystals of which are shown in (b)2 . Contact growth twins: (c)1 is a schematic example of twinning in cassiterite (tinstone) while an actual twinned crystal is shown in (c)2 . (d) shows a typical contact twin in gypsum Figs. (b)2 , (c)2 , (d) courtesy of C. Brogiato, Curator of the Museo di Mineralogia e Petrologia, Universit` a di Padova; the crystal in (a)2 is a gift of M. Senechal to M.P.

Mathewson (1928), Buerger (1945), (1960), Elliss and Treuting (1951), Cahn (1954) pp. 262, 381–383, 409–411, Hall (1954), Hartman (1956), Holser (1958), Donnay (1960), Dana (1962) vol. 3 p. 75, Klassen-Nekliudova (1964), Bilby and Crocker (1965), Barrett and Massalsky (1966) p. 406, Bevis and Crocker (1968), Kelly and Groves (1970) pp. 296–304, Santoro (1974), Zoltai and Stout (1984), and others. We report here the deﬁnition by Friedel (1926) and Cahn (1954), which exampliﬁes the standard attitude: ‘A twin is a polycrystalline ediﬁce built up of two or more homogeneous portions of the same crystal species in juxtaposition and oriented with respect to one another according to well deﬁned laws’. This covers the phenomenon in general. A classiﬁcation of twins according to their physical origin is then very common in the literature, where growth, deformation and transformation twins are often distinguished. Growth twins are usually produced in crystals during the whole crystallization process, in the course of vapor-to-solid or liquid-to-solid phase changes. Sometimes they can also be formed through bonding of already formed macroscopic individuals, at the end of the crystallization. Growth twins are in general divided into contact twins, in which the interface is a single, well individualized plane, and penetration twins, which are inter-

30

INTRODUCTION

(a)

(b)

Figure 1.2 The Baumhauer experiment; (a) schematic description; (b) picture of an actual twinned calcite crystal From Klassen-Nekliudova (1964), courtesy of Kluwer Academic/Plenum Publishers.

(b) (a)

(c)

(d)

Figure 1.3 X-ray diﬀraction of a hexagonal metal, along (a) the sixfold hexagonal axis; (b) one of the twofold axes in the basal plane; (c) the sixfold axis after the twinning shear sketched in (d) has occurred; notice the twofold-like spots From Klassen-Nekliudova (1964), courtesy of Kluwer Academic/Plenum Publishers.

grown in such a way that they have more than one composition plane or surface. Fig. 1.1 shows some examples of growth macrotwins found in pyrite and staurolite (penetration), and in tinstone and gypsum (contact). Alternatively, a single crystal may become twinned by mechanical actions, for instance by impact; these are called deformation twins, and are much less abundant than the twins spontaneously produced during growth. An illustrative example is the so-called Baumhauer experiment in calcite, shown in Fig. 1.2: a sharp edge is carefully pressed onto the calcite crystal, which eventually yields and partially ﬂips to a mirror-related shape. In deformation twins there is not only a change of the external shape of the crystal; rather, this change reﬂects a rearrangement of the crystal lattice, as can be seen, for instance, in the change of the X-ray pattern shown in Fig. 1.3. Both Figs. 1.2 and 1.3(d) show the existence of a macroscopic simple-shear deformation acting as a mechanical twinning mechanism. The existence of

1.2 SOME EXPERIMENTAL OBSERVATIONS

31

R2 R1 (c)

(b) (a)

Figure 1.4 (a) A wedge of orthorhombic martensite grown in cubic austenite of a CuAlNi alloy. (b) A blowup of an apparently homogeneous region of martensite, which actually consists of twin bands, each one made of other twin bands. (c) A cartoon of the deformation relating two individuals of a (compound) twin (a) and (b) from Tan and Xu (1990), courtesy of Springer Verlag.

such a twinning shear, described below and exempliﬁed in Figs. 1.4(c) and 1.7(a), implies a restriction on the possible twin laws which is called the twinning equation and is analyzed in chapter 8. This describes the deformation of crystals modelled by simple lattices, which are the object of most of this book. We stress that in many cases the details of the lattice units, or the motif, do matter for twinning, and require a corresponding kinematics which is only partly understood, as detailed in chapter 11. Notice that there are twins in complex crystals with no macroscopic shear but only a rearrangement of the motif in the crystal structure. These are called shufﬂe twins, and a typical example is the Dauphin´e twinning mode in quartz (chapter 11). Transformation twins develop as the consequence of the mechanical stress arising in the crystal because of a phase transition, that is, due to the structural reorganization and symmetry change in the lattice that may be caused by the variations in the pressure or temperature conditions. An example

32

INTRODUCTION

(a)

(b) (c)

Figure 1.5 Patches of twins in (a) Rochelle salt and (b), (c) CuAlNi shape memory alloy. (c) shows a detail of the twin bands in one of the patches of (b) (a) from Klassen-Nekliudova (1964), courtesy of Kluwer Academic/Plenum Publishers; (b), (c) courtesy of C. Chu and R.D. James.

of the low-symmetry (martensitic) twins in CuAlNi arranged in a wedge formation immersed in the high-symmetry phase (austenite) is shown in Fig 1.4. Notice the rather sharp interface between austenite and twinned martensite, which is called a habit plane, and must satisfy kinematic compatibility restrictions resulting in a certain simple shear condition; this is introduced in the crystallographic theory of martensite, which is widely used by metallurgists to determine habit planes and will be detailed §10.4. A shear condition also relates the twin-related bands in the martensite, as well as the twin bands within the bands visible in Fig 1.4(b). Unlike InTl alloys, which often show twins in a simple laminar structure over most of the specimen (Basinski and Christian (1954)), this alloy is an example in which the microstructure accommodating the stress is not made of a single set of twinned lamellae, but rather of a double laminate, consisting of twinned layers within twinned layers. Another similar example, in NiMn, is given by Baele et al. (1987). Also more complicated patches of twinned regions can easily occur, as exempliﬁed in two diﬀerent materials in Fig. 1.5. Notice that in Fig. 1.5(c) the twin bands clearly taper when they approach a laminate of a diﬀerent orientation, to accommodate the interfacial stress. The width of two neighboring twin bands like the ones in Figs. 1.4(b) and 1.5(b)-(c) depends on the applied loads, which also aﬀect the tapering and produce branching of the needles at the edge of the laminate; see James et al. (1995). In that case one can choose the loading in such a way that one twin variant disappears in favor of the other. The phenomenon is largely reversible, at least in a statistical sense: by reversing the loads both variants come back with the same volume fraction, the bands not necessarily appearing at the same place as before. Also, by cycling the temperature around the transition value, we can make the twins disap-

1.2 SOME EXPERIMENTAL OBSERVATIONS

33

pear when martensite transforms to austenite, and reappear in the reverse transformation, with the same volume fraction but generally diﬀerent details in the patterns. The reversibility of both the phase transition and the formation of twins makes it reasonable to look for a suitable thermoelastic model for these phenomena. On the other hand, such a model should be also able to explain the presence of hysteresis in typical stress-strain and strain-temperature experimental curves; as anticipated, one of the leading ideas is that dynamics and metastability may play an important role in this phenomenon. Since transformation and deformation twins are both induced by means of either stress- or temperature-driven deformations in crystals, they are grouped under the common denomination of mechanical twins. Most transformation twins have the characteristics mentioned above, and are typically found in the low-symmetry variants produced in the so-called displacive phase transformations (Buerger (1960)). Their presence, usually in the form of ﬁnely twinned microstructures as in Figs. 1.4(b) and 1.5, can considerably change the macroscopic behavior of the crystal, and is thought to be the basis for the remarkable behavior of shape memory alloys, two of which appear in Figs. 1.4-1.5. A qualitative description of the shape memory eﬀect has been given at the beginning of this chapter. This and similar behaviors make the investigation of transformation twins particularly interesting also from the point of view of the applications, which take advantage of the built-in switching mechanisms of these alloys. The technologies used in almost all aspects of the new electronic, biomedical, environmental, energy, and transportation systems, ask for materials whose properties are far diﬀerent from the conventional ones used in the past; the new materials (active or smart materials) must allow for a wide variety of applications based on their ability to respond actively to the environmental conditions, manifesting associated functions such as intelligent sensing, processing, actuation and feedback. Moreover, these materials are the perfect choice for microdevices of the aforementioned kind because any small amount of them has the required properties. It is thus very natural to also study second-order or weak ﬁrst-order phase transitions, and the related possible twins, in the context of nonlinear elasticity theory, and this will be done in chapters 7 to 9. By second-order we mean that the thermodynamic state, which in our case is a measure of deformation, changes continuously across the transition, while weak ﬁrstorder means, roughly, that the deformation has a jump at the transition, but not a large one, in a suitable sense. For these phase changes and the related twins we are able to produce a satisfactory model. Also, since the behavior of active crystals depends on the number of the coexisting phases and on the multiplicity of the symmetry-related variants, the performance should be expected to be best for a one component multiphase crystalline solid near a triple point in the phase diagram. Various multiphase crystalline solids exhibiting a solid-state triple point are known; particularly

34

INTRODUCTION

Figure 1.6 Experimental phase diagram of zirconia (ZrO2 ) showing the progressive tetragonal-orthorhombic-monoclinic symmetry reduction, and the related triple point. The holohedries are detailed in §§5.1.1, 5.1.3.1, and 5.1.4.1

interesting for its toughening action on ceramics is zirconia (ZrO2 ), whose relevant part of the phase diagram is shown in Fig. 1.6. An energy function ﬁt to describe the various phase transitions and the triple point has been recently proposed by Truskinovsky and Zanzotto (2002). Other transitions that are not weak are of both theoretical and applicative interest. An example is the face-centered cubic to body-centered cubic phase change in iron, of utmost importance in metallurgy. Another class of nonweak transitions consists of those in which the elementary cell of the crystalline lattice doubles, triples, etc, at the transition, as in the body-centered cubic to hexagonal close-packed transformation. In a diﬀerent framework, physicists treat these transitions by means of wave vectors, irreducible representations of the space groups, etc. Such weak transitions are in the process of being studied within the continuum theory presented here, but for a satisfactory model more work is needed. Nevertheless, many microstructures can be already well described in the present theory ﬁt for weak transformations. We emphasize that mechanical twinning and slip are the main mechanisms in the so-called plastic ﬂow of metals (see for instance Lemaitre and Chaboche (1988)), because they are the processes through which crystals can accommodate large deformations. Still, twinning is more often encountered than slip because twins are produced in abundance also during the growth process from melt. This is one of the reasons why mechanical twinning is so important in crystal mechanics. Examination of a single crystal after it has been subjected to plastic deformation reveals that when this takes place by slip, lamellae glide over each other on dense crystallographic planes (that is, planes in the lattice containing nets of atoms) called the slip planes, which are characteristic for each material. During the slip process the nearest-neighbor relations are disrupted. This does not happen

1.2 SOME EXPERIMENTAL OBSERVATIONS

(a)

35

(b)

Figure 1.7 (a) twinning and (b) slip in simple lattices. Compare slip along the plane represented by a dotted line in (b) with twinning of the same lattice along the same plane, drawn in Fig. 1.4(c)

in mechanical twinning, which is produced by a shearing movement over one another of the planes of atoms parallel to some composition plane (the twin interface), which is often but not always a crystallographic plane. The twinned regions are usually in the form of plates, often very thin lamellae, on each one of which the shearing appears homogeneous under the microscope; for instance, in iron these twin lamellae are known as Neumann bands. During gliding, adjacent planes of atoms slide past one another, usually aided by the movement of dislocations. In both deformation mechanisms the lattice structure is preserved, either in the same orientation as the original one of the undeformed single crystal, for slips, or in a diﬀerent orientation, for twins. In more detail, the conﬁguration of a mechanically twinned crystal, near a twin interface and away from other twins or the surface of the crystal, can be described as follows (see Fig 1.4(c)): two regions R1 and R2 , in each one of which the deformation is homogeneous, are separated by a planar interface. On the one hand, the lattice structure in R2 can be obtained from the one in R1 by means of a period-two orthogonal transformation (the twinning operation or twin law): in most cases a reﬂection across the planar interface but also, less often, a rotation by π with axis in that plane. According to a classiﬁcation rather common in crystallography and mineralogy, twins of the ﬁrst and second kinds are of (conventional) Type 1 and Type 2, respectively, and are precisely described in §8.3.5. The one in Fig 1.4(c) is actually of either kind, and is called compound. For the Type-1 twins the twin interface is a crystallographic (rational) plane, while for the Type-2 twins that are not compound the interface is typically not rational.

36

INTRODUCTION

On the other hand, the lattice in R2 can be also obtained from the one in R1 by means of a simple shear, as indicated in the picture. This fact can be related to the observation that the displacement remains continuous across the twin interface, on which its gradient suﬀers a jump discontinuity. The twinning equation, that is, the requirement that a lattice congruent to the original one be reconstructed across the interface by a simple shear, is the key restriction on the possible mechanical twinning operations. In Fig 1.4(c) we have not considered the common and physically important case in which the shear alone is not capable of describing all the atomic movements, and additional shuﬄe atomic displacements have to be introduced. A schematic example of this case is given in Fig 1.7(a). As mentioned above, an extreme example is given by the Dauphin´e domains in quartz, which are related by shuﬄe displacements alone, with no shear of the skeletal lattice. Except for Dauphin´e and other similar twins, in twinning a shear is always observed and often measured (Cahn (1954)). The presence of a simple shearing deformation is a distinctive feature of mechanical twins; it introduces a basic distinction between mechanical and growth twins, which is clearly reﬂected in the theory. Indeed, the requirement that the twin reorientations be always obtained through a shearing deformation of a homogeneous parent crystal provides a deﬁnite condition restricting the allowed twinning operations in mechanical twins. Such a condition is per se meaningless for growth twins, which occur during the actual formation of the (parent) crystal itself. Still, for mechanical twins the existence of a simple shear is not enough to make the deﬁnition unambiguous. In the case of growth twins the requirements are even less deﬁnite and precise; and only on some key points the literature seems to agree and be assertive: the same reorientations have to be found in a great number of specimens, frequency and reproducibility of a given pattern being a quality of genuine twins as opposed to random events or twins of the imagination (Friedel (1926), p. 422). It is instructive to quote the deﬁnition given by Dana (1962): ‘A twin is a geometrical position of intergrowth of two or more crystals of the same species with which is associated a frequency of occurrence greater than that of chance’. Along with the problem of deﬁning true twins as opposed to random intergrowths, goes the question of their classiﬁcation, which is tackled in various diﬀerent ways, the same being true for the great variety of twinning mechanisms proposed to explain the experimental observations. For both mechanical and growth twins alike, earlier geometric theories, notably the one by Friedel (1926) and the complemented and weakened (1933) version by himself, still hold a ﬁrm place in the mineralogical and metallurgical literature (see for instance Cahn (1954), Donnay and Donnay (1959), or Santoro (1974)). Here what matters is not the shear, which is not mentioned, but rather the purely geometric idea of coincidence-site lattices: equivalent conditions for twinning are (i) a lattice row and a lattice plane are orthogonal, or nearly so; (ii) there is a superlattice left invariant, or

1.2 SOME EXPERIMENTAL OBSERVATIONS

37

nearly so, by the twinning operation. An algebraic version of either requirement is given by Santoro (1974): if ea , a = 1, 2, 3, is a basis for the lattice, and M = (M ra ) is a matrix of rationals, then superlattices left invariant, or almost so, by a twinning operation are characterized by the condition that M ra er · es M sb be approximately equal to ea · eb . On the contrary, based on early contributions of the german mineralogists detailed by Niggli (1928), for instance, other authors, among which Hartman (1956) and Holser (1958), (1960), appreciate the importance of the twinning shear as a way to lower energy, and relate twinning to the crystal structure; also, shear becomes a fundamental ingredient of the crystallographic theory of martensite mentioned above. Another important feature that genuine twins possess is very early clear to crystallographers: twins must have interfaces of very low, or comparatively low energy. For instance, Buerger (1945) p. 470–471, 475, (1960), Barrett and Massalsky (1966) and Bollmann (1970) argue that for twin interfaces to have low energy it is necessary that they be rational (crystallographic), with low indices (deﬁned in §§3.7.3–3.7.4), because in this way the coordination and the interatomic distances at the interface are not too diﬀerent from those of the homogeneous crystal; Cahn (1954) p. 392–393, regards the twin boundaries to be surfaces of less stability and lower cohesion; Hartman (1956) associates low interfacial energy with high matching at the interface. The latter author, together with Evans (1912) and Holser (1958) among others, follows ideas of the aforementioned german mineralogists in giving additional structural conditions for twinning in which diﬀerent matching conditions at the (crystallographic) twin interface are used in order to deﬁne the allowed twin laws. As an example, Barrett and Massalsky (1966) and Zoltai and Stout (1984) assume the twin interface to be crystallographic and to ‘be the same plane’ on both of its sides, that is, to have the same indices in respective lattice bases. Another assumption is that every crystallographic direction in the twin interface be ‘the same direction’ as above. This second assumption is in some cases alternative, in others added to the ﬁrst, and sometimes one imposes that in either one of the previous requirements the two bases involved be identical. Other restrictions on the possible twinning modes are expressed in terms of the twinning operation. The theoretical predictions delivered by the twinning equation leave open a wide range of possibilities, other than the conventional Type-1 and Type-2 twins mentioned above, and a number of special nonconventional solutions are by now known. Still, there is a common opinion reported by Buerger (1960), but not shared by him, according to which the only orientations allowed for the individuals in a twin pair can be the conventional ones; the others are denied or considered as mere intergrowths rather than twins. Also, in many mineralogy and metallurgy textbooks only Type-1 and Type-2 twins are analyzed, but this seems to be a matter of nomenclature rather than a real questioning about the polycrystalline conﬁgurations actually observed.

38

INTRODUCTION

The aforementioned restrictions, structural and otherwise, are also attempts to solve one of the main problems with the theories proposed in the past, like the one of Friedel (1926) mentioned above: the general rules are too broad, in that they allow for many a priori possibilities that have never been actually observed – see Buerger (1945) p. 475, and Cahn (1954) p. 378–379. This is the case also with the thermoelastic theory of mechanical twinning proposed and elaborated by various authors in recent years. A great deal of work has been done in trying to understand which further rules, besides the general necessary ones, have to be given to complement the general theories. This in order to have them better agree with the observations, and to render them precise enough to give an understanding, for instance, of why some conﬁgurations occur much more often than others, or why certain possibilities have never been observed. Among the rules reported in the literature we mention the following: a) Various hypotheses of rationality for the interfaces and for other shear elements (see for instance Buerger (1945)). b) Minimum or ‘small’ shear hypotheses (see for instance Friedel (1926) p. 487, Hall (1954), and Bevis et al. (1968)). c) Finite period for the relative orientations of the various individuals: generally 2, 3, 4 or 6 (see for instance Friedel (1926) and Barrett and Massalsky (1966)). As we mentioned, the occurrence of periods other than 2 is denied by some authors, originating a sort of prejudice according to which true twins can only exhibit the conventional orientations given in Section 8.3.5 (see Buerger (1960)). d) Some further, less common rules; for instance, that the mean adjustments of atoms in twinning deformations should be minimized; or that all the atoms should move roughly the same distance (see Hall (1954) p. 85). Other conditions are stated by Cahn (1954) p. 381, and for growth twins by Evans (1912), Hartman (1956), and Holser (1958). There is no common agreement among workers about the range of validity of the hypotheses above, which all have no theoretical basis and admit counterexamples. They are helpful criteria which are used in practice to select one among the various twinning modes that could ﬁt the observed experimental data collected analyzing twinned specimens of crystals. Concerning nonconventional mechanical twins, Friedel (1926) p. 490 admits the possibility of their existence; still, the analysis of a transformation twinning mode in leucite (also considered by Curien (1960)), which he describes by means of a fourfold operation, is not a proper example because it can be also described by a twofold twin operation (Zanzotto (1988)). For metals, two important facts emerge from the experimental reports:

1.2 SOME EXPERIMENTAL OBSERVATIONS

39

(1) There is no conclusive evidence of twins with nonconventional orientations3 (Bevis and Crocker (1969), p. 527). (2) In addition to (1), also Type-2 twins, not of the compound type, are indeed very rare. Cahn (1953), studying the twinning deformations in orthorhombic uranium, presented the ﬁrst example of a mode of Type 2 ever observed in metals. Only a few other cases are reported in the literature: again in uranium, by Daniel et al. (1971), and in crystalline mercury, by Crocker et al. (1966) and Guyoncourt and Crocker (1968). These two observations agree with, and perhaps are a basis for the common aforementioned assumption of conventionality. Also, they may explain why rationality of the shear elements plays such an important role in all the presentations of twinning in the mineralogical and metallurgical literature. The rationality assumptions seem to have been dropped for the ﬁrst time in the research program carried out in the 1960-70s by a number of British metallurgists (Crocker (1962), (1982), Bilby and Crocker (1965), Bevis (1968), Bevis and Crocker (1968), (1969), Acton et al. (1970), and other references listed therein). They formulated a purely kinematical theory of twinning with the explicit aim of studying possible nonconventional mechanical twins in metals. However, as mentioned above, they were unable to observe without doubt any of the predicted nonconventional modes. Interestingly, what is stated in (1) and (2) above seems also to be true for mechanical twinning in minerals, although in this case the situation is a little more confused than with metals. Therefore the prejudice mentioned above, that true twins only involve period-two conventional operations, has possible roots in the fact that only these conﬁgurations seem to be actually observed, at least for mechanical twinning. The reasons why this happens are not fully clear yet, and are a matter of current investigation. A few related remarks will be made in §9.4. In spite of all the eﬀorts, there is not yet a clear understanding of the phenomenon, and we do not know the best way to set some deﬁnite conditions restricting the possible twin reorientations so that they give the conﬁgurations usually described as genuine twins by mineralogists and metallurgists. The general features of a twin aggregate everyone agrees upon are: (i) in crystals the constitutive elements can adopt many relative orientations in equilibrium, and twins are equilibria with the same stability quality as a homogeneous crystal; (ii) the twin law must be frequent and reproducible; (iii) the surface energy associated with a twin interface is low. Of these, we embrace the prejudice (iii) that the energy associated to a twin interface is low; and actually, to keep the analysis simple, we assume throughout this book, as a ﬁrst approximation, that there is no surface 3

Only recently a nonconventional transformation twinning mode has been found in LaNbO4 (Jian and James (1997)).

40

INTRODUCTION

energy associated to those interfaces. Adding such an energy in the analysis of equilibria is the subject of ongoing research. For instance, the tapering of twin bands when approaching a diﬀerently oriented patch of twins, as shown in Fig 1.5, requires energy to be associated to the twin interface (see James et al. (1995). We will not study matching conditions or such, although further work in this direction is likely to give interesting results regarding the conditions that restrict the possible twin laws, especially the ones for growth twins. As we mentioned, in the opinion of almost all authors the key feature of growth twinning is a low-energy junction, possibly approximated by a stress-free joint. This, together with an idea of stability – the energy of the junction must be able to remain low during growth – further reduces the allowed morphologies, as is detailed by Zanzotto (1989), (1990). Matching conditions of a rather general kind are also considered by Ericksen (1983) and James (1984b). We do not address much issue (ii) either, since no surface energy is considered in the analysis. Nevertheless some qualitative arguments on frequency and reproducibility of transformation twins, based on low interfacial energy and suitable genericity conditions, will be presented in §9.4. Rather, in agreement with (i) above, the basis of our work is to regard twins as stable equilibria, in the absence of interfacial energy for convenience. Concentrating on modelling mechanical twins, we use thermoelasticity to make this notion precise, at least from the phenomenological point of view given by the framework of continuum mechanics. In particular, for a class of transformation twins the elastic approach allows one to make explicit and quantitative predictions.

CHAPTER 2

Preliminaries Here we brieﬂy recall some deﬁnitions and results of group theory, linear algebra and continuum mechanics that will be used in the following chapters. The mathematical apparatus and the details are kept at the minimum. 2.1 Basic notation Throughout this volume the symbol := [=:] indicates that an equality deﬁnes its left-hand [right-hand] side; however, this notation will be mostly used to stress the points in which important notions are introduced. As usual, → means ‘maps to’; the arrow → indicates a function or map, while ⇒, ⇔ denote implication and double implication, respectively. We also use the standard notation ∪, ∩, \, . . ., of set theory for union, intersection, diﬀerence of sets, etc. The symbols N, Z, R, denote the sets of natural, integral and real numbers, respectively, and Rn denotes the vector space of ordered n-tuples of real numbers (vectors); the latter are denoted in lowercase italic boldface or by their n ordered coordinates in a speciﬁc basis, as usual. The space Rn is endowed with the Euclidean scalar product, denoted by ‘ · ’, and with the associated Euclidean norm given by v 2 = v · v for any v ∈ Rn . The angle between the directions of the vectors u and , and u1 , . . . , un denotes the span of the vectors v is indicated by uv u1 , . . . , un , that is, the linear subspace generated by them. Furthermore, ∧ denotes the usual wedge (or cross) product of two vectors in R3 , and B denotes the 9-dimensional space of all bases of R3 , or triples of linearly independent vectors, with the usual Euclidean norm. Given any two subspaces of Rn , H and K say, such that H ∩ K = {0}, their (direct) sum H⊕K is deﬁned as the smallest subspace of Rn containing both H and K. A basis for H ⊕ K is given by a union of bases for H and K. When all vectors in a basis of H are orthogonal to all vectors in a basis for K (and vice versa), H and K are said to be orthogonal and their direct sum, called their orthogonal sum, is denoted by H K. A cone in Rn is a subset closed under multiplication by a scalar. The summation convention over repeated indices is adopted, for instance in linear combinations of vectors and in representations of vectors and tensors in a basis. Also, when there is no danger of confusion, we use ‘running indices’ without specifying explicitly their range: for instance, we write the basis ea in place of the basis {ea , a = 1, 2, 3} or the basis {e1 , e2 , e3 }. 41

42

PRELIMINARIES

2.2 Some notions of elementary group theory 2.2.1 Basic deﬁnitions We presume the reader to be familiar with the notions of group, subgroup and group homomorphism and isomorphism (for instance in Mac Lane and Birkhoﬀ (1967), Yale (1968), or Miller (1972)). The group operation is usually indicated as a product, which in general is not commutative, and the identity element by 1. When the group operation is commutative, the group itself is called commutative, and the operation is often indicated as a sum, with identity 0. The notation H ≤ G means that H is a subgroup of a group G and H < G that H is a proper subgroup. The order, or cardinality, of G, denoted by #G, is the number of elements of G when this is ﬁnite. A basic result is Lagrange’s theorem: the order of a ﬁnite group is an integral multiple of the order of each of its subgroups. If X is a subset of a group G, the subgroup of G generated by X is the intersection of all the subgroups of G containing X. If this subgroup is actually G itself, X is called a set of generators for G, and all the elements of G can be obtained by suitably multiplying elements of X. A group G generated by one element g diﬀerent from the identity is called cyclic, and #G is called the period or order of g. Each element of G determines a left and a right coset of H in G, which are respectively the disjoint subsets of G of the form: Hg := {hg : h ∈ H},

gH := {gh : h ∈ H} .

(2.1)

Cosets are not subgroups when g ∈ / H. Each coset has the same number of elements as H, and the union of the cosets is the group itself. If G is ﬁnite, the index of the subgroup H in G is the number of (right or left) cosets of H in G – see Proposition 2.1 below. 2.2.2 Conjugacy For any elements h and g of G, the conjugate of h by g is the element ghg −1 ∈ G, and the set of all the distinct conjugates of h is called its conjugacy class. Analogously, for any subgroup H of G and any g ∈ G, the conjugate of H by g is the subgroup gHg −1 := {ghg −1 : h ∈ H},

(2.2)

which is isomorphic to H. The family of all the subgroups gHg −1 that are conjugate to H in G is called the conjugacy class of H. In general, not all the subgroups that are isomorphic are conjugate; also, the conjugates gHg −1 of H may not be all distinct (see (2.3) below). If gHg −1 = H for all g ∈ G, H is called a invariant or normal or self-conjugate subgroup of G: in symbols, H $ G. In this case the right and left cosets are identical; the set of cosets has a natural group structure induced by that of G, and

2.2 SOME NOTIONS OF ELEMENTARY GROUP THEORY

43

is called the quotient group of G by H, to be denoted by G/H (Mac Lane and Birkhoﬀ (1967)). Given a group G and H ≤ G, the following subgroups of G give information on how h ∈ H, or H itself, behave with respect to conjugacy in G: the centralizer CG (h) [ CG (H) ] of h [ of H ] in G, and the normalizer NG (H) of H in G, deﬁned by CG (h) := {g ∈ G : ghg −1 = h}, CG (H) := ∩h∈H CG (h) = {g ∈ G : gh = hg for all h ∈ H}, NG (H) := {g ∈ G : gHg −1 = H}.

(2.3)

Thus CG (h) collects all the elements of G commuting with h (or leaving it invariant under conjugacy), CG (H) all those that commute with all the elements of H, and NG (H) all those that conjugate H to itself. Both H and CG (H) are normal in NG (H) and, of course, NG (H) = G when H $ G. For the following useful results on ﬁnite groups see, for instance, Mac Lane and Birkhoﬀ (1967) p. 102, 467, 470; statement (1) is an immediate consequence of Lagrange’s theorem. Proposition 2.1 Let G be a ﬁnite group, g ∈ G and H ≤ G; we have: (1) The index of H in G is the ratio #G/#H. (2) The number of distinct conjugates of g is the index of the centralizer of g in G : #G/#CG (g). (3) The number of distinct conjugates of H in G is given by the index #G/#NG (H) of the normalizer of H in G. Given a group G and a subset S ⊆ G (possibly not a subgroup), it is useful to consider the elements of S that commute with all the elements of a subgroup H ≤ G, with H ⊆ S. Such elements of S form a subset of S which, for simplicity, we will still call the centralizer of H in S, and for which we still use the notation CS (H); the latter is not a subgroup of G if S is not such. The case in which G = Aut and S = Sym> (these spaces are deﬁned in §2.3.1 below) gives a typical example of this situation. 2.2.3 Group actions and symmetry In general the notion that an object has a certain symmetry is expressed by the fact that it is invariant under the action of a suitable group. Formally, if N is a mathematical structure whose set of automorphisms is Aut N, an action A of a group G on N is a group homomorphism A : G → Aut N. This means that to each g ∈ G corresponds a bijective map N → N that transforms in some prescribed way (g acts on ) the elements of N, in such a manner that to the product of two elements of G is associated the composition of the corresponding maps. The transform of n ∈ N by the automorphism A(g) is simply denoted by g(n). The G-orbit G(n) of n under the

44

PRELIMINARIES

action of G collects all the transforms of n under the action of elements of G; this and the transform g(M) of a subset M ⊆ N are given explicitly by G(n) := {g(n) : g ∈ G},

g(M) := {g(n) : n ∈ M}.

(2.4)

The elements of G transforming n ∈ N to itself, that is, leaving n invariant, form a subgroup of G called the stabilizer of n; sometimes, this is also called the symmetry group of n. There are two ways of stabilizing a subset M of N, that is, by looking at the elements g ∈ G that either: (1) stabilize each element of M; in this case the subgroup {g ∈ G : g(n) = n for all n ∈ M}

(2.5)

of G is usually called the stabilizer of M; or (2) stabilize M as a whole, that is, leave M invariant, transforming any n ∈ M to g(n) ∈ M; in this case the subgroup {g ∈ G : g(M) = M}

(2.6)

is often called the symmetry group of M. When N is G itself, a natural example of action of G on G is given by conjugacy (§2.2.1). In this case the G-orbit of h is the conjugacy class of h, and the stabilizer of h is the centralizer CG (h). The two subgroups stabilizing a subgroup H in the sense of (2.5) and (2.6) are the centralizer CG (H) and the normalizer NG (H), respectively. A subset of a metric space N is called discrete if its intersection with any bounded subset of N is a ﬁnite set. A group G (acting on N) is discrete if for any n ∈ N the G-orbit of n is a discrete subset of N. A very important example of a discrete (arithmetic) group, often used in this volume, is given by GL(3, Z), which is the inﬁnite group of invertible 3 by 3 matrices with integral entries. Notice that necessarily any element m ∈ GL(3, Z) has determinant ±1, because the inverse of an integral matrix is again an integral matrix if and only if its determinant is unimodular. Hua and Reiner (1949) show that a choice of generators for this group is       0

g1 =  1 0

0

1

0

0 ,

1

0

1

g2 =  0 0

1

0

1

0 ,

0

1

g3 = 

−1

0

0

1

0 .

0

0

1

0

(2.7)

The following property will be useful in chapter 3: Lemma 2.2 Any m ∈ GL(3, Z) has period 1, 2, 3, 4, 6, or ∞. When a group G acts on a vector space V , the elements of Aut V are invertible linear maps, and a subspace W of V is said to be invariant under the action of G if g(W ) = W for all g ∈ G. An invariant subspace W is irreducible for the action of G if the only invariant subspaces of W are W itself and {0}, that is, if W does not have any nontrivial invariant subspace.

2.3 LINEAR AND ORTHOGONAL TRANSFORMATIONS

45

2.3 Linear and orthogonal transformations Of the tensor algebra based on R3 we mostly need the second-order tensors, so in this volume the term tensor will mean linear transformation from R3 to R3 , unless otherwise speciﬁed. Tensors are denoted in capital bold italics, exceptions being the null and identity tensors, 0 and 1 respectively, and tensor products of vectors, for instance a ⊗ n. Remember that, for any vector v , (a ⊗ n)v := a (n · v ). We assume familiarity with the basics on linear maps of ﬁnite-dimensional real vector spaces, in particular with the concepts of domain, kernel, image, and with the Euclidean structure associated with the scalar product A · B := tr AB t ,

(2.8)

where tr denotes the trace of a tensor or matrix. Also, recall that the rank of a tensor is the dimension of its image. We often associate with any tensor A the matrix A representing it in a given basis, and recall that any such representation1 is an isomorphism between the algebras of tensors and of real matrices. Since the properties of diﬀerent matrices representing the same linear transformation in diﬀerent bases will be relevant for us, we keep linear transformations and their groups clearly distinct from matrices and their groups. The 9-dimensional space Aut R3 of all invertible tensors is denoted for short by Aut ; the latter contains the 6-dimensional subspace of symmetric tensors and the convex cone of symmetric, positive deﬁnite tensors, denoted by Sym and Sym> , respectively. Aut is a group with respect to the usual noncommutative tensor multiplication, with unit element 1;2 Sym is not a multiplicative subgroup of Aut, only a vector subspace. In general, if P is any set of tensors or matrices, P + denotes the subset of the elements with positive determinant. If A is any matrix, A = (Aab ) indicates that its entries are the numbers Aab ; diag(n1 , . . . , nr ) denotes the r by r diagonal matrix with entries n1 , . . . , nr , and the diagonal matrix with all entries k is simply denoted k. A superscript t indicates the transpose of any matrix or tensor, and −t the inverse transpose. The space of symmetric matrices (or quadratic forms) on Rn is denoted by Qn and is n(n + 1)/2-dimensional; its subset of positive deﬁnite symmetric matrices is a convex cone, denoted by Q> n . The spaces Sym and Q3 are clearly isomorphic, as are Sym> and Q> . 3 The metric C of a basis ea of R3 , a = 1, 2, 3, is deﬁned by C = (Cab ), 1

2

Cab := ea · eb

⇒

C ∈ Q> 3.

(2.9)

This is a special instance of group representations, the general theory of which will not be used here. See for instance Weyl (1946), Hamermesh (1960), Lyubarskii (1960), Bhagavantam and Venkatarayudu (1969), Bradley and Cracknell (1972). Aut is in fact isomorphic to the algebra GL(3, R) of invertible 3 by 3 real matrices.

46

PRELIMINARIES

The following relations characterize the dual basis e a : ea · e b = δab ,

ea = Cab e b , e b = C ba ea ,

where

C ba Cac = δcb .

(2.10)

The dual basis is the natural basis for dual vectors (typically normals to vector subspaces), or covectors, which here, through the Euclidean scalar product, are identiﬁed with vectors. Only the diﬀerence of their transformation properties under linear maps will distinguish them, as the amplitude a and normal n in chapter 8. For instance, if F ∈ Aut has representative matrix f = (f ba ) in the basis ea , and we set ˜ ea := Fea = f ba eb ,

(2.11)

then all the following equalities hold: ˜ e a = F −t e a ,

˜ e k · eb = (f −1 )kb ,

˜ e k = (f −1 )kb e b ,

(2.12)

with an obvious analogue for a typical covector. If C˜ is the metric of the basis ˜ ea , the transformation (2.11) deﬁnes a natural action of the group of invertible 3 by 3 real matrices, GL(3, R), on Q> 3: C → C˜ = f t Cf .

(2.13)

We denote by O the set of orthogonal tensors (or linear isometries), which by deﬁnition preserve the scalar product in R3 . Any Q ∈ O satisﬁes QQ t = 1 = Q t Q, and if the matrix q = (q ba ) represents Q in the basis ea (Qea = q ba eb ), q leaves the metric C invariant under the action (2.13): q t Cq = C.

(2.14)

Also, any two bases ea and ˜ ea in B have the same metric if and only if they are related by an orthogonal transformation; in obvious notation C˜ = C

⇔

˜ ea = Qea for some Q ∈ O.

(2.15)

The elements of O+ are called rotations; each rotation has a 1-dimensional eigenspace for the eigenvalue 1, called the rotation axis. The other two eigenvalues are exp(±iω), where ω is the angle of rotation in radians. If v is a vector parallel to the axis, the counterclockwise rotation of ω about the oriented direction of v is denoted by Rvω .

(2.16) Rvω

The plane orthogonal to v is an invariant plane for which, in an orthonormal basis whose third axis is along v , is represented by the matrix   cos ω

 sin ω 0

− sin ω cos ω 0

0

0.

(2.17)

1

The matrices satisfying (2.14) for C = 1 are called orthogonal, and their group is denoted by O(3) (orthogonal group); SO(3) is the standard notation for O(3)+ (special orthogonal group).

2.3 LINEAR AND ORTHOGONAL TRANSFORMATIONS

47

The tensor groups Aut, O, etc., induce natural actions on R3 , for which the deﬁnitions in §2.2.2 apply. For instance, for any set V of vectors OV := {Qv : Q ∈ O, v ∈ V}

(2.18)

denotes the orbit of V under the standard action of O. Analogously, for any subset U of Aut, the O-orbit is deﬁned by OU := {QU : Q ∈ O, U ∈ U } ⊂ Aut.

(2.19)

In §5.4.1 we consider the action of O on Sym, which is an analog of (2.13). The following decomposition will often be used (Truesdell (1977)): Proposition 2.3 (Polar decomposition) Any element F of Aut can be uniquely written in the form F = RU , with R ∈ O and U 2 = F t F , U ∈ Sym > .

(2.20)

We will also need the following version of the well known theorem of simultaneous diagonalizability of two quadratic forms in Rn : Proposition 2.4 In Rn , let C = U 2 , with U a positive deﬁnite symmetric tensor, and let C2 be a symmetric tensor. Then linearly independent vectors a1 , . . . , an and real numbers λ1 , . . . , λn exist such that n n C2 = r=1 λr ar ⊗ar and C = r=1 ar ⊗ar , or ar ·C −1 as = δrs . (2.21) In particular, by (2.21)3 the vectors a r := C −1 ar are the duals of the ar . Indeed, by standard linear algebra, any symmetric tensor has a spectral decomposition; hence so does U −1 C2 U −1 , being symmetric: n n U −1 C2 U −1 = r=1 λr cr ⊗ cr , (2.22) r=1 cr ⊗ cr = 1, the cr being an orthonormal basis. Then (2.21) holds for ar = Ucr . By known properties of the spectral decomposition, the basis ar is unique if all the eigenvalues λr are diﬀerent, whereas inﬁnitely many bases can be chosen in any eigenspace corresponding to a multiple eigenvalue. Notice that the λr and ar in (2.21) solve the eigenvalue problem (C2 C −1 − λ1)a = 0,

(2.23)

whose characteristic equation can be written in the form det(C2 − λC ) = 0.

(2.24)

The last two equations are preferable to (2.22) for determining λr and ar if C rather than U is given, for they avoid taking the square root of C . The rank rk of C2 − C is often of importance in twinning problems. Based on (2.21) and on the action of C2 − C on the dual vectors a r , one can check that rk is the dimension of the span (λ1 − 1)a1 , . . . , (λn − 1)an ;

(2.25)

therefore rk is the number of the λr s that are diﬀerent from 1. Also

48

PRELIMINARIES

(Sylvester’s inertia theorem, see for instance Parlett (1980)), the number of positive and negative eigenvalues of C2 − C is the number of positive and negative (λr − 1)s, respectively. 2.3.1 Tensors with period two The tensors H ∈ Aut whose period is two, that is, such that H 2 = 1, are of particular interest in the theory of twinning. Proposition 2.5 A tensor H ∈ Aut such that H 2 = 1 has determinant 1 or −1 and, if H $= ±1, it can be written in the form H = det H (2b ⊗ n − 1),

(2.26)

where b and n are suitable vectors such that b · n = 1. When H has determinant 1, b belongs to the 1-dimensional eigenspace associated to the eigenvalue 1 of H , while n is orthogonal to the 2-dimensional eigenspace associated to the double eigenvalue −1. In general b and n are not parallel, but they are when H ∈ O+ ; in this case H is a rotation of π about the axis n, and can be written explicitly: H = Rnπ := 2n−2 n ⊗ n − 1.

(2.27)

Similarly, if det H = −1, b in (2.26) is in the 1-dimensional eigenspace associated to the eigenvalue −1 and n is orthogonal to the 2-dimensional eigenspace of H associated to the double eigenvalue 1 (invariant plane of H ). In this case H in (2.26) is called a linear reﬂection. If H ∈ O, again b is parallel to n; H can then be written in the form H = 1 − 2n−2 n ⊗ n

(2.28)

and is called a reﬂection, whose mirror plane is orthogonal to n. It is not diﬃcult to check that any tensor H of period two is conjugate in Aut to some period-two elements of O (see Proposition 2.12 below). Furthermore H , which by (2.26) is diagonalizable with entries ±1, is represented in suitable bases by matrices in GL(3, Z). When this happens, the components of the eigenvectors of H in such bases are solutions of a linear homogeneous system with integral coeﬃcients, hence can be chosen to be integers up to a factor. 2.3.2 Simple shears Also simple shears are linear transformations of great importance in the theory of mechanical twinning. Here we give some of their basic properties, partially following the nomenclature used in the experimental literature. In a simple shear S any plane parallel to a given plane K1 is moved onto itself at a distance that is proportional to its distance from K1 , and in a direction which is the same for all planes. If n is the normal to K1 , there

2.3 LINEAR AND ORTHOGONAL TRANSFORMATIONS

49

Figure 2.1 The ‘shear elements’ K1 , K2 , η1 , η2

is a vector a orthogonal to n such that S takes the form S = 1 + a ⊗ n,

with a · n = 0 ,

(2.29)

where the vectors a and n are deﬁned up to reciprocal multiplicative factors. Since det S = 1 + a · n, (2.29)2 implies det S = 1. In an orthonormal basis vi with v1 and v2 parallel to a and n, respectively (hence v1 , v3 ∈ K1 ), we have (s being called the amount of shear)   1

s

0

Svi = sji vj , with (sji ) =  0

1

0

0

0

1

and s = an .

(2.30)

2a 2n + n, η2 := − a, a2 n2

(2.31)

For S is as in (2.29), consider the vectors K1 := n, η1 := a, K2 :=

the ﬁrst two being introduced for later convenience. It is common practice in the mineralogical and metallurgical literature to specify simple shears by means of their elements K1 , K2 , η1 , η2 (see Fig. 2.1). The plane K1 orthogonal to K1 is called the invariant plane of the shear; the oriented direction of the shear amplitude vector η1 is indicated by η1 ; the plane of shear containing a and n is denoted by S; the oriented direction of the vector η2 is indicated by η2 , and K2 denotes the plane containing η2 and n ∧ a. K2 is called the second undistorted plane of S because the length of neither η2 nor n ∧ a is aﬀected by the shearing deformation, so that K2 is the unique plane which is only rotated by S . It is not diﬃcult to see that the vector K2 in (2.31) is normal to K2 and forms an acute angle with n. We will analyze the common twins called conventional (§8.3.5) based on

50

PRELIMINARIES

the following deﬁnitions – see (2.16) – and consequent identities: Rnπ = 2n−2 n ⊗ n − 1,

Raπ = 2a−2 a ⊗ a − 1,

(2.32)

2η2 ⊗ K1 − 1, (2.33) η2 · K 1 2η ⊗ K2 Raπ S = η1 ⊗ K2 − 1 = 1 − 1. (2.34) η1 · K2 By their last expressions, Rnπ S and Raπ S are independent of the lengths of η2 , K1 , and η1 , K2 , respectively; hence these tensors only depend on the directions of the respective vector pairs. In crystallography these directions are usually characterized by their crystallographic indices – see §3.6. Since Rnπ S only depends on K1 and η2 by (2.33), and Rnπ is determined by K1 , also S is determined by K1 and η2 , the analog for K2 and η1 being true by (2.34). This proves a well known result: a simple shear S is completely determined by either the pair (K1 , η2 ) or (K2 , η1 ) of its elements. Notice that both Rnπ S and Raπ S have the form (2.26), hence have period two; this can also be veriﬁed by a direct computation. One last property of shears is important in the analysis of twinning in crystals. Given any shear S as in (2.29), the equation Rnπ S = η2 ⊗ K1 − 1 =

¯ S, Sr = R

S r = 1 + a r ⊗ nr ,

ar · nr = 0,

¯ ∈ O+ , R

(2.35)

has a unique nontrivial solution Sr $= S , the reciprocal shear to S , with 1 η , nr = s2 K2 , sr = s, (Sr )r = S , (2.36) 4 + s2 2 as follows, for instance, by a direct computation. This shows the remarkable property of the two reciprocal shears: K1 and η2 in one coincide with K2 and η1 in the other, and vice versa, and the amounts of shear are equal. ar =

2.3.3 Finite groups of tensors or matrices In this subsection we recall a few elementary results about some useful groups of tensors; they have obvious counterparts for the analogous matrix groups, which we do not mention explicitly. In general, we are interested in having information on the groups of tensors commuting with (that is, centralizing) a given tensor or group of tensors. Finding the group CAut (F ) of tensors commuting with a given F ∈ Aut is a classical problem in linear algebra – see for instance Lancaster and Tismenetsky (1985) or Ortega (1987). We will not need such generality here, and only consider the simpler case of an orthogonal F . The ﬁrst proposition is a consequence of the uniqueness of the polar decomposition (2.20). ¯ R ∈ O, U ∈ Sym> , F = RU ; then Proposition 2.6 Let Q, Q, ¯ = FQ QF

⇔

¯ = RQ . QU = UQ and QR

(2.37)

2.3 LINEAR AND ORTHOGONAL TRANSFORMATIONS

51

¯ = Q in (2.37) we see that the tensors F = RU , R ∈ By choosing Q > O, U ∈ Sym , commuting with a given Q ∈ O are those for which both R and U commute with Q: Corollary 2.7 The group of tensors F commuting with a given Q ∈ O, that is, the centralizer CAut (Q), is given by CAut (Q) = CO (Q) CAut (Q) ∩ Sym > (2.38) = {RU : RQRt = Q, UQU −1 = Q}. The tensors U ∈ Sym > (Truesdell (1977) p. 198) [R ∈ O+ ] commuting with a tensor Q ∈ O are characterized as follows : Proposition 2.8 A tensor U ∈ Sym > commutes with Q ∈ O, that is, U ∈ CAut (Q) ∩ Sym > or Q ∈ CAut (U ) ∩ O, if and only if Q leaves the eigenspaces of U invariant. If Q ∈ O+ , its axis is an eigenvector of U . Proposition 2.9 A rotation R ∈ O+ commutes with another rotation Q ∈ O+ , that is, R ∈ CO (Q) and Q ∈ CO (R), if and only if either they share their axes or they are rotations of π with orthogonal axes. Since −1 commutes with all tensors, it is straightforward to extend these commutation conditions to any two elements in O. ¯ and Q in P one has For any P < O, by choosing in formula (2.37) Q Corollary 2.10 The normalizer NAut (P ), that is, the group of tensors conjugating P to P for P < O, is given by NAut (P ) = NO (P ) CAut (P ) ∩ Sym > (2.39) = {RU : Q ∈ P ⇒ RQRt ∈ P and UQU −1 = Q}. By choosing R = 1 in (2.37) we see that U ∈ Sym > conjugates P to P if and only if it commutes with each element of P : NAut (P ) ∩ Sym > = CAut (P ) ∩ Sym > .

(2.40)

Also, an immediate consequence of Proposition 2.8 is: Corollary 2.11 The tensor U belongs to CAut (P ) ∩ Sym > if and only if each R ∈ P leaves the eigenspaces of U invariant. Truesdell (1977), p. 201, lists, as we do in Table 6.1 below, the forms of the tensors in CAut (P ) ∩ Sym > when P either describes transverse isotropy or is one of the groups called the crystallographic point groups (§3.2.1). Another problem of great interest in crystallography is establishing the O- (and Aut -)conjugacy classes of ﬁnite groups of tensors. We will return with more details on this point in Proposition 3.2; in Proposition 3.5 we will also give some important results regarding the GL(3, Z)-conjugacy classes of ﬁnite groups of matrices. Here we mention two basic useful results: Proposition 2.12 Any ﬁnite subgroup L = {L1 , . . . , Ln } of Aut is conjugate in Aut to a ﬁnite subgroup of O.

52

PRELIMINARIES

Indeed, consider the natural action C → Lt CL of Aut on Sym> , which is ¯ say,3 which is analogous to (2.13). There certainly is a tensor in Sym> , C ¯ is the square left invariant under the action of all the Li in L; then, if U −1 ¯ ¯ ¯ is a subgroup of O. root of C , U LU Proposition 2.13 Two subgroups P and P¯ of O are conjugate in Aut if and only if they are conjugate in O. This useful result roughly says that, in O, orthogonal and linear conjugacy ¯ ∈ P¯ and Q ∈ P : if are ‘equivalent’; this follows from (2.37) by choosing Q −1 t ¯ ¯ P = F P F , then P = RP R also, where F = RU in (2.20), R ∈ O. Of course R is not the only element of O conjugating P¯ to P : RQ does too if and only if Q ∈ NO (P ). Let v be an arbitrary (unit) vector in R3 and introduce the following notation: O2 is the group of orthogonal transformations in R2 , and O2+ its subgroup of proper orthogonal transformations; for any integer n ≥ 2, Zn is 2π/n the cyclic subgroup of O+ , of order n, generated by Rv (the rotational symmetries of the n-pyramid); also for n ≥ 2, Dn is the dihedral group, 2π/n of order 2n, generated by Rv and by a twofold rotation whose axis is orthogonal to v (the rotational symmetries of the n-prism). The order is 2n because the group contains n − 1 other twofold rotations whose axes are all orthogonal to v and evenly spaced. Clearly Zn < Dn . Finally, let T, O, I denote the groups of rotational symmetry of the ﬁve Platonic solids: a regular tetrahedron, a cube (or octahedron), and a regular dodecahedron (or icosahedron), all centered at the origin in R3 . The following holds: Proposition 2.14 Any subgroup of O+ which is closed in the topology induced by the scalar product (2.8) is orthogonally conjugate to one of the groups in the following list: {1, Zn , Dn , T, O, I, O2+ , O2 , O+ }

(n ≥ 2).

(2.42)

For the proof of this result, which will be useful in the analysis of crystallographic point groups in chapter 3, see Golubitsky et al. (1988). 2.4 Aﬃne transformations We denote by A3 the 3-dimensional real aﬃne space (with typical points P, Q, etc.) whose associated vector space of translations is R3 . The group of invertible aﬃne transformations of A3 is denoted by Aﬀ (3); its subgroup of isometric aﬃne transformations, or isometries, which preserve the aﬃne Euclidean structure of A3 , is the Euclidean group, denoted by E(3). We use the Grassmann notation for the points in A3 and its translation 3

Given an arbitrary C ∈ Sym> , the desired property holds for the average ¯ := C

1 #L

Li ∈L

Lti CLi .

(2.41)

2.5 CONTINUUM MECHANICS

53

vectors; for instance, v = P − Q, P = Q + v , or v = QP indicate that the point P diﬀers from Q by the vector v ∈ R3 . Usually, we will ﬁx a point O in A3 as origin, and replace the typical point P of A3 by its position vector x = OP. Once the origin O is ﬁxed, it is well known that any a ∈Aﬀ (3), in particular any isometry e ∈ E(3), can be uniquely represented by means of a pair (t, A),4 with t ∈ R3 and A ∈ Lin , or in O for isometries, t and A being deﬁned by t := Oa(O),

AOP := a(O)a(P ).

(2.43)

Then the action of an aﬃne transformation a is: ¯ := a(P) = AOP + t + O. P

(2.44)

The following rules hold for the representatives of the product and of the inverse of aﬃne maps: a ≈ (t, A), a ≈ (t , A ) ⇒

aa ≈ (t + At , AA ) and a−1 ≈ (−A−1 t, A−1 ).

(2.45)

The representation of the aﬃne transformation a through the pair (t, A) ˜ to a ∈Aﬀ (3) is associated a depends on the point O; if we replace O by O, ˜ for which the analogue of (2.44) holds. As is known, new pair (˜t, A) ˜ ˜t = t + (A − 1)OO

˜ = A. and A

(2.46)

Thus both A and the projection of t onto the kernel of A − 1, to be called the essential translation of a, are independent of O, hence are properties of the aﬃne map a itself. By (2.45) the translation subgroup T (3) of E(3), consisting of the pairs (t, 1), t ∈ R3 , is a maximal commutative (and therefore normal) subgroup of E(3) and it is a vector space isomorphic to R3 . Unlike for the general elements of Aﬀ (3), the representation of the translations in Aﬀ (3) as pairs (t, 1) is independent of the choice of the origin O. Notice that the central inversion (0, −1) does not commute in general with other elements of Aﬀ (3). t

2.5 Continuum mechanics We recall some basic notions in the theory of crystalline solids as suitable continua, by modelling their deformation, thermomechanical response, etc.; for details see for instance Truesdell and Toupin (1960), Truesdell and Noll (1965), Jaunzemis (1967), Wang and Truesdell (1973), Truesˇ dell (1977), Gurtin (1981), Maugin (1993), Silhav´ y (1997), M¨ uller and Ruggeri (1998). 4

We follow the notation (t, A ) proposed by Michel (1996), ﬁnding it more convenient than the usual (A, t ).

54

PRELIMINARIES

Figure 2.2 Finite deformation

2.5.1 Deformation An open subset R of R3 which is topologically a ball, is regarded as the reference conﬁguration (or reference state) for a body whose macroscopic deformation is described by an invertible function χ : R → R3 . We denote by x and y , respectively, the typical point in R (material point) and in the present conﬁguration χ(R) with respect to some given observer (that is, in a given Cartesian coordinate system). The deformation y = χ(x ) can then be represented as follows: y r = χr (xs ), r, s = 1, 2, 3.

(2.47)

In the theory of inﬁnitesimal deformations one also introduces the displacement vector u := y − x , which will be seldom needed here. We assume χ to be invertible, continuous and piecewise continuously diﬀerentiable, with deformation gradient F (x ) := Dχ(x ) such that J := | det F | > 0 ,

(2.48)

D denoting the gradient operator. In Cartesian coordinates: ∂χr . (2.49) ∂xs If F does not depend on x the deformation is called homogeneous. For the purposes of this volume it is suﬃcient to assume that the deformation gradient F (x ) may fail to be continuous on a ﬁnite number of F rs =

2.5 CONTINUUM MECHANICS

55

regular oriented surfaces in R, on which F suﬀers a jump discontinuity.5 Let us ﬁx any point x on one of these surfaces, whose normal at x is n ∗ ,6 and consider a suitably small sphere centered at x and contained in R. We label ‘2’ the part of the sphere that the positive normal n ∗ points into, and ‘1’ the other; and denote by F2 and F1 the ﬁnite limits of F at x from the 2 and from the 1 part of the sphere, respectively. Then, by the Hadamard compatibility conditions (Truesdell and Toupin (1960) eq. (175.9)), which guarantee the continuity of χ, the jump in F is a tensor of rank 1: [[F ]] := F2 − F1 = a ⊗ n ∗ ,

(2.50)

where the amplitude a of [[F ]] can be any vector. A continuous, piecewise smooth deformation is called coherent in the experimental literature, and so is called any surface on which (2.50) holds. The tensor U ∈ Sym > appearing in the polar decomposition (2.20) of the deformation gradient F is called stretch. One can also deﬁne the right [left] Cauchy–Green tensor C [B]: C := F t F = U 2 ,

B := FF t

C , B ∈ Sym > ,

(2.51)

and the (nonlinear) strain tensor E as follows: 1

E = (C − 1).

(2.52)

2

If the displacement u is small in some sense, so that quadratic terms in the derivatives of u are negligible, one obtains the expression 1

E = (Du + (Du)t ), 2

1

Esr = ( 2

∂ur ∂us + ) s ∂x ∂xr

(2.53)

which is used in linear elasticity. If χτ is a one-parameter family of deformations, a surface in χτ (R) whose inverse image in R does not depend on τ is called a material surface, because it does not sweep through the material points in R when the deformation χτ varies with τ . Recall that vectors in the reference state R are transformed into vectors in the present conﬁguration χτ (R) by means of the deformation gradient F , while the normal n ∗ to a plane in R is transformed into the normal n to the corresponding surface in χτ (R) by F −t . Vectors that vary as above with the deformation are called material. 5

6

These assumptions allow us to describe the simplest cases of twins and microstructures in crystals undergoing solid-state phase transformations. See the literature quoted in §2.5.2 and in chapter 10 for the technical conditions arising in the variational models used for the more complex cases of microstructure formation. Whenever necessary, reference vectors (and covectors) are distinguished from their present counterparts by an added asterisk.

56

PRELIMINARIES

2.5.2 Thermodynamic potentials and their invariance Later on we will adopt nonlinear elasticity theory to describe the behavior of crystalline substances that can change phase. The basic model is the one of a homogeneous thermoelastic material, which admits an internal energy density 7 ;, an entropy density η and a (Helmholz) free energy density φ. The constitutive equations for these quantities have the form: ˜ , θ) := ;˜(F , θ) − θ η˜(F , θ), ; = ;˜(F , θ), η = η˜(F , θ), φ = φ(F

(2.54)

with F ∈ D and θ ∈ I. Here θ is the absolute temperature, which varies in a suitable interval I, and D ⊆ Aut is a suitable open set of allowed deformation gradients. The constitutive functions in (2.54) are supposed to be smooth enough over their domain to allow for all the operations that will be necessary. The Clausius-Duhem inequality, a form of the second principle of thermodynamics, requires φ˜ to be a potential for the Piola–Kirchhoﬀ stress tensor P and the entropy density η (for such constitutive restrictions see for instance Truesdell and Noll (1965), who also introduce the Cauchy stress tensor T and the second (symmetric) Piola-Kirchhoﬀ stress tensor σ): ∂ φ˜ ∂ φ˜ P= (2.55) and η = − , P = JTF −t = F σ . ∂F ∂θ In continuum mechanics one imposes two types of invariance on the constitutive functions (Truesdell (1977)). The ﬁrst invariance, called objectivity or frame indiﬀerence, in its standard form relates to any isometric change of coordinate system, which aﬀects the present conﬁguration (see (2.59) below); the other, called material symmetry, relates to the changes in the reference conﬁguration that are mechanically undetectable (see (2.60) below). Neither change aﬀects the scalar thermal variable θ. In the case of thermoelastic materials the ﬁrst invariance reduces to the Galilean invariance and, indeed, to the Euclidean invariance, that is, to the invariance under changes of observer through any rigid-body displacement.8 The invariance of the free energy function φ˜ of homogeneous thermoelastic materials due to both objectivity and material symmetry reduces to the equalities in (2.65) below. To see this, introduce a new reference conﬁguration R by means of an invertible aﬃne map κ of R onto R , with gradient H := Dκ ∈ Aut. In addition, consider any new observer with same space-time units as the original one, and label with a prime geometric and kinematic quantities related to the new observer. Since the change 7 8

All the densities are intended per unit reference volume. As in Truesdell and Noll (1965), here we admit motions which do not preserve the orientation of space, so that Q in (2.56) below can vary in O and not only in O+ . There is some debate about this issue; see for instance Ericksen (2000c). However, this does not aﬀect the ﬁnal invariance properties (2.65) of the constitutive function φ˜ for the class of elastic crystals considered here, which are centrosymmetrical by material symmetry. A word of caution in favor of O+ should be said for the invariance of constitutive equations of multilattices (chapter 11).

2.5 CONTINUUM MECHANICS

57

of observer preserves the space-time distance and the orientation of time, the positions y and y and times t and t, representing the same event-point for the new and old observer, are related in this way: y = Qy + y0 ,

t = t + a ,

(2.56)

for some Q ∈ O, y0 ∈ R , and a ∈ R. Classical arguments (see for instance Truesdell (1977)) then show that the response function φ˜ giving the free energy of the body in deformations relative to the new reference conﬁguration R and with respect to the new observer, is related to the original free energy function φ˜ as follows: 3

˜ t FH , θ) . φ˜ (F , θ) = (det H )−1 φ(Q

(2.57)

On the one hand, Euclidean invariance requires the response functions φ˜ and φ˜ to coincide for any change of observer; that is, in (2.57) φ˜ must coincide with φ˜ when H = 1, for any Q ∈ O and F ∈ D: ˜ , θ) = φ(QF ˜ φ(F , θ).

(2.58)

The domains of φ˜ and φ˜ must also coincide, and this in turn forces the domain D of φ˜ to be invariant under left multiplication by any orthogonal tensor. An important consequence of (2.58) is that the constitutive function is independent of the orthogonal part R in the polar decomposition (2.20) of F , so that φ only depends on U , or on C introduced in (2.51): ˜ , θ) = φ(U ˜ , θ) φ(F

⇔

ˆ , θ). φ = φ(C

(2.59)

On the other hand, material symmetry requires that there be a (possibly trivial) group G of unimodular tensors H such that ˜ , θ) = φ˜ (F , θ) φ(F

⇔

˜ , θ) = φ(FH ˜ φ(F , θ),

(2.60)

˜ any H in G, and any for any F in the common domain D of φ˜ and φ, θ; thus the domain D must be invariant also under right multiplication by any H ∈ G. The group G is called the material symmetry group relative to the reference conﬁguration R. Its elements are called material symmetries; by (2.60) they are the gradients of aﬃne maps giving new reference conﬁgurations for which the response function φ˜ remains unchanged. Such new conﬁgurations are mechanically indistinguishable from the old reference state, because they give rise to the same mechanical response to any subsequent deformation; for this reason the group G greatly contributes to the characterization of the speciﬁc mechanical properties of a material.9 The material symmetry group depends on the reference conﬁguration 9

In the classical extension to nonlinear elasticity of the assumptions made in linear elasticity, it is assumed (Truesdell and Noll (1965)) that a body is solid if there is a reference conﬁguration, called undistorted, such that its material symmetry group G is contained in the orthogonal group: G ≤ O. In particular, the solid is isotropic if G = O, and is crystalline if G coincides with certain subgroups of O (Laue groups) that we describe in §3.2.3. In chapter 6 we will see how these assumptions are interpreted within the elastic model for crystal mechanics developed in this volume.

58

PRELIMINARIES

˜ is obtained from R by an aﬃne transformation κ ˜ with as follows: if R ˜ ˜ = H ∈ Aut, then, in obvious notation, Dκ ˜=H ˜ GH ˜ −1 . G

(2.61)

˜ φ, ˆ D and G For simplicity we do not indicate the explicit dependence of φ, on the reference conﬁguration R. Notice that, by (2.51), ¯ = RFH F

¯ := F ¯ tF ¯ = H t CH , ⇔ C

(2.62)

hence the analogue of (2.65) for the function φˆ in (2.59) is ˆ , θ) = φ(H ˆ t CH , θ) φ(C

(2.63)

for all H ∈ G, all C = F t F with F ∈ D, and all θ. Since (2.63) trivially holds for H = −1, in the classical theory of nonlinear elasticity the material symmetry group of any reference conﬁguration always contains the central inversion (see for instance Truesdell and Noll (1965), §31): − 1 ∈ G.

(2.64)

To summarize, the invariance of the constitutive function φ˜ in nonlinear thermoelasticity theory is given by ˜ , θ) = φ(QFH ˜ φ(F , θ)

(2.65)

for all θ, all F ∈ D, all Q ∈ O and all H in a suitable group G of unimodular tensors containing at least −1. Similarly for the functions ;˜ and η˜ in (2.54). In chapter 6 we will propose a choice of the group G for multiphase crystals, based on the molecular model of simple lattices. 2.5.3 Stability of equilibrium We will mostly be concerned with static aspects of certain solid-state phase transitions in crystalline materials. We assume that equilibria and their stability may be analyzed through the classical energy method, which is presented and put in a critical perspective in relation to dynamical stability by Knops and Wilkes (1973), for instance; see also Ericksen (1966), (1981b), James (1982), Marsden and Hughes (1983), Ball (1990). Therefore we deﬁne the equilibrium states of a body to be the critical points of the function or functional giving the value of a suitable thermodynamic potential, as discussed below.10 An equilibrium will be called stable when it is an absolute minimizer, and metastable when it is a relative minimizer in some appropriate sense. Since we mostly consider the simplest minimizers, we do not detail the technical aspects of the nonconvex variational 10

There are also approaches to equilibrium in which the existence of a potential is not assumed; for instance the ones originating from the work of Stoppelli (1957) (see Wang and Truesdell (1973), Valent (1988)).

2.5 CONTINUUM MECHANICS

59

problems originating from crystal mechanics, in particular the need to consider minimizing sequences in addition to minimizers; we give a sketchy outline in chapter 10, and otherwise refer to Dacorogna (1989), Ball and James (1992), (2002), Bhattacharya et al. (1994), Luskin (1996a,b), Bhatˇ tacharya and Kohn (1997), Silhav´ y (1997), M¨ uller (1998), Pedregal (2000), and the literature quoted therein. The choice of the potential to be minimized depends, among other things, upon the choice of boundary conditions. Here we consider the case of a crystalline solid immersed in a heat-bath whose temperature θ0 and pressure p0 are considered as control parameters. This is the problem addressed by Gibbs (1878) in his famous memoir on the equilibrium of heterogeneous substances. In this case the appropriate potential to be minimized is the availability A, sometimes also called ballistic free energy, whose constitutive function is – see (2.48), (2.54): ˜ , θ, θ0 , p0 ) := ;˜(F , θ) − θ0 η˜(F , θ) + Jp0 ; A = A(F (2.66) thus the total availability functional of the body is the volume integral

˜ Dχ(x ), θ(x ), θ0 , p0 dx . (2.67) A[χ, θ, θ0 , p0 ] := A R

The motivation for this choice comes from the earlier work of Duhem (1911) and the thermokinetic approach of Ericksen (1966), who show that the sum of A and of the total kinetic energy is a nonincreasing function of time along processes for the solid immersed in a heat-bath with parameters θ0 and p0 , in the absence of body forces and heat sources for the solid itself (see also Pippard (1957), Knops and Wilkes (1973), Landau et al. (1980), Erickˇ sen (1991a); Silhav´ y (1997) in particular presents an extension including arbitrary external forces deriving from a potential). For the purpose of establishing a simpler minimization problem, we restrict our attention to the special but interesting minimizers (χ, θ) of A that are homogeneous (Ericksen (1996b)). Therefore we look for minimiz˜ in the (F , θ)-space Aut × I. The usual critical point ers of the integrand A ˜ in (2.66) gives: necessary condition for a minimum of the function A ∂˜ ; ∂ η˜ ∂ = θ0 and (˜ ; − θ0 η˜) = −p0 J F −t . ∂θ ∂θ ∂F If the speciﬁc heat (at constant conﬁguration)

(2.68)

∂ 2 φ˜ (2.69) ∂θ2 is not zero, then11 (2.70)2 and the relation between the functions ;˜(F , θ) and ;¯(F , η) in footnote 11 imply that (2.68)1 is equivalent to κ := −θ

θ = θ0 . 11

(2.71)

¯ The condition κ = 0 also guarantees that (2.55)2 can be inverted to give θ = θ(F, η), ¯ so that the internal energy density = ¯(F, η) := ˜(F, θ(F, η)) is a potential for P and

60

PRELIMINARIES

This says that in equilibrium the solid has everywhere the same temperature as the environment. Granted this, (2.68)2 becomes ∂ φ˜ P(F , θ0 ) = = −p0 JF −t ⇔ T = −p0 1 , (2.72) ∂F θ=θ0 so that the equilibrium stress of the body is necessarily everywhere hydrostatic, with the same pressure as the environment. Formula (2.72)1 is exactly the critical point condition for the Gibbs free energy φ˜ + Jp evaluated at the heat-bath temperature θ0 and pressure p0 , that is, for the potential ˜ , θ0 ) + Jp0 , γ˜ (F ; p0 , θ0 ) := φ(F (2.73) in which θ0 and p0 are regarded as control parameters. Furthermore, the second diﬀerential test for a minimum of the availability ˜ applied to the critical points satisfying (2.71), reads A, 2 ˜ = dF · ∂ γ˜ dF + κ dθ2 ≥ 0 ; d2 A (2.74) ∂F ∂F θ0 so, if we also assume κ to be positive, (2.74) holds if and only if the second diﬀerential of the potential γ˜ in (2.73) is nonnegative. In the rest of this volume we will assume κ > 0; the discussion above then shows that mini˜ in Aut × I is equivalent to minimizing γ˜ in Aut. The potential γ˜ mizing A is regarded as the appropriate one for loading by hydrostatic pressure by Ericksen (1981b) and James (1982), (1986b), for instance. We call natural state any stable or metastable, unstressed or hydrostatically stressed, equilibrium state corresponding to assigned values of the control parameters θ0 and p0 and to no other loads. For simplicity, in most cases we will consider the hydrostatic load to be negligible, and will set p0 = 0. Then γ˜ reduces to the Helmoltz free energy density φ˜ evaluated at the environmental temperature θ0 . In the hypotheses above the equilibria of the solid are minimizers of the (free) energy functional

Φ[χ, θ] = φ˜ Dχ(x ), θ) dx (2.75) R

at ﬁxed temperature, which here and hereafter we just indicate by θ. Consequently, in what follows, we will look for the body conﬁgurations that minimize the energy functional at a given temperature. We assume φ˜ to be bounded below, and choose the arbitrary additive constant in such a way that φ˜ ≥ 0, whence also Φ[χ, θ] ≥ 0. ˜ γ˜ , and φ˜ all have the same invariance under Notice that the potentials A, both objectivity and material symmetry. In particular, their dependence on F is only through the stretch U . θ, the two constitutive restrictions (2.55) being equivalent to P=

∂¯ ∂F

and

θ=

∂¯ . ∂η

(2.70)

CHAPTER 3

Simple lattices From the molecular point of view, crystalline substances are assumed to have a discrete, triply periodic structure, the most basic example of which is a (3-dimensional) simple lattice; that is, a set L of points obtained as integral combinations of three independent lattice vectors (see (3.1). As will be discussed in detail in chapter 11, actually most crystals found in nature need to be described by the more complex structures called multilattices, which are ﬁnite collections of translates of a given simple lattice. In many circumstances, however, it is possible to disregard the internal details of the crystalline structure, and describe it as a simple lattice in which the molecular units are regarded as points.1 The notion of a simple lattice is thus central in any theory of crystalline behavior, and is the basis for the study of crystal symmetry which plays a fundamental role in this volume. In this chapter we give the basic deﬁnitions and discuss the essential properties of simple lattices, their symmetry, their classiﬁcation and their kinematics. Most of the considerations we make on lattice symmetry are classical, but we present the material by stressing the relation between kinematics and symmetry of lattices, in a way (Ericksen (1979)) that is suitable for the nonlinear elastic model developed in later chapters. We omit the lengthy proofs of the main classiﬁcation results, which can be found, for instance, in Miller (1972) or Sternberg (1994). Schwarzenberger (1972), Engel (1986), Michel (1995) can also be consulted for more abstract introductions to the subject of mathematical crystallography, an introduction to which is given by Senechal (1990). One of the central notions introduced in this chapter is that of global symmetry of a simple lattice L; this is connected with the indeterminateness in the choice of the lattice vectors for L, and is described by the action of a global symmetry group – in this case, the group GL(3, Z) of §1.2 – on a suitable conﬁguration space, which for simple lattices is the set B of all bases, or the set Q> 3 of all lattice metrics. On the other hand, the symmetry of simple lattices is usually described by the crystallographic point groups, which are the ﬁnite groups of isometries leaving a lattice invariant; this is also called the geometric symmetry of simple lattices. The point groups are classically studied and classiﬁed in 1

A simple-lattice model for crystals, with binary interactions by central forces between lattice points, was used for instance by Cauchy (1828a,b), (1829), in his molecular approach to crystal elasticity. 61

62

SIMPLE LATTICES

crystallography into orthogonal conjugacy classes; in turn, all 3-dimensional simple lattices are divided into the well known seven crystal systems. A more reﬁned description of the symmetry properties of simple lattices is obtained by representing the point groups as ﬁnite groups of integral matrices, and by investigating their conjugacy properties in GL(3, Z). This gives the arithmetic symmetry of simple lattices, and leads to their classiﬁcation into the well known fourteen Bravais lattice types. We will see in §4.1.1 that the classical point-group symmetries are but a ‘local’ version of the global symmetry of simple lattices; indeed, the latter symmetry will be shown to be compatible with the former in the range of ‘small but ﬁnite’ deformations, that is, when the action of the global group GL(3, Z) is restricted to suitable neighborhoods in the conﬁguration spaces B or Q> 3 . This will be very important when we will develop a model for phase transitions in elastic crystalline materials. In this chapter we also begin to study the kinematics of deformable simple lattices, as they undergo a (homogeneous) deformation, and study the symmetry-preserving stretches, which keep the symmetry of a simple lattice constant. This constitutes the ﬁrst step towards the analysis of symmetry breaking in deformable simple lattices, which will be studied in detail in chapters 4 and 5. 3.1 Deﬁnitions and main properties. Global symmetry of simple lattices A 3-dimensional simple lattice L(ea ) in R3 is given by: L(ea ) := {x ∈ R3 : x = M a ea , a = 1, 2, 3, M a ∈ Z};

(3.1)

the three linearly independent vectors ea , a = 1, 2, 3, in R3 are called the lattice basis or the lattice vectors; they belong to the conﬁguration space B of simple lattices, which is the 9-dimensional space of all triples of linearly independent vectors.2 Henceforth the word ‘lattice’ will always mean ‘simple lattice’; also, when there is no danger of confusion, the lattice L(ea ) is indicated by L only. Each point of L gives the position,3 in a crystalline conﬁguration, of an atom or of a selected point in a cluster of atoms, the details of which we decide to disregard; for instance, the selected point may be the center of mass of the cluster. In any case, all points of a lattice are regarded as physically indistinguishable. 2

3

A simple lattice as in (3.1) is an example of a freely generated module over a principal ideal domain, in this case the ring Z. The theory of such structures is useful in many crystallographic questions. For details we refer for instance to Jacobson (1974), chapter 3. This is the actual, or the most probable, or the average position; the presence of atomic vibrations above absolute zero is only accounted for by the entropic part of the free energy density, to be introduced in chapter 6.

3.1 DEFINITIONS AND GLOBAL SYMMETRY

63

As in (2.9), the lattice basis ea determines the lattice metric C: Cab := ea · eb , C = (Cab ) ∈ Q> 3,

(3.2)

where Q> 3 is the 6-dimensional set of all the symmetric positive-deﬁnite 3 by 3 real matrices – see §2.3.1. C is also called the Gram matrix or the gramian. Two sets of lattice vectors have the same metric if and only if they are orthogonally related – see (2.15). The parallelepiped P(ea ) = {λa ea , 0 ≤ λa ≤ 1} is called an elementary or unit cell for L, with volume given by (volP(ea ))2 = det C .

(3.3)

The dual basis vectors e a , deﬁned by the relations (2.10), are sometimes called reciprocal lattice vectors.4 In crystallography the fundamental question regarding the symmetry of lattices is the following: (a) Given a lattice L, to ﬁnd all the orthogonal transformations Q that map L onto itself, that is, are such that QL = L. By (3.1) and the linearity of Q, this amounts to ﬁnding all the transformations Q such that L(Qea ) = L(ea ) , (3.4) that is, such that the orthogonally transformed vectors Qea are again lattice vectors for L(ea ). This is a special case of the more general problem: (b) Given L(ea ), to ﬁnd all the linearly transformed vectors Hea which generate L(ea ) (H ∈ Aut). We answer question (b) ﬁrst – see Proposition 3.1 and formulas (3.9)– (3.10) – and return to (a) later, in §3.2. Problem (b) refers to the observation that a lattice L does not uniquely determine its lattice basis, or its unit cell. Fig. 3.1 shows a simple 2-dimensional example of such lack of uniqueness, which is fully described as follows: Proposition 3.1 A lattice L(ea ) determines its lattice basis up to transformations in GL(3, Z): L(mba eb ) = L(ea ) 4

⇔

m = (mba ) ∈ GL(3, Z).

(3.5)

Often in the literature – see for instance Landau et al. (1980), ch. 13 – the reciprocal vectors are deﬁned by the analogue of (2.10) in which the Kronecker delta is multiplied by 2π. This is because the lattice generated by the reciprocal vectors, which is called reciprocal lattice, is useful in describing interplanar lattice spacing, crystal growth, X-ray diﬀraction, irreducible representations of space groups, lattice vibrations and waves, and Fourier-series expansion of L-periodic functions. The chosen normalization of the reciprocal basis is convenient, as the addition of the scalar product of a lattice vector and a reciprocal vector in the exponent of an imaginary exponential does not aﬀect the value of the exponential due to its 2π-periodicity. We adopt the standard deﬁnition (2.10) because the topics mentioned above are outside the scope of this book, and dual vectors only play here a secondary, purely geometric role. For introductory remarks on some of the topics above see Senechal (1990).

64

SIMPLE LATTICES

Recall from §2.2.2 that GL(3, Z) is the inﬁnite, discrete group of invertible (that is, with determinant ±1, or unimodular) 3 by 3 matrices with integral entries. This group plays a very important role in crystallography and will be often encountered below. To prove Proposition 3.1, notice ﬁrst that, by (3.1), any triple va of linearly independent vectors in L(ea ) has the form va = v ba eb ,

v ba ∈ Z ,

det(v ba ) $= 0 ;

(3.6)

in general, three such vectors va generate a sublattice of L(ea ) and are called sublattice vectors – see for instance Ericksen (1982a). Any new triple of lattice vectors ¯ ea generating L(ea ) necessarily satisﬁes (3.6) for some choice, say mba , of the integral coeﬃcients v ba constituting an invertible matrix. In order for the ¯ ea to generate L, it must also be possible to express the vectors ea as integral combinations of the ¯ ea ; the coeﬃcient matrix is then given by (mba )−1 . In conclusion, ¯ ea is a new basis for L(ea ) if and only if it is related to the original basis by an invertible integral matrix m: ¯ ea = mba eb

with m ∈ GL(3, Z);

(3.7)

equivalently, (3.5) holds. From (3.7) we also see that (3.6) deﬁnes the generators of a proper sublattice S(va )L(ea ) if and only if |det v| > 1. The change of lattice basis (3.7) induces a natural action of the global symmetry group GL(3, Z) on the conﬁguration space B of lattices. Correspondingly, from (3.7) we obtain, in obvious notation, the following transformation of the lattice metric: C → C¯ = mt Cm , m ∈ GL(3, Z), (3.8) which induces an action of GL(3, Z) on the space Q> 3 . We will return to these actions in more detail in §§3.3, 3.5. Due to (2.15), and since the elastic theory developed later will be required to be Galilean invariant, the space Q> 3 of metrics will often be considered in place of the space B of bases. For this reason we call also Q> 3 the conﬁguration space of lattices. Notice that, by (3.3), (3.5) and (3.8), any triple of vectors ¯ ea in the lattice L is a new basis for L if and only if the volume of the cell P(¯ ea ) is the same as that of P(ea ): all unit cells in a lattice have the same volume. Based on (3.7), we can answer problem (b) above by introducing the global symmetry group G(ea ) of a given lattice L(ea ) as the group of unimodular tensors that leave L(ea ) invariant, indeed the maximal subgroup of Aut having this property: G(ea ) := {H ∈ Aut : L(Hea ) = L(ea ) } < Aut = {H ∈ Aut : Hea =

mba eb ,

(3.9)

m ∈ GL(3, Z)}.

Thus the invariance of all lattices is abstractly the same, being given by groups that in suitable bases are all represented by GL(3, Z); this too will often be called the global symmetry group of lattices.

3.1 DEFINITIONS AND GLOBAL SYMMETRY

65

¯ e2 e2

e1 e1 = ¯

Figure 3.1 Lattice bases ea and ¯ ea = mba eb , m ∈ GL(2, Z), and elementary cells for a 2-dimensional lattice

In general, the subgroups of Aut or O that leave a lattice invariant, or, equivalently, that can be represented as groups of integral matrices in suitable bases, are called crystallographic.5 Thus, for any ea , G(ea ) is a maximal crystallographic subgroup of Aut, and all such subgroups are conjugate. Indeed, by deﬁnition, any maximal crystallographic group has the form G(ea ), ea = Fea = f ba ea , for a suitable F ∈ Aut. Since the elements of G(ea ) have all the form H = mba eb ⊗ e a , m ∈ GL(3, Z) ,

(3.10)

by (3.10), (2.11) and (2.12) the elements of G(Fea ) can be written as H = hba eb ⊗ e a = (f hf −1 )ba eb ⊗ e a .

(3.11)

So, as stated above, the group G(Fea ) is conjugate to G(ea ) in Aut: G(Fea ) = F G(ea )F −1 .

(3.12)

e a

When the vectors are a new basis for L(ea ), both f and h are in GL(3, Z), so that in this case G(Fea ) = G(ea ), and thus: G(mba eb ) = G(ea ) for all m ∈ GL(3, Z) .

(3.13)

Therefore the group G(ea ) depends only on the lattice L itself, rather than on the choice of the lattice basis. Notice that G(ea ) contains operations that are neither orthogonal nor of ﬁnite period. A remarkable class of the latter is given by the simple shears S leaving L(ea ) invariant, or lattice-invariant shears:6 Sea = sba eb , with S = 1 + a ⊗ n, a · n = 0, (sba ) ∈ GL(3, Z) . 5

6

(3.14)

As sometimes Aut is identiﬁed with the group GL(3, R) of invertible 3 by 3 real matrices, the crystallographic subgroups of Aut are deﬁned as those that are conjugate to subgroups of GL(3, Z). See Proposition 2.12 and Proposition 3.2 below for some information on crystallographic groups; see also Humphreys (1990). Often in the metallurgical literature this name is given to the transformations we call twinning shears in chapter 8.

66

SIMPLE LATTICES

For any such transformation there are suitable lattice vectors for which (3.14)1 holds with   1

(sba ) =  0 0

n

0

1

0

0

1

= (g2 )n for some n ∈ Z,

(3.15)

g2 being one of the generators of GL(3, Z) listed in (2.7). We address to Ericksen (1984) for a proof of (3.15) and a discussion of other properties of lattice-invariant shears. These transformations play an important role in the ambiguities that are sometimes met in estimating the macroscopic deformation of crystalline solids when only the lattice conﬁgurations can be observed experimentally, for instance by X-ray techniques as in Fig. 1.3 (see Hall (1954) for some examples). 3.2 The geometric symmetry of lattices and their classiﬁcation into crystal systems 3.2.1 Crystallographic point groups and holohedries We now go back to to the original question (a) asked in the previous section, the issue being to study all the orthogonal transformations leaving a lattice invariant. Given the basis ea , we look for the orthogonal tensors Q such that Qea is still a set of lattice vectors for L(ea ). These transformations are called symmetry operations for L, and their invariant 1- or 2-dimensional spaces are called the symmetry elements of L; speciﬁcally, the invariant plane for a symmetry operation Q with detQ = −1 is called a symmetry plane for L, while the axis of Q when detQ = 1 is called a symmetry axis for L. Some basic properties of the symmetry elements of lattices are given in Proposition 3.12 in §3.6.1. By deﬁnition, the symmetry operations of L(ea ) are the orthogonal elements of the group G(ea ) deﬁned in (3.9), and, due to (3.7), they can be explicitly deﬁned as the tensors Q ∈ O such that Qea = mba eb

for some

m ∈ GL(3, Z).

(3.16)

The (geometric) holohedry P (ea ) < O of the lattice L(ea ), also called the (holohedral ) point group of L(ea ), collects all its symmetry operations: P (ea ) := G(ea ) ∩ O = {Q ∈ O : L(Qea ) = L(ea ) } = {Q ∈ O : Qea =

mba eb

(3.17)

, m ∈ GL(3, Z)}.

As G(ea ) is the maximal subgroup of Aut leaving L(ea ) invariant, the holohedry P (ea ) is the maximal subgroup of O mapping L(ea ) onto itself. In general, any subgroup of O leaving some lattice invariant is called a crystallographic point group. It is important to notice that not all the crystallographic point groups are holohedral because, by (3.17), a crystal-

3.2 GEOMETRIC SYMMETRY AND CRYSTAL SYSTEMS

67

lographic subgroup P of O is a holohedry if and only if there exists a basis ea such that P = P (ea ), that is, P must be the maximal group of symmetry operations of some lattice7 . We will sometimes omit the adjective ‘crystallographic’ for point groups, but this, in our context, should always be understood. The holohedries formally answer question (a) of §3.1; later in this chapter we will study their classiﬁcation and thus obtain a clearer picture of the symmetry properties of lattices.8 Hereafter we only point out some main features of the crystallographic point groups and their elements, which can be readily obtained from the deﬁnition (3.17).9 Firstly, the analogue of (3.13) holds for G(ea ) ∩ O, that is, P (ea ): P (mba eb ) = P (ea ) for all m ∈ GL(3, Z).

(3.18)

This means that, like G(ea ), also the holohedry P (ea ) of L(ea ) depends only on the lattice itself and not on the choice of its basis. Furthermore, by (3.12) and (3.17), for any Q ∈ O we have P (Qea ) = G(Qea ) ∩ O = QG(ea )Q t ∩ O = QP (ea )Q t .

(3.19)

Therefore, applying an orthogonal transformation to a lattice changes its holohedry by conjugacy in O. Secondly, it is immediate to see from (3.17) that all the holohedries contain the central inversion: − 1 ∈ P (ea ) for all bases

ea ∈ B,

(3.20)

which is not always true for a nonholohedral point group.10 Thirdly, the crystallographic point groups are all ﬁnite groups. Indeed, by (3.16), for any a, b = 1, 2, 3, we have mba = e b · Qea ⇒ |mba | ≤ e b ea ,

(3.21)

and, given the lattice basis, the last inequality only admits a ﬁnite number of integral solutions. Moreover, since by (3.16) Q is represented in the basis 7

8 9

10

However, a crystallographic point group is always a subgroup of some holohedry. Remark that, by (3.17), only the holohedral crystallographic point groups can be realized as full symmetry groups of lattices; the nonholohedral groups can only be realized as symmetry groups of the more complex crystal structures called multilattices in the introduction to this chapter, and detailed in chapter 11. In chapter 11 we brieﬂy discuss the groups of aﬃne isometries of lattices regarded as sets of points in the aﬃne space A3 , and their classiﬁcation (Bieberbach (1912)). For other general results on the structure and properties of the point groups, see Seitz (1934), Yale (1968), Kelly and Groves (1970), Miller (1972), Sirotin and Shaskolskaya (1982), Grove and Benson (1985), Humphreys (1990), Forte and Vianello (1996), and the International Tables (1996). For instance, (3.20) says that no point group consisting only of rotations, such as the 2π/n cyclic group generated by a rotation Rv , with n = 2, 3, 4, 6, can be holohedral.

68

SIMPLE LATTICES

ea by m ∈ GL(3, Z), trQ ∈ Z

for all Q ∈ P (ea ) .

(3.22)

Hence the so-called crystallographic restriction (which is also called the law of rational indices, see Somigliana (1894) or Love (1927)) holds for any symmetry operation Q of a lattice: the period of the operation is 1, 2, 3, 4 or 6 .

(3.23)

Indeed, if Q is a rotation, we can always choose an orthonormal basis in such a way that the third coordinate axis is parallel to the axis of rotation, so that the representative matrix of Q is (2.17), ω being the angle of rotation. Then, due to (3.22), tr Q = 2 cos ω + 1 is an integer, which implies restrictions on ω that are equivalent to (3.23). It is not diﬃcult to use this result to prove that (3.23) is also true when the orthogonal tensor Q has a negative determinant. The crystallographic restriction characterizes the ﬁnite crystallographic subgroups of Aut among all its ﬁnite (or all its crystallographic) subgroups: Proposition 3.2 Let G be a ﬁnite subgroup of Aut; then the following statements are equivalent: (1) G is crystallographic; (2) G can be represented by matrices in GL(3, Z); (3) G is linearly conjugate to a crystallographic point group; (4) the elements of G all satisfy the crystallographic restriction (3.23). The ﬁrst two statements are equivalent also if G is not ﬁnite. The remaining equivalences can be proved with the help of Proposition 2.13, Lemma 2.2 and Proposition 2.14. See Humphreys (1990), for instance, for more information on crystallographic groups. We remark that the matrix m representing an element H ∈ G(ea ) in the basis ea – see (3.10) – does not in general leave the metric C of ea invariant under (2.13). However, by deﬁnition, this happens if H ∈ O; so, for any H ∈ P (ea ), the representative matrix m satisﬁes the equality mt Cm = C.

(3.24) Q> 3

The converse is also true: if (3.24) holds for a matrix C ∈ and m ∈ GL(3, Z), then there exist lattice vectors ea and a tensor Q ∈ O such that C is the metric of ea , and (3.16) holds for m, so that Q ∈ P (ea ). Notice, however, that, by (3.7), two lattices L(ea ) and L(¯ ea ) can be congruent (or even coincide) without necessarily having the descriptive parameters ea and ¯ ea related by an orthogonal transformation. 3.2.2 Crystal classes and crystal systems In order to better understand the symmetry properties captured by the holohedral (and nonholohedral) point groups, it is necessary to introduce

3.2 GEOMETRIC SYMMETRY AND CRYSTAL SYSTEMS

69

a classiﬁcation criterion for these subgroups of O. The classiﬁcation principle adopted in crystallography is conjugacy within O. Indeed, according to (3.19), transforming orthogonally a lattice L(ea ) changes P (ea ) to an orthogonal conjugate, and since we should not regard as essentially diﬀerent the symmetries of two lattices that are congruent, the natural suggestion is then to divide the point groups into orthogonal conjugacy classes.11 The following theorem is a classical result in crystallography; it gives a classiﬁcation of the conjugacy classes of all the crystallographic point groups,12 and, in particular, of all the holohedries: Theorem 3.3 There are thirty-two conjugacy classes of crystallographic point groups in O. Each such conjugacy class is called a (geometric) crystal class. The lengthy proof of Theorem 3.3, which is done by construction, can be found in many references on crystallography, for instance the ones quoted in footnote 9. It is immediate to see that any orthogonal conjugate of a holohedral group is itself holohedral. Since here we only consider simple lattices, only the orthogonal conjugacy classes of the holohedries are important for us: such holohedral classes are called geometric Bravais classes, or crystal systems. With a common abuse of language, a crystal class is said to belong to the smallest crystal system containing it. As already mentioned, a crystallographic point group P is holohedral if and only if P = P (ea ) for some basis ea . A direct check of this condition for all the thirty-two crystal classes leads to the following classical corollary of Theorem 3.3, which gives the classiﬁcation of the holohedral groups and answers question (a) in §3.1: Corollary 3.4 There are seven crystal systems in R3 : triclinic, monoclinic, orthorhombic, rhombohedral, tetragonal, hexagonal, and cubic. Their partial ordering is shown in Fig. 3.2. We will see later that the information contained in Fig. 3.2, that is, the inclusion relations of the holohedries in O up to orthogonal conjugacy, is 11

12

Other possible classiﬁcation criteria are group isomorphism or linear conjugacy, that is, conjugacy in Aut. Group isomorphism is a weaker criterion than (linear or) orthogonal conjugacy, and actually provides a classiﬁcation that is too coarse. Conjugate groups are isomorphic – see §1.2.2 – but the converse need not hold, as can be checked in Aut, or O, with the cyclic subgroups {1, −1} and {1, Rπ v }. The diﬀerent geometric nature of these two groups is reﬂected in the fact that there are crystalline structures, for instance potassium persulfate and tartaric acid, that are invariant under one of them but not under the other – see Sternberg (1994). The diﬀerence in symmetry of the structures of these substances is entirely missed by the identical multiplication tables of their point groups, which is the only information given by group isomorphism. However, notice that for holohedral point groups alone isomorphism and orthogonal conjugacy do give the same classiﬁcation. Linear and orthogonal conjugacy, on the other hand, provide the same classiﬁcation of point groups: by Proposition 2.13 two subgroups of O are linearly conjugate only if they are orthogonally conjugate. By Propositions 2.12, 2.13 and 3.2, this result also gives a count of the linear conjugacy classes of any ﬁnite crystallographic subgroup of Aut. There also exists analogous classiﬁcation theorems for the conjugacy classes of ﬁnite subgroups of O or Aut, and – see Proposition 2.14 – of closed subgroups of O+ . References are given in footnote 9.

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SIMPLE LATTICES

cubic (48) hexagonal (24) tetragonal (16) rhombohedral (12) orthorhombic (8) monoclinic triclinic

(4)

(2)

Figure 3.2 The names, International Symbols, and symmetry hierarchy of the seven crystal systems. The number of elements in each holohedry is indicated in parenthesis. Table 3.1 gives a complete list of elements in the groups representing the systems. Another widely used notation for the cubic system is m3m. See Remark 5.6 for the inclusion ¯ 3m → 6/mmm

not suﬃcient for an elastic theory of phase transitions and twinning. For the latter we actually need the detailed group-subgroup relations for the holohedries, as in Figs. 5.1 and 5.2. The lists of elements in Table 3.1 give an explicit description of the symmetry axes and planes characteristic of each system (see also §3.4). The cubic and hexagonal holohedries are maximal crystallographic subgroups in O, thus giving the most symmetric systems. The classiﬁcation of holohedries into crystal systems can be used to classify also the bases in B and the lattices they generate, which are considered (geometrically) equivalent and are again said to belong to the same crystal system when their holohedries are conjugate in O. A natural extension of the same criterion also gives a classiﬁcation into crystal systems for the lat> tice metrics in Q> 3 : two elements of Q3 are equivalent if they correspond to bases whose holohedries are orthogonally conjugate. While two congruent lattices necessarily belong to the same system, it is important to stress that this can also happen for noncongruent lattices, as we will see in detail in §3.3. 3.2.3 Laue groups We have seen in §3.2.1 that the point group of a simple lattice is always holohedral. However, as we mentioned earlier, the structure of most crystalline solids cannot be described by a simple lattice; in this case the point group need not be holohedral. Also these crystals are considered in the classical theory of nonlinear elasticity, where, as prescribed by (2.64), the

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71

Table 3.1 The 7 crystal systems and the related 14 lattice types (§3.3.2) International symbols contracted ¯ 1

Name (order) Triclinic (2)

Elements of the rotational subgroup

Lattice types

1

P

1, Rπ k

P C

π π 1, Rπ i , Rj , Rk

P C F I

2/m

Monoclinic (4)

mmm

Orthorhombic (8)

¯ 3m

Rhombohedral (12)

4/mmm

Tetragonal (16)

π π π 2 2 1, Rπ i , Rj , Rk , R i±j , Rk , Rk

6/mmm

Hexagonal (24)

π π 3 3 ,R 3 , 1, Rπ i , Rj , Rk , Rk , Rk k

π 1, Rπ i ,R

i±

2π √

3j

4π

, Rk 3 , Rk 3 π

π

m¯ 3m

Cubic (48)

2π

R 3π

P I

4π

P

5π Rk 3

√ , Rπ √ , Rπ 3i±j i± 3j π 3π π 3π 2 , R 2 , Rπ , R 2 1, Ri2 , Rπ , R i j i j j π 3π 2π 4π 2 ,R 3 ,R 3 , Rk2 , Rπ vn vn k , Rk π π Ri±j , Rj±k , Rπ i±k

,

P F I

The vectors i, j, k form an orthonormal basis, and vn is any one of the four vectors (i ± j ± k). As in (2.16), Rω v denotes the rotation by the angle ω about the direction of v. Add the negative of the listed tensors to obtain the full list of elements of each holohedral group. In the last column the notation is: P=primitive, C=base-centered, F=face-centered, I=body-centered, R=rhombohedral.

constitutive equations are always invariant under the central inversion −1. Accordingly, an elastic crystalline substance is required to possess, relative to some reference conﬁguration called undistorted, a material symmetry group generated by −1 together with a point group in any one of the thirtytwo crystal classes; see for instance Coleman and Noll (1964) or Truesdell and Noll (1965). Also the symmetry of the X-ray diﬀraction pattern of a crystal is in general higher than its point group, as such patterns are always centrosymmetrical. The symmetry groups of the patterns are again obtained by adding −1 to the point groups in any one of the thirty-two crystal classes. The subgroups of O generated in this way, called Laue groups, thus have great relevance, and for completeness we brieﬂy describe them here. Clearly, due to (3.20), all the holohedral point groups are Laue groups. Based on Theorem 2.5 in Miller (1972), for instance, it is possible to show that the Laue groups belong to eleven conjugacy classes13 among the thirty-two mentioned in Theorem 3.3, that is, to the classes of groups that already contain 13

The term Laue groups is often used also to indicate such conjugacy classes.

72

SIMPLE LATTICES

Table 3.2 The four nonholohedral crystal classes to be added to the seven crystal systems in Table 3.1 to give the complete list of the eleven Laue groups. The generators are given in terms of the same orthonormal basis i, j, k, as in Table 3.1 International symbol contracted ¯ 3

Crystal Class (Crystal System)

Order

2π rhombohedral

(rhombohedral)

4/m

dipyramidal (tetragonal)

6/m

dipyramidal (hexagonal)

m¯ 3

Generators of the rotational subgroup

didodecahedral (cubic)

Rk 3 π Rk2 π Rk3

Rπ i ,

6 8 ,

12 2π 3 Ri+ j+k

24

the central inversion. We list in Table 3.2 the four crystal classes that must be added to the seven crystal systems to complete the list of the (classes of) Laue groups. 3.3 The arithmetic symmetry of lattices and their classiﬁcation into Bravais lattice types 3.3.1 Lattice groups The classiﬁcation criterion considered so far, based on the equivalence of the holohedral groups, gives a very useful description of the symmetry properties of the bases in B and of the lattices they generate. However, a better characterization of such symmetry is given by the groups of matrices in GL(3, Z) that appear in the deﬁnition (3.17) of the holohedries. As we will see, these groups, which are integral representations14 of the holohedries, can discriminate, through their conjugacy properties in GL(3, Z), diﬀerent symmetry types of lattices in a sharper way than the point groups in O; this gives a ﬁner classiﬁcation of lattices than the one based on the crystal systems. Roughly, this is because conjugacy in GL(3, Z) is a more stringent condition than conjugacy in O or Aut (see formulas (3.29)–(3.33) below). Here the classical question, answered by Bravais (1850) (but see footnote 17), is the following: given a holohedral group P , to ﬁnd all distinct types of lattices that are compatible with P . To make this statement precise, we introduce the lattice group15 L(ea ) of a basis ea or of its lattice L(ea ), as the ﬁnite subgroup of GL(3, Z) constituted by all the matrices 14 15

See the footnote 1 in §2.3. Also the terms integral or arithmetic holohedries are used in the literature for our lattice groups. In fact the latter name is used (see for instance Yale (1968) or Miller (1972)) to indicate the lattices themselves, regarded as groups of translations of the aﬃne space R3 . Following Ericksen (1979), we use the term lattice groups for the integral matrix groups because, as we will see in §3.3.2, they determine the distinct lattice types within each system.

3.3 ARITHMETIC SYMMETRY AND BRAVAIS LATTICE TYPES

73

m appearing in (3.17): L(ea ) := {m ∈ GL(3, Z) : mba eb = Qea , Q ∈ P (ea )} .

(3.25)

Thus, by deﬁnition, the lattice groups, being the integral representations of the holohedries in their own lattice bases, are the maximal (ﬁnite) subgroups of GL(3, Z) acting orthogonally on some lattice. Their main properties have been investigated by many authors; we mention Niggli (1928), Niggli and Nowacki (1935), Seitz (1935), Burckhardt (1947), Miller (1972), Janssen (1973), Ericksen (1979), Engel (1986), Opechowski (1986), Sternberg (1994), Michel (1995), Parry (1998), Adeleke (2000). It is easy to check that a change of lattice basis changes the lattice group to a conjugate in GL(3, Z): L(mba eb ) = m−1 L(ea )m

if

m ∈ GL(3, Z).

(3.26)

Also, an orthogonal transformation does not change the lattice group: L(Qea ) = L(ea ) if

Q ∈ O.

(3.27)

Due to (3.27), L(ea ) can be equivalently deﬁned as the stabilizer of the metric C of the basis ea , that is, as the maximal subgroup of GL(3, Z) whose elements leave C invariant under the action (3.8): L(ea ) = {m ∈ GL(3, Z) : mt Cm = C} =: L(C).

(3.28)

Since L(ea ) actually depends only on the lattice metric, we will often denote it by L(C) as in (3.28)2 . Due to its deﬁnition, L(ea ) shares with P (ea ) a number of properties (also recall Propositions 2.12, 2.13 and 3.2). Proposition 3.5 The following holds: (1) Any lattice group contains the inversion −1 ∈ GL(3, Z). (2) A subgroup of GL(3, Z) is a lattice group or a subgroup thereof if and only if it is ﬁnite. (3) All the elements of a lattice group satisfy the crystallographic restriction (3.23). This in turn characterizes the ﬁnite subgroups of GL(3, Z). 3.3.2 Conjugacy in O (crystal systems) and in GL(3, Z) (Bravais lattice types) Due to the transformation law (3.26) for the lattice groups under a change of lattice basis, the lattice group L(ea ), unlike the holohedry (see (3.18)), is determined by the basis ea and not by the lattice L(ea ) itself. By (3.5) and (3.26), L(ea ) indeed determines an entire conjugacy class in GL(3, Z) of lattice groups. The GL(3, Z)-conjugacy class of a ﬁnite subgroup of GL(3, Z) is called

74

SIMPLE LATTICES

an arithmetic class, and the groups in it arithmetically equivalent. In particular, a GL(3, Z)-conjugacy class of lattice groups, as the one identiﬁed by L(ea ), is called an arithmetic Bravais class. In order to study how the construction above helps in the classiﬁcation of lattices, let us ﬁrst notice that each arithmetic Bravais class determines a unique crystal system. Roughly, consider an arithmetic class and select a lattice group L in it; select then a basis ea such that L = L(ea ), as is possible by deﬁnition; then use (3.16) to construct the holohedry P = P (ea ). Since any element of GL(3, Z) also belongs to GL(3, R), hence represents an element of Aut, we conclude that changing the representative L in the class considered, or the basis ea related to L as above, or both, produces a holohedry which is linearly conjugate to P above, hence belongs to the same system as P by Proposition 2.13. On the contrary, a given holohedry P < O, or a given crystal system, in general determine various arithmetically inequivalent (classes of) lattice groups, depending on the bases ea that verify P = P (ea ). The basic reason is that the same symmetry operation in P < O can be represented in diﬀerent bases by integral matrices which need not be related by a transformation in GL(3, Z). As an example, consider the two bases indicated explicitly in formulas (3.39) and (3.40) below, which here will be denoted by ea and ˜ ea , respectively. They have the same monoclinic holohedry P := P (ea ) = P (˜ ea ) = {1, −1, Rjπ , −Rjπ },

(3.29)

where j is the unit vector along the y axis in Fig. 3.4. With respect to such bases, the twofold rotation Rjπ has, respectively, the following representative matrices in GL(3, Z):     m=

−1

0

0

1

0

0

0

0 −1

0

and m ˜ =  −1 0

−1 0 0

0

0;

(3.30)

−1

these are not GL(3, Z)-conjugate. Indeed, by contradiction, if mm ¯ =m ˜m ¯ for some m ¯ ∈ GL(3, Z), an elementary calculation gives: m ¯ 32 = 0, m ¯ 21 = m ¯ 11 , m ¯ 13 = m ¯ 23 , and m ¯ 12 = −m ¯ 22 ,

(3.31)

which in turn implies

1 3 det m ¯ = 2m ¯ 22 m ¯ 3−m ¯ 13 m ¯ 31 $= ±1 , ¯ 1m

(3.32)

against the hypothesis. This means that the lattice groups L(ea ) = {1, −1, m, −m}

and L(˜ ea ) = {1, −1, m, ˜ −m} ˜

(3.33)

are not conjugate in GL(3, Z); they are inequivalent integral representations of the same holohedral group P in (3.29). This example shows that GL(3, Z)-conjugacy is a stricter condition than conjugacy in O or Aut, as mentioned at the beginning of §3.3.1, and that each crystal system can

3.3 ARITHMETIC SYMMETRY AND BRAVAIS LATTICE TYPES

75

correspond to more than one arithmetic Bravais class. A result in mathematical crystallography answers the next natural question. Theorem 3.6 Seventy-three are the conjugacy classes of ﬁnite subgroups in GL(3, Z) (arithmetic classes); among these fourteen are arithmetic Bravais classes (conjugacy classes of lattice groups). The geometric meaning of this theorem will be clariﬁed in §§3.3.3 and 3.4. Concerning the proof, we recall that the number of conjugacy classes of ﬁnite subgroups of GL(n, Z) is ﬁnite for any n by a theorem of Jordan (1880). In the 3-dimensional case their number turns out to be seventy-three by a direct computation, which also shows the Bravais classes to be fourteen. We address the reader to the references in §3.3.1 for details. The Bravais classes in GL(3, Z) counted in Theorem 3.6 give us a new arithmetic criterion for classifying the bases in the space B and the lattices they generate, a criterion which is ﬁner than the geometric one (§3.2) giving the crystal systems. Since, as implied by (3.26), any lattice L(ea ) corresponds to an entire arithmetic Bravais class of lattice groups, it is natural to consider as equivalent all the lattices or bases whose lattice groups are in the same arithmetic Bravais class. Explicitly, this new classiﬁcation regards two bases ea and ea in B, and the lattices they generate, as equivalent when their lattice groups are arithmetically equivalent: L(ea ) = m−1 L(ea )m

for some

m ∈ GL(3, Z).

(3.34)

Bases or lattices for which (3.34) holds have or belong to the same Bravais lattice type.16 Thus two lattices have the same Bravais type if and only if for some choice of their bases the associated lattice groups coincide. The following classical corollary of Theorem 3.6 classiﬁes lattices and their bases into Bravais lattice types, and indicates how many distinct types belong to each crystal system: Corollary 3.7 There are fourteen distinct Bravais lattice types in three dimensions. They are divided among the seven crystal systems as indicated in Table 3.1, and their symmetry hierarchy is given in Fig. 3.12, p. 89. The fourteen Bravais lattice types are described in detail in §3.4, and their construction is lengthy but instructive; see the references in §3.3.1.17 16

17

Often the Bravais lattice types are simply called ‘lattice types’ or ‘Bravais lattices’; also the term ‘symmetry types of lattices’ is used. In the literature, the lattices themselves are sometimes called ‘Bravais lattices’ – see for instance Landau et al. (1980), Ericksen (1989). The classiﬁcation of lattices into Bravais lattice types is based on deﬁnitions that are sometimes diﬀerent from (3.34). For instance, lattices can be equivalently partitioned into Bravais lattice types based on the aﬃne equivalence (or isomorphism) of their aﬃne symmetry groups (see chapter 11). It is interesting to notice that the criterion originally used by Bravais (1850) and Cauchy for the classiﬁcation of lattices in distinct types was based on certain properties maintained by the lattices themselves along continuous deformations. This criterion is actually not equivalent to the arithmetic criterion based on (3.34), which emerged as the ‘correct’ one only in the 1930s. So, Bravais’ celebrated 1850 memoir actually gave the right answer to the ‘wrong’ question. See Pitteri and Zanzotto (1996a) and Ericksen (1996b) for some details.

76

SIMPLE LATTICES

Table 3.1 reports the standard notation and nomenclature for the lattice types; standard names as face-centered cubic or ‘f.c.c.’, body-centered cubic or ‘b.c.c.’, etc. are explained in §3.3.3. The lattice groups of the three cubic types (primitive, face- and bodycentered) and the hexagonal lattice groups are maximal ﬁnite subgroups in GL(3, Z), and give the most symmetric lattice types. The partial ordering of the arithmetic Bravais classes, which gives the inclusions relations for the lattice groups up to conjugacy in GL(3, Z), is given in Fig. 3.12 (compare with Fig. 3.2). As is the case with the crystal systems, the subdivision of the space B into Bravais lattice types naturally carries over to the space Q> 3 of lattice metrics: two metrics C and C are of the same Bravais type when their lattice groups are arithmetically equivalent. Notice that, by (3.26), the action (3.8) of GL(3, Z) on Q> 3 necessarily produces metrics of the same Bravais type; the metrics C¯ and C in (3.8), whose GL(3, Z)-orbits coincide, are themselves called arithmetically equivalent, or symmetry-related. Any bases ¯ ea and ea in B whose metrics are in the same GL(3, Z)-orbit are also called symmetry-related; by (2.15) this happens if and only if there are Q ∈ O and m ∈ GL(3, Z) such that ¯ ea = mba Qeb

or ¯ ea ∈ mba Oeb .

(3.35)

Notice that not all the bases or metrics belonging to the same Bravais type are symmetry-related; a general description of the Bravais lattice types 18 in B or Q> 3 will be given in §3.6. 3.3.3 Centerings From the geometric point of view, the subdivision into Bravais types within each crystal system given in Table 3.1 corresponds to the existence of (arithmetically) inequivalent lattice centerings for the lattices belonging to that system. Roughly, given a lattice, called primitive, with lattice vectors along as many symmetry axes as are available in that system, it is often possible to preserve both the simple lattice structure and the holohedry by suitably adding lattice points in suitable positions within the unit primitive cell. Centering of a lattice can occur, for instance, by adding two points, each at the center of a suitable pair of parallel faces of the unit primitive cell. In this way, a base-centered version of the original lattice is obtained. Another way in which centering occurs is by adding to the unit primitive cell a point at the center of each face, in which case the new lattice is called face-centered. 18

Referring to the action (3.8) of GL(3, Z), a Bravais type in Q> 3 is what is usually called a stratum of a group action (see for instance Michel (1995)): this is a subset of the space on which the group acts, whose points have conjugate stabilizers – see §2.2.2 – under the action. Also, the abstract space of lattices is sometimes identiﬁed with the double quotient O\GL(3, R)/GL(3, Z) – see for instance Senechal (1990), (1992), Michel (1995).

3.3 ARITHMETIC SYMMETRY AND BRAVAIS LATTICE TYPES

77

Finally, centering can occur by adding to the primitive unit cell a point at the center of the unit cell itself, thus obtaining a body-centered lattice. Direct computations show that not all the centerings do indeed produce inequivalent lattices within each system; some however do, in such a way that fourteen distinct lattice types are found within the seven crystal systems, as is summarized by Table 3.1 (the adjective ‘primitive’ is usually omitted if a system only contains one type). The example in (3.29)–(3.33) gives a proof that there are (at least) two inequivalent Bravais types in the monoclinic system; these types are called primitive monoclinic and basecentered monoclinic, and one can show that they are the only inequivalent types in the monoclinic system. Another interesting example of inequivalent centerings within the same system is given by the primitive cubic, the face-centered cubic (f.c.c.), and the body-centered cubic (b.c.c.) lattices – see (3.52)–(3.56). Metallurgists analyze a classical ﬁrst-order phase transformation in iron, in which the crystalline structure changes from b.c.c. to f.c.c. – see §3.6.3. In this transition the crystal system of both the iron phases is cubic, so that the symmetry change cannot be described in terms of the holohedries, which are equivalent; it is only captured by the change in the phases’ (inequivalent) lattice groups. Table 3.3 lists the f.c.c. and b.c.c. lattice groups L1 and L2 representing the cubic holohedry – see Table 3.1 – with respect to the bases ea in (3.53) and (3.56), respectively. To reduce the size of the table the following convention – see International Tables (1996) – is adopted: any integral matrix m is represented by means of the way it transforms (through usual matrix multiplication) a generic (column) triple of coordinates x, y, z; and the minus sign is represented by an overbar. For instance, the matrix representing Rkπ in L1 , which is indicated by y, x, x ¯ + y¯ + z¯ in the table, has rows (0, 1, 0), (1, 0, 0) and (−1, −1, −1). By inspection, L1 and L2 are diﬀerent and have a common lattice subgroup; the corresponding common holohedry is rhombohedral, is denoted by Ri +j +k and detailed in §5.1.2. We show by contradiction that L1 and L2 are not equivalent. As is geometrically clear, any threefold rotation 2π/3 can be transformed into Ri +j +k by conjugation in the cubic holohedry, and then, similarly, any twofold rotation about a face diagonal normal to i + j + k can be transformed into, say, Riπ−j . Since the analogue is true for the integral representations through conjugation in GL(3, Z), L1 and L2 are equivalent only if some m ¯ ∈ GL(3, Z) conjugates the representative 2π/3 π matrices of Ri +j +k and Ri −j in L1 and L2 . We thus obtain the following nececessary restrictions on m: ¯ all the diagonal elements are equal, say to m ¯ 11 , and all the oﬀ-diagonal elements have the same value, say m ¯ 12 . Then det m ¯ = (m ¯ 12 − m ¯ 11 )2 (2m ¯ 12 + m ¯ 11 ),

(3.36)

which can be 1 if and only if m ¯ = 1, against the fact that the two lattice groups in Table 3.3 are distinct.

78

SIMPLE LATTICES

Table 3.3 The f.c.c. and b.c.c. lattice group elements with positive determinant associated with the bases (3.53) and (3.56), respectively. See the text for the notation. Notice the common rhombohedral lattice subgroup rotation

1

L1 (fcc) x, y, z

L2 (bcc)

rotation

L1 (fcc)

L2 (bcc)

x, y, z

Rπ i+j

x ¯, y, ¯ x+y+z

x ¯ + z, y ¯ + z, z

Rπ i−j 3π Rk 2 π Rk2 3π Ri 2

Rπ k

y, x, x ¯+y ¯+z ¯

y+z ¯, x + z ¯, z ¯

Rπ j

z, x ¯+y ¯+z ¯, x

y ¯ + z, y, ¯ x+y ¯

Rπ i

x ¯+y ¯+z ¯, z, y

x ¯, x ¯ + z, x ¯+y

2π 3 Ri+j+k 2π 3 R−i+j−k 2π 3 Ri−j−k 2π 3 R−i−j+k 4π 3 Ri+j+k 4π 3 Ri−j−k 4π 3 R−i−j+k 4π 3 R−i+j−k

y, ¯ x ¯, z ¯

y, ¯ x ¯, z ¯

z ¯, x + y + z, y ¯

x+z ¯, x, x + y ¯

x + y + z, z ¯, x ¯

y, y + z ¯, x ¯+y

z, x, y

z, x, y

z ¯, x ¯, x + y + z

y+z ¯, x ¯ + y, y

x ¯+y ¯+z ¯, y, x

z ¯, y + z ¯, x + z ¯

Rπ j+k

x + y + z, y, ¯ z ¯

x, x + y, ¯ x+z ¯

x, z, x ¯+y ¯+z ¯

x + y, ¯ y ¯ + z, y ¯

Rπ j−k

x ¯, z ¯, y ¯

x ¯, z ¯, y ¯

y, ¯ x + y + z, x ¯

y ¯ + z, z, x ¯+z

y, x ¯+y ¯+z ¯, z

x ¯ + y, x ¯, x ¯+z

π Ri2 π Rj2

y, z, x

y, z, x

y, ¯ z ¯, x + y + z

x + y, ¯ x+z ¯, x

x, x ¯+y ¯+z ¯, y

x+z ¯, z ¯, y + z ¯

Rπ i+k

x ¯, x + y + z, z ¯

x ¯ + y, y, y + z ¯

y, ¯ x + y, ¯ y ¯+z

3π Rj 2

x + y + z, x ¯, y ¯

z, x ¯ + z, y ¯+z

x ¯ + z, x ¯ + y, x ¯

Rπ −i+k

z ¯, y, ¯ x ¯

z ¯, y, ¯ x ¯

x ¯+y ¯+z ¯, x, z z, y, x ¯+y ¯+z ¯

3.4 The fourteen Bravais lattices Here we give a detailed description of the fourteen Bravais lattices types mentioned in Corollary 3.7. As usual, we group the lattice types according to the crystal system to which they belong, and indicate the most commonly used bases and metrics for each type. The ﬁgures show the corresponding unit cells, which in some cases have same symmetry as the lattices themselves. The lattice vectors are written in terms of their components in an orthonormal basis, chosen along as many symmetry axes as possible. In metallurgy and mineralogy various conventions are used on the standard coordinate axes for the lattices in the various crystal systems. For each system one ﬁrst considers the primitive Bravais type belonging to it. The lattice parameters in this type are deﬁned as the side lengths a, b, c, and the interaxial angles α, β, γ, in a standard basis ea for such a primitive Bravais lattice in that system: a = e1 , b = e2 , c = e3 , α = e 2 e3 , β = e 1 e3 , γ = e 1 e2 .

(3.37)

The standard basis is deﬁned for each system, and, conventionally, these are also considered the lattice parameters for any centered lattices in that system. The quantities in (3.37), restricted as speciﬁed below, are thus distinctive of each crystal system, except for the rhombohedral one, for which the same lattice parameters as for the hexagonal system are often

3.4 THE FOURTEEN BRAVAIS LATTICES

79

Figure 3.3 The triclinic lattice, with no symmetry axes or planes

used – see Fig. 3.8. The entries of the lattice metrics of the given lattice bases are suitable functions of the components of the lattice vectors, or of the lattice parameters (3.37). Their expression can be obtained from (2.9); we only indicate the main restrictions that characterize them. For a summary of various information on the crystal systems, the lattice types, their symmetry groups and symmetry hierarchies we refer to Table 3.1 and Figs. 3.2, 3.12.

Triclinic system (a, b, c all diﬀerent; α, β, γ all diﬀerent and not π/2). The lattices in this system have no symmetry axes or planes. Triclinic type (Fig. 3.3) e1 = (e11 ,e21 ,e31 ) e2 = (e12 ,e22 ,e32 ) , e3 = (e13 ,e23 ,e33 )



C11

C =  C12 C13

C12

C13

C22

C23

C23

C33

 .

(3.38)

Monoclinic system (a $= b $= c = $ a, α = γ = π/2 $= β). The lattices in this system are characterized by the presence of one twofold symmetry axis. The standard coordinate axes for lattices in this system are always such that the y axis is along the (unique) twofold symmetry axis. Primitive monoclinic type (Fig. 3.4(a)) e1 = (a,0,0) e2 = (0,b,0) , e3 = (c cos β,0,c sin β )



C=

C11

0

C13

0

C22

0

C13

0

C33

 .

(3.39)

80

SIMPLE LATTICES

Figure 3.4 The two monoclinic lattice types, whose twofold axis is along the y axis. (a) The primitive monoclinic lattice. (b) The base-centered monoclinic lattice. The extra lattice point in this type are always at the center of two opposing rectangular faces of the standard nonunit cell

Base-centered monoclinic type (Fig. 3.4(b)) Two bases are often used for this lattice: one is the rhombic basis   e1 = ( a2 , − 2b , 0) C11 C12 C13 , C =  C12 C11 C13  e2 = ( a2 , 2b , 0) C13 C13 C33 e3 = (c cos β,0,c sin β )

(3.40)

in Fig. 3.4 (b). The other is obtained by replacing the vector e1 in (3.40) by ˆ e1 = (a, 0, 0) = e1 + e2 . This Bravais type can be also described as body-centered monoclinic by changing the lattice vectors appropriately. Also, recall formulas (3.29) and (3.33), which give the holohedry and the inequivalent lattice groups of the monoclinic bases in (3.39) and (3.40). Orthorhombic system (a $= b $= c $= a, α = β = γ = π/2). The lattices in this system are characterized by the presence of three mutually orthogonal twofold symmetry axes. The coordinate axes in this system are always taken along the three twofold symmetry axes. Primitive orthorhombic type (Fig. 3.5(a))  e1 = (a,0,0) C11 e2 = (0,b,0) , C= 0 e3 = (0,0,c) 0

0

0

C22

0

0

C33

 .

(3.41)

Base-centered orthorhombic type As in the base-centered monoclinic case, a basis often used for this lattice

3.4 THE FOURTEEN BRAVAIS LATTICES

81

Figure 3.5 (a) The primitive orthorhombic lattice. (b) The base-centered orthorhombic lattice. The three twofold axes are the coordinate axes

Figure 3.6 (a) The face-centered orthorhombic lattice. (b) The body-centered orthorhombic lattice. The three twofold axes are the coordinate axes

is the following rhombic basis (Fig. 3.5(b)):  e1 = ( a2 , − 2b , 0) C11 C =  C12 e2 = ( a2 , 2b , 0) , 0 e3 = (0,0,c)

C12

0

C11

0

0

C33

 .

Face-centered orthorhombic type A basis often utilized for this lattice is (Fig. 3.6(a)):   e1 = (0, 2b , 2c ) C12 +C13 C12 C13 . e2 = ( a2 , 0, 2c ) , C =  C12 C12 +C23 C23 a b C C C +C e3 = ( 2 , 2 , 0 ) 13 23 13 23 Body-centered orthorhombic type (Fig. 3.6(b))  e1 = (a,0,0) C11 0 e2 = (0,b,0) , C22 C= 0 1 1 e3 = ( a2 , 2b , c) 2 C11 2 C22

1 2 C11 1 2 C22

(3.42)

(3.43)

 .

(3.44)

C33

Also the lattice basis formed by the position vectors of three ‘body-center’

82

SIMPLE LATTICES

Figure 3.7 The two tetragonal lattice types, whose fourfold axis is along the z axis. (a) The primitive tetragonal lattice. (b) The body-centered tetragonal lattice

points is considered in this case. An analogous choice in the cubic system is shown in Fig. 3.11(b). Tetragonal system (a = b $= c, α = β = γ = π2 ). The lattices in this system are characterized by one fourfold symmetry axis and, orthogonal to it, four twofold axes spaced π/4 apart. The coordinate axes for the lattices in this system are always along three mutually orthogonal twofold axes, with the z axis along the fourfold axis. Primitive tetragonal type (Fig. 3.7(a)) e1 = (a,0,0) e2 = (0,a,0) , e3 = (0,0,c)



C=



C11

0

0

C11

0

0

0

C33

Body-centered tetragonal (b.c.t.) type (Fig. 3.7(b))  e1 = (a,0,0) C11 0 e2 = (0,a,0) , C11 C= 0 1 1 e3 = ( a2 , a2 , 2c ) 2 C11 2 C11

0

.

1 2 C11 1 2 C11

(3.45)

 .

(3.46)

C33

This lattice can also be described as face-centered tetragonal by means of the basis and metric (compare the latter with C in (3.43)):   ˜ e1 = ( a2 , a2 , 2c ) C12 +C13 C12 C13 (3.47) ˜ e2 = ( a2 , − a2 , 2c ) , C =  C12 C12 +C13 C13  . C13 C13 2C13 ˜ e3 = (a,0,0)

3.4 THE FOURTEEN BRAVAIS LATTICES

(a)

83

(c)

(b)

Figure 3.8 (a) The rhombohedral lattice, with basis given by (3.49). The vectors (3.50) generating the hexagonal sublattice are indicated by ˜ ea . The threefold axis is the z axis, along ˜ e3 , while the three twofold axes are along ˜ e1 , ˜ e2 and −˜ e1 −˜ e2 , respectively. (b) The hexagonal lattice, in which the sixfold axis is the z axis. Also the shaded base-centered orthohexagonal double cell is put in evidence. (c) The basal plane in (b), showing the lattice vectors and the twofold symmetry axes

Rhombohedral system The lattices in this system have one threefold symmetry axis, and a triplet of twofold symmetry axes orthogonal to it, spaced 2π/3 apart. Rhombohedral type The usual cell for these lattices, which has the same symmetry as the lattice itself, is given by a rhombohedral basis ea with vectors of equal length and equal interaxial angles: e1 = e2 = e3 , e 2 e3 = e 1 e3 = e 1 e2 ; for instance, in the orthonormal basis of Fig. 3.8: √  e1 = ( a2 , 63a , 3c ) C11 C12 √ e2 = (− a2 , 63a , 3c ) , C =  C12 C11 e3 = (0, −

√

3a c 3 , 3)

C12

C12

C12 C12

(3.48)

 .

(3.49)

C11

As is customary, the lattice parameters a and c of the hexagonal nonunit cell for the rhombohedral lattice, shown in Fig. 3.8(a), are utilized in (3.49). Such nonunit cell is the unit cell of a hexagonal sublattice generated by the vectors in (3.50) below, denoted by ˜ ea in Fig. 3.8(a). Notice that ˜ e1 = e1 −e2 , ˜ e2 = e2 −e3 , ˜ e3 = e1 +e2 +e3 . Sometimes the rhombohedral lattice is also regarded as a doubly centered √ version of the hexagonal lattice, with √ centers placed at ( a2 , a 6 3 , 3c ) and (0, a 3 3 , 2c 3 ) of the hexagonal cell in (3.50), referred again to the orthonormal basis in Fig. 3.8(b).

84

SIMPLE LATTICES

As we will see in detail in §5.1, there are four rhombohedral holohedral subgroups in any cubic holohedry; their threefold axes are aligned with one of the main cubic diagonals each. We have already considered one of them in the presentation of Table 3.3, which shows a common rhombohedral lattice group for the f.c.c. and b.c.c. lattices. Hexagonal system (a = b $= c, α = β = π/2, γ = 2π/3). The lattices in this system are characterized by the presence of a sixfold symmetry axis with six twofold symmetry axes orthogonal to it, spaced π/3 apart. The standard rectangular coordinate axes in this lattice always have the z axis along the sixfold symmetry axis, and the x axis along one of the twofold symmetry axes. Hexagonal type (Fig. 3.8(b)) e1 = (a,0,0)√ e2 = (− a2 , 23a , 0) , e3 = (0,0,c)



C11

C =  − 12 C11

− 12 C11

0

C11

0

0

C33

0

 .

(3.50)

In the crystallographic literature the plane of e1 and e2 is called the basal plane, and the direction of e3 is called the optic axis. We also give explicitly the sublattice vectors and metric for the nonunit base-centered orthohexagonal cell sometimes used in the literature (Otte and Crocker (1965)); it is a special case of base-centered orthorhombic cell, and has twice the unit-cell volume:   ˜11 ˜ e1 = (a,0,0) C 0 0 √ (3.51) ˜ e2 = (0, 3a,0) , C˜ =  0 C˜11 0 . ˜33 0 0 C ˜ e3 = (0,0,c) Neither this cell nor the elementary cell (3.50) have the symmetry of the lattice. See §3.7.4 for the indexing used in hexagonal lattices. Cubic system (a = b = c, α = β = γ = π/2). The lattices in this system all share the symmetry elements of a cube, which include three mutually orthogonal fourfold symmetry axes along the cubic edges, four threefold axes along the main cubic diagonals, and six further twofold axes along the face diagonals. The standard coordinate axes for this system are always along the three fourfold axes. Primitive cubic type (Fig. 3.9) e1 = (a,0,0) e2 = (0,a,0) , e3 = (0,0,a)



C=

C11

0

0

0

C11

0

0

0

C11

  , C11 = a2 .

(3.52)

3.4 THE FOURTEEN BRAVAIS LATTICES

85

Figure 3.9 The primitive cubic lattice

Figure 3.10 The face-centered cubic (f.c.c.) lattice. (a) The rhombohedral unit cell given by (3.53). (b) The close-packed nets in the f.c.c. lattice – see §3.7.2 – and the lattice vectors (3.54)

Face-centered cubic type The standard cell is rhombohedral, with lattice vectors (Fig. 3.10):   1 1 e1 = (0, a2 , a2 ) C11 2 C11 2 C11 a2 e2 = ( a2 , 0, a2 ) , C =  21 C11 C11 12 C11  , C11 = . (3.53) 2 a a 1 1 e3 = ( 2 , 2 , 0 ) C11 2 C11 2 C11 Another useful lattice basis is the following (Fig. 3.10(b)):   1 ˜ ˜11 c1 = ( a2 , a2 , 0) C 0 2 C11 2 1 ˜  , C˜11 = a . ˜11 c2 = (− a2 , a2 , 0) , C˜ =  0 C 2 C11 2 1 ˜ 1 ˜ ˜11 C c3 = (0, a2 , a2 ) 2 C11 2 C11

(3.54)

Body-centered cubic type A rather common lattice basis is given by:  C11 0 e1 = (a,0,0) e2 = (0,a,0) , C =  0 C11 1 1 e3 = ( a2 , a2 , a2 ) 2 C11 2 C11

1 2 C11 1 2 C11 3 4 C11

 , C11 = a2 .

(3.55)

86

SIMPLE LATTICES

Figure 3.11 The body-centered cubic (b.c.c.) lattice. (a) The unit cell of the lattice vectors in (3.55). (b) The rhombohedral unit cell with basis (3.56)

In Fig. 3.11(a), for drawing convenience, the third lattice vector is e3 − e2 . Also for this lattice a rhombohedral unit cell, shown in Fig. 3.11(b), is often used; it is generated by the lattice vectors ˜ e 1 = e 3 − e2 , ˜ e 2 = e 1 + e2 − e3 , ˜ e 3 = e3 − e1 :   ˜11 ˜11 − 1 C ˜ ˜ e1 = (− a2 , a2 , a2 ) C − 13 C 3 11 2 ˜11 = a . (3.56) ˜11 ˜11  , C C ˜ e2 = ( a2 , − a2 , a2 ) , C˜ =  − 13 C˜11 − 13 C 4 ˜11 − 1 C ˜ ˜11 C ˜ e3 = ( a2 , a2 , − a2 ) − 13 C 3 11 Notice that the three cubic lattices are special cases of the rhombohedral lattices, obtained for suitable values of the lattice parameters a and c in (3.49), that is for suitable values of the interaxial angle in (3.48). See also Table 3.3, where a common rhombohedral lattice group for the f.c.c. and b.c.c. lattices is shown, and Fig. 3.19, where the rhombohedral and the tree cubic sets of lattices up to GL(3, Z)-equivalence – see §3.5 – are represented. In §3.6.3 we describe the Bain stretch, which relates the face- and bodycentered cubic lattices, and in §3.7.2 we discuss in more detail the closepacked structures, of which the face-centered cubic lattice is an example. Remark 3.1 Only six of the fourteen Bravais lattice types are realized in nature as such, and are listed in the Strukturberichte (1915-1940): rhombohedral (β-Po, α-Hg), centered tetragonal (protoactinium, In), hexagonal (metastable Si), primitive cubic (α-Po), f.c.c (Al, Cu, Ag, Au, ...), b.c.c. (Li, Na, Fe, ...). The analogue holds also for multilattices, see Remark 11.3.

3.5 FIXED SETS OF LATTICE GROUPS

87

3.5 Fixed sets of lattice groups Having determined how many lattice types there are in three dimensions, we now study for which (homogeneous) deformations the symmetry of these lattices ‘remains the same’. First of all, we examine how a basis ea can be deformed in such a way that its lattice group stays constant; in this case, according to (3.34), the lattice remains of the same Bravais type as L(ea ). This question leads to studying the structure of sets of bases or metrics sharing the same lattice group. Knowledge of these subsets of the spaces B or Q> 3 is very interesting for the study of phase transitions. In this section we discuss the main properties of these sets, thereby investigating also the symmetry-preserving stretches of simple lattices. Any subgroup L < GL(3, Z) determines a connected set I(L) in the space Q> 3 , called the ﬁxed set of L, consisting of all the metrics stabilized by the elements of L: t I(L) := {C ∈ Q> 3 : m Cm = C for all m ∈ L}.

(3.57)

The main properties of the ﬁxed sets in Q> 3 are given in Proposition 3.8 below. The group L also determines an O-invariant subset E(L) of the space B of bases, again called the ﬁxed set of L, consisting of all the bases on which the elements of L act orthogonally: E(L)

:= {ea ∈ B : for any m ∈ L, mba eb = Qea , Q ∈ O} = {ea ∈ B : ea · eb ∈ I(L)}. (3.58)

Given a basis ea with metric C, the ﬁxed sets E(L(ea )) and I(L(C)) of the lattice group L(ea ) = L(C) in (3.25) and (3.28) are also called the ﬁxed sets of ea and C, respectively. The classical names given in Fig. 3.12 for the Bravais lattice types and the lattice groups are also used for the associated ﬁxed sets in B or Q> 3. Remark 3.2 A lattice group and some of its subgroups, not themselves lattice groups, may have coincident ﬁxed sets; unless otherwise speciﬁed, when speaking of the ﬁxed set of a group, we always refer to the unique lattice group that deﬁnes it. Also, we denote by I(L)∗ , and call the proper ﬁxed set of L, the subset of I(L) consisting of the elements that are stabilized exactly by the matrices in L and no others (likewise for the bases and the proper ﬁxed sets E(L(ea ))∗ in B). We will see that I(L)∗ almost coincides with I(L).19 By deﬁnition, the metrics in I(L(C))∗ all share the same lattice group 19

In general a ﬁxed set I(L(C)) is constituted by all the metrics in I(L(C))∗ , which have exactly the same lattice group as C, plus the metrics that are ‘more symmetric’ than C; that is, are stabilized by more matrices than just those in L(C). Such more symmetric metrics belong to the ‘smaller’ ﬁxed sets contained in I(L(C)). Indeed, as indicated in statement (3) of Proposition 3.8 below, the ﬁxed set I(L(C)) contains the lower-dimensional ﬁxed sets of any lattice groups containing L(C). See also the example in (3.62)–(3.63), and Fig. 3.13.

88

SIMPLE LATTICES

L(C), hence correspond to lattices of the same Bravais type. By analyzing the structure and arrangement of the (proper) ﬁxed sets in B and Q> 3 we then keep track of how a lattice changes its symmetry when its basis undergoes a homogeneous deformation, or when a path is given in the conﬁguration spaces B or Q> 3. Some general properties of the lattice groups and their ﬁxed sets, which are very useful in the analysis of symmetry changes, are summarized in the following Proposition (see also Ericksen (1979) or Michel (1995), where some proofs can be found). Proposition 3.8 The following statements hold: (1) The ﬁxed set of a subgroup L of GL(3, Z) is nonempty if and only if L is ﬁnite. (2) For any L < GL(3, Z) the analogue of (3.57) in Q3 deﬁnes a vector subspace; in Q> 3 , (3.57) deﬁnes a linear convex cone. (3) The ﬁxed set I(L) of a lattice group L contains, as a submanifold of strictly smaller dimension, the ﬁxed set of any lattice group L larger than L: I(L )I(L) ⇔ L < L . (3.59) Thus, with L any lattice group strictly containing L,

I(L)∗ = I(L) \ L >L I(L ).

(3.60)

(4) The ﬁxed sets I(L1 ) and I(L2 ) of any two ﬁnite subgroups L1 and L2 of GL(3, Z) have a nonempty intersection if and only if L1 and L2 together generate a ﬁnite subgroup of GL(3, Z), whose ﬁxed set is I(L1 ) ∩ I(L2 ).20 (5) For any C ∈ Q> 3 and any m ∈ GL(3, Z) the lattice groups and ﬁxed sets transform as follows (recall (3.26)): L(mt Cm) = m−1 L(C)m,

I(m−1 L(C)m) = mt I(L(C))m.

(3.61)

Analogous properties hold for the ﬁxed sets in B deﬁned in (3.58). Remark 3.3 Statement (3) above says that, for any lattice group L, I(L) is constituted by all the metrics C whose lattice group is L, plus all the 20

We remark that if L1 and L2 are lattice groups that generate a ﬁnite group, the latter is not necessarily itself a lattice group. As an example, take the hexagonal basis ea in (3.50) and consider L(ea ) and P (ea ) (the group Hk collecting the proper elements of P (ea ) is listed in Table 3.1). The monoclinic holohedries P1 = {±1, ±Rπ i } and P2 = {±1, ±Rπ √ } are both contained in P (ea ) and correspond, through the basis i− 3j

ea , to lattice subgroups L1 and L2 , respectively, of L(ea ). One can check that the lattice groups L1 and L2 generate a proper subgroup, R say, of L(ea ). Notice that R is isomorphic to a rhombohedral lattice group, and yet is not itself a lattice group, because any metric that is stabilized by R is actually stabilized by the entire L(ea ). This is shown in Remark 4.1.

3.5 FIXED SETS OF LATTICE GROUPS

89

Figure 3.12 The inclusion relations for the 3-dimensional Bravais lattice types (or, the lattice groups up to GL(3, Z)-conjugacy). The denominations combine the notation for holohedries and centerings in Table 3.1

metrics that are more symmetric than C. Now, the problem of ﬁnding21 the ﬁxed set I(L) of a given lattice group L is straightforward; however, obtaining an explicit description of the proper ﬁxed set I(L)∗ is far from trivial: formula (3.60) formally describes I(L)∗ , but ﬁnding all the lattice groups L containing L is in general a complex task. Not much is known, in fact, about the inclusion relations among all the lattice groups in GL(3, Z). A diagram of such relations up to GL(3, Z)-conjugacy is given in Fig. 3.12 – see Michel (1995) – and some remarks on the simpler 2-dimensional case are added in §3.8. We will give a local description of the proper ﬁxed sets I(L)∗ in chapters 4 and 5. Statement (5) in Proposition 3.8 indicates how a ﬁxed set transforms under a change of lattice basis; formulas (3.60)–(3.61)2 tell us that each Bravais lattice type in Q> 3 is constituted by the union of an inﬁnite number of arithmetically equivalent proper ﬁxed sets. Indeed, from (3.61)2 , by varying the matrix m in GL(3, Z) and by excluding the more symmetric metrics as speciﬁed in (3.60), one obtains all the metrics that are in the Bravais lattice type of a given metric C. Analogous statements hold for the ﬁxed sets in B. A precise description of the structure of the Bravais types in Q> 3 and B is given in part (3) of Proposition 3.10 below. 21

Finding the ﬁxed set of a given lattice group is indeed a linear problem in Q3 , whose solutions must be restricted by suitable inequalities in order that they be positivedeﬁnite, according to (3.57).

90

SIMPLE LATTICES

1

1

1

Figure 3.13 One of the primitive orthorhombic ﬁxed sets, including (parts of ) three 2-dimensional primitive tetragonal ﬁxed sets, whose intersection is a 1dimensional primitive cubic ﬁxed set. The forms of the related metrics are also shown. The intersections of the tetragonal ﬁxed sets with the triangle intercepting each axis at 1 are fully drawn

3.5.1 An example It is useful to consider the six independent entries Cab , a ≤ b of a typical lattice metric C as coordinates in the space Q3 and, according to statement (2) above, to describe the ﬁxed sets of lattice groups by means of linear equations on these six variables, to which suitable inequalities guaranteeing positive-deﬁniteness are then added. For instance, the (1-dimensional) primitive cubic ﬁxed set of the metrics C in (3.52)2 is given by the metrics whose entries satisfy the following equations and inequalities: C11 = C22 = C33 > 0,

Cab = 0 for a < b,

(3.62)

as is geometrically evident. By (3.45), a (2-dimensional) primitive tetragonal ﬁxed set containing the primitive cubic ﬁxed set (recall Proposition 3.8(3) and formula (3.45)) is given by: C11 = C22 > 0,

C33 > 0,

Cab = 0 for a < b.

(3.63)

There exist exactly two other primitive tetragonal ﬁxed sets containing the primitive cubic set (3.62): they are obtained by substituting, respectively, equation C11 = C33 or C22 = C33 for the one in (3.63)1 . In turn, all these ﬁxed sets are contained in the (3-dimensional) primitive orthorhombic ﬁxed set given by C11 > 0, C22 > 0, C33 > 0, Cab = 0

for a < b,

(3.64)

3.6 SYMMETRY-PRESERVING STRETCHES FOR SIMPLE LATTICES

91

as in (3.41). See also Fig. 3.13, where for convenience only part of the tetragonal ﬁxed sets are drawn. This example shows that almost all the metrics or bases belonging to a given ﬁxed set have the same lattice group, except for lower-dimensional submanifolds on which the lattice group can increase – see footnote 19. For instance, the ﬁxed set given by (3.62) is contained as a lower-dimensional submanifold in the one given by (3.63), and the primitive cubic lattice group stabilizing the metrics (3.62) contains as a subgroup the primitive tetragonal lattice group stabilizing the metrics (3.63). Notice also that the aforementioned three primitive tetragonal ﬁxed sets considered above are in the conditions of statement (4) of Proposition 3.8: their common intersection is the primitive cubic set (3.62) and their lattice groups together generate the primitive cubic lattice group stabilizing the metrics in (3.62). See §5.1.1 for more details. 3.6 Symmetry-preserving stretches for simple lattices 3.6.1 Commutation relations As mentioned earlier, a main point of interest is studying the deformations that maintain each basis or metric in its own ﬁxed set. This section discusses the main properties of such deformations. For any basis ea , let us consider: (a) the set C(ea ) which collects the stretches22 U ∈ Sym > centralizing (that is, commuting with, see (2.3)) all the elements of the holohedry P (ea ): C(ea ) := {U ∈ Sym > : QU = UQ for all Q ∈ P (ea )},

(3.65)

and (b) the set U (ea ) collecting the stretches U that preserve the lattice group of ea : U (ea ) := {U ∈ Sym > : L(Uea ) = L(ea )} ⊆ C(ea ).

(3.66)

Below we study in some detail the diﬀerence between C(ea ) and U (ea ), and show why the inclusion relation (3.66)2 holds. For the moment we recall that, by Proposition 2.8, a stretch U ∈ Sym> commutes with Q ∈ O if and only if Q leaves the eigenspaces of U invariant: this gives one way to determine the set C(ea ) when the group P (ea ) is given. Some main properties of the stretches in C(ea ) and U (ea ) are summarized in the following Lemma 3.9 Let ea be a basis in B with metric C, and let F ∈ Aut have polar decomposition F = RU as in (2.20); then (1) U ∈ C(ea ) if and only if the lattice group of the deformed basis Fe a coincides with or is larger than the lattice group of ea : L(Fea ) ≥ L(ea ). Therefore 22

It is straightforward to check that U belongs to C(ea ) if and only if so does C = U 2 or the strain tensor E – see (2.52).

92

SIMPLE LATTICES

U ∈ C(ea ) ⇔ Uea ∈ E(L(ea )) ⇔ (Uea · Ueb ) ∈ I(L(C)),

(3.67)

and, since L(ea ) = L(Fea ) if and only if U ∈ U (ea ), U ∈ U (ea ) ⇔ Uea ∈ E(L(ea ))∗ ⇔ (Uea · Ueb ) ∈ I(L(C))∗ ;

(3.68)

in particular, the inclusion relation (3.66)2 holds. (2) If U ∈ C(ea ) then P (Uea ) ≥ P (ea )

and

P (Fea ) ≥ RP (ea )Rt .

(3.69)

P (Fea ) = RP (ea )Rt .

(3.70)

The converse does not hold. (2’) If U ∈ U (ea ) then23 P (Uea ) = P (ea )

and

The converse is false: either of the following can hold: P (Uea ) = P (ea )

with

P (Fea ) = P (ea )

L(Uea ) $= L(ea ) with

or

(3.71)

L(Fea ) $= L(ea ).

(3.72)

Proof. Consider any two bases ea and ea = RUea such that L(ea ) ≥ L(ea ).

(3.73)

By the deﬁnition (3.16), for any Q ∈ P (ea ) there is m ∈ L(ea ) such that Qea = mba eb . By (3.73), the equality (3.16) also holds for m, ea and some orthogonal Q ∈ P (ea ). Then Q ea = Q RUea = mba ea = RU (mba eb ) = RUQea ,

(3.74)

hence Q RU = RUQ. By the uniqueness of the polar decomposition, as in (2.37), this equality holds if and only if Q = RQRt

and Q t UQ = U ,

(3.75)

the second of which implies U ∈ C(ea ). Vice versa, if U ∈ C(ea ) and Q ∈ P (ea ), we can write QUea = UQea = mba Uea , which means that L(Uea ) ≥ L(ea ).

24

23

24

(3.76)

This proves the ﬁrst statement in (1)

Notice that a lattice-group-preserving deformation F = RU as in this statement does not in general preserve the holohedry P (ea ), for F transforms the latter to a conjugate in O through R due to (3.19). To see that it can be either L(Uea ) > L(ea ) or L(Uea ) = L(ea ) depending on the U ∈ C(ea ) considered, one can take a primitive tetragonal basis ea as in (3.45), in which the three vectors are mutually orthogonal and two have the same length, and check that a stretch in C(ea ) can transform ea into a primitive cubic basis with all the vectors mutually orthogonal and of the same length – see also formulas (3.62)–(3.63).

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93

above, and thus implies (3.67) by (3.57) and (3.58). The second statement in (1) just recalls the deﬁnition of U (ea ), which is equivalent to (3.68). Finally, due to (3.60), formulas (3.67) and (3.68) imply relation (3.66)2 . Regarding the ‘if’ part of statement (2), if U ∈ C(ea ), statement (1) gives (3.73), hence also (3.75)1 , which in turn implies (3.69) with the help of (3.19). Analogously for the ‘if’ part of statement (2 ), which is an immediate consequence of statement (2) applied to both the bases ea and Uea . As stated in (2) and (2 ), the converse implications do not hold: indeed, a tensor F ∈ Aut, or even a stretch U ∈ Sym > , can preserve the holohedry of a basis ea without doing so for the lattice group. As far as the stretch is concerned, consider an arbitrary basis ea in (3.39) and the basis Uea , with   1

U = Uab e a ⊗ e b ∈ Sym > ,

(Uab ) =  1 0

1

0

2

0.

0

1

(3.77)

One can check that P (ea ) = P (Uea ) = {±1, ±Rjπ }; however, the lattice groups L(ea ) and L(Uea ) are conjugate through the inverse of (Uab ) in (3.77) (and have disjoint ﬁxed sets). Therefore these lattice groups are distinct, and not in a group-subgroup relation, so that U ∈ / C(ea ), proving (3.71) as well as the last statement of (2).25 The stretch U in this counterexample preserves both the crystal system and the Bravais type of the basis ea . Examples of deformations F = RU proving (3.72) can be obtained, with suitable Rs, from the stretches U that preserve the crystal system while changing the Bravais type of the basis ea : by deﬁnition, these stretches maintain the holohedry P (ea ) up to conjugacy in O, as required by (3.72)1 , but do not preserve L(ea ), not even up to arithmetic equivalence. Such stretches exist by the fact that the crystal systems in B do not in general coincide with the Bravais lattice types, as we have seen in Corollary 3.7. A well known example of a stretch which preserves the system but changes the Bravais type (that is, the centering) of a lattice is given by the Bain stretch, which transforms a b.c.c. basis into an f.c.c. one, as detailed in §3.6.3. Remark 3.4 (a) Lemma 3.9(2 ) says that any stretch U ∈ U (ea ), which by deﬁnition preserves the lattice group of the basis ea , necessarily preserves also its holohedry, while the converse is not true. This indicates a clear diﬀerence between the holohedral point groups in O and the lattice groups in GL(3, Z), showing once again how the symmetry properties they describe are quite distinct.26 (b) In general U (ea ) ⊆ C(ea ), as indicated in (3.66)2 . These two sets of stretches, however, almost coincide. Indeed, by Proposition 3.8(3), they do 25 26

This counterexample shows that the subsets of B on which the holohedry is constant have a nontrivial structure which has not been much investigated in the literature. One can investigate these properties more in detail by studying the structure and arrangement of subsets of bases in B which share the same holohedry, versus those which share the same lattice group (proper ﬁxed sets, as in (3.60)); see also footnote 25.

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SIMPLE LATTICES

except for the special stretches belonging to suitable low-dimensional submanifolds of C(ea ) (not in U (ea )), which actually increase the lattice group of the basis ea . As mentioned earlier, it is not too diﬃcult to determine the set C(ea ) once the group P (ea ) is given; one can use directly the deﬁnition (3.65) with the aid of Proposition 2.8, or relations (3.67) once the ﬁxed set I(L(ea )) ⊂ Q> 3 is determined. It is much harder to ﬁnd all the stretches in U (ea ) from (3.68), as is mentioned in Remark 3.3. (c) The diﬀerences between U (ea ) and C(ea ), as well as the diﬃculties about U (ea ) mentioned in (b), disappear when only ‘small’ stretches (U close to 1) are considered.27 Indeed, in §4.3.1 we show that the sets C(ea ) and U (ea ) coincide in suitable neighborhoods of 1 in Sym> ; this will make it manageable, in chapter 5, to ﬁnd the explicit forms of the small stretches in U (ea ). The coincidence of C(ea ) and U (ea ) near 1 means that statement (2 ) in Lemma 3.9 becomes the same as statement (2) for small stretches; we will see in §4.3.1 that also the converse to the direct statements in (2)–(2 ) holds for U near 1. Preserving the holohedry of a basis is thus the same, for small stretches, as preserving the lattice group; so, in the conﬁguration spaces of simple lattices, the diﬀerences between the lattice groups and the holohedries are locally less relevant than in the large. For these reasons such groups are sometimes not clearly distinguished in the literature.28 (d) The stretches in U (ea ), which preserve both the holohedry and the lattice group of a basis ea – and a fortiori its Bravais type – are called symmetry-preserving for ea . These nontrivial deformations are very important from the thermomechanical point of view, for they describe the free thermal expansion of a lattice, or its deformation under varying pressure, away from phase transitions – see §6.6.1. We will make the small symmetry-preserving stretches explicit in chapters 4 and 5. 3.6.2 Structure of the ﬁxed sets Lemma 3.9 says that a lattice group does not uniquely determine a basis (or a lattice). It only does so up to a symmetry-preserving stretch in (3.66) and an orthogonal transformation. Precisely, a ﬁxed set contains all the bases obtained from a given basis ea by means of any product RU , with R ∈ O and U ∈ C(ea ) (or U ∈ U (ea ) if we consider a proper ﬁxed set). By (3.9) we also see that all the bases belonging to the Bravais lattice type of ea in B diﬀer from ea by a deformation RU as above, but with a possible 27 28

This case is physically interesting as many (weak) solid-state phase transitions in crystalline solids can be modelled under this hypothesis (chapters 6 and 7). Notice that here we are only considering small stretches (U near 1) and not necessarily small orthogonal parts R of the polar decomposition F = RU of any tensor F ∈ Aut. Even for small U ∈ U (ea ) the holohedry of a basis Fea is thus in general conjugate to P (ea ), and not coincident with it, as remarked in footnote 23. For this reason in B there is never a complete local equivalence of P (ea ) and L(ea ) in the description of lattice symmetry. This problem does not arise in the space of metrics Q> 3 , whose elements are invariant under orthogonal transformation of the lattice basis.

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95

initial deformation in the global symmetry group G(ea ). Indeed, the ﬁxed sets and the Bravais lattice types in B and Q> 3 are explicitly described in the following Proposition (recall Proposition 3.8): Proposition 3.10 Let ea be a basis with metric C. Then: (1) The ﬁxed set E(L(ea )) [The proper ﬁxed set E(L(ea ))∗ ] in B has elements given by all the bases RUea , for R ∈ O and U ∈ C(ea ) [U ∈ U (ea )]: in the notation of (2.18) E(L(ea )) = OC(ea )ea ,

E(L(ea ))∗ = OU (ea )ea .

(3.78)

(2) The elements of the ﬁxed set I(L(C)) of C are the matrices (C¯ab ) such ¯ab e a ⊗ e b in C(ea ), ¯ =U that, for some stretch U ¯ )ab . ¯ C −1 U ¯ 2 = C¯ab e a ⊗ e b ⇔ C¯ab = U ¯ ea · U ¯ eb = (U U (3.79) The analogue holds if I(L(C))∗ and U (ea ) are substituted for I(L(C)) and C(ea ), respectively. (3) The Bravais type of ea in B is given by all the bases RUHea , where R ∈ O, U ∈ U (ea ) and H ∈ G(ea ).29 In Q> 3 the Bravais type of C is ¯ given by all the metrics mt Cm, where m ∈ GL(3, Z) and C¯ ∈ I(C)∗ (see (3.61)). Remark 3.5 According to (3.79)1 , the matrices C¯ab ∈ I(L(C)) represent, in the given basis ea , the squares of the stretches in C(ea ). However, it is often convenient to represent such stretches in an orthonormal basis, ci say, relative to which the lattice vectors ea have components given by the matrix E = (eia ) = (ci · ea ). For instance, we will do so in chapter 5. In this case the metrics C¯ ∈ I(L(ea )) in (3.79) have the expression ˜ ¯ 2 = C˜ij ci ⊗ cj , C¯ = E t CE, (3.80) where U 2 ¯ in the basis ci . Thus, while for in terms of the matrix C˜ representing U two bases ea and ea with the same holohedry the sets of stretches C(ea ) and C(ea ) – but not in general U (ea ) and U (ea ) – clearly coincide (and so do the matrices that represent them in an orthonormal basis), the sets of matrices C¯ in (3.79) and (3.80), that is, the ﬁxed sets I(L(ea )) and I(L(ea )), in general are neither the same nor are they equivalent. They are the same, or equivalent, when so are the lattice groups L(ea ) and L(ea ). 3.6.3 The Bain stretch in the centered cubic lattices A noteworthy example of how a stretch does not preserve symmetry in the cubic system is the Bain stretch,30 which transforms a body-centered into 29

30

Here the stretch U is in U (ea ) and not in C(ea ) because, from its deﬁnition, the Bravais type of ea in B is the set of all bases in B whose lattice group is equivalent to L(ea ), and thus does not include any of the more symmetric bases whose lattice group strictly contains any group L(mba eb ), m ∈ GL(3, Z). Therefore the Bravais type of ea is given by the union of all the proper ﬁxed sets E(L(Hea ))∗ , for H ∈ G(ea ). This denomination is sometimes used in the literature also for other phase-transformation stretches in crystalline solids.

96

SIMPLE LATTICES

Figure 3.14 The body-centered tetragonal cell for the f.c.c. lattice – see (3.54) – obtained from an equally oriented b.c.c. cell by the Bain stretch (3.81)

a face-centered cubic lattice. This transformation preserves the crystal system (that is, the holohedry of the lattice up to conjugacy in O) but not the Bravais type, and is very important in the theory of phase transformations in cubic crystals; for instance, one of the most important phase changes in metallurgy, the α-γ (that is, b.c.c. to f.c.c.) transformation observed in iron, can be described in terms of the Bain stretch (Nishiyama (1978)). Referring to Fig. 3.14, consider the face-centered cubic lattice whose components in the orthonormal basis (i , j , k ) are given by (3.54) for a = 1. This lattice contains a body-centered tetragonal basis whose √ √ components in the basis (c1 ,√c2 , k ), for c1 = (i + j )/ 2, c2 = (j − i )/ 2, are given by (3.46)1 for a = 2/2, c = 1. This cell is obtained by applying the stretch √ UB = 1 + ( 2 − 1)k ⊗ k , UB ∈ Sym > , (3.81) to the b.c.c. cell whose vectors ea are represented in the basis (c1 , c2 , k ) as √ in (3.55) for a = 2/2. The Bain stretch UB , applied to the b.c.c. lattice vectors ea , produces lattice vectors UB ea for the f.c.c. lattice we started with. The holohedries of the two bases are orthogonally conjugate: π

7π

P (ea ) = Rk4 P (UB ea )Rk4 ,

(3.82)

that is, ea and UB ea belong to the same (cubic) crystal system in B. However – see §3.3.3 – L(ea ) and L(UB ea ), corresponding to diﬀerent centerings, are inequivalent cubic lattice groups. This is an example of the stretches for which (3.72) holds. Other examples of stretches that change the type but preserve the system of a lattice can be obtained from the analysis of Pitteri and Zanzotto (1996a). The linear transformation of the f.c.c. basis, or of the related primitive cubic (nonunit) basis, into the b.c.t. basis above, which produces the b.c.c. lattice in the inverse Bain deformation, are examples of the so-called lattice correspondence; this speciﬁes the structural unit (possibly made of more than one unit cell) in the parent phase which is transformed into a unit of the product phase. More comments on this point are given in §7.6.3.

3.7 LATTICE SUBSPACES, PACKINGS AND INDICES

97

3.7 Lattice subspaces, packings and indices 3.7.1 Lattice rows and lattice planes One often needs to consider the lower-dimensional sublattices of a given lattice L in R3 , namely the sets of lattice points belonging to 1- and 2-dimensional subspaces of R3 . Clearly, not any such subspace contain points of L other than 0. The ones that do are called rational or crystallographic; they are generated by one vector in L, v say, or by two linearly independent vectors in L, u, v say, respectively. The set of integral multiples of v , or of the integral combinations of u and v , are indicated by ≺v/ and ≺u, w/, respectively. The cosets e + ≺ v / and e + ≺ u, w / for any e in L are called the lattice rows and the lattice planes or nets of L. Among their properties, the following are very useful (Bravais (1850) pp. 4 and 61): Proposition 3.11 Given any net ν of a lattice L and any row r of ν, it is always possible to ﬁnd lattice vectors ea for L such that e1 is parallel to r and e2 is parallel to ν. The original proof is constructive. Alternatively, one can use the well known properties of the elementary operations on rectangular matrices with entries in Z (for instance in Jacobson (1974), p. 175). Proposition 3.12 Any symmetry axis of L is a lattice row contained in a symmetry plane. Furthermore, any symmetry axis is orthogonal to a lattice net; in particular, any symmetry plane is a lattice net orthogonal to a twofold symmetry axis. Indeed, let R be a rotational symmetry of L of period r, and v ∈ L. Then (1 + R + . . . + Rr−1 )v is in L and belongs to the axis of R by construction; thus the axis is a lattice row. By similar reasonings, the fact that L is centrosymmetrical and that the negative of a reﬂection is a rotation of period 2, one can prove the remaining assumptions. The assertion that the axis of any rotation of period 3, 4, 6, belongs to a symmetry plane is proved, for instance, by Yale (1968). 3.7.2 Close-packed structures A simple lattice can be viewed as a periodic repetition in the third dimension, usually called a stacking, of a 2-dimensional planar pattern. For instance, from the description in §3.4 we see that ﬁve of the Bravais lattices admit a threefold symmetry axis (that is, the three cubic, the hexagonal and the rhombohedral lattices) and are thus obtained by stacking triequiangular, or centered hexagonal, nets of points. In particular, we already remarked that all the three cubic lattices are special cases of the rhombohedral lattice, obtained for speciﬁc ratios of the parameters a and c in the rhombohedral basis (3.49). It is noteworthy that, among them, the f.c.c. lattice is realized by the centers of equal spheres in a close-packed

98

SIMPLE LATTICES

stacking, representing the ‘atoms’ in a densest31 possible arrangement, with approximate density 0.7405 – see Fig. 3.10(b). Suppose a hexagonal net is given by the centers of close-packed spheres on a plane. When a second layer of close-packed spheres is deposited on the ﬁrst one, their centers project on the ﬁrst hexagonal net as shown in Fig. 3.15(a). If a third layer is deposited on top of the second one, displaced in the same way with respect to the one underneath, and so on, the centers of all the spheres make up an f.c.c. lattice – see Figs. 3.15(b) and 3.10(b). However, there is another possibility of close-packing spheres in space: the third layer can be deposited directly above the ﬁrst one, in such a way that the centers of the ﬁrst and third layers project orthogonally onto each other, so as to form a primitive hexagonal cell. If this procedure is repeated, the structure one obtains is called hexagonal close-packed, or h.c.p., which also gives the maximum density of sphere-packing in space. It is important to remark that in an h.c.p. structure the centers of all the spheres do not form a simple lattice as in the case of the f.c.c. stacking pattern; the h.c.p. structure is in fact a 2-lattice constituted by two suitably translated and compenetrating hexagonal simple lattices. This is a ﬁrst example of the structures called multilattices, that we study in chapter 11. More details on the h.c.p. structure can be found in §11.5.1. Besides f.c.c. and h.c.p. there are other, more complex ways of tiling the close-packed layers which will not be considered here. Both f.c.c. and the h.c.p. structures are of great importance in metallurgy: for instance, the austenitic steels, aluminum, nickel, copper, silver and gold crystallize in simple f.c.c. lattices at normal conditions, while zinc, cobalt, zirconium and titanium are h.c.p. Structures which are built up by the uniform stacking of atomic layers often exhibit various kinds of irregularities in their sequence, called stacking faults. These have sometimes a clear relation with the phenomena of deformation and growth twinning mentioned in the Introduction – see for instance Bollmann (1970). 3.7.3 Miller indices and crystallographic equivalence It is usual in the metallurgical and mineralogical literature to use a threeindex system of notation, called Miller indices, to indicate the rows and planes in R3 (and in a lattice L). Let ca be a basis for R3 , with its reciprocal basis c a ; the indices [u1 u2 u3 ] in square brakets indicate the direction of the vector u = ua ca , while indices (v1 v2 v3 ) in parentheses indicate the family of planes orthogonal to the vector v = va c a . The Miller indices are deﬁned up to a constant common factor, and integral values of the indices characterize the crystallographic elements of R3 with respect to the basis 31

That no arrangement of spheres of equal radius in 3-dimensional space can have density greater than that of an f.c.c. packing is actually a famous, for long open problem of classical geometry, called the Kepler conjecture (see Hales (1994), (1998)).

3.7 LATTICE SUBSPACES, PACKINGS AND INDICES

99

(a)

(b)

Figure 3.15 Close-packing of spheres. The stacking of two close-packed nets (gray is lower) is schematically represented in (a). By placing the third layer on top of the ﬁrst and repeating this arrangement of the two layers we obtain the h.c.p. structure. The other possibility to place the third layer, giving the f.c.c. structure when repeated, is shown in (b)

ca , that is, the rows and nets of the lattice L(ca ). Negative indices are usually indicated by an overbar: for instance, ¯ 1 means −1. This convention has already been used in Table 3.3. It is not diﬃcult to see that the Miller indices of a plane π equal, up to a common factor, the reciprocals of the intercepts on the coordinate axes of any plane parallel to π not containing the origin. Remark 3.6 In the literature the indices in a lattice are not always given with respect to an actual lattice basis. There are a number of conventions for indexing, and much care is needed when interpreting and comparing the experimental reports. Some general guidelines are the following: • The indices for a primitive lattice in a system are usually given with respect to a standard basis as given in §3.4 (for instance, in the monoclinic system the twofold axis is always along the y-axis (so that β $= π/2, and actually the obtuse β is the standard choice), or in the tetragonal system the fourfold axis is always along the z-axis). • For the centered lattices in a system the same standard primitive basis is used for indexing (thus ‘skipping’ the centering); for instance, the indices in a b.c.c. or an f.c.c. lattice are given with respect to the primitive cubic sublattice basis along the three perpendicular fourfold axes (see the primitive cubic sublattices outlined in Figs. 3.10 and 3.11).

100

SIMPLE LATTICES

A

B OA = OB

e2

O

e1

Figure 3.16 Traces of congruent but not crystallographically equivalent nets (the lattice vector e3 is orthogonal to the page)

• In the hexagonal system the two lattice vectors in the basal plane orthogonal to the sixfold axis are always spaced 120◦ apart; the commonly used sets of indices are described in §3.7.4. • In the rhombohedral system indices are often taken with respect to the associated hexagonal nonunit cell – see the hexagonal basis ˜ ea in Fig. 3.8(a) – and are thus given as in §3.7.4. The other common choice is to take indices with respect to the rhombohedral basis (3.49).

The notion of crystallographic equivalence under the holohedry P (ea ) of L(ea ) can be introduced: two vectors u and v in L(ea ) are crystallographically equivalent if v = Qu for some symmetry operation Q ∈ P (ea ). Analogously, two lattice planes are said to be crystallographically equivalent when they can be brought into coincidence by some Q ∈ P (ea ), that is, when their normals are crystallographically equivalent. Therefore crystallographically equivalent rows [planes] have the same indices relative to suitable congruent sets of [reciprocal] lattice vectors. Parallel lattice rows or nets are also considered crystallographically equivalent. We remark that while two crystallographically equivalent planes in a lattice carry congruent nets of lattice points, the converse is in general not true. For instance, if in the primitive orthorhombic lattice generated by the lattice vectors (3.41) the lattice parameters a and b satisfy the special relation a2 +9b2 = 4a2 +b2 , the nets ≺e2 +2e1 , e3 / and ≺e1 +3e2 , e3 / are congruent but do not belong to crystallographically equivalent planes, as is easily seen with the help of Fig. 3.16. However, this property is nongeneric, in the sense that it is destroyed by (small) symmetry-preserving variations of the lattice parameters – see also §8.3.4.

3.7 LATTICE SUBSPACES, PACKINGS AND INDICES

101

3.7.4 Miller-Bravais indices for hexagonal lattices In the hexagonal lattice with basis ea given in (3.50) the reciprocal basis e a in (2.10) is explicitly given by: √ √ e 1 = a1 , 3a3 , 0 , e 2 = 0, 23a3 , 0 , e 3 = 0, 0, 1c . (3.83) In this case the usual Miller indices are often replaced by the Miller-Bravais indices – see for instance Otte and Crocker (1965) or Sirotin and Shaskolskaya (1982). This is a four-index notation which is useful in many crystallographic computations and is widely used in the literature. The Miller-Bravais indices [u1 u2 u4 u3 ] for the direction of a vector d are obtained by introducing a fourth axis e4 = −(e1 + e2 ) and resolving d into four components: 4 d = i=1 ui ei , with the condition u1 + u2 + u4 = 0 . (3.84) The index u4 is of course redundant and can be omitted, so the direction [u1 u2 u4 u3 ] is often indicated by [u1 u2 · u3 ]. One of the main advantages of the four-index system is that all the crystallographically equivalent directions have permutations of the ﬁrst three indices, since the three axes on the basal plane e1 , e2 are all crystallographically equivalent. The Miller and Miller-Bravais indices of a direction are related as follows: M.: [u1 u2 u3 ] ⇒ M.-B.: [2u1 −u2 2u2 −u1 −(u1 +u2 ) 3u3 ] M.-B.: [u1 u2 u4 u3 ] ⇒ M.: [2u1 +u2 u1 +2u2 u3 ] .

(3.85)

An analogous Miller-Bravais four-index notation is used for the planes: the indices (v1 , v2 , v4 , v3 ) are deﬁned as the coeﬃcients of the expansion, analogous to (3.84), of a normal n in terms of the four reciprocal vectors 2 3

1

2

3

3

e 1 − e 2,

1

1

1

3

3

3

e 2 − e 1 , − e 1 − e 2 , and e 3 ,

(3.86)

with the restriction v1 + v2 + v4 = 0. In this way, crystallographically equivalent planes all have permutations of v1 , v2 , v4 , and the indices of a plane are again proportional to the reciprocals of its intercepts with the four lattice vectors in (3.84). The relation of the Miller-Bravais to the Miller indices for planes is the following: M.: (v1 v2 v3 ) ⇒ M.-B.: (v1 v2 −(v1 +v2 ) v3 ) M.-B.: (v1 v2 v4 v3 ) ⇒ M.: (v1 v2 v3 ) .

(3.87)

In the hexagonal lattices also a three-index notation based on the orthohexagonal double cell in (3.51) and Fig. 3.8(b) is sometimes used (see Otte and Crocker (1965) for details).

102

SIMPLE LATTICES

3.8 Lattice groups and ﬁxed sets for planar lattices As we have anticipated in Remark 3.3, the global structure of the ﬁxed sets in Q> 3 is important for the study of phase transitions in simple lattices but is not well investigated. What is known is that structure up to GL(3, Z)-equivalence, this being a way of representing the space of all lattices (Schwarzenberger (1972), Parry (1976), Engel (1986), Michel (1995)). This representation is also related to the classical problem of reduction of quadratic forms, ﬁrst attacked by Lagrange and Dirichlet. The identiﬁcations forced by GL(3, Z)-equivalence introduce topological complexities in the space of 3-dimensional lattices, which is also diﬃcult to visualize because of its being 6-dimensional. Everything is much easier, with no compromise on the conceptual framework, for the nets (2-dimensional lattices), represented by their metrics in Q> 2 which is a 3-dimensional cone. Since the ﬁxed sets are also conical in Q> 2 , we simplify their description by selecting one element for each conical line in Q> 2 . Analytically, a convenient way of doing so is the following: for any metric C consider the metric C = (tr C )−1 C . Since now C11 + C22 = 1, we can represent all such matrices in the C11 -C12 plane. Geometrically, this procedure is given in Fig. 3.17(a): for any 1-dimensional subspace in Q2 take the intersection with the aﬃne plane C11 + C22 = 1 and then project it onto the C11 -C12 plane. As a result, in this plane linear subspaces of Q2 are represented by aﬃne subspaces of one less dimension. It is then easy to check that all the elements of Q> 2 are represented by internal points of the circle of center (1/2, 0) and radius 1/2, as described in Fig. 3.17(b). There, also a number of the (inﬁnitely many) projected ﬁxed sets are indicated by dashed or dotted lines in the lower half circle. The lines obtained by symmetry about the C11 axis, which also exist, are not all drawn to avoid cluttering the ﬁgure. The shadowed triangular region, F, and the rhombus, Di (after Dirichlet), of which F is one fourth, have a special meaning to be clariﬁed shortly. In three dimensions, the tr C = 1 condition can again be used to section Q> 3 and the ﬁxed sets, thus producing aﬃne subsets of a 5-dimensional space. This is indicated in Fig. 3.13: tr C = 1 on the triangle having intercepts 1 on the three axes. On this triangle the primitive cubic ﬁxed set becomes a point, where three primitive tetragonal lines intersect. Here the remaining parameters in C have been set equal to zero. To have a glimpse at the complexity of the 3-dimensional case, in Fig. 3.19 we show another slice of the 5-dimensional section of Q> 3 , in which C11 , C22 , C33 are set equal to 1/3 and the other parameters in C vary. Only the sliced analogue of the triangle F is drawn, and this again contains the primitive cubic ﬁxed set. For some of the lines in Fig. 3.17 we give the deﬁning equation, and for one, the vertical diameter of the circle, the nontrivial generator of the corresponding lattice group (the 2 by 2 inversion is always included). Also, some of the ﬁxed sets have a graphical indication of the Bravais lattice

3.8 LATTICE GROUPS AND FIXED SETS FOR PLANAR LATTICES

103

C12 1/2

C12

P

C11 = 1/2 1/4

C11 = 2C12 P

C11

P

C12 = 0

1/2

1

C22 C11 + C22 = 1 C11

(a)

(b)

Figure 3.17 Representation of Q> 2 modulo dilations in the C11 –C12 plane. Traces of various ﬁxed sets are shown

type to which they belong: the analogue of Corollary 3.7 states that nets exhibit ﬁve distinct Bravais types within four crystal systems; details are given in Fig. 3.18. So, for instance, starting from the origin and travelling clockwise along the border of the shaded triangle F we ﬁnd: an edge of rhombbi (centered rectangles), a vertex of hexagons, an edge of rhombbi, a vertex of squares, and an edge of rectangles. Notice how the lattice bases change on the various rhombic ﬁxed sets drawn. In particular, along the oblique edge of F we ﬁnd rhombbi whose acute angle is between zero and π/3, while along the vertical edge we ﬁnd the remaining ones, with acute angle between π/3 and π/2. Fig. 3.17(b) also provides examples for the general statements of Proposition 3.8. For instance, the rectangular ﬁxed set C12 = 0 contains as a submanifold of lower dimension the square ﬁxed set. Also, the two rhombic ﬁxed sets bounding F intersect at a lower-dimensional manifold which is the ﬁxed set of nets having the symmetries of both rhombbi, that is, of hexagonal nets. From that ﬁgure one can also begin to understand how the knowledge of the arrangements of the ﬁxed sets in the space of metrics can be important for the description of changes of symmetry along a continuous deformation of a lattice. For instance, consider how an hexagonal net can change (actually, lower) its symmetry along a continuous path in Q> 2: unless it moves in the interior of F, thus reducing the symmetry to the trivial one, it has to exit the hexagonal ﬁxed set and move into one of the rhombic ﬁxed sets intersecting in the hexagons; thus, by pure geometry, we know that in no way the hexagon can directly deform, in a continuous way, into a primitive rectangle. This holds no matter which energetics we may assume to govern the changes of the metric C in Q> 2 . This result is almost obvious, but in 3 dimensions similar reasonings provide nontrivial infor-

104

SIMPLE LATTICES

hexagon

square

C

P

rectangle

parallelogram

Figure 3.18 The four crystal systems and the ﬁve Bravais types of nets

mation on which systems and which centerings can or cannot be obtained by symmetry breaking along continuous paths; as an additional item, any smooth path joining f.c.c. and b.c.c. must pass through ﬁxed sets of lower symmetry – see Fig. 3.19. Information on symmetry breaking will be given in chapters 4 and 5. Any simple lattice is represented by inﬁnitely many lattice metrics or lattice bases, belonging to inﬁnitely many, arithmetically equivalent, ﬁxed sets. For classiﬁcation purposes one ﬁrst tries to reduce such an inﬁnity by suitably selecting, for each lattice, a best or straightest lattice cell: this can be described as the one formed with the shortest linearly independent lattice vectors existing in the lattice. In Q> 2 one can proceed as follows: > 1) A best cell always exists: for any C ∈ Q2 there is only a ﬁnite number of matrices m ∈ GL(2, Z) such that tr (mt C m) ≤ tr C; as a consequence, there is a ﬁnite number of matrices C¯ which minimize the trace among all the ones arithmetically equivalent to C. 2) Any C¯ as in (1) satisﬁes the inequalities C¯11 ± 2C¯12 ≥0 and C¯22 ± 2C¯12 ≥ 0;

(3.88)

these give a convex, polyhedral cone C in Q> 2 which projects into the rhombus Di in Fig. 3.17(b) having the shaded triangle as one of its fourths. By construction, C or Di contain an arithmetic equivalent of any metric, and in this case possibly more than one. Indeed 3) For any C¯ in the interior of C, mt C¯ m ∈ C if and only if m belongs

3.8 LATTICE GROUPS AND FIXED SETS FOR PLANAR LATTICES

105

to the following ‘square’ group, where all the signs should be chosen independently; this is the lattice group of the square lattice referred to an orthogonal lattice basis, and it induces in the circle of Fig. 3.17 symmetries about its horizontal and vertical diameters: ±1 0 0 ±1 , . (3.89) 0

±1

±1

0

4) Show that, if C belongs to the boundary of C, then other matrices m may leave it on that boundary. Explicitly, corresponding to any one of the inequalities (3.88) that C satisﬁes as an equality there is one of the following matrices: 1 1 1 0 and ; (3.90) 0 ±1 ±1 1 for instance, the ﬁrst matrix with the + or − sign corresponds to equality in (3.88)1 with the same sign, etc. By the properties above, the lattice groups for nets represented by metrics in C (or projected in Di ) are the ﬁnite groups that can be constructed with matrices in (3.89) or (3.90). To eliminate the ambiguities still present in Di one can, for instance, choose the ﬁrst lattice vector to be the shortest, that is, C11 ≤ C22 , and the angle of the lattice vectors not to be obtuse, that is, C12 ≥ 0. These equalities select the shaded triangle F (without the origin) within the rhombus Di. F is called a fundamental domain ; it has the property, which can be checked by inspection, of containing one and exactly one representative for each net, and actually, for each family of nets diﬀering by a dilation. It can be regarded as a way of representing the space of nets (a double quotient like the one mentioned in footnote 18). Fundamental domains in the space B of bases are also called Wigner-Seitz domains, and Brillouin zones in the space of reciprocal lattices. Bases represented by metrics in a 3-dimensional analogue of F, called reduced bases, have been classiﬁed by Seeber (1824) and Niggli (1928), and used as descriptors of 3-dimensional lattices – see also Engel (1986). Special slices of a 3-dimensional fundamental domain are shown in Figs. 3.13 and 3.19. As is shown by Parry (1976), the knowledge of a function – for instance the one representing the energy density of a simple lattice – on the fundamental domain or on the set of reduced bases, if subject to certain restrictions on the boundary of such sets, is suﬃcient to extend that function to the whole of Q> 3 or B-space in such a way that the extended function is invariant under GL(3, Z). A similar description of fundamental domains is introduced by Fosdick and Hertog (1990), for similar purposes.

106

SIMPLE LATTICES

C23 1/6

F R2 P 1/6

C13

1/6

C12

R3

T I3 −1/6

OI1

I T I2

Figure 3.19 Section C11 = C22 = C33 = 1/3 of the Wigner-Seitz domain of 3-dimensional lattices, showing the primitive cubic (P ), f.c.c. (F ) and b.c.c. (I) reduced cells; and, among the others, two rhombohedral (R2 and R3), two centered tetragonal (T I2 and T I3) and one body-centered orthorhombic (OI1) ﬁxed sets

CHAPTER 4

Weak-transformation neighborhoods and variant structures In the previous chapter we have seen how the global symmetry of simple lattices is described by the actions (3.7) and (3.8) of the global symmetry group GL(3, Z) on the conﬁguration spaces B and Q> 3 . In this chapter we show that this global symmetry is locally compatible with the notions on crystal symmetry usually considered in crystallography and physics. Indeed, we show that, for any given conﬁguration of a simple lattice described by a basis ea ∈ B, there exists a suitable orthogonally invariant neighborhood, to be called a weak-transformation neighborhood or, in short, wt-nbhd,1 in which the global symmetry reduces to the lattice-group symmetry dictated by that crystal conﬁguration. This is because the action of the group GL(3, Z) on B, when restricted to a wt-nbhd of ea in B, reduces to the action of the (ﬁnite) lattice group L(ea ). In this way the global symmetry of crystals is reconciled with their crystallographic symmetry, in the range of ‘small but ﬁnite deformations’. An analogous result also holds for multilattices (chapter 11). We will use the wt-nbhds in chapter 6 to reduce in a rational way the domain and invariance of the energy function of elastic crystalline solids. This procedure helps modelling the behavior of a crystal that undergoes symmetry-breaking phase transitions in the solid state; indeed, one of the main properties of the wt-nbhds is that in them symmetry cannot increase. In §3.6 the deformations that preserve the (arithmetic) symmetry of a given lattice L(ea ) are described by the symmetry-preserving stretches that keep a basis ea ∈ B in its own ﬁxed set. Similarly, the analysis in this chapter considers the possible diﬀerent types of small (near 1) symmetrybreaking deformations that lower the symmetry of ea and move it oﬀ its ﬁxed set, while keeping it inside the wt-nbhd. In a wt-nbhd of ea there are only bases with smaller symmetry than ea , and thus one ﬁnds all the possibilities of symmetry breaking for ea by listing the subgroups of L(ea ). Since the latter is a ﬁnite group, it has only a manageably small number of relevant subgroups; these give the local symmetry hierarchies that dictate 1

Sometimes in the literature the wt-nbhds are called Ericksen-Pitteri neighborhoods. 107

108

WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

a priori in how many distinct ways symmetry can be lowered by deforming ea not too severely. Based on Zanzotto (1996b), we give some general properties of the small symmetry-breaking stretches, of the wt-nbhds, and of the local orbits they contain (variant structures); the systematic and explicit description of all these orbits is postponed to chapter 5.2 This analysis makes it possible to detail the symmetry changes that can take place in simple lattices undergoing deformations that are not too large, thus enabling one to consider a wide range of solid-state phase transformations in crystals, to be called weak3 – see Ericksen (1989). This property of wt-nbhds justiﬁes their name. The weak phase changes are of great experimental relevance because they include, as special cases, the symmetry-breaking phase transitions; to these we will mostly conﬁne our attention in the following chapters. 4.1 Reconciliation of the local and global symmetries of crystals: weak-transformation neighborhoods According to the results in §3.1, a tensor F ∈ Aut applied to a basis ea deforms the lattice L(ea ), and if F belongs to G(ea ) the deformation restores L(ea ), that is, L(ea ) = L(Fea ) – see (3.9). From a physical point of view, however, it may not be realistic to consider all the tensors in G(ea ) (or in Aut) as allowable lattice deformations: stretches that are too large may have to be excluded, as they may lead to damage or fracture in the crystal, or to plastic phenomena for which the elastic model we are aiming at is not suitable. Here we show how a convenient restriction of the deformation range can be made, which also gives a reduction of the global symmetry of crystal conﬁgurations to their usual crystallographic symmetry. Proposition 4.1 below4 (Ericksen (1980b), Pitteri (1984)), which is purely geometric in character, shows that there exists a suitable orthogonally invariant weak-transformation neighborhood (wt-nbhd) Nea ⊂ B of any given basis ea , on which the action (3.7) of the global symmetry group GL(3, Z) on the whole of B coincides with the action of the lattice subgroup L(ea ). Equivalently, in a suitable wt-nbhd NC ⊂ Q> 3 of any metric C the action (3.8) of GL(3, Z) reduces to the one of the lattice group L(C). In order to state this result, we ﬁrst recall that, for any basis ea ∈ B ¯e is a neighborhood of ea in B if and only if there is a with metric C, N a ¯ neighborhood N1 of 1 in Aut such that ¯1 ea = {Fea : F ∈ N ¯1 }. ¯e = N N a 2 3

4

(4.1)

Some results are available also for the simplest multilattices; see chapter 11. Precisely, we call weak any phase transition in which the high- and low-symmetry phases are described by lattices admitting bases that both belong to the same wtnbhd in B as in Proposition 4.1 below. This result can also be phrased in terms of slices for the action of a Lie group on a manifold (Duistermaat and Kolk (1999) §2.3): the action of GL(3, Z) on the spaces B or Q> 3 is proper at any one of their points.

4.1 RECONCILIATION OF GLOBAL AND LOCAL SYMMETRIES

109

¯e is O-invariant, that is, Q N ¯e = N ¯e or Q N ¯1 = N ¯1 for all Q ∈ O, when N a a a ¯1 = {F = RU : R ∈ O, U 2 = C¯ab e a ⊗ e b , C¯ ∈ N ¯C } N

(4.2)

¯C ⊂ Q> of C.5 for some neighborhood N 3 Proposition 4.1 For any basis ea with metric C there is a neighborhood NC of C in the 6-dimensional space Q> 3 such that: (1 )

mt NC m = NC for all m ∈ L(C) ≡ L(ea ), mt NC m ∩ NC = ∅ for all m ∈ GL(3, Z)\L(C).

Equivalently, there is an orthogonally invariant neighborhood Nea = N1 ea in the 9-dimensional space B such that for N1 ⊂ Aut the following hold: (2 )

N1 H = N1 for all N1 H ∩ N1 = ∅ for all

H ∈ P (ea ), H ∈ G(ea )\P (ea ).

The basis ea is called the center of Nea ; so is C for NC , and 1 for N1 . Proposition 4.1 is also proved by Fosdick and Hertog (1990), Ball and James (1992) and Cermelli and Mazzucco (1996). Here we give a proof of the part concerning Nea , based on the following results: ¯1 ea ⊂ B, N ¯1 is compact and so is also the ¯e = N (a) For any compact N a subset of Aut given by: ¯1 H ∩ N ¯1 $= ∅}. Σ := {H ∈ Aut : N (4.3) −1 ˆ ¯ ˜ ˆ ˜ Indeed, if H ∈ Σ, there are F , F ∈ N1 such that H = F F . Since the inverse, where deﬁned, and the product of linear maps are continuous, ¯1 . Σ is compact because so is N (b) If a continuous function f from a metric space to itself maps x to f (x ) $= x , there is a neighborhood N of x disjoint from its image f (N ). For, otherwise, there would exist a sequence {xn } which converges to x together with the sequence {f (xn )}, and this implies f (x ) = x , against the hypothesis. ˇ1 of 1 and any ﬁnite group Pˇ ≤ Aut, the set (c) For any neighborhood N ˇ1 H N1 = ∩H ∈Pˇ N (4.4) ˇ1 which is right-invariant is the maximal (possibly empty) subset of N ˇ ˇ under P , that is, N1 H = N1 for all H ∈ P . Proof of Proposition 4.1. Let us choose any orthogonally invariant and ¯e = N ¯1 ea of ea in B. By (a) the intersection compact neighborhood N a S := G(ea ) ∩ Σ is a ﬁnite subset of Aut, because G(ea ) is discrete and Σ is compact; notice that the set S contains at least the holohedry P (ea ). Then, ¯C of C which corresponds to N ¯1 as in (4.2), consider the neighborhood N and apply (b) a ﬁnite number of times to the linear maps in Q> 3 deﬁned 5

¯e is O-invariant when N ¯e = {¯ ¯C }. Equivalently, N ea : (¯ ea · ¯ eb ) ∈ N a a

110

WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

by C¯ → mt C¯ m, the matrices m ∈ GL(3, Z) representing the elements of S\P (ea ) in the basis ea . This produces a (possibly smaller) neighborhood ˇC ⊂ N ˇC m ∩ N ¯C such that mt N ˇC = ∅ for all m ∈ N / L(C); in turn, (4.2) ¯1 of 1 such that N ˇ1 H ∩ N ˇ1 = ∅ ˇ1 ⊆ N gives an O-invariant neighborhood N ˇ ˇ for all H ∈ G(ea )\P (ea ). Then, by applying (c) to N1 with P = P (ea ), we obtain an O-invariant neighborhood N1 of 1 that satisﬁes requirement (2) of Proposition 4.1 and is nonempty because it necessarily contains 1.6 The details about the structure of the neighborhoods and the properties of the orbits of the elements in them are studied in this and the following chapter. The problem of characterizing the maximal neighborhoods satisfying the requirements of Proposition 4.1 is still under attack; based on the analysis of the 2-dimensional lattices (nets), their ﬁxed sets and fundamental domain, which have been sketched in §3.8, Conti and Zanzotto (2002) study the maximal neighborhoods in the case of nets, and apply this knowledge to modelling certain nonweak phase transformations of nets. In the rest of this chapter we analyze how symmetry can change (or, better, be reduced ) in the wt-nbhds of Proposition 4.1. Given a basis ea with metric C, Proposition 4.1 implies that any C in a wt-nbhd NC is mapped outside NC by any m ∈ GL(3, Z)\L(C). This implies that the metrics in NC are necessarily at most as symmetric as C. That is, in NC or Nea the symmetry does not increase: Corollary 4.2 Given a basis ea with metric C and a wt-nbhd NC as in Proposition 4.1, we have L(C ) ≤ L(C) for all C ∈ NC . Analogously, for any ea ∈ Nea we have L(ea ) ≤ L(ea ). Due to statement (2) in Proposition 3.8, a neighborhood NC with center C always contains a nonempty portion of the ﬁxed set I(L(C)), whose elements are the only metrics in NC with same lattice group L(C) (and same Bravais lattice type) as C. Corollary 4.2 implies that any other element of NC is stabilized by a lattice group strictly smaller than L(C). More precisely, as a consequence of the deﬁnitions, of statements (3) and (4) in Proposition 3.8, and of Corollary 4.2, we have: Corollary 4.3 A neighborhood NC ⊂ Q> 3 with the properties in Proposition 4.1 contains a nonempty part of a ﬁxed set I(L) if and only if L ≤ L(C);7 also, the ﬁxed sets I(L), for L < L(C), all intersect at the ﬁxed set I(L(C)) containing the center C. Analogously for the ﬁxed sets E(L) in B intersecting the wt-nbhds Nea . 6

7

The set S can contain a group P˜ > P (ea ); if so, by applying (b) to the elements of ˜1 S\P˜ only, and (c) with Pˇ := P˜ , we obtain an orthogonally invariant neighborhood N which is right-invariant under P˜ and satisﬁes requirement (2) of the Proposition for P˜ rather than P (ea ). For instance, if ea is a tetragonal basis, P˜ may be a cubic holohedry, ˜1 contains also a ‘nearby’ cubic basis and (two) other tetragonal ‘variants’ so that N of ea (see formulas (3.62)–(3.63) and Fig. 3.13). Also such larger neighborhoods may be of interest in the modelling of weak phase transitions. We only consider the unique lattice group deﬁning any given ﬁxed set, see Remark 3.2.

4.2 SYMMETRY-BREAKING STRETCHES FOR SIMPLE LATTICES

111

These corollaries formalize the well known fact that no small deformation of a lattice will increase its symmetry – see for instance Landau et al. (1980) – and specify which symmetry hierarchies are present locally in the conﬁguration spaces B or Q> 3. 4.2 Symmetry-breaking stretches for simple lattices Corollary 4.3 tells us that for any given basis ea , the ﬁxed sets E(L) intersecting a wt-nbhd Nea are those with L ≤ L(ea ); such ﬁxed sets all contain the ﬁxed set E(L(ea )). In order to better detail the (local) structure of the neighborhoods it is useful to study the properties of the various kinds of stretches in Sym > that lower the lattice group of ea , deforming ea ∈ E(L(ea )) into Uea ∈ E(L), L < L(ea ), E(L(ea ))E(L). To be precise: if ea ∈ B and L ≤ L(ea ) is a lattice group, any stretch U ∈ Sym > for which L(Uea ) = L ≤ L(ea ) (4.5) is called an L-symmetry-breaking stretch for ea , or simply a symmetrybreaking stretch when there is no danger of confusion. We indicate by UL (ea ) the set of L-symmetry-breaking stretches for ea . We also consider the symmetry-preserving stretches U ∈ U (ea ) in §3.6.1, for which L(Uea ) = L(ea ), as a special case of the symmetry-breaking stretches. By deﬁnition, any stretch U in UL (ea ) produces a basis Uea whose lattice group is exactly L; thus, for any lattice group L ≤ L(ea ) we have (compare with (3.68)): U ∈ UL (ea ) ⇔ Uea ∈ E(L)∗ ⊂ B ⇔ (Uea · Ueb ) ∈ I(L)∗ ⊂ Q> 3.

(4.6)

The following analogue of Lemma 3.9 holds: Lemma 4.4 Let ea be a basis, and F ∈ Aut have polar decomposition F = RU with U ∈ Sym > and R ∈ O; then: (1) U ∈ UL (ea ) if and only if the lattice group L of the deformed basis Fea is a subgroup of L(ea ): L(Fea ) = L ≤ L(ea ).

(4.7)

(2) If U ∈ UL (ea ), then – see also footnote 15: P (Uea ) ≤ P (ea )

and

P (Fea ) = RP (Uea )Rt ≤ RP (ea )Rt .

(4.8) (4.9)

The converse is false; that is, either one of the following can hold: P (Uea ) ≤ P (ea ) P (Fea ) ≤ P (ea )

with with

L(Uea ) L(ea ), or

(4.10)

L(Fea ) L(ea ).

(4.11)

112

WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

The proof of this Lemma is similar to that of Lemma 3.9, and the counterexamples for (4.10) and (4.11) can be deduced from the ones given for formulae (3.71) and (3.72). Remark 4.1 It is instructive to give another explicit example for (4.11), which also illustrates a peculiar property of some holohedries and lattice groups. Consider the hexagonal basis ea in (3.50), whose sixfold axis is k , and the rhombohedral basis in (3.49), whose threefold axis is also k . Let us denote the latter basis by ea , and by F the linear transformation such that ea = Fea . One can verify that the holohedries of ea and Fea are in a groupsubgroup relation, but the lattice groups are not, as stated in (4.11). Indeed, in this case the subgroup L < L(ea ) constituted by the integral matrices representing the elements of P (Fea ) < P (ea ) in the basis ea is in fact not a lattice group, for no hexagonal lattice group contains a rhombohedral 2π/3 lattice group: the action of the rotation Rk on the hexagonal basis (3.50) is represented by the matrix   2π

0

mk3 =  1 0

−1 −1 0

0

0,

(4.12)

1

which belongs to L as well as L(ea ); and one can check that a metric C satisﬁes the condition mt Cm = C for this m if and only if it has the form in (3.50). Therefore any metric left invariant by the rhombohedral subgroup L is also left invariant by the entire hexagonal lattice group L(ea ), as anticipated in chapter 3, footnote 20. So, L is just a ﬁnite subgroup of GL(3, Z) which is linearly conjugate to the rhombohedral lattice subgroups of GL(3, Z) but is not itself a lattice group; in particular, L is linearly (but not arithmetically) conjugate to the rhombohedral lattice group L(Fea ).8 Remark 4.2 (a) By Lemma 4.4 (2), not all stretches which reduce the holohedries are symmetry-breaking, that is, also reduce the lattice groups. As already noticed in Remark 3.4(a), this shows the diﬀerence between the symmetry properties described by the holohedries and the lattice groups. Many of these diﬀerences disappear if only stretches near 1 are considered; indeed (§4.3.2), for small stretches P (Uea ) ≤ P (ea ) ⇔ L(Uea ) ≤ L(ea ), unlike what is stated in (4.10) for general stretches. (b) Analogously to Lemma 3.9, Lemma 4.4(2) states that any deformation F = RU with U ∈ UL (ea ) reduces the lattice group to L but transforms the holohedry to a subgroup that is orthogonally conjugate, through R, to a subgroup of P (ea ) representing L. Thus the holohedries are in a groupsubgroup relation only when R = 1, and stretches in UL (ea ) are considered – see (4.8)–(4.9). 8

L(Fea ) is contained in the lattice groups of suitable cubic bases with a main diagonal along k . Indeed, only the cubic lattice groups contain rhombohedral lattice groups as proper subgroups. See also Remark 4.4(d) and Fig. 5.1.

4.3 SMALL DEFORMATIONS AND WEAK PHASE TRANSFORMATIONS

113

4.3 Small deformations in simple lattices and weak phase transformations By (4.6), ﬁnding all the symmetry-breaking stretches for a basis ea is equivalent to determining the proper ﬁxed sets I(L)∗ for any lattice group L contained in L(ea ). As already mentioned in Remark 3.3, there is no general procedure that allows one to do so, as the structure of the inclusion relations for the ﬁnite subgroups of GL(3, Z) is not known.9 From now on we will mostly concentrate on the case of stretches close to the identity (which we call for brevity small stretches), and deformations F belonging to a wt-nbhd N1 of 1 in Aut as in Proposition 4.1.10 As anticipated, due to the properties of the wt-nbhd in that Proposition, in this case we will be able to give a complete and explicit description of the small symmetry-breaking stretches, and thus of the local structure of the wt-nbhds.11 Remark 4.3 As already mentioned, the case of small stretches is important in the thermoelastic modelling of the weak phase transformations in crystalline solids, to be considered in detail in chapter 7, and the rest of this chapter and chapter 5 can thus be regarded as a study of the kinematics of weak phase transitions. By deﬁnition, the weak phase changes are characterized by lattice deformations belonging to some wt-nbhd in Aut. Therefore the lattice groups of the weak phases must have a common supergroup which is also a lattice group. So, for instance, a phase transformation from an f.c.c. to a b.c.c. conﬁguration as the one mentioned in §3.6.3 is never weak. This is immediate from Corollary 4.2, which says that no two cubic bases (or metrics) of diﬀerent Bravais type can be in the same wt-nbhd, because their lattice groups are never in a group-subgroup relation. Also the nonweak phase changes, such as the mentioned f.c.c.-to-b.c.c. transformation in ferrous alloys, constitute a very interesting research subject for crystal mechanics. For these phase transformations some work has been done by physicists (see for instance Tol´edano and Dmitriev (1996) and references therein). Complications arise because the structure of the conﬁguration spaces B or Q> 3 in the large is quite complex, and its analysis requires considering all the ﬁnite subgroups of GL(3, Z). For this reason Conti and Zanzotto (2002) propose a model of nonweak transitions for 2-dimensional elastic crystals (nets), whose conﬁguration spaces are indeed easier to analyze. 9 10 11

Indeed, in order to ﬁnd all the symmetry-breaking stretches of the bases in B one must also know all the lattice groups containing any lattice group. Notice that we still are in the range of ﬁnite (albeit not too large) deformations; we are not developing a theory involving inﬁnitesimal strains. We notice that any small stretch U is necessarily symmetry-breaking for ea : due to Corollary 4.2 the relation L(Uea ) ≤ L(ea ) necessarily holds. When we say that we give an explicit description, we mean an explicit description of any set UL (ea ) near 1 for all the lattice groups L < L(ea ). Equivalently, we describe all the ﬁxed sets I(L) near the metric C of ea .

114

WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

Also, and more importantly, irreversible plasticity phenomena, such as the creation and motion of dislocations, play a central role in the nonweak transformations, and a purely elastic approach is likely to prove much less eﬀective than with the weak transitions. Similar remarks hold for more complex phase changes, such as the f.c.c.-to-h.c.p. or the b.c.c.-to-h.c.p. transitions mentioned in the introduction to chapter 7, for which even the most basic kinematical framework is at present incomplete. From now on, given a basis ea with metric C, we indicate by N1 , Nea = N1 ea , and NC , suitable wt-nbhds of 1 in Aut, of ea in B, and of C in Q> 3, respectively, for which the properties indicated in Proposition 4.1 hold. 4.3.1 Small symmetry-preserving stretches Given ea ∈ B, statement (1) of Lemma 3.9 asserts that the stretches U in C(ea ) keep ea in the ﬁxed set E(L(ea )), possibly increasing its lattice group: L(Uea ) ≥ L(ea ). However, if U is close enough to 1, that is, if U ∈ C(ea ) ∩ N1 , then Uea ∈ Nea = N1 ea and, by Corollary 4.2, the deformed basis Uea cannot have a larger lattice group than L(ea ). Therefore, for any small U ∈ C(ea ), L(Uea ) = L(ea ) so that, by deﬁnition, U ∈ U (ea ). The small stretches in C(ea ) and in U (ea ) thus coincide, as was anticipated in Remark 3.4(c): C(ea ) ∩ N1 = U (ea ) ∩ N1 =: U (ea )1 .

(4.13)

Equivalently, by (3.67) and (3.68), either one of the following holds: I(L(C)) ∩ NC = I(L(C))∗ ∩ NC =: I(L(C))C , ∗

or

(4.14)

E(L(ea )) ∩ Nea = E(L(ea )) ∩ Nea = E(L(ea ))ea . (4.15) This means that near C the ﬁxed set I(L(C)) and the proper ﬁxed set I(L(C))∗ coincide, making it feasible to describe the latter at least locally, near C (analogously for E(L(ea ))∗ near the set Oea ). We have set (4.13)2 – (4.15)2 for ease of notation. Formula (4.13)1 suggests an explicit procedure to ﬁnd all the small symmetry-preserving stretches for a basis ea , based on the commutativity properties deﬁning C(ea ) in (3.65). Alternatively, formulas (3.67), (3.68), and (4.14)1 can be used, once the ﬁxed set I(L(C)) containing the metric C of ea is calculated. We give explicitly the small symmetry-preserving stretches in chapter 5. As was also anticipated in Remark 3.4(c), if only stretches in the set (4.13) are considered, the two statements in part (1) of Lemma 3.9 coincide, as do statements (2) and (2 ); the converse of both also holds. For small stretches the diﬀerences between the holohedries and the lattice groups, put in evidence after Lemma 3.9, are thus considerably less relevant than for arbitrary stretches, being reduced to the sole fact that the holohedries, unlike the lattice groups, are not invariant under orthogonal transformations of the basis, but transform to an orthogonal conjugate.

4.3 SMALL DEFORMATIONS AND WEAK PHASE TRANSFORMATIONS

115

4.3.2 Small symmetry-breaking stretches As with the symmetry-preserving stretches, also for the symmetry-breaking ones the diﬀerences between holohedries and lattice groups are greatly reduced if only small stretches are considered, and we will obtain an explicit procedure to calculate them. The stretches in the sets (4.13) maintain both the holohedry and the lattice group of a basis ea . Any other small stretch U in Sym > ∩ N1 but not in U (ea )1 breaks the symmetry of ea by reducing its lattice group and holohedry, thus taking ea to a larger ﬁxed set which contains E(L(ea )) as a proper subset. As with Lemma 3.9, in the case of small stretches there is a great simpliﬁcation of the statements of Lemma 4.4. The simpliﬁcation is based on Corollary 4.2, and to describe it in detail we give some deﬁnitions. If ea ∈ B and L is a lattice group, L ≤ L(ea ), we introduce the subgroup PL (ea ) ≤ P (ea ) collecting all the orthogonal transformations in P (ea ) that are represented in the basis ea by the matrices in L: PL (ea ) := {Q ∈ P (ea ) : Qea = mba eb with m ∈ L ≤ L(ea )} ⊆ P (ea ).

(4.16)

In analogy with the deﬁnition of C(ea ) in (3.65), we denote by CL (ea ) the set of stretches12 U ∈ Sym > commuting with all the elements of PL (ea ): CL (ea ) := {U ∈ Sym > : QU = UQ for all Q ∈ PL (ea )}.

(4.17)

Notice that if L and L are two lattice groups contained in L(ea ) and L ≤ L , we clearly have CL (ea ) ⊆ CL (ea ), (4.18) while in general, by deﬁnition, UL (ea ) UL (ea ).

(4.19)

We now discuss how these sets are related when only small stretches are considered. For ease of notation we set, for L ≤ L(ea ) ≡ L(C), CL (ea )1 := CL (ea ) ∩ N1 , UL (ea )1 := UL (ea ) ∩ N1 , E(L)ea := E(L) ∩ Nea , I(L)C := I(L) ∩ NC ,

(4.20)

and analogously for E(L)∗ and I(L)∗ . Thus E(L)ea denotes the O-invariant portion of the ﬁxed set E(L) ⊂ B near Oea , I(L)C denotes the portion of I(L) near C, etc. Proposition 4.5 Let ea ∈ B have metric C, and let L, L be lattice groups contained in L(ea ) = L(C); furthermore, let N1 ⊂ Aut, Nea ⊂ B, and NC ⊂ Q> 3 be wt-nbhds. 12

As in footnote 22, one can check that U belongs to CL (ea ) if and only if so does C = U 2 or the strain E – see (2.52).

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(1) In the notation of (2.18) we have (compare with (3.78)) E(L) = OCL (ea )ea ,

E(L)∗ = OUL (ea )ea ,

so that UL (ea ) ⊆ CL (ea ). (2) We also have13

CL (ea )1 = L≤L ≤L(ea ) UL (ea )1 = L≤L ≤L(ea ) CL (ea )1 Equivalently: E(L)ea = or also I(L)C =

L≤L ≤L(C)

L≤L ≤L(C)

E(L )ea = I(L )C =

L≤L ≤L(C)

(4.22)

E(L )∗ea ,

(4.23)

I(L )∗C .

(4.24)

L≤L ≤L(C)

(4.21)

Thus the following equivalent statements hold: ∗ E(L)∗ea = E(L)ea L
(4.25) (4.26)

(3) Let F ∈ N1 have polar decomposition F = RU as in (2.20) (this happens if and only if U ∈ Sym > ∩ N1 ), and let L ≤ L(ea ) be a lattice group. Then the following relations L(Fea ) = L(Uea ) = L ≤ L(ea ),

(4.27)

and P (Uea ) = PL (ea ) ≤ P (ea ) ⇔

(4.28)

P (Fea ) = RPL (ea )R ≤ RP (ea )R , t

t

can only hold together and, by deﬁnition, they are true if and only if U ∈ UL (ea )1 – see (4.20) and below (4.5). We leave the proof to the reader. Remark 4.4 (a) Among other things, statement (1) above says that, in general, UL (ea ) ⊆ CL (ea ). This is similar to what happens with the sets U (ea ) and C(ea ) – see (3.66)2 . Statement (2) shows the diﬀerence between UL (ea ) and CL (ea ) for small stretches, or, by (4.21)1 , the diﬀerence between E(L) and E(L)∗ near ea . Unlike the sets U (ea ) and C(ea ), whose intersections with N1 coincide (see (4.13)), for UL (ea ) and CL (ea ) in general UL (ea )1 ⊆ CL (ea )1 ; indeed, (4.22)1 holds, that is, in CL (ea )1 there are also all the stretches belonging to UL (ea )1 for L ≤ L ≤ L(ea ). In other words, the local portion E(L)ea of E(L) near ea contains (and is actually constituted by) all the portions E(L )ea of the ﬁxed sets E(L ) whose symmetry is intermediate between L and L(ea ) – see (4.23) and (4.25). Recall 13

This is not true in the large. In general (4.18) holds, and in addition one has UL (ea ) ⊆ CL (ea ) for all L ≤ L ≤ L(ea ) and ∪L≤L ≤L(ea ) UL (ea ) ⊆ CL (ea ). Only locally does the equality hold, as stated by (4.22)–(4.26).

4.3 SMALL DEFORMATIONS AND WEAK PHASE TRANSFORMATIONS

117

that by (4.21) and statement (3) in Proposition 3.8, the sets CL (ea ) and UL (ea ), for L ≤ L , are lower-dimensional submanifolds in CL (ea ). (b) As a consequence of (4.5) and Corollary 4.2, only small L-symmetrybreaking stretches up to orthogonal transformations (for any L ≤ L(ea )) can be applied to a basis ea in order for it to remain in a wt-nbhd Nea . Proposition 4.5 speciﬁes how such stretches describe, up to orthogonal operations, the structure of the wt-nbhds: the bases in Nea are given by RUea , with R ∈ O and U ∈ UL (ea )1 , for any L ≤ L(ea ). In the notation (2.19), for any basis ea ∈ B with metric C we have, referring to the wt-nbhd N1 of Proposition 4.1:14

N1 = L≤L(ea ) OUL (ea )1 = L≤L(ea ) OCL (ea )1 . (4.30) (c) As a consequence of Lemma 4.4 and Proposition 4.5 one can phrase Corollary 4.2 also in terms of the holohedral point groups of the bases ea and ea there considered; however, as was noticed in Remark 4.2(b), for the holohedries the inclusion relation must be intended up to conjugacy in O: if F = RU ∈ N1 , and ea = Fea ∈ Nea , in general P (Fea ) is not a subgroup of the holohedry P (ea ) of the center, but is only conjugate to a subgroup of P (ea ) through the orthogonal operation R appearing in the polar decomposition of F . By statement (3) of Proposition 4.5 only if we consider the bases ea = Uea in Nea obtained from ea by means of a stretch U ∈ Sym > , are the holohedries necessarily in a group-subgroup relation as are the lattice groups in Corollary 4.2. The stretches U such that Uea is in Nea , which are all the small symmetry-breaking stretches as remarked above, thus reduce together the lattice group and the holohedry of ea (‘in the small’ (4.10) or (4.11) cannot be true). (d) Notice that, given a lattice group L ≤ L(ea ), by (4.27)–(4.28) the subgroup PL (ea ) ≤ P (ea ) deﬁned in (4.16) is always a holohedry.15 On the contrary (and here is another instance of the diﬀerences in the properties of the holohedries and the lattice groups, as was exempliﬁed below Lemma 4.4), if we pick any holohedry P contained in a given holohedry P (ea ), the subgroup of the matrices of L(ea ) that represent the elements of P in the basis ea need not be a lattice group. Thus the holohedries that are 14

Equivalently, the wt-nbhds Nea ⊂ B and NC ⊂ Q> 3 have the following structures: Ne a NC

15

= =

L≤L(ea )

E(L)∗ea =

I(L)∗C L≤L(C)

=

L≤L(ea ) L≤L(C)

E(L)ea ,

(4.29)

I(L)C .

This means that in order to describe NC completely one needs to ﬁnd all the metrics C close to C and belonging to I(L) – that is, such that mt C m = C for all m ∈ L – for some lattice group L ≤ L(C). That is, PL (ea ) is the holohedral point group of some lattice; indeed, PL (ea ) = P (Uea ) for any U ∈ UL (ea ), not only for U ∈ UL (ea )ea . This completes the information given in (4.8).

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WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

contained in P (ea ) are of two kinds, the ones representing lattice groups and the others; in what follows we will refer to a holohedral subgroup PL (ea ) of any holohedry P (ea ) whose corresponding integral matrices m form a lattice subgroup of L(ea ) as to the L-holohedral subgroup of P (ea ).16 4.4 Procedure for constructing the small symmetry-breaking stretches Although the lattice groups in GL(3, Z) are in general better suited than the holohedries in O to study the kinematics and the symmetry properties of deformable crystals, in various cases it is nonetheless useful to consider also the holohedries; for instance in the explicit determination of the small symmetry-breaking stretches of a basis. Indeed, as we stressed at the beginning of §4.3, for any basis ea and any lattice group L ≤ L(ea ), in order to ﬁnd all the L-symmetry breaking stretches U ∈ UL (ea ) one should describe the proper ﬁxed sets I(L)∗ , – see (4.6) – which is in general too hard a problem. However, when only small stretches are considered, the task becomes manageable; for it reduces to ﬁnding the (small) stretches in CL (ea ) for all the lattice groups L ≤ L(ea ). For this reason we ﬁrst summarize the link between the small L-symmetry-breaking stretches for a basis ea and the L-holohedral subgroups PL (ea ) of the holohedry P (ea ), as follows: Proposition 4.6 For any basis ea and any lattice group L ≤ L(ea ), the small L-symmetry-breaking stretches U ∈ UL (ea )1 for ea are the small stretches U in CL (ea )1 , which commute with all the elements of the holohedral subgroup PL (ea ), minus those commuting also with the elements of any holohedral subgroup PL (ea ) that may exist between PL (ea ) and P (ea ). Accordingly, the following procedure will be used in chapter 5 to determine all the sets UL (ea )1 , thereby also obtaining an explicit description of the local structure of the wt-nbhds. Let a basis ea be given. 1. Since L(ea ) is ﬁnite and has known subgroups, we have a ﬁnite procedure to determine all the lattice subgroups L ≤ L(ea ) and their groupsubgroup relations. 2. More conveniently, we can ﬁnd all the holohedral subgroups PL (ea ) ≤ P (ea ) of the holohedry P (ea ), and their group-subgroup relations. 3. For any L ≤ L(ea ) we select the small stretches U ∈ CL (ea )1 , that is, the ones commuting with all the elements of PL (ea ). By (4.21) U ∈ CL (ea ) ⇔ Uea ∈ E(L) ⇔ (Uea · Ueb ) ∈ I(L)

(4.31)

(compare with the analogous description (4.6) for the stretches in UL (ea ) and with (3.67)–(3.68)). Conditions (4.31) or, directly, the commutativity properties in (4.17), can thus be used to determine the stretches 16

The example below Lemma 4.4 shows a case of a holohedry P , contained in a holohedry P (ea ), which is not an L-holohedral subgroup of P (ea ).

4.4 CONSTRUCTING THE SMALL SYMMETRY-BREAKING STRETCHES

119

in CL (ea ). Then Proposition 4.6 guarantees that, for small stretches, this procedure does not take us too far from the original goal of ﬁnding UL (ea )1 : the small stretches in CL (ea ) coincide with the small stretches in UL (ea ) except for the presence of the small stretches in UL (ea ) or, actually, in CL (ea ), if any lattice groups L exist between L and L(ea ). Therefore we get UL (ea )1 by eliminating those among the stretches in CL (ea ) that also commute with PL (ea ) for some L < L ≤ L(ea ). Remark 4.5 The procedure above, which is trivial if the basis ea has triclinic or monoclinic symmetry, is otherwise best applied by ﬁrst determining the smallest nontrivial (that is, the monoclinic) lattice subgroups L of L(ea ) and the corresponding sets I(L ) or CL (ea ). Then, for any monoclinic subgroup L, UL (ea ) is obtained by eliminating from CL (ea ) all the intersections with the monoclinic CL (ea ), L $= L. The same construction can be repeated for the intersections themselves, etc., thus producing a complete local description of the wt-nbhds in B or Q> 3 . In spite of this being the best way of constructing the local ﬁxed sets, in chapter 5 we prefer to present them ordered by progressively lowering the symmetry from the one of the center, the latter being of maximal symmetry, that is, cubic or hexagonal. Visually, we start from the top and proceed downwards in the trees of holohedries in Figs. 5.1 and 5.2. Remark 4.6 The description of the wt-nbhds in terms of small symmetrybreaking stretches allows for a more compact classiﬁcation than the one based on the ﬁxed sets. To exemplify, consider three cubic lattice bases ear , r = 1, 2, 3, of the primitive, face-centered and body-centered types, respectively, given by (3.52), (3.53) and (3.55) in the same orthonormal basis. These three lattices share the same cubic holohedry P (ear ) = Cijk given in Table 5.1 below,17 while the lattice groups are diﬀerent and not arithmetically equivalent, and thus inequivalent are the related ﬁxed sets. On the other hand, in all three cases the tree of the lattice subgroups is represented by the common tree of the holohedral subgroups of Cijk , shown in Fig. 5.1. Also, for any triple of lattice groups Lr representing the same holohedral subgroup, say P , of Cijk , the distinct ﬁxed sets are described by identical sets of symmetry-breaking stretches U = ULr (ear ), r = 1, 2, 3, which are constructed from the sets of stretches commuting with the holohedral subgroups of Cijk as indicated in the scheme above, once for all the three centerings (see also footnote 19). Still, the diﬀerence of lattice groups and ﬁxed sets is not lost: as is shown in Fig. 5.1, the centering of the basis ear dictates the centering of all the deformed bases in its wt-nbhd. Remark 4.7 In chapter 5 the above scheme is used to obtain the explicit form of the small symmetry-breaking stretches in the sets CL (ea ) in case 17

About the notation, see Remark 5.2: since the central inversion is irrelevant for all matters concerning simple lattices, we denote each holohedry by the same symbol used for its subgroup of rotations.

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WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

ea is a suitable cubic or hexagonal basis, for all the lattice (sub)groups L ≤ L(ea ). As we have seen in §4.3.2, this amounts to giving the elements of Sym> ∩ N1 commuting with all the holohedries PL (ea ) ≤ P (ea ). In fact, these holohedries belong to all the crystal systems in B as L varies among all the lattice subgroups of a cubic and a hexagonal lattice group. Therefore the results in chapter 5 also give the sets of (small) symmetrypreserving stretches for suitable representative lattice bases in all the seven ¯ ∈ CL (ea ) systems. Explicitly, if ea is a cubic or a hexagonal basis and U ¯ ¯ ea := U ea , then P (¯ ea ) = P (U ea ) = PL (ea ) is a for L < L(ea ), and if ¯ holohedry in one of the less symmetric systems, and C(¯ ea ) = {V ∈ Sym > : QV = VQ, Q ∈ P (¯ ea )} = CL (ea ).

(4.32)

4.5 Variant structures (local orbits) in the wt-nbhds 4.5.1 An example When a small L-symmetry-breaking stretch U ∈ UL (ea )1 is applied to a basis ea ∈ B, its lattice group and its holohedry are lowered to subgroups L(Uea ) = L ≤ L(ea ) and P (Uea ) = PL (ea ) ≤ P (ea ). As an example, take ea to be a (primitive cubic) orthonormal basis (i , j , k ). The holohedry P (ea ) is the cubic group Cijk given in Table 5.1 below, and L(ea ) is the associated group of integral matrices representing the operations of P (ea ) in the basis ea ; these can be easily calculated from the deﬁnition (3.25). One of the symmetry-breaking stretches for ea is the tensor, U1 say, whose matrix representation in the basis ea is diag(α, α, δ), with α and δ $= α positive numbers – see (3.62)–(3.63), or Table 5.2 below. If U1 is close to the identity, that is, if |α − 1| and |δ − 1| are small enough, the deformed basis U1 ea belongs to a wt-nbhd Nea ⊂ B of ea as in Proposition 4.1. The action of U1 on ea maintains all the ea mutually orthogonal while elongating (or shrinking) anisotropically the vector e3 = k , thereby producing a primitive tetragonal basis U1 ea whose symmetry is described by the tetragonal holohedry P (U1 ea ) = Tk < Cijk shown in Table 5.2 below. The associated lattice group is L = L(U1 ea ) < L(ea ), and U1 is an L-symmetry-breaking stretch for ea . The deformed basis U1 ea , while remaining in Nea , is now in the primitive tetragonal ﬁxed set E(L), which contains the cubic E(L(ea )) as a subset. The cubic point group P (ea ), which by Proposition 4.1 dictates the symmetry in Nea , forces the presence in Nea of other primitive tetragonal bases that are symmetry-related to U1 ea . Indeed, all the tensors Rt U1 R, R ∈ P (ea ) are symmetry-breaking for ea , producing primitive tetragonal bases in Nea which belong to the same GL(3, Z)-orbit as U1 ea (see (3.35)). In general not all the stretches Rt U1 R are distinct; in this case only three are: the tensors U1 , U2 , U3 represented in the basis ea by the matrices diag(α, α, δ), diag(δ, α, α), diag(α, δ, α),

(4.33)

4.5 VARIANT STRUCTURES (LOCAL ORBITS) IN THE WT-NBHDS

121

B

Q> 3

Figure 4.1 GL(3, Z)-orbits in the spaces B and Q> 3 . A wt-nbhd of a cubic basis (and metric) and the included tetragonal variants are indicated schematically

respectively – compare with Table 5.2. Notice how U2 and U3 deform the primitive cubic basis ea : they both maintain the lattice vectors mutually orthogonal, while elongating anisotropically e1 and e2 , respectively (compare with the action of U1 ). So, the bases U1 ea , U2 ea and U3 ea in Nea , whose holohedries are all distinct tetragonal subgroups of P (ea ) having fourfold symmetry axes along k , i and j , respectively, belong to three distinct primitive tetragonal ﬁxed sets E(L(Ui ea )), i = 1, 2, 3, respectively; these all have nonempty intersection with Nea , as well as with the primitive cubic ﬁxed set E(L(ea )) of the center basis ea – see also Fig. 3.13. Due to its orthogonal invariance, Nea contains the cubic basis ea and the primitive tetragonal bases Ui ea , i = 1, 2, 3 above, together with their entire O-orbits – see (2.18): Oea

and OUi ea ;

(4.34)

this is schematically represented in Fig. 4.1. The orbits OUi ea are called the tetragonal variants of the orbit OU1 ea in the cubic wt-nbhd Nea ; such variants collect all the bases in Nea that are symmetry-related to U1 ea (see (3.35)). The set of O-orbits OU1 ea ,

OU2 ea ,

OU3 ea ,

(4.35)

is called a variant structure in Nea . We can also describe the corresponding variants in the space Q> 3 . Let C be the metric of the center basis ea (in this particular case we have C = 1 because ea is an orthonormal basis), and let C1 = diag(α2 , α2 , δ 2 ) be the metric of the deformed basis U1 ea considered above. The wt-nbhd

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Nea ⊂ B corresponds through (4.2) to a wt-nbhd NC ⊂ Q> 3 , and the orbit Oea corresponds to the center metric C = 1. The variants OUi ea ⊂ Nea , i = 1, 2, 3, in turn correspond to the following three variants in NC :18 C1 = diag(α2, α2, δ 2 ), C2 = diag(δ 2, α2, α2 ), C3 = diag(α2, δ 2, α2 ).

(4.36)

As the variants OUi ea ⊂ Nea , which collect all the bases in Nea that are symmetry-related to U1 ea , the variants Ci , i = 1, 2, 3 are the only metrics in NC that are symmetry-related to C1 , for they are the only distinct metrics among those given by mt C1 m

for m ∈ L(C).

(4.37)

The metrics Ci indeed constitute the L(C)-orbit of C1 (local orbit, in NC ), which is the intersection of the GL(3, Z)-orbit of C3 (global orbit) with the wt-nbhd NC – see Fig. 4.1.19 4.5.2 General deﬁnitions As the foregoing example shows, by applying a symmetry-breaking stretch to a basis ea one gets several variants of less symmetric, symmetry-related bases in a wt-nbhd Nea . In that case the symmetry breaking is cubic-totetragonal but there are other kinds of weak symmetry changes. We have seen above that the orbits of the bases (or metrics) generated by the local symmetry groups are one of the most important aspects that must be taken into account for the description of the weak symmetry changes. In the rest of this chapter we study some main properties of the local orbits, these being described in detail in chapter 5. Given any C ∈ Q> 3 and a wt-nbhd NC as in Proposition 4.1, let us consider any element C1 ∈ NC and all the metrics in Q> 3 that are symmetryrelated to C1 , that is, those belonging to the GL(3, Z)-orbit of C1 . The properties of the wt-nbhd NC imply that among all such metrics only those belonging to the L(C)-orbit of C1 (local orbit of C1 near C) are contained 18

19

Notice that the matrices in (4.36) are the squares of those in (4.33), as indicated by Proposition 3.10 (2) and formula (3.80) with E = (eia ) = 1, because the center of NC is C = 1 (that is, the basis ea is orthonormal). In general the metrics have the more complex expressions in (3.80). In this case the variants Ci in (4.36) belong to three distinct primitive tetragonal ﬁxed sets I(L(Ci )), respectively, which have a nonempty intersection with NC and all meet at the primitive cubic ﬁxed set I(L(C)) = I(L(1)) = {diag (α, α, α), α > 0} to which the center C = 1 belongs (see Fig. 3.13 and §5.1.1). These ﬁxed sets coincide with those described by (3.62)–(3.63) in the example of §3.5.1. However, notice that the same stretches U1 , U2 , U3 as in (4.33) would produce, if applied to a diﬀerent cubic basis, for instance to an f.c.c. basis as in (3.53), three centered tetragonal ﬁxed sets; the matrices in them do not have expressions like those in (4.36), but can be obtained through (3.80) with the correct matrix E. This shows why, based on (4.29), it is often advantageous to work with stretches and holohedries, as discussed in Remark 4.6, rather than directly with the metrics and the lattice groups. Working with stretches, in fact, gives a uniﬁed way of describing wt-nbhds in B or Q> 3 that are linearly but not arithmetically equivalent. We will take full advantage of this fact in chapter 5.

4.5 VARIANT STRUCTURES (LOCAL ORBITS) IN THE WT-NBHDS

123

in NC . These elements are called the C-variants of C1 , given by: {mt C1 m : m ∈ L(C)} =: {C1 , . . . , CN }.

(4.38)

A set of variants as in (4.38) is called a C-variant structure. Due to the properties of orbits of group actions, given any elements C¯ and Cˆ in NC their variant structures either are disjoint or coincide. Also, the variants in a structure (4.38) are all of the same Bravais type; their number N depends on both C and C1 (see Proposition 4.7 below). Usually it is clear what the center C of a wt-nbhd is, and there is no danger of confusion when just speaking about the variants or the variant structures in the wt-nbhds. Let ea be any basis with metric C, and U1 be a small symmetry-breaking stretch for ea , so that the metric C1 of U1 ea is in NC . By analogy with (4.38) the variants of U1 are the stretches {Q t U1 Q : Q ∈ P (ea )} =: {U1 , . . . , UN }.

(4.39)

It is immediate to see that in (4.39) N is the same as in (4.38). The O-orbits OU1 ea , . . . , OUN ea

(4.40)

of the bases Ur ea that are obtained from the center ea by means of the symmetry-breaking stretch variants in (4.39), are called the variants of OU1 ea . They are all contained, together with the center orbit Oea , in the orthogonally invariant wt-nbhd Nea corresponding to NC through (4.1)– (4.2). Also in this case a set of variants is called a variant structure. Notice that, due to (2.15), each variant orbit OUr ea ⊂ Nea corresponds to one and only one of the variant metrics Cr ∈ NC listed in (4.38) (see Fig. 4.1) and the bases belonging to the orbits in (4.40) are all the bases in Nea that are symmetry-related to U1 ea – recall (3.35). Also the variants – see (2.19) OU1 , . . . , OUN

(4.41)

in N1 ⊂ Aut can be considered; they correspond to the orbits (4.40) in Nea , while the orbit O1 = O ⊂ N1 of the identity corresponds to the center orbit Oea ⊂ Nea . 4.5.3 Variants and cosets Let C, C1 and NC be as in §4.5.2. To each variant Cr of C1 in (4.38) is associated in a natural way the lattice (sub)group L(Cr ) ≤ L(C) that stabilizes Cr under the action (3.8). By deﬁnition, Cr $= C1 and there is at least an element mr ∈ L(C) such that Cr = mtr C1 mr ,

mr ∈ L(C);

(4.42)

however, remark that the variant Cr can be obtained from C1 also by means of any element of the left coset L(C1 )mr of L(C1 ) in L(C): Cr = m ¯ t C1 m ¯ for all m ¯ ∈ L(C1 )mr . Thus the variants of a given C1 are in a

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WEAK-TRANSFORMATION NEIGHBORHOODS AND VARIANTS

one-to-one correspondence with the left cosets of the lattice (sub)group L(C1 ) in L(C).20 Analogously, each variant OUr ea ⊂ Nea has the naturally associated holohedry P (Ur ea ) = PL(Cr ) (ea ) ≤ P (ea ), and each variant Ur = Qrt U1 Qr ,

Qr ∈ P (ea ),

(4.44)

can be obtained by conjugacy through any element Q in the left coset P (U1 ea )Qr of P (U1 ea ) in P (ea ). The following is a consequence of Lagrange’s theorem (§2.2). Proposition 4.7 Let ea be a basis with metric C, and let U1 be an Lsymmetry-breaking stretch for ea . Then L(U1 ea ) = L ≤ L(ea ) and P (U1 ea ) = PL (ea ) ≤ P (ea ); furthermore, if C1 is the metric of U1 ea , the number N of distinct variants of C1 (or of U1 , or of the orbit OU1 ea ) deﬁned in (4.38)–(4.40) is given by: N = #L(C)/#L(C1 ) = #P (ea )/#P (U1 ea ).

(4.45)

4.5.4 Variant structures and conjugacy classes Let C, C1 and NC be as in §4.5.2. By (3.61) and (4.42) the lattice groups L(Cr ) and L(C1 ) of two variants Cr and C1 are conjugate in L(C): L(Cr ) = m−1 r L(C1 )mr for some mr ∈ L(C). It is immediate to check that L(C1 ) is conjugate to L(Cr ) also (but in general not only) through any elements of the coset L(C1 )mr . Analogously for the holohedries P (U1 ea ),..., P (UN ea ), which form a conjugacy class of holohedral subgroups of P (ea ). The conjugacy action of L(C) on its lattice subgroups gives a natural criterion for distinguishing the variant structures that are essentially diﬀerent in a wt-nbhd NC : there are in NC as many distinct variant structures as there are distinct conjugacy classes of lattice subgroups in L(C). Analogously with the variant structures in the wt-nbhds Nea ⊂ B and the conjugacy classes of holohedral subgroups in P (ea ). Each distinct variant structure corresponds to an essentially distinct way of breaking the symmetry of the center basis: in each structure only certain arrangements of symmetry axes and planes remain after the center’s symmetry is reduced (see Figs. 5.1 and 5.2). Therefore not only is it important to know all the lattice (sub)groups of the lattice groups (or, equivalently, the holohedral subgroups of the holohedries), as we have seen in §4.4; it is also necessary to determine all their conjugacy classes. For the maximal holohedries or lattice groups this is discussed in chapter 5. It is important to notice that the lattice groups L(Cr ), r = 1, . . . , N , in some cases may not be all distinct subgroups of L(C). Indeed, in general L(C1 ) may have a nontrivial normalizer in L(C): there may be elements 20

We recall that a group coincides with the union of all the cosets of any of its subgroups: in this case we have L(C) = ∪r L(C1 )mr . (4.43) This gives a way of calculating N in (4.38) or (4.40) – see Proposition 4.7.

4.5 VARIANT STRUCTURES (LOCAL ORBITS) IN THE WT-NBHDS

125

of L(C)\L(C1 ) that leave the subgroup L(C1 ) invariant under conjugacy – see §2.2.1. When this happens, in the conjugacy class {m−1 L(C1 )m, m ∈ L(C)} of L(C1 ) some subgroups coincide; this means that the ﬁxed set I(L(C1 )) is stabilized as a whole by elements of L(C) not in L(C1 ).21 The number N ≤ N of distinct subgroups among L(C1 ), . . . , L(CN ), which is also the number of distinct ﬁxed sets I(L(Cr )) whose intersection with NC is nonempty, is always a divisor of the number N of variants, due to Proposition 2.1. When N < N , the N variants in a structure are distributed among the N ﬁxed sets, so that more than one variant belongs to each ﬁxed set. For instance it can happen, as in §5.1.3.1 below, that L(C1 ) is a normal subgroup of L(C): in this case all the conjugates of L(C1 ) coincide with L(C1 ), whose conjugacy class is constituted only by L(C1 ) itself. Then the distinct variants C1 , . . . , CN all share the same lattice group L(C1 ), and they belong to the same ﬁxed set I(L(C1 )) intersecting NC . These variants correspond to lattices sharing the same symmetry axes and planes, as they all have the same lattice group and holohedry. Also among the variants Ur = Qrt U1 Qr of a stretch U1 , those for which the bases Ur ea and U1 ea have holohedries that are not distinct, and which belong to the same ﬁxed set intersecting Nea , are given by operations Qr in the normalizer of P (U1 ea ) in P (ea ) if such normalizer is nontrivial. The numbers N and N can be obtained by applying statement (3) of Proposition 2.1 to L(C1 ) and its normalizer in L(C) – or to P (U1 ea ) and its normalizer in P (ea ). However, in chapter 5 we will obtain N and N for each variant structure in the wt-nbhds from the direct analysis of the variants and the ﬁxed sets.

21

By deﬁnition, the elements of L(C1 ) stabilize I(L(C1 )) pointwise – see §2.2.2.

CHAPTER 5

Variant structures and kinematics of symmetry breaking Based on Zanzotto (1996b),1 in this chapter we make explicit the local structure of the wt-nbhds and of the variant structures in them, and we give all the (small) symmetry-breaking stretches that are possible in the conﬁguration space B. To do so, we follow the procedure outlined in §4.4. We give all the information in the case of wt-nbhds of bases ea of maximal symmetry, that is, whose holohedry P (ea ) is in the cubic or in the hexagonal system and, as anticipated, we proceed by progressively breaking the symmetry of the center. This information can be used also to cover the cases in which the symmetry of ea is nonmaximal, although the matter is not completely trivial (see Zanzotto (1996b) for some details). For the purposes of chapters 8 and 9, here we also indicate explicitly the left cosets of a suitable symmetry group for each variant structure, as discussed in §4.5.3. Recall that in each variant structure there is the permanence of certain speciﬁc sets of symmetry axes or planes in the lattice after the symmetry of the center ea has been reduced. The nomenclature and notation we use for the groups, the conjugacy classes and the variant structures are meant to help recall such speciﬁc symmetry properties. Much of the information regarding the phase variants, their symmetry groups and centerings, and the symmetry-breaking stretches is summarized in Tables 5.1–5.14 and in Figs. 5.1 and 5.2. The following remarks simplify the treatment of the next sections, and should be borne in mind as well as the procedure described explicitly in §4.4. The last section sets itself apart; it deals with certain geometric consequences of the structure of the wt-nbhds described here, to be used in the analysis of bifurcations in chapter 7. Remark 5.1 In what follows any stretch U is given by means of its representative matrix in an orthonormal basis (i , j , k ) speciﬁed in each case. In order to obtain the expression of the metric (in the ﬁxed set) corresponding to each symmetry-breaking stretch, formula (3.80) must be used. Remark 5.2 We have seen in chapter 3 that all the point groups of simple lattices contain the central inversion −1 and that, correspondingly, all 1

Part of the following information is also given by Hane and Shield (1998), (1999a,b), (2000a,b), Hane (1999), James and Hane (2000). 127

128

EXPLICIT VARIANT STRUCTURES

lattice groups contain −1. Since these commute with any tensor or matrix, the presence of −1 or −1 does not change the ﬁxed sets, nor the symmetrybreaking stretches or the variants. We thus restrict our discussion to the elements with positive determinant of any symmetry group that we will consider later on. We do so without changing nomenclature or symbols. Remark 5.3 When we indicate a certain form for the typical symmetrybreaking stretch U in a variant structure, we always understand a number of conditions to hold for the stretch components: 1. They must satisfy inequalities guaranteeing that U be positive-deﬁnite and also belong to a wt-nbhd, so it must be suﬃciently near 1. 2. All the components with diﬀerent names can take diﬀerent values; moreover, the given expression of the stretch is intended to hold generically, in the sense speciﬁed in Remark 3.4(b): we do not indicate explicitly that for speciﬁc values of these components the symmetry can actually increase, as discussed in Proposition 4.5. Indeed, according to the procedure introduced in §4.4, in a wt-nbhd of the basis ea , for any lattice group L ≤ L(ea ) the small stretches in UL (ea )1 are obtained by taking all those in CL (ea )1 , except the ones that also commute with PL (ea ) for some L , L < L ≤ L(ea ). In practice, one looks at the trees of holohedries in Tables 5.1 or 5.2. For any chosen holohedry P in one of the trees (P being one of the PL (ea ) for ea of maximal symmetry at the vertex of the tree), the form of the stretches in CL (ea )1 is found in the tables below. Among these, one excludes the stretches given for the holohedries that are above P in the tree and directly connected to it (so, these are the smallest holohedries strictly containing P ). Remark 5.4 Recall that, based on Remark 4.7, these Tables can also be interpreted as giving the explicit form of the (small) symmetry-preserving stretches for suitable bases in all of the seven crystal systems. 5.1 Variant structures in cubic wt-nbhds Let ea be any cubic basis in B, and let the orthonormal vectors i , j , k be chosen along the fourfold axes of symmetry of L(ea ). Then the holohedry P (ea ) coincides with the cubic group Cijk – see Remark 5.2 – given in Table 5.1 below (recall that Rvω indicates the rotation of an angle ω about the direction of v ). The basis ea can be either of the face-centered cubic (f.c.c.), body centered cubic (b.c.c.), or primitive cubic type. Only for the last type can the basis ea coincide with the orthonormal basis (i , j , k ). In any case, in L(ea ) there is a (possibly nonunit) cubic cell. The lattice groups related to the bases (3.53) and (3.56) have been listed in Table 3.3, while the one for the primitive type related to the basis (3.52) can be easily calculated: this group is the 3-dimensional analogue of the 2dimensional square group in (3.89); it consists of the unimodular matrices

5.1 VARIANT STRUCTURES IN CUBIC WT-NBHDS

129

PF I

PI I R RR

CI F

PF I

PCC C C C

PP P

Figure 5.1 The tree of holohedral subgroups PL (ea ) when ea is a cubic basis. Vertical and oblique lines indicate inclusion, while horizontal lines indicate conjugacy within Cijk . Corresponding to the centering of the cubic basis ea indicated under Cijk , and in the same position, one ﬁnds, under the name of each holohedral subgroup in the table, the centering of the bases which belong to the wt-nbhd Nea and have that holohedry as symmetry group. For simplicity only the inclusions of the monoclinic holohedries in two of the rhombohedral holohedries are shown, the others can be easily reconstructed

which have only one nonzero element in each row and in each column, the nonzero elements taking the values ±1 in all possible ways. By Propositions 3.10 or 4.5, the symmetry-preserving stretches for the basis ea , regardless of its centering, are those commuting with Cijk , that is, the dilations of the cubic lattice L(ea ), as in Table 5.1: U = diag(α, α, α),

α > 0.

(5.1)

This implies that the cubic ﬁxed sets are all 1-dimensional submanifolds of the 6-dimensional space Q> 3. We now need to discuss the holohedral subgroups of Cijk . In order to do so, it is useful to familiarize oneself with the list of its elements in Table 5.1, taking notice of the arrangement and period of the symmetry axes in this holohedry: this counts three fourfold (and thus also twofold) axes along the cubic edges, four threefold axes along the main cubic diagonals,

130

EXPLICIT VARIANT STRUCTURES

Table 5.1 Cubic holohedry matrix of transformation stretch

α 0 0

0 α 0

variant

U

generating coset from variant U point group

0 0 α

Cijk Cijk =

π π π 2 {1, Rπ i , Rj , Rk , Ri

3π

π

3π

π

3π

, Ri 2 , Rj2 , Rj 2 , Rk2 , Rk 2 , 2π

4π

π π π π π 3 3 Rπ i−j , Ri+j , Ri−k , Ri+k , Rj−k , Rj+k , Ri+j+k , Ri+j+k , 2π

4π

2π

4π

2π

4π

3 3 3 3 3 3 Ri+ , Ri+ , Ri−j , Ri−j , Ri−j−k , Ri−j−k } +k +k j−k j−k

and six further twofold axes along the cubic face-diagonals (abstractly the rotational group Cijk coincides with the group S4 of the permutations of four objects, in this case the four main diagonals of a cube). As we can read from Fig. 3.2, taking also Remark 5.2 into account, the possible orders of the holohedral rotational subgroups are 2, 4, 6 and 8; 12 is excluded because Cijk does not contain any sixfold element. In the next subsections we describe each family of conjugate holohedral subgroups of P (ea ) = Cijk , to each of which is associated a set of variants in the wt-nbhd Nea according to the discussion in §4.5. Fig. 5.1 gives a graphic representation of the relevant inclusion and conjugacy relations, the latter represented by horizontal lines, for Cijk . This ﬁgure, together with Fig. 5.2 for the hexagonal case, gives the details missing from the classical diagram in Fig. 3.2, where the same inclusions are given only up to isomorphism (or orthogonal conjugacy), which is too little information for the purposes of an elastic model in which the variants (not taken into account in Fig. 5.1) play an essential role – see chapters 8–10. 5.1.1 Tetragonal conjugacy class and variant structure In P (ea ) = Cijk there is one conjugacy class consisting of the three tetragonal holohedral subgroups Ti , Tj , Tk , listed in Table 5.2. These are all dihedral subgroups of order 8 of Cijk , characterized by the presence of a unique fourfold axis (indicated as an index in the group’s symbol) along one edge of the cubic cell of the reference lattice P (ea ), together with two twofold axes along the other two edges and two further twofold axes along the facediagonals orthogonal to the fourfold axis. Such are the symmetry axes of each tetragonal variant. There are no other subgroups of order 8 in Cijk : this can be seen by means of a direct computation or as a consequence of Sylov’s theorem – see for instance Mac Lane and Birkhoﬀ (1967). From (4.39), the three tetragonal variants corresponding to this con-

5.1 VARIANT STRUCTURES IN CUBIC WT-NBHDS

131

Table 5.2 Tetragonal holohedries matrix of transformation stretch variant

α 0 0

0 α 0

0 0 δ

U1

δ 0 0

0 α 0

0 0 α

α 0 0

U2 4π

Tk

point group

Tk = {1, Rπ i , π π Rπ j , Rk , Ri−j , π

3π

2 2 } Rπ i+j , Rk , Rk

0 0 α

U3 2π

3 Tk,2 = {Ri+ , j+k

3 Tk,3 = {Ri+ , j+k

Ti = {1, Rπ i , π π Rπ j , Rk , Rj−k ,

Tj = {1, Rπ i , π π Rπ j , Rk , Ri−k ,

4π 4π 3 3 Ri+ , Ri−j−k , j−k 4π π 3 Ri−j , R , i−k +k π 3π Rj2 , Rj 2 , Rπ i+k }

generating coset from variant U1

0 δ 0

π

3π

2 2 } Rπ j+k , Ri , Ri

2π 2π 3 3 Ri−j , Ri+ , +k j−k 2π 3 Ri−j−k , Rπ , j−k 3π π 2 Ri 2 , R π , j+k Ri }

π

3π

2 2 } Rπ i+k , Rj , Rj

jugacy class of subgroups of Cijk by Proposition 4.7, are given by the symmetry-breaking stretches in Table 5.2. In this table we also give the cosets Tk ,2 and Tk ,3 which collect the rotations relating the variants U2 and U3 , respectively, to U1 according to (4.44). Recall that the analogue of (4.43) holds: Cijk = Tk ∪ Tk ,2 ∪ Tk ,3 , and that the number of cosets equals the number of variants. The symmetry-breaking stretches Ur maintain the three fourfold cubic axes mutually orthogonal, and each of them elongates anisotropically one axis. The three primitive tetragonal variants so obtained belong each to a distinct 2-dimensional ﬁxed set I(L(Ur ea )) in Q> 3 , because the three holohedries P (Ur ea ), r = 1, 2, 3, and the corresponding lattice groups, are all distinct. These three ﬁxed sets intersect at the cubic ﬁxed set I(L(ea )) (see §4.5.1). One can check that these variants always maintain the centering of the cubic parent basis, in the following sense: they are primitive tetragonal if ea is primitive cubic, while they are of the centered tetragonal type if ea is either face-centered or body-centered cubic. 5.1.2 Rhombohedral conjugacy class and variant structure There is in Cijk one rhombohedral conjugacy class constituted by the four subgroups in Table 5.3. Each subgroup is a dihedral group of order 6; it contains one of the four threefold rotations in Cijk about the main cubic diagonals, and three twofold rotations whose axes are along certain facediagonals of the cubic reference cell. Thus each rhombohedral variant has a triplet of coplanar twofold axes spaced π/3 apart and orthogonal to a threefold axis which is indicated as an index in the group’s symbol. These subgroups are all conjugate – see Table 5.3; abstractly, they correspond to the subgroups of permutations in S4 with one ﬁxed point.

132

EXPLICIT VARIANT STRUCTURES

Table 5.3 Rhombohedral holohedries matrix of transformation stretch

α β β

β α β

α −β −β −β α β −β β α

β β α

variant

U1

U2

generating coset from variant U1

Ri+j+k

2π

π

3π

π 2 2 } Rπ i , Rj+k , Rj , Rk 2π

point group

4π

3 3 Ri+j+k,2 = {Ri+ , Ri−j , j−k +k

2π

3 Ri+j+k = {1, Ri+ , j+k

4π π π 3 Ri+ , Rπ i−j , Rj−k , Ri−k } j+k

3 Ri−j−k = {1, Ri−j−k ,

4π π π 3 , Rπ Ri−j−k i+j , Rj−k , Ri+k }

matrix of transformation stretch

α −β β −β α −β β −β α

α β −β β α −β −β −β α

variant

U3

U4

2π

generating coset from variant U1

2π

4π

3π

π

2 , Rπ , , R 2 } Rπ j , Ri i+k k

π 2 Rπ k , Ri

2π

point group

4π

3 3 3 3 Ri+j+k,3 = {Ri−j−k , Ri+ , Ri+j+k,4 = {Ri−j+k , Ri−j−k , j−k

3 Ri−j+k = {1, Ri−j , +k

4π π π 3 Ri−j , Rπ i+j , Rj+k , Ri−k } +k

3π , Rj 2

, Rπ i+j }

2π

3 Ri+j−k = {1, Ri+ , j−k 4π

π π 3 Ri+ , Rπ i−j , Rj+k , Ri+k } j−k

Since any other subgroup of order 6 in Cijk must contain one of the threefold operations, such a subgroup coincides necessarily with one of the groups given in Table 5.3, so no other rhombohedral holohedries exist in Cijk . Corresponding to this conjugacy class there exists a rhombohedral variant structure in Nea which, by Proposition 4.7, is constituted by four variants; their symmetry-breaking stretches and holohedries are shown in Table 5.3. Each one of the stretches U1 , . . . , U4 gives one of the four rhombohedral variants by elongating the cubic parent lattice along one of the four main diagonals in the cubic cell. The corresponding metrics belong to four distinct 2-dimensional ﬁxed sets in Q> 3 . The centering is (primitive) rhombohedral regardless of the Bravais type of the parent cubic lattice, because this is the only Bravais type in the rhombohedral system. 5.1.3 Orthorhombic conjugacy classes and variant structures There are two distinct conjugacy classes of orthorhombic subgroups (dihedral subgroups of order 4) in Cijk and, correspondingly, two distinct orthorhombic variant structures in the cubic wt-nbhd Nea . By Proposition 4.7, they are constituted by six variants each. There are no other

5.1 VARIANT STRUCTURES IN CUBIC WT-NBHDS

133

Table 5.4 Orthorhombic ‘cubic edges’ holohedries matrix of transformation stretch

α 0 0 0 γ 0 0 0 δ

variant

U1

generating coset from variant U1

Oijk

γ 0 0

0 δ 0

0 0 α

U2

4π

2π

2π

4π

4π

2π

4π

3 3 , Ri−j } Ri−j−k +k

3 3 , Ri−j−k } Ri+ j−k

π π Oijk = {1, Rπ i , R j , Rk }

γ 0 0

variant

point group

0 0 γ

3 3 3 3 , Ri+ , Oijk,2 = {Ri+ , Ri−j , Oijk,3 = {Ri+ j+k j−k j+k +k

point group

generating coset from variant U1

0 α 0 U3

2π

matrix of transformation stretch

δ 0 0

0 α 0

0 0 δ

δ 0 0

U4

π , Rk2

0 0 α

α 0 0

U5

Oijk,4 = {Rπ i+j , 3π Rk 2

0 γ 0

, Rπ i−j }

3π 2 , Rπ i−k , Rj

0 0 γ

U6 Oijk,6 = {Rπ j+k ,

Oijk,5 = {Rπ i+k , π Rj2

0 δ 0

}

3π

π

2 ,R2 } Rπ j−k , Ri i

π π Oijk = {1, Rπ i , Rj , Rk }

conjugacy classes of orthorhombic subgroups in Cijk : the point groups in Tables 5.4 and 5.5 exhaust all the possibilities of combining three twofold elements of Cijk so as to obtain a dihedral group of order 4. Geometrically this corresponds to all the possible choices of three mutually orthogonal twofold axes, which characterize the orthorhombic symmetry, among the nine twofold axes of the cubic reference lattice L(ea ). 5.1.3.1 Orthorhombic ‘cubic edges’ variants The ﬁrst orthorhombic conjugacy class in Cijk consists in the normal orthorhombic subgroup Oijk in Table 5.4. Its three perpendicular twofold axes are along the three edges of the cubic cell in the lattice L(ea ), whence the title name for this variant structure. In this structure there are six variants; their symmetry-breaking stretches are given in Table 5.4. Notice that the stretches Ui generate the six variants by stretching anisotropically the original cubic lattice along the directions of the three perpendicular fourfold cubic axes, while maintaining them orthogonal to each other. Since Oijk is normal in Cijk , the metrics of the six variants all belong to the same 3-dimensional ﬁxed set in Q> 3 and all have the same three twofold symmetry axes i , j , k (which are fourfold axes in the cubic lattice). It is immediate to check that the centering of these orthorhombic variants is always the same as the centering of the parent cubic basis ea – see Fig. 5.1.

134

EXPLICIT VARIANT STRUCTURES

Table 5.5 Orthorhombic ‘mixed axes’ holohedries matrix of transformation stretch

α β 0

β α 0

variant

U1

generating coset from variant U1

Ok,i±j

point group

matrix of transformation stretch

variant generating coset from variant U1 point group

Ok,i±j,2 = {Rπ i , 3π

π

2 ,R2 } Rπ j , Rk k

π π Ok,i±j = {1, Rπ k , Ri−j , Ri+j }

δ 0 0

0 α β

0 β α

δ 0 0 0 α −β 0 −β α

U3

U4

4π 3 Ok,i±j,3 = {Ri+ , j+k 4π π π 3 2 Ri−j+k , Ri−k , Rj

3π

2 , Ok,i±j,4 = {Rπ i+k , Rj

}

4π

4π

3 3 Ri+ , Ri−j−k ,} j−k

π π Oi,j±k = {1, Rπ i , Rj−k , Rj+k }

point group

matrix of transformation stretch

0 0 δ

U2

variant generating coset from variant U1

α −β −β α 0 0

0 0 δ

α 0 β

0 δ 0

β 0 α

α 0 −β

U5 2π 3 Ok,i±j,5 = {Ri+ , j+k 2π 3π π 3 2 Ri−j−k , Rj−k , Ri

0 −β δ 0 0 α U6 π

2 Ok,i±j,6 = {Rπ j+k , Ri ,

}

2π

2π

3 3 Ri+ , Ri−j ,} j−k +k

π π Oj,k±i = {1, Rπ j , Rk−i , Rk+i }

5.1.3.2 Orthorhombic ‘mixed axes’ variants The second conjugacy class of orthorhombic subgroups in Cijk is constituted by the three point groups in Table 5.5. Each one of these is also a subgroup of one of the three conjugate tetragonal groups, and contains a triplet of twofold axes (indicated as indices in the group’s symbol) that are ‘mixed’ in the cubic reference lattice L(ea ): two are along orthogonal face-diagonals and one along the edge orthogonal to that face, in the cubic cell of L(ea ); thus the title name for the variant structure of this conjugacy class. The symmetry-breaking stretches giving the six variants in this structure are given in Table 5.5. Notice how U1 acts on the cubic cell of the reference lattice: one of the cubic fourfold axes (in this case the k axis) is stretched anisotropically, while being maintained orthogonal to the plane of the other two fourfold axes, whose lengths are kept equal and whose angle is changed.

5.1 VARIANT STRUCTURES IN CUBIC WT-NBHDS

135

This structure consists of six variants falling into three distinct pairs; U1 and U2 have the same symmetry axes, and so on. The metrics of stretches in the same pair belong to the same ﬁxed set. It is not diﬃcult to see that the centering of these orthorhombic variants depends on the centering of the parent cubic lattice as follows: if the parent basis ea is primitive, facecentered, or body-centered, these orthorhombic variants are of the basecentered, body-centered, or face-centered type, respectively (see Fig. 5.1). 5.1.4 Monoclinic conjugacy classes and variant structures There are nine period-two operations, hence nine monoclinic subgroups (cyclic subgroups of order 2) in Cijk . They are divided into two distinct conjugacy classes; correspondingly, there are two distinct monoclinic variant structures in the cubic wt-nbhd Nea . By Proposition 4.7, these structures are constituted by twelve variants each. 5.1.4.1 Monoclinic ‘cubic edges’ variants The ﬁrst conjugacy class of monoclinic subgroups in Cijk consists of the three subgroups Mi , Mj , Mk – see Table 5.6 – of the normal orthorhombic subgroup Oijk . In each of these subgroups the unique twofold symmetry axis, indicated as an index in the group’s symbol, is along one of the edges of the cubic cell of the parent lattice, whence the title name for the twelve variants in this conjugacy class. Their symmetry-breaking stretches are also given in Table 5.6. Notice that U1 deforms the parent cubic lattice by stretching it anisotropically along a fourfold cubic axis while maintaining it orthogonal to the plane of a cubic face on which the stretch is arbitrary. Equivalently, it can be thought of stretching anisotropically the three fourfold cubic axes and slanting one of them in one of the coordinate planes containing it. The metrics of these monoclinic variants belong to three distinct 4dimensional ﬁxed sets in Q> 3 . The variants maintain the centering of the reference lattice: if this is primitive cubic, the monoclinic variants are also primitive; if it is f.c.c. or b.c.c., the variants are all of the centered monoclinic type – see Fig. 5.1. 5.1.4.2 Monoclinic ‘face-diagonals’ variants The second conjugacy class of monoclinic subgroups in Cijk is constituted by six point groups in Table 5.7, which also gives the symmetry-breaking stretches. The unique twofold symmetry axis in each of these subgroups is one of the face diagonals in the cubic cell of the parent lattice. Notice how U1 acts on the cubic parent lattice: up to an overall rotation of the lattice, U1 anisotropically stretches one of the fourfold cubic axes (in this case k ) and slants it on the deformed i , j plane, in such a way that its projection be along the deformed face diagonal. On the i , j plane the

136

EXPLICIT VARIANT STRUCTURES

Table 5.6 Monoclinic ‘cubic edges’ holohedries matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group

α 0 β

0 β δ 0 0 γ U1

α 0 −β

Mj

0 −β δ 0 0 γ U2

π Mj,2 = {Rπ k , Ri }

Mj = {1, Rπ j } γ 0 β

0 β δ 0 0 α U3

γ 0 −β

0 −β δ 0 0 α U4 3π

π Mj,3 = {Rπ i−k , Ri+k }

π

Mj,4 = {Rj 2 , Rj2 }

Mj = {1, Rπ j } γ β 0

γ −β −β α 0 0 U6

β 0 α 0 0 δ U5 4π

4π

4π

3 3 Mj,5 = {Ri−j−k , Ri+ } j+k

α β 0

0 0 δ 4π

3 3 Mj,6 = {Ri+ , Ri−j } j−k +k π Mk = {1, Rk }

α −β −β γ 0 0 U8

β 0 γ 0 0 δ U7 π

2 Mj,7 = {Rπ j−k , Ri }

0 0 δ

3π

Mj,8 = {Ri 2 , Rπ j+k }

Mk = {1, Rπ k } δ 0 0

0 0 γ β β α U9 2π

δ 0 0 0 γ −β 0 −β α U10 2π

2π

2π

3 3 3 3 Mj,9 = {Ri+ , Ri+ } Mj,10 = {Ri−j , Ri−j−k } j+k j−k +k π Mi = {1, Ri }

δ 0 0

0 0 α β β γ U11

δ 0 0 0 α −β 0 −β γ U12 3π

2 } Mj,11 = {Rπ i−j , Rk

π

2 Mj,12 = {Rπ i+j , Rk }

Mi = {1, Rπ i }

5.1 VARIANT STRUCTURES IN CUBIC WT-NBHDS

137

Table 5.7 Monoclinic ‘face diagonals’ holohedries α β −$ β α −$ −$ −$ δ U2

matrix of transformation stretch variant generating coset from variant U1 point group

α β $

matrix of transformation stretch variant generating coset from variant U1 point group

α −β −$ −β α $ −$ $ δ U3

matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group matrix of transformation stretch variant generating coset from variant U1 point group

β $ α $ $ δ U1 Mi−j

π Mi−j,2 = {Rπ i+j , Rk }

Mi−j = {1, Rπ i−j } α −β $ −β α −$ $ −$ δ U4

3π

Mi−j,3 = {Rk 2 , Rπ i }

π

2 Mi−j,4 = {Rπ j , Rk }

Mi+j = {1, Rπ i+j } α $ β

α −$ β −$ δ −$ β −$ α U6

$ β δ $ $ α U5 2π

3 Mi−j,5 = {Ri+ , Rπ j−k } j+k

2π

3π

3 Mi−j,6 = {Ri−j−k , Ri 2 }

Mi−k = {1, Rπ i−k } α −$ −β −$ δ $ −β $ α U7

α $ −β $ δ −$ −β −$ α U8

2π

2π

π

3 3 Mi−j,7 = {Ri+ , Rπ Mi−j,8 = {Ri−j , Ri2 } j+k } j−k +k Mi+k = {1, Rπ } i+k

δ $ $

δ −$ −$ −$ α β −$ β α U10

$ $ α β β α U9 4π

3 Mi−j,9 = {Rπ i−k , Ri+j+k }

4π

π

3 Mi−j,10 = {Ri−j , Rj2 } +k

Mj−k = {1, Rπ j−k } δ −$ $ −$ α −β $ −β α U11 4π

δ $ −$ $ α −β −$ −β α U12 4π

3π

3 3 Mi−j,11 = {Ri+ , Rπ Mi−j,12 = {Ri−j−k , Rj 2 } i+k } j−k Mj+k = {1, Rπ } j+k

138

EXPLICIT VARIANT STRUCTURES

deformation is such that the angle between the vectors i and j is changed, but their lengths are maintained equal. According to Table 5.7 there are six distinct holohedries, hence the twelve monoclinic variants have metrics in six distinct 4-dimensional ﬁxed sets in Q> 3 . It is not diﬃcult to check that the variants are always of the centered monoclinic type, regardless of the centering of the parent cubic phase. Remark 5.5 We have seen in §3.3.2 that conjugacy in GL(3, Z) is a stricter condition than conjugacy in O or Aut. Here we see by an example that conjugacy in the lattice group L(ea ) of the reference basis is in general stricter than conjugacy in GL(3, Z). Indeed, if the parent basis ea is either f.c.c. or b.c.c., the variants in this monoclinic structure have the same Bravais lattice type (centering) as the variants in the previous monoclinic structure. By deﬁnition, this means that the two distinct local conjugacy classes (in L(ea )) actually belong to the same conjugacy class in GL(3, Z). This shows that for the lattice groups of the centered monoclinic type conjugacy in an f.c.c. (or b.c.c.) lattice group L(ea ) is stricter than conjugacy in GL(3, Z). Therefore, indicating the Bravais lattice type of the variants (that is, the conjugacy classes of their lattice groups in GL(3, Z)), let alone their crystal system, does not give enough information regarding the nature of a symmetry-breaking transition: only the conjugacy class in the lattice group L(ea ) of the reference basis does. 5.1.5 Triclinic conjugacy class and variant structure There is obviously only one (normal) triclinic subgroup, i. e. {1}, in Cijk . Correspondingly, there is a unique triclinic variant structure in Nea , constituted by twenty-four variants, whose centering is (primitive) triclinic regardless of the centering of the reference cubic basis ea . In terms of the positive-deﬁnite symmetry-breaking stretch U1 in Table 5.8, the variants U1 , . . . ,U24 are obtained through conjugacy with each rotation in Cijk , since the twenty-three cosets of {1} in Cijk are each given by a singleton {R}, where R is a nonunit element of Cijk . Some of the variants Ur are obtained by permuting the indices 1, 2, 3 in U1 and, independently, by changing the sign in two among its oﬀ-diagonal entries δ, ;, η. The stretch U1 deforms the parent cubic lattice in an arbitrary fashion, thereby destroying all its symmetry, except for the central inversion. All the variants have the same holohedry and the corresponding metrics belong to the 6-dimensional triclinic ﬁxed set in Q> 3. 5.2 Variant structures in hexagonal wt-nbhds In this section we describe the variant structures in a wt-nbhd Nea of any reference basis ea ∈ B in the hexagonal system, which can only be of the primitive hexagonal Bravais type. For deﬁniteness, given an orthonormal

5.2 VARIANT STRUCTURES IN HEXAGONAL WT-NBHDS

139

Table 5.8 Triclinic holohedry matrix of transformation strain

α β $

β γ η

$ η δ

variant

Ui

generating coset from variant U1

Ri {1}

point group

Table 5.9 Hexagonal holohedry matrix of transformation stretch

α 0 0

0 α 0

0 0 δ

variant

U

generating coset from variant U

Hk

point group

π π 3 3 ,R 3 ,R 3 , Hk = {1, Rπ i , Rj , Rk , Rk , Rk k k

π

Rπ

i+

√

3j

, Rπ

i−

√

3j

√ , Rπ

2π

3i+j

4π

√ , Rπ

3i−j

5π

}

basis (i , j , k ), we choose ea as follows (see Fig. 3.8(b), (c) and (3.50)): √ e1 = i , 2e2 = −i + 3j , e3 = k . (5.2) The holohedry P (ea ) is then given – see Remark 5.2 – by the group Hk in Table 5.9, which is a dihedral group of order 12. The unique sixfold optic axis characterizing this system has the direction k of the axis of the hexagonal prism (which is not a unit cell) in the lattice L(ea ); thus the notation for Hk . The unit vectors i and j span the basal plane perpendicular to k . On this plane there are six equally spaced twofold axes, along the diagonals and side-axes of the centered hexagons in the basal plane i , j of L(ea ). The √ directions of the three diagonal axes (full lines in Fig. 3.8(c)) are √ i , i ± 3j , while the side-axes (dashed lines in that ﬁgure) are given by j , 3i ± j . We express the symmetry-breaking stretches U for ea as matrices referred to the orthonormal basis (i , j , k ). The form of U in this basis2 can be read in Table 5.9; the corresponding metrics constitute the 2-dimensional hexagonal ﬁxed set I(L(ea )) in Q> 3. Notice how U deforms the hexagonal lattice: it stretches anisotropically the optic axis while maintaining it orthogonal to the basal plane on which the stretch is isotropic. 2

Notice that the matrix U in Table 5.9 has the same form as U1 in Table 5.2. However, due to (3.80), the tetragonal and hexagonal ﬁxed sets in Q> 3 are of course distinct.

140

EXPLICIT VARIANT STRUCTURES

P

C

M√3i−j C

C

P

P

Figure 5.2 The tree of holohedral subgroups PL (ea ) when ea is a hexagonal basis

In the rest of this section we analyze the holohedral subgroups PL (ea ) of Hk whose corresponding groups of integral matrices are lattice groups, according to the deﬁnition in (4.16). Fig. 5.2 summarizes the inclusion and conjugacy relations. Remark 5.6 As explained in §4.4, in order to describe the variant structures in Nea we need to ﬁnd all the L-holohedral subgroups of P (ea ) = Hk . We stress that there are no holohedral subgroups in Hk whose corresponding L is a rhombohedral lattice group. Indeed, no such groups are contained in any hexagonal lattice group such as L(ea ) for the hexagonal basis ea in (5.2) (see also footnote 20 in chapter 3 and Remark 4.1). The group L(ea ) does contain two subgroups L and L , say, that are isomorphic to rhombohedral lattice groups, but are not themselves lattice groups; that is, they are not GL(3, Z)-conjugate of any rhombohedral lattice group. As mentioned in Remark 4.1, the reason is that any lattice that is invariant under the action of L or L according to (3.25), is actually a hexagonal lattice, invariant under the full hexagonal group L(ea ).3 3

This means that no symmetry-breaking stretch for a hexagonal basis can produce a rhombohedral basis, and there are no rhombohedral variant structures in the hexagonal wt-nbhd Nea . Notice that the two rhombohedral subgroups of Hk , say P and P , corresponding to the aforementioned subgroups L and L of L(ea ), are indeed themselves holohedries; however, they are not the holohedries of any basis in Nea .

5.2 VARIANT STRUCTURES IN HEXAGONAL WT-NBHDS

141

Table 5.10 Orthorhombic holohedries matrix of transformation strain

α 0 0

0 γ 0

0 0 δ

variant

U1

generating coset from variant U1

Oi j k

point group

π π Oi j k = {1, Rπ i , Rj , Rk }

√

α+3γ

matrix of transformation strain

√

4 3(γ−α) 4

3(γ−α) 4 3α +γ 4

0

point group

π

√ Oi j k,2 = {Rk3 , Rπ

3i−j

Oi−√3j,√3i+j,k = {1, Rπ

√

√

4 3(α−γ ) 4

variant

point group

, Rπ

i−

4π

i+

√

3j

√

3j

√ , Rπ

3(α−γ ) 4 3α +γ 4

0

generating coset from variant U1

δ

U2

α+3γ

matrix of transformation strain

0

0

variant generating coset from variant U1

0

0

, Rk 3 }

3i+j

, Rπ k }

0 0 δ

U3 Oi j k,3 =

2π {Rk 3

, Rπ

i−

√

3j

Oi+√3j,√3i−j,k = {1, Rπ

i+

√ , Rπ

√

3j

5π

3i+j

√ , Rπ

, Rk 3 }

3i−j

, Rπ k }

5.2.1 Orthorhombic conjugacy class and variant structure In Hk there is one conjugacy class of orthorhombic subgroups, constituted by the point groups in Table 5.10. The three orthorhombic axes in each of these subgroups come from triples of twofold axes in the parent lattice, two of which are perpendicular to each other on the basal plane (one is along a hexagonal diagonal, while the other is along a hexagonal side-axis), plus the optic axis. As usual, the axes are indicated as indices in the groups’ symbols. There are no other orthorhombic subgroups in Hk , as can be seen by a direct computation or as a consequence of Sylov’s theorem mentioned in §5.1.1. Corresponding to this conjugacy class in Hk , by Proposition 4.7 there are three orthorhombic variants in Nea , whose symmetry-breaking stretches are also given in Table 5.10. Since all the holohedries P (Ur ea ) are distinct, these three orthorhombic

142

EXPLICIT VARIANT STRUCTURES

variants have metrics belonging each to a distinct 3-dimensional ﬁxed set in Q> 3 whose Bravais type is always base-centered orthorhombic. Notice that U1 deforms the hexagonal parent lattice by anisotropically stretching the optic axis and a pair of orthogonal twofold axes in the basal plane, while maintaining them mutually orthogonal. 5.2.2 Monoclinic conjugacy classes and variant structures There are seven twofold rotations in Hk , and it can be veriﬁed by a direct computation that the cyclic subgroups of order two they generate split themselves into three conjugacy classes (the monoclinic subgroup generated by the twofold rotation Rkπ is normal in Hk ). Correspondingly, there exist three distinct monoclinic variant structures in the hexagonal wt-nbhd Nea . Due to Proposition 4.7, they are constituted by six variants each. 5.2.2.1 Monoclinic ‘basal diagonals’ variants The ﬁrst conjugacy class of monoclinic subgroups in Hk consists of the three point groups in Table 5.11. These have one twofold symmetry axis each, along the diagonals of the hexagons tiling the basal plane in the reference hexagonal lattice, whence the title name for the variants. Their six symmetry-breaking stretches are also given in Table 5.11. The six variants split in three pairs sharing their monoclinic axis and the 4-dimensional ﬁxed set in Q> 3 for the corresponding metrics. Notice that U1 deforms the parent hexagonal lattice by slanting the optic axis in such a way that its projection on the basal plane be along the twofold side axis j , while keeping the latter orthogonal to the twofold basal diagonal monoclinic axis i ; thus the hexagonal net in the basal plane is deformed into a net of centered rectangles. Since the analogue holds for the other stretches, the Bravais type of all these variants is centered monoclinic. 5.2.2.2 Monoclinic ‘basal side-axes’ variants The second conjugacy class of monoclinic subgroups in Hk is also constituted by three point groups, listed in Table 5.12. These subgroups have one twofold symmetry axis each, along the side-axes of the hexagons tiling the basal plane in the reference lattice, whence the title name for the variants. The symmetry-breaking stretches are also given in Table 5.12. Thus, analogously to §5.2.2.1, these variants split in three pairs sharing their monoclinic axis and having metrics that belong to three 4-dimensional ﬁxed sets in Q> 3. The deformations imposed by the stretches Ui to the reference hexagonal lattice are analogous to those of the previous set of symmetry-breaking stretches, interchanging the role of the two types of basal axes. These variants are of the centered monoclinic Bravais type. As at the end of §5.1.4.2,

5.2 VARIANT STRUCTURES IN HEXAGONAL WT-NBHDS

Table 5.11 Monoclinic ‘basal diagonals’ holohedries matrix of transformation stretch variant generating coset from variant U1 point group

α 0 0

0 0 γ β β δ U1

matrix of transformation stretch

√

Mi

α 0 0 0 γ −β 0 −β δ U2 π Mi,2 = {Rπ j , Rk }

Mi = {1, Rπ i } √

α+3γ

4 3(γ−α) 4 √

− 23β variant generating coset from variant U1 point group

3j

, Rk 3 } i−

√

√

4 3(γ−α) 4 √

√

3j

}

√

3(γ−α) 4 3α+γ 4

3β 2

β

2

β

δ

2

U4 √ Mi,4 = {Rπ

π

3i−j

, Rk3 }

Mi−√3j = {1, Rπ

i−

√

√

4 3(α−γ ) 4 √

√

3j

}

√

3(α−γ ) 4 3α+γ 4

3β 2

3β 2

−β 2

−β 2

δ

U5 2π

Mi,5 = {Rk 3 , Rπ

√

Mi+√3j = {1, Rπ

√

i−

i+

√

α+3γ √

4 3(α−γ ) 4

− variant generating coset from variant U1 point group

√

Mi−√3j = {1, Rπ

α+3γ

matrix of transformation stretch

δ 4π

i+

variant generating coset from variant U1 point group

variant generating coset from variant U1 point group

−β 2

U3 Mi,3 = {Rπ

3β 2

matrix of transformation stretch

− 23β

−β 2

α+3γ

matrix of transformation stretch

√

3(γ−α) 4 3α+γ 4

√

3(α−γ ) 4 3α+γ 4

3β 2

β

2

3j

3j

}

} √

− 23β β

2

δ

U6 5π

√ Mi,6 = {Rk 3 , Rπ

Mi+ 3j = √

}

3i+j {1, Rπ √ } i+ 3j

143

144

EXPLICIT VARIANT STRUCTURES

Table 5.12 Monoclinic ‘basal side-axes’ holohedries matrix of transformation stretch variant generating coset from variant U1 point group

matrix of transformation stretch

α 0 β

0 β γ 0 0 δ U1 Mj

π Mj,2 = {Rπ k , Ri }

Mj = {1, Rπ j } 3γ +α 4 √ 3(γ−α) 4 β

2

variant generating coset from variant U1 point group

matrix of transformation stretch

√

matrix of transformation stretch

2

− 23β

√

Mj,3 = {Rπ

i+

δ π

√

3j

, Rk3 }

√ M√3i+j = {1, Rπ

3γ +α 4 √ 3(γ−α) 4

√

3i+j

3(γ−α) 4 γ +3α 4

}

−β 2

√

3β 2

√

3β 2

δ

3i−j

, Rk 3 }

U4 √ Mj,4 = {Rπ

4π

√ M√3i+j = {1, Rπ

3γ +α 4 √ 3(α−γ ) 4

variant generating coset from variant U1 point group

√

3i+j

3(α−γ ) 4 γ +3α 4

} −β 2 √

− 23β

√

− 23β

δ

U5 √ Mj,5 = {Rπ

2π

3i+j

, Rk 3 }

√ M√3i−j = {1, Rπ

√

3γ +α 4 3(α−γ ) 4 β

√

3i−j

3(α−γ ) 4 γ +3α 4

}

β

2 3β 2

√

√

3β 2

2

variant generating coset from variant U1 point group

β √

U3

−β 2

matrix of transformation strain

3(γ−α) 4 γ +3α 4

− 23β

−β 2 variant generating coset from variant U1 point group

0 −β γ 0 0 δ U2

α 0 −β

δ

U6 5π

Mj,6 = {Rk 3 , Rπ

i−

√ M√3i−j = {1, Rπ

√

3j

3i−j

}

}

5.2 VARIANT STRUCTURES IN HEXAGONAL WT-NBHDS

Table 5.13 Monoclinic ‘optic axis’ holohedries matrix of transformation strain variant generating coset from variant U1

α β 0

β 0 γ 0 0 δ U1 Mk

0 0 δ

π Mk,2 = {Rπ i , Rj }

Mk = {1, Rπ k }

point group

α−2

matrix of transformation stretch

−β γ 0 U2

α −β 0

√

−

3β +3γ 4 √ − 3α+2β + 3γ 4 √

√

√

3α+2β + 3γ 4 √ 3α+2 3β +γ 4

0

0

variant Mk,3

√

√

√

4 √ 3α−2β− 3γ 4

3α−2β− 3γ 4 √ 3α+2 3β +γ 4

0

0

δ

0 variant generating coset from variant U1 point group

U4 2π

5π

Mk = {1, Rπ k } √

−

3β +3γ 4 √ √ − 3α−2β + 3γ 4

√

√

3α−2β + 3γ 4 √ 3α−2 3β +γ 4

0 variant generating coset from variant U1 point group

0

π

4π

Mk,5 = {Rk3 , Rk 3 } Mk = {1, Rπ k }

√

√

√

3β +3γ 4 √ 3α+2β− 3γ 4

√

3α+2β− 3γ 4 √ 3α−2 3β +γ 4

0

0

δ

0 U6 Mk,6 = {Rπ

i−

Mk =

√

3j

√ , Rπ

{1, Rπ k }

3i+j

0 0 δ

U5

α+2

variant generating coset from variant U1 point group

0

Mk,4 = {Rk 3 , Rk 3 }

α+2

matrix of transformation stretch

δ

√ = {Rπ √ , Rπ } i+ 3j 3i−j π Mk = {1, Rk }

√ α−2 3β +3γ

matrix of transformation stretch

0

U3

generating coset from variant U1 point group

matrix of transformation stretch

0

}

0

145

146

EXPLICIT VARIANT STRUCTURES

Table 5.14 Triclinic holohedry matrix of transformation strain

α β $

β γ η

variant

Ui

generating coset from variant U1

Ri

point group

$ η δ

{1}

we notice that the Bravais lattice type of this variant structure always coincides with the type of the structure in §5.2.2.1. Similar remarks apply. 5.2.2.3 Monoclinic ‘optic axis’ variants The third conjugacy class of monoclinic subgroups in Hk is constituted by the normal subgroup Mk in Table 5.13; the monoclinic axis for these six variants is along the optic axis k , whence the title name. The six symmetrybreaking stretches are also given in Table 5.13, and the related metrics all belong to the same 4-dimensional ﬁxed set in Q> 3 , as mentioned above. These variants are always of the primitive monoclinic Bravais type. The symmetry-breaking stretch U1 keeps the optic axis orthogonal to the basal plane, while on the latter the deformation is arbitrary; this lowers the periodicity of the axis k from 6 to 2. 5.2.3 Triclinic conjugacy class and variant structure As for the cubic holohedry, there is obviously only one (normal) triclinic subgroup, i. e. {1}, in the hexagonal holohedry Hk . Correspondingly, there is a unique triclinic variant structure, constituted by twelve variants, whose centering is (primitive) triclinic. The symmetry-breaking stretch U1 is given in Table 5.14, and the twelve variants U1 , U2 , . . . , U12 , are obtained by conjugacy with each element of Hk , because the eleven cosets of {1} in Hk are each given by the sets {R}, where R is any nonunit element of Hk listed in Table 5.9. In this case U1 imposes an arbitrary deformation on the hexagonal reference lattice, thereby destroying all the lattice symmetries except −1. All the variants share the same holohedry and have metrics belonging to the same 6-dimensional triclinic ﬁxed set in Q> 3. 5.3 Kinematics of weak phase transformations and symmetry breaking in simple lattices Based on Zanzotto (1996b), in this section we discuss some consequences of the results just obtained on the local structure of the wt-nbhds in the

5.3 KINEMATICS OF WEAK PHASE TRANSFORMATIONS

147

spaces B and Q> 3 . With this information in hand we can give a kinematical analysis of the weak phase transformations in simple lattices, and in particular of their symmetry-breaking transformations. These results give a detailed background against which one can analyze the experimental data on phase transitions, and in which no possibility is a priori excluded. We do not discuss all such kinematic possibilities in detail; they can be worked out by means of the information given in chapter 4 and earlier in this chapter. Rather, we give examples that help in understanding the procedure. Previous partial analyses can be found in Ericksen (1989), (1996a,b), Bhattacharya and Kohn (1996), Efendiev and Luskin (2001). Analogous models for monatomic 2-lattices are given by Fadda and Zanzotto (2001b) and Ericksen (2002a,c). Let us consider ﬁrst the case of an f.c.c. basis, whose holohedry P (ea ) and lattice group L(ea ) are maximal. Inspection of Fig. 5.1 gives us all the paths through which the symmetry of ea can be reduced within a wtnbhd Nea .4 There is essentially one kind of cubic-to-tetragonal symmetry reduction (see §5.1.1), for which we introduce the convenient notation Cijk → {Ti , Tj , Tk } =: Tijk ;

(5.3)

by applying the stretches U1 , . . . , U3 in Table 5.2 to the cubic reference lattice ea one obtains three variants which are the only ones with tetragonal symmetry one can get in Nea . Likewise, and in the same notation, there is one kind of cubic-to-rhombohedral paths (see §5.1.2) Cijk → {Ri +j +k , Ri −j −k , Ri −j +k , Ri +j −k } =: Rijk .

(5.4)

This is because there exists only one tetragonal and one rhombohedral variant structure, respectively, in Nea . Fig. 5.1 also shows that there are two essentially distinct paths Cijk → Oijk and Cijk → {Ok ,i ±j , Oi ,j ±k , Oj ,k ±i } =: Oijk

(5.5)

for cubic-to-orthorhombic symmetry breaking because in Nea there are two distinct orthorhombic variant structures, given in §5.1.3.1 and §5.1.3.2 respectively. Recall how these two mechanisms act: the ﬁrst stretches the three edges of the reference cubic cell anisotropically, while keeping them mutually orthogonal; the second deforms one of the faces of the cube into a rhombus, while maintaining its plane orthogonal to the cubic edge not contained in it. Also, the two sets of variants have diﬀerent centering.5 Analogously, since in Nea there exist the two sets of variants in §§5.1.4.1 and 5.1.4.2, in which the monoclinic axes are former edges or former face 4

5

The situation is less well understood if one does not restrict the stretches to belong to a wt-nbhd. For instance – see §3.6.3 – we can continuously deform a b.c.c. lattice into a body-centered tetragonal one which, if stretched further to reach the Bain stretch, becomes f.c.c. Similar as well as more complicated nonweak transformations, usually called reconstructive, are considered in the literature. We address the reader to Tol´ edano and Dmitriev (1996) for details and extensive references. All such centerings are speciﬁed in Fig. 5.1. Also recall Remark 4.6.

148

EXPLICIT VARIANT STRUCTURES

diagonals in the cubic lattice L(ea ), respectively, there are two essentially distinct cubic-to-monoclinic symmetry-breaking paths: Cijk → {Mi , Mj , Mk } =: Mijk

and

(5.6)

˜ ijk . Cijk → {Mi −j , Mi +j , Mi −k , Mi +k , Mj −k , Mj +k } =: M (5.7) For the purpose of studying phase transitions in crystals, one is also often interested in the intermediate symmetries along the symmetry-breaking paths. These are interesting for the modelling of crystals whose stable phases exhibit progressive symmetry reduction, as in Fig. 1.6. See also §7.6.4. Notice that the progressive symmetry breaking is described in terms of suitably chosen cells in the various phases. All the needed information is given, again, by Figs. 5.1 and 5.2: there are ﬁve essentially distinct paths along which symmetry can be stepwise lowered from cubic to monoclinic: Cijk → Tijk → Oijk → Mijk Cijk → Tijk → Oijk → Mijk ˜ ijk Cijk → Tijk → Oijk → M Cijk → Rijk → Mijk ˜ ijk Cijk → Rijk → M

(5.8)

Along the same lines of thought we can analyze the case when the basis ea is hexagonal, based on the information given by Fig. 5.2. In this case, unlike with the cubic bases, there is only one (primitive hexagonal) Bravais type and therefore the centering of the variants is uniquely determined. The information in Figs. 5.1 and 5.2 also allows one to determine all the symmetry-breaking paths and the local structure of wt-nbhds of bases whose symmetry is not maximal (cubic or hexagonal). As an example, we consider the case of a primitive tetragonal basis ea such that P (ea ) = Tk as in §5.1.1. It is not diﬃcult to obtain the diagram in Fig. 5.3 for the inclusion relation of the holohedral subgroups of Tk and their Tk -conjugacy classes. There we see that in the neighborhood of the tetragonal basis ea one has the following transition paths: Tk → Oijk → {Mi , Mj } Tk → Oijk → Mk Tk → Ok ,i ±j → {Mi +j , Mi −j } Tk → Ok ,i ±j → Mk .

(5.9)

For instance, zirconia (ZrO2 ) is a material that exhibits tetragonal-orthorhombic-monoclinic symmetry breaking, following path (5.9)2 according to Truskinovsky and Zanzotto (2002) (see also §7.6.4). It must be remarked that, when the basis ea is not of maximal symmetry, the centering of a given low-symmetry variant structure is not always determined uniquely by the centering of ea . For instance, if ea is centered tetragonal, the orthorhombic variants whose holohedries are Oijk < Tk can either be face-centered or body-centered, depending on whether the

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

149

PII

CIF

C CC

PFI

P CC

P CC

PPP

Figure 5.3 The tree of holohedral subgroups PL (ea ) when ea is a tetragonal basis. Vertical and oblique lines indicate inclusion, while horizontal lines indicate conjugacy within Tk . Corresponding to the centering of the tetragonal basis ea indicated under Tk , and in the same position, one ﬁnds, under the name of each holohedral subgroup in the table, the centering of the bases which belong to the wt-nbhd Nea and have that holohedry as symmetry group

tetragonal basis ea generates a face-centered or a body-centered cell, respectively. Likewise, the variants whose symmetry is Ok ,i ±j < Tk can either be body-centered or face-centered, respectively. Fig. 5.3 summarizes these statements. Besides showing all the possibilities for weak transformations, the discussion above also gives explicit necessary conditions that indicate which weak (in particular symmetry-breaking) phase transitions are kinematically impossible in simple lattices. For instance, the fact – see Remark 5.6 – that no rhombohedral holohedries are contained in a hexagonal holohedry implies that any hexagonal-to-rhombohedral transition cannot be weak. In the same way, the transition, say, from the face-centered cubic lattice structure to the base-centered orthorhombic one cannot be weak. Again, inspection of Figs. 5.1 and 5.2 gives all the possibilities. 5.4 Irreducible invariant subspaces of Sym for the action of the holohedries In this section we study the decompositions of the space Sym of symmetric tensors into orthogonal sums of irreducible invariant subspaces under the action of the holohedries. This analysis, which takes advantage of the results in §§5.1–5.2, will be used in §6.8, where we give the fourth-order elasticity tensors of crystalline materials, and in chapter 7, where we classify the bifurcation patterns for the elastic equilibria of simple lattices. We address

150

EXPLICIT VARIANT STRUCTURES

the reader to Golubitsky et al. (1988) for a general treatment of irreducible invariant subspaces of Rn under the action of a compact Lie group. Consider the 6-dimensional vector space Sym equipped with the standard scalar product ‘·’ given by (2.8). This deﬁnes an orthogonality relation and an orthogonal sum of subspaces analogous to the ones deﬁned in §2.1 for R3 . For any subspace V of Sym, V ⊥ denotes the orthogonal complement of V, given by V ⊥ = {U ∈ Sym : U · V = 0 for all V ∈ V};

(5.10)

⊥

thus V collects all the tensors that are orthogonal to all the tensors in V, and one has, for any V: Sym = V V ⊥ . (5.11) Sometimes in this section we will refer to the tensors in Sym as ‘vectors’; recall that we denote by V1 , . . . , Vn the subspace generated by the vectors V1 , . . . , Vn . As usual, a basis V1 , . . . , V6 of Sym is called orthonormal if all its vectors are of unit norm and mutually orthogonal. 5.4.1 General properties Recall from §2.2.2 that, given an action of a group G on a vector space, a nontrivial subspace W is called invariant if g(W) = W for all g ∈ G; W is called irreducible invariant – for brevity i.i. – if it is invariant and the trivial subspace {0} is the only invariant subspace it contains properly. Given a holohedry P < O, consider the linear action V → Q t VQ

(5.12)

of P on Sym. For brevity we call (5.12) the action of P on Sym. We will determine all the possible ways in which Sym can be decomposed into an orthogonal sum of i.i. subspaces for such an action, for any holohedry P . It is not diﬃcult to see that we only need to indicate the possible decompositions6 of Sym corresponding to one holohedry in each crystal system, because two orthogonally conjugate holohedries produce orthogonally conjugate decompositions: Lemma 5.1 Let P < O be a holohedry and P = RP Rt for some R ∈ O. Then i Vi is a decomposition of Sym into irreducible P -invariant subspaces if and only if i RVi Rt is a decomposition of Sym into irreducible P -invariant subspaces. Remark 5.7 As we did earlier in this chapter, also in this section we will only consider the elements with positive determinant of any holohedry, since this does not aﬀect the resulting decompositions. We will do so without changing nomenclature or notation. 6

Hereafter ‘decomposition’ will mean decomposition of Sym into an orthogonal sum of i.i. subspaces under the action of some holohedry. The latter will be understood from the context.

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

151

Given a holohedry P < O, the ﬁrst step in determining the related decomˆ ) of Sym which is pointwise positions of Sym is ﬁnding the subspace C(P stabilized by P under the action (5.12), that is, whose elements commute ˆ ) are important because of with all the elements of P . The subspaces C(P the following ˆ ) be the subspace of the Lemma 5.2 Let P < O be a holohedry and C(P elements of Sym stabilized by the group action (5.12) associated with P . Then ˆ ) and C(P ˆ )⊥ are both invariant subspaces of Sym under the action (1) C(P of P . ˆ ) or (2) Any nontrivial i.i. subspace W of Sym is contained in either C(P ˆ )⊥ . C(P ˆ ), with h = dim C(P ˆ ), (3) Given any orthonormal basis V1 , . . . , Vh of C(P 7 one can write the decomposition ˆ ) = hi=1 Vi C(P

(5.13)

ˆ ) is 1-dimensional, that is, unless which is clearly not unique unless C(P P is a cubic holohedry. ˆ ) is stabilized by the action Proof. (1) By deﬁnition, each vector in C(P ˆ ) is pointwise invariant under this (5.12) associated with P , hence C(P action, and its i.i. subspaces are those and only those of dimension one. ˆ ) is invariant, so is also its orthogonal complement C(P ˆ )⊥ . Since C(P ˆ ) $= {0} or W ∩ (2) Statement (2) certainly holds when either W ∩ C(P ⊥ ˆ C(P ) $= {0}. Indeed, assume W to be an invariant subspace of Sym which ˆ ) [C(P ˆ )⊥ ]; then W is irreducible if contains a nontrivial element of C(P ˆ ˆ and only if it is a subspace of C(P ) [C(P )⊥ ] because its intersection with ˆ ) [C(P ˆ )⊥ ] is also a nontrivial invariant subspace. C(P Assume now W to be a nontrivial invariant subspace such that ˆ ) = {0} = W ∩ C(P ˆ )⊥ . W ∩ C(P

(5.14)

ˆ ) and 0 $= W ∈ Then W contains a sum V = U + W , with 0 $= U ∈ C(P ⊥ ˆ C(P ) . By deﬁnition, there is at least an element R of P which, while ˆ )⊥ . leaving U unchanged, transforms W into a diﬀerent vector still in C(P t ˆ Therefore R VR − V is a nonzero vector belonging to both C(P )⊥ and W, contradicting (5.14). ˆ ). (3) This too is an immediate consequence of the deﬁnition of C(P Due to Lemma 5.2, in order to ﬁnd the decompositions of Sym related ˆ ) and its straightto a given P we ﬁrst need to determine the subspace C(P forward decompositions (5.13). To this end we can take advantage of the 7

ˆ ) has the following dimension, indicated Recall, from the results in §§6.1–6.2, that C(P in parenthesis, depending on the system of the holohedry P : cubic (1), hexagonal (2), tetragonal (2), rhombohedral (2), orthorhombic (3), monoclinic (4), triclinic (6).

152

EXPLICIT VARIANT STRUCTURES

results obtained earlier in this chapter. Suppose a holohedry P and any basis ea ∈ B such that P = P (ea ) be given. We have deﬁned in (3.65) the set C(ea ) of elements of Sym> stabilized by the action (5.12) of P (ea ), that is, commuting with all the elements of P (ea ), and in §§5.1–5.2 we have described explicitly the forms of all the symmetry-breaking tensors in Sym> for a cubic and a hexagonal basis. As pointed out in Remark 4.7, this also gives the explicit forms of the tensors in C(ea ) for ea belonging to the various crystal systems – see also Remark 5.1. For our present purposes we only need to ‘extend’ such sets C(ea ) ⊂ Sym > to the subspaces ˆ a ) = C(P ˆ ) ⊂ Sym collecting the elements of the entire space Sym that C(e ˆ ) are stabilized by (5.12) for Q ∈ P = P (ea ). This gives us the spaces C(P ˆ ˆ mentioned in Lemma 5.2. The elements of C(ea ) = C(P ) have the same matrix representation given in §§5.1–5.2 for the elements of C(ea ), with respect to the same basis, except that their entries are not restricted by the requirement of positive-deﬁniteness holding for the elements of C(ea ). ˆ ) makes us only concentrate, in the The knowledge of the spaces C(P following sections, on the possible decompositions of the orthogonal comˆ )⊥ : plements C(P ˆ )⊥ = na=1 Va , C(P (5.15) ⊥ ˆ where the number n of the i.i. subspaces Va in C(P ) , as well as their dimensions, depend on the holohedry P .8 For the purpose of searching a decomposition as in (5.15) it is useful to ˆ )⊥ , then the orthogonal notice that if W is an invariant subspace of C(P ⊥ ˆ complement of W within C(P ) is also invariant under the group action. ˆ )⊥ Depending on the given holohedry P , a decomposition (5.15) of C(P need not be unique. The following Lemma is useful in the search of further ˆ )⊥ , once one is found. decompositions of C(P ˆ )⊥ as in (5.15) be given. If there Lemma 5.3 Let a decomposition of C(P exists a further nontrivial i.i. subspace V with a nontrivial projection on some of the Va , say V1 , . . . Vr , r ≤ n, then V1 , . . . Vr and V all have the same dimension. Proof. Consider one of the i.i. subspaces in the decomposition (5.15), say V1 , ˆ )⊥ which intersects and assume V to be any nontrivial i.i. subspace of C(P V1 only at 0 (for otherwise it cannot be irreducible). The kernel and the image of the orthogonal projection π of V on V1 are invariant subspaces of V and V1 , respectively, because π commutes with the group action. Then: (i) If the dimension of V is bigger than that of V1 , the kernel of π is necessarily nontrivial, and thus is V itself. Therefore V is orthogonal to V1 , that is, the projection of V on V1 is trivial. (ii) If the dimension of V is smaller than that of V1 , the image of π is a 8

ˆ ) and of all the subspaces Va is 6, the Of course, the sum of the dimensions of C(P dimension of Sym.

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

153

proper subspace of V1 and must thus be trivial. We conclude again that the projection of V on V1 is trivial. Now the statement of the Lemma follows at once from the assumption that V is not orthogonal to any one of the spaces V1 , . . . Vr . The actual existence of a subspace V as in Lemma 5.3 depends on the speciﬁc action of the holohedry P on each of the V1 , . . . , Vr , and will be checked case by case. 5.4.2 Reduced actions and reduced symmetry groups on the i.i. subspaces Given a holohedry P < O and a related decomposition of Sym, it is important to determine how P acts through (5.12) on the elements of each i.i. subspace. The action of P on any i.i. subspace is called reduced. To study it, we represent the vectors of Sym in some orthonormal basis VA , each element of which belongs to one of the i.i. subspaces of the given decomposition. If we write any V ∈ Sym in terms of the VA : V = 6A=1 y A VA , (5.16) each transformation V → Q t VQ of the group action (5.12) induces a linear transformation y A → ρAB y B (5.17) of the coordinates (y A ) ∈ R6 deﬁned by (5.16). It is straightforward to verify that the matrix ρ = (ρAB ) in (5.17) is actually orthogonal, ρ ρt = 1, because (5.12) preserves the scalar product of tensors. Moreover, since each vector VA belongs to an i.i. subspace, the matrix ρ is a block matrix: each block submatrix, say ρ , is itself orthogonal and represents (5.12) when reduced to one of the i.i. subspaces in the decomposition, say V, producing an orthogonal transformation of the coordinates y A that pertain to V. Given a basis VA and an i.i. subspace V as above, when Q varies in P the block submatrices relative to V form a group P called the reduced (symmetry) group of P on V. We use the notation ρ ≈ Q to indicate that an orthogonal submatrix ρ represents the reduced action of a transformation Q ∈ P on a given i.i. subspace; thus ρ belongs to a reduced symmetry group. We will see that the i.i. subspaces of Sym can be of dimension 1, 2 or 3; correspondingly, we have 1-, 2- and 3-dimensional matrices ρ . For the 2-dimensional case we use for brevity the notation cos ϑ − sin ϑ r(ϑ) := . (5.18) sin ϑ cos ϑ Recall that, for each V ∈ Sym, the action (5.12) of a holohedry P ˆ ), by deﬁnition generates a set of variants of V as in (4.39). If V ∈ C(P ˆ )⊥ the number of variants depends the number of variants is 1. If V ∈ C(P ˆ )⊥ to which V on the speciﬁc action of P on the i.i. subspace V ⊂ C(P belongs: all such variants belong to V, and their number is equal to the

154

EXPLICIT VARIANT STRUCTURES

number of elements of the reduced group P relative to V. This group can be shown to be a faithful representation of the quotient P \P0 , the latter being the normal subgroup of P that stabilizes the subspace V. This point of view is adopted by Tol´edano and Dmitriev (1996), p. 33, for instance. Remark 5.8 Notice that the reduced group P depends in general on the basis VA used in (5.16), and changes by conjugacy under a change of this basis. Since our analysis of bifurcations in chapter 7 is qualitatively unaffected by this change – see Remark 7.3 – we will only consider the reduced groups related to some suitable basis VA that we will specify explicitly for each system.9 5.4.3 Decompositions of Sym under the action of the holohedries Here we present the decompositions of Sym under the action of the holohedries and the reduced group on each i.i.-subspace. By Lemma 5.1 we choose one basis ea per system, the latter containing P (ea ); then we specify a set of orthonormal vectors (i , j , k ) for that system, whose directions are taken along as many symmetry axes as possible in the lattice L(ea ). Remark 5.9 We use the basis (i , j , k ) rather than ea because the former gives the simplest matrix expression for the elements in the i.i. subspaces of Sym. Besides, the basis (i , j , k ) only depends on the system of L(ea ), and neither on the basis ea in that system nor on its Bravais lattice type (centering). The choice of (i , j , k ) follows the practice common in the literature for the orientation of the basis giving the crystallographic indices in each system. For instance, regardless of the actual lattice vectors ea , in the orthorhombic system the basis (i , j , k ) is always chosen along the three twofold symmetry axes of L(ea ); in the tetragonal system k is along the fourfold axis of L(ea ), and so on (Remark 3.6). We will explicitly mention when there is more than one standard choice in the literature, for this is a common source of confusion when comparing results from diﬀerent authors. As in §§5.1–5.2 we will express all tensors in Sym for the decomposition pertaining to the holohedry P (ea ) through their representative matrices in the basis (i , j , k ) mentioned above. Once ea , P (ea ), and (i , j , k ) are given for any system, the ﬁrst step ˆ a) in determining the decompositions of Sym is ﬁnding the subspace C(e and its (trivial) decompositions (5.13). We do so by using the results of §§5.1–5.2, as discussed after the proof of Lemma 5.2. Then we indicate the ˆ a )⊥ . We possible decompositions (5.15) of the orthogonal complement C(e also specify the reduced groups P for each i.i. subspace with respect to suitable bases VA , A = 1, . . . , 6. 9

We will see below that all but one of the i.i. subspaces are 1- or 2-dimensional. It is not too diﬃcult to check that in all the latter cases the matrix group P is actually independent of the chosen basis VA .

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

155

5.4.3.1 Triclinic decompositions If ea is a triclinic basis, any V ∈ Sym is stable under the action of the ˆ a ) = Sym. Each 1-dimensional triclinic group P (ea ) = {1}, hence C(e subspace of Sym is irreducible invariant, with a trivial reduced symmetry group: P = {1}, with 1 ≈ 1 . (5.19) A decomposition of Sym is obtained by selecting any six mutually orthogonal vectors VA in (5.13). 5.4.3.2 Monoclinic decompositions Let ea be any monoclinic basis. As in (3.39) or (3.40) we choose the unit vector j along the unique twofold axis;10 thus the holohedry is P (ea ) = ˆ a ) is Mj = {1, Rjπ }. According to Tables 5.6 or 5.12, the subspace C(e four-dimensional, given by tensors represented in the basis (i , j , k ) by the matrices of the form   0

α

0

β

γ

0

β

0

δ

.

(5.20)

ˆ a )⊥ is given by all the tenThe 2-dimensional orthogonal complement C(e sors with matrices   0

η

0

η

0

0

ϕ.

ϕ

0

(5.21)

ˆ a )⊥ to its opposite. Therefore any The rotation Rjπ maps each V ∈ C(e ˆ a )⊥ is irreducible invariant, and inﬁnitely 1-dimensional subspace of C(e ˆ many decompositions of C(ea )⊥ as in (5.15) can be obtained by choosing ˆ a )⊥ . The reduced any pair of orthogonal 1-dimensional subspaces in C(e symmetry group on any such subspace is P = {1, −1},

(5.22)

where 1 ≈ 1 and −1 ≈ Rjπ ; the action of the monoclinic group on the ˆ a )⊥ thus produces two variants, both belonging to typical element of C(e the same i.i. subspace. ˆ a )⊥ is left invariant under the Remark 5.10 Any vector V $= 0 in C(e map (5.12) if and only if Q belongs to the (trivial) triclinic subgroup of Mj . We summarize this fact by saying that V has triclinic symmetry, and we will do the analogue below for the other holohedries P (ea ). Based on footnote 12 in chapter 4, this deﬁnition can be shown to be consistent with the description of the symmetry-breaking stretches U in a wt-nbhd of ea 10

Love (1927) and other elasticians typically choose the monoclinic axis along k. We follow the standard choice in the metallurgical and mineralogical literature, in which this axis is always along j (see also §3.4 and Remark 3.6).

156

EXPLICIT VARIANT STRUCTURES

if we interpret any symmetric tensor V in a neighborhood of 0 as a strain tensor: V = 1/2(U 2 − 1) – see (2.52). We also specify below how the centering of L(Uea ) depends on the one of L(ea ); this can be obtained by rotating, if necessary, the basis used for L(ea ) in Figs. 5.1 or 5.2 into the basis (i , j , k ) used below for the system of L(ea ). 5.4.3.3 Orthorhombic decompositions Let ea be any orthorhombic basis. We choose the basis (i , j , k ) along the three twofold axes, so that the holohedry is P (ea ) = Oijk – see §5.1.3.1. ˆ a ) is 3-dimensional, given According to Tables 5.4 or 5.10, the subspace C(e by the tensors with matrices of the form:   α

0

0

0

γ

0.

0

0

δ

(5.23)

ˆ a )⊥ there are three mutually orthogIn the orthogonal complement C(e onal 1-dimensional i.i. subspaces, which we denote by V4 , V5 , V6 , each generated by one of the (orthonormal) tensors       V4 =

1 √ 2

0

0 0

0

0

0

1 , V5

1

0

=

1 √ 2

0

0 1

0

1

0

0 , V6

0

0

=

1 √ 2

0

1 0

1

0

0

0 ,

0

0

respectively. One thus obtains the decomposition ˆ a )⊥ = 6 Vr . C(e

(5.24)

(5.25)

r=4

The action of Oijk on the tensors in (5.24) is Riπ V4 Riπ =

V4 , Rjπ V4 Rjπ = −V4 , Rkπ V4 Rkπ = −V4

Riπ V5 Riπ = −V5 , Rjπ V5 Rjπ = Riπ V6 Riπ

= −V6 ,

Rjπ V6 Rjπ

V5 , Rkπ V5 Rkπ = −V5

= −V6 ,

Rkπ V6 Rkπ

=

(5.26)

V6 .

Thus the reduced symmetry group related to the 1-dimensional subspaces Va above is given by (5.22), the same for all three subspaces; nevertheless Oijk acts in a diﬀerent way on each subspace: for instance, on V6 one has 1 ≈ 1 ≈ Rkπ and Rjπ ≈ −1 ≈ Riπ , while on V5 one has 1 ≈ 1 ≈ Rjπ and Rkπ ≈ −1 ≈ Riπ . The action of Oijk on any nonzero V ∈ Vr generates another element in Vr distinct from V , producing 2 variants. Based on Remark 5.10, any such V has monoclinic symmetry. For instance, on V6 the holohedry is Mk , etc. If the orthorhombic centering is P, F, I, then the monoclinic centering is P, C, C, respectively, and can be read directly in Fig. 5.1 for the monoclinic subgroups of Oijk . If the orthorhombic centering is C, then we are dealing with an orthorhombic ‘mixed axes’ variant in the cubic tree, and Fig. 5.1 indicates that two of the subspaces Vr are of centered monoclinics, while the third is of primitive monoclinics. For instance, if the centering of the

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

157

orthorhombic basis ea is in the i , j plane, then P (ea ) corresponds to Ok ,i ±j in the cubic tree of Fig. 5.1, and thus V6 , whose holohedry is Mk , is made of primitive monoclinics, while the other two subspaces consist of centered monoclinics. This same conclusion can be directly read from the hexagonal tree in Fig. 5.2 under the holohedry Oijk . If the centering of ea is in the i , k plane, then P (ea ) corresponds to Oj ,i ±k in Fig. 5.1, and thus the primitive monoclinics are in V5 , and so on. ˆ a )⊥ diﬀerent from To check whether there exist decompositions of C(e (5.25), we ﬁrst use Lemma 5.3, which in this case tells us that any other ˆ a )⊥ must be 1-dimensional. However, by inspection of i.i. subspace in C(e the diﬀerent actions (5.26) of Oijk on the diﬀerent Vr , we immediately see ˆ a )⊥ diﬀerent from the Vr cannot that any 1-dimensional subspace in C(e be invariant, so that (5.25) is unique. 5.4.3.4 Rhombohedral decompositions Let ea be any rhombohedral basis. We choose the basis (i , j , k ) in such a way that the holohedry is P (ea ) = Ri +j +k – see §5.1.2. Recall that Ri +j +k 2π/3 is generated by its elements Riπ−j and Ri +j +k . ˆ a ) is 2-dimensional, According to Table 5.3, in this case the subspace C(e given by the tensors with matrices of the form   β

α

β

β

α

β

β

β

α

,

(5.27)

ˆ a ) is for example and a basis for C(e   V1 =

1 √ 3

1

0 0

0



0

1

0

0

1

and V2 =

1 √ 6

0

1 1

1



1

0

1.

1

0

(5.28)

ˆ a )⊥ there are two mutually orthogonal 2-dimensional i.i. subIn C(e spaces V1 = V3 , V4 and V2 = V5 , V6 , where the orthonormal tensors V3 , . . . , V6 are given by     V3 =

1 √ 2

 V5 =

1 2

1

0

0

−1

0

0

0

0

0

0

0,

0

1

0

1 √ 6



0 −1  ,

1 −1

V4 =

1

0 0

 V6 =

1 √

2 3

0 1

0,

(5.29)

0 −2

0

2

2

0

−1

−1



0

−1



−1  ;

(5.30)

0

this can be checked directly. On both V1 and V2 the reduced symmetry

158

EXPLICIT VARIANT STRUCTURES

group P has six elements and is generated by the matrices11 4π 2π −1 0 3 ≈ Riπ−j and r( ) ≈ Ri +j f := +k . 0 1 3

(5.31)

By Remark 5.10, any vector V in either V1 or V2 , except those in suitable 1-dimensional centered monoclinic subspaces,12 has triclinic symmetry. ˆ a )⊥ must be By Lemma 5.3, any possible further13 i.i. subspace W of C(e ˆ a )⊥ ; 2-dimensional. Consider an arbitrary vector W1 of unit norm in C(e it can always be written in terms of three angles σ, ϑ, τ as follows: W1 = cos σ cos ϑV 3 + sin ϑV4 + sin σ cos(ϑ + τ )V5 + sin(ϑ + τ )V6 . (5.32) 4π/3

By (5.31), the action of Ri +j +k on W1 increments ϑ by 2π/3 (producing a vector not parallel to W1 ), while the action of Riπ−j changes the sign of V3 and V5 ; one can verify this also through a direct computation. Then the following assertions can be proved: (a) Any unit vector W1 belongs to the 2-dimensional subspace V whose unit vectors are obtained by varying ϑ in (5.32) while keeping σ and τ ﬁxed. The space V is i.i. under the action of the cyclic group R0 < 2π/3 Ri +j +k generated by Ri +j +k . An orthonormal basis of V is for instance constituted by W1 and the unit vector along ˜ 1, W2 = W1 + 2W

2π

4π

3 ˜1 =R 3 W i +j +k W1 Ri +j +k ,

(5.33)

˜ 1 = −1/2. Thus σ and τ can be used to parametrize because W1 · W ˆ a ) that are i.i. under the action of R0 .14 all the subspaces in C(e (b) Given the space V generated by W1 and W2 as in (a), the vector Riπ−j W1 Riπ−j belongs to V – which makes V i.i. under the full rhombohedral holohedry Ri +j +k – if and only if W1 has the form (5.32) with τ = 0. ˆ a )⊥ a one-parameter family, By statements (a) and (b) there exists in C(e say V(σ), of i.i. subspaces under the action of Ri +j +k : V(σ) := {cos σ(xV3 + yV4 ) + sin σ(xV5 + yV6 ) : x, y ∈ R}.

(5.34)

Clearly, σ also parametrizes a 1-dimensional family of pairs of orthogonal i.i. 11 12 13 14

See (5.18) for the notation; f is sometimes called the (2-dimensional) ﬂip – see Golubitsky et al. (1988). We do not detail this assertion here. An analogous one, corresponding to the same reduced invariance, is treated explicitly in §5.4.3.6 for that subspace V1 . In a diﬀerent way than here, the search for additional i.i. subspaces is carried on by Ericksen (1996b), correcting the earlier treatment of Ericksen (1993). Consider the decompositions of Sym under the action of the subgroup R0 < Ri+j+k : the space of symmetric tensors commuting with all the elements of R0 coincides ˆ a ) in (5.27). Then, by the discussion in (a), any 2-dimensional subspace of with C(e ˆ ˆ a )⊥ can be decomposed as the orthogonal sum V V of C(ea )⊥ is i.i., so that C(e ˆ a )⊥ . any 2-dimensional subspace V with its orthogonal complement V within C(e

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

159

subspaces V(σ) and V (σ), where V (σ) denotes the orthogonal complement ˆ a )⊥ : of V(σ) within C(e π V (σ) = V(σ + ). (5.35) 2 The pairs V(σ) and V (σ) give inﬁnitely many decompositions ˆ a )⊥ = V(σ)V (σ). C(e

(5.36)

The pair V1 , V2 deﬁned by (5.29)–(5.30) corresponds to the choice σ = 0 in (5.36). The reduced symmetry group P is the same for all the i.i. subspaces V(σ), and has the generators (5.31). The action of the rhombohedral holohedry on the typical element of V(σ) produces six variants. The choice of ‘cubic’ basis (i , j , k ) above, which has the threefold axis along i + j + k and a twofold axis along i − j , reﬂects the fact that rhombohedral lattice groups are subgroups of cubic lattice groups. On the other hand, for the rhombohedral system other choices of the orthonormal basis are also common in the literature – see for instance Love (1927), Landau and Lifˇsits (1959), Gurtin (1972), Sirotin and Shaskolskaya (1982), Ericksen (1993), (1996b). In all the aforementioned references the basis, to be denoted by (c1 , c2 , c3 ), has the threefold axis along c3 . Moreover, one of twofold axes is along c1 in all15 but Love (1927) and Landau and Lifˇsits (1959), who take it along c2 (probably because that was the choice in the International Tables at that time). To compare with Ericksen (1993) it is convenient to discuss the choice of basis (c1 , c2 , c3 ) with twofold axis along c1 . The rotation R relating the two bases is given by c1 = Ri =

1 √ (i 2

− j ) and c3 = Rk =

1 √ (i 3

+ j + k ).

(5.37)

The matrix representation of any V ∈ V(σ) in the basis (c1 , c2 , c3 ) is conjugate to the representative matrix of V in the basis (i , j , k ) through the matrix representing the rotation R in the latter basis. By means of the convenient deﬁnition σ ¯ = σ − σ0 ,

for

cos σ0 = 3−1/2 , sin σ0 = −(2/3)1/2 ,

(5.38)

¯ σ ); here and by straightforward trigonometry, we have V(σ) = V(¯ ¯ σ ) = {cos σ ¯ 3 + yV ¯ 4 ) + sin σ ¯ 5 + yV ¯ 6 ) : x, y ∈ R}, V(¯ ¯ (xV ¯ (xV

(5.39)

¯ 3 = cos σ0 V3 + sin σ0 V5 , etc.. In the basis (c1 , c2 , c3 ) the represenwith V 15

This basis is used for the rhombohedral lattice in §3.4, where we regard it as a doubly centered hexagonal lattice – see also Fig. 3.8. This choice of basis will produce in §6.7 an elasticity tensor diﬀerent from, but easy to reduce to, the one given by Love (1927) and Landau and Lifˇsits (1959). Remember that all the choices of bases above correspond to one and the same holohedry: P (ea ) = Ri+j+k = Rc3 , so that 2π/3

R c3

2π/3

= Ri+j+k , etc.

160

EXPLICIT VARIANT STRUCTURES

tative matrices are: ¯3 = V



1 √ 2

0

1

0





0

0,

0

0

0

0

0

1

 1 √ 2

¯5 = V

1

0 1

1 √ 2

¯4 = V



0

0,

0

0

1

¯6 = V

0

0

−1

0

0

0

0

0

0

 1 √ 2

0

0 0



0,

(5.40)



0

1.

1

0

(5.41)

¯ 1, V ¯ 2 represented in the basis To these we can add the unit vectors V (c1 , c2 , c3 ) by the matrices in (5.50). These are the basis vectors and the ¯ σ ) used by Ericksen (1996b). parametrized family V(¯ It is instructive to compare these expressions with the hexagonal decomˆ a )⊥ are V(0) ¯ position in §5.4.3.6 below: there the i.i. subspaces in C(e and π ¯ V( 2 ), but the reduced symmetry group on the latter subspace is not fully generated by (5.31). 5.4.3.5 Tetragonal decompositions Let ea be any tetragonal basis and choose the vectors i , j , k in such a way that the holohedry P (ea ) is Tk – see §5.1.1. According to Table 5.2, the ˆ a ) is given by tensors with matrices of the form 2-dimensional subspace C(e   α

0

0

0

α

0;

0

0

δ

(5.42)

ˆ a ) is given for instance by the orthonormal vectors16 a basis for C(e     V1 =

1 √ 3

1

0

0 0

0

1

0,

0

1

V2 =

1 √ 6

1

0 0

0

0

1

0

0

−2

.

(5.43)

ˆ a )⊥ there are two mutually orthogonal 1-dimensional i.i. subspaces In C(e V1 = V3 , V2 = V6 , and a 2-dimensional subspace V3 = V4 , V5 , with     V3 =

1 √ 2

1

V4 = 16

0

−1

0

0

0

0

0

0

 1 √ 2

0

0

0 0

0,

1 √ 2



0

1

1

0

Of course we can also choose ˜ 1 = √1 V 2

V6 =

1

0

0

0

1

0

0

0

0

 , V5 =

1 √ 2

1

0 0

˜2 = V

1

0

0

0,

0

0

0

0

and

0

1

1

(5.44)



0

0

0

0

0

0

0

0

0

0

0

0

1

.

(5.45)

ˆ a ). The choice (5.43) is more convenient for the as an orthonormal basis for C(e analysis in §§6.7 and 7.5.

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

161

The reduced group on V1 and V2 is P = {1, −1}; in detail, restricting our π/2 attention to the generators Riπ , Rk of Tk , we have 1 ≈ 1 ≈ Riπ and −1 ≈ π/2 π/2 Rk on V1 , while on V2 we have 1 ≈ 1(≈ Riπ−j ) and Rk ≈ −1 ≈ Riπ . The action of the tetragonal holohedry on the typical element of V1 [V2 ] thus produces two variants which, based on Remark 5.10, have orthorhombic symmetry, with holohedry Oijk [Ok ,i ±j ]. By inspection of Fig. 5.1, if the tetragonal basis ea has centering P, I, I, the centering of V1 [V2 ] is P, F, I, [C, I, F,] respectively. The reduced group P on the 2-dimensional i.i. subspace V3 has 8 elements and is generated by – see (5.31)1 : f ≈ Riπ

3π π and r( ) ≈ Rk2 . 2

(5.46)

The action of the tetragonal holohedry on the typical tensor of V3 thus produces eight variants. By Remark 5.10, these have triclinic symmetry, except on suitable 1-dimensional monoclinic subspaces. These correspond to subspaces of R2 that are left invariant by some subgroup of the reduced group P. Since the latter is the group of symmetries of a square with center at the origin and edges parallel to the coordinate axes, one can see easily that invariant subspaces are: the origin itself (under the central inversion representing the monoclinic holohedry Mk but also, of course, under the whole of P); the two axes x and y in R2 , corresponding to the tetragonally conjugate holohedries Mj and Mi , respectively; and the two diagonals x = y and x = −y, corresponding to the tetragonally conjugate holohedries Mi −j and Mi +j , respectively. Excluding the origin, the centerings of the monoclinic variants corresponding to the P, I, I centering of the tetragonal basis ea can be read from Fig. 5.3: the variants on the x and y axes are P, C, C, while those on x = ±y are all centered. We use Lemma 5.3 to show the uniqueness of the decomposition ˆ a ) = V1 V2 V3 ; C(e

(5.47)

ˆ a )⊥ , if any, are reindeed, the possibilities of further i.i. subspaces in C(e stricted to the 1-dimensional subspaces of V1 V2 diﬀerent from V1 and V2 . Any nontrivial element W of such a subspace W has the form W1 + W2 , where Vr 6 Wr $= 0, r = 1, 2. Since Riπ W1 Riπ = W1

and Riπ W2 Riπ = −W2 ,

(5.48)

ˆ a )⊥ is (5.47).17 W cannot be invariant, hence the only decomposition of C(e 17

Consider the decomposition of Sym under the action of the cyclic subgroup T0 < Tk π/2 generated by Rk : the space of tensors commuting with all the elements of T0 coˆ a ) in (5.42), and the subspaces Vi , i = 1, 2, 3, are also i.i. under the incides with C(e action of T0 . However the reduced actions of T0 on V1 and V2 imply that any 1dimensional subspace W of V1 V2 is i.i. under T0 , with reduced group P = {1, −1}. ˆ a )⊥ = W W V3 , Therefore in this case there are inﬁnitely many decompositions C(e

162

EXPLICIT VARIANT STRUCTURES

5.4.3.6 Hexagonal decompositions Let ea be any hexagonal basis and the unit vectors i , j , k be such that the holohedry P (ea ) is Hk – see §5.2.18 According to Table 5.9, the 2-dimenˆ a ) is given by the tensors with matrices of the form sional set C(e   α

0

0

0

α

0;

0

0

δ

(5.49)

ˆ a ) is given for instance by an orthonormal basis for C(e    V1 =

1 √ 2

1

0 0

0

0

1

0

0

0

0

and V2 =  0 0

0

0



0

0.

0

1

(5.50)

ˆ a )⊥ there are two mutually orthogoIn the orthogonal complement C(e nal 2-dimensional i.i. subspaces V1 = V3 , V4 and V2 = V5 , V6 , these ¯3 tensors having in the basis (i , j , k ) the same matrix representation as V ¯ to V6 in (5.40)–(5.41), respectively. The reduced group P1 on V1 has order six and is generated by f ≈ Riπ

and r(

π 2π ) ≈ Rk3 3

(5.51)

(recall (5.31)1 ). Thus the action of Hk on the typical element of V1 produces six monoclinic variants, the holohedry being Mk by (5.51)2 . In addition, (5.51)1 implies that V1 contains an orthorhombic subspace (corresponding to x = 0 in R2 ) whose holohedry is Oijk , the one containing Mk and Mi according to Table 5.2; also, the latter implies that the orthorhombics are base-centered. The action of the hexagonal holohedry produces √ two more subspaces of centered orthorhombics in V1 , of equation y = ±x 3/3; the holohedries are again given in Table 5.2. The reduced group P2 on V2 has order twelve; generators are 5π π (5.52) f ≈ Riπ and r( ) ≈ Rk3 . 3 Thus the action of the hexagonal holohedry on the typical element of V2 produces twelve triclinic variants. By (5.52)1 the subspace of V2 of equation x = 0 is made of centered monoclinics with holohedry Mi . Since P2 is the group of symmetries of a regular hexagon centered at the origin, there √ are in V2 two more subspaces of centered monoclinics, of equation y = ±x 3/3, whose holohedries are hexagonally conjugate to Mi as in Table 5.11. The symmetries of the hexagon also imply that there is in V2 another triple of subspaces of centered monoclinics with hexagonally conjugate holohedries;

18

where W is any 1-dimensional subspace in V1 V2 and W is its orthogonal complement within V1 V2 . This basis (i, j, k) coincides with the basis (c1 , c2 , c3 ) in (5.37).

5.4 IRREDUCIBLE INVARIANT SUBSPACES FOR THE HOLOHEDRIES

163

one√has equation y = 0 and holohedry Mj ; the others have equation y = ±x 3 and holohedries given in Table 5.12. To check the uniqueness of the decomposition ˆ a )⊥ = V1 V2 C(e

(5.53)

we use Lemma 5.3, according to which any other nontrivial i.i. subspace ˆ a )⊥ must be 2-dimensional, with any element W $= 0 of the form W of C(e W1 + W2 , where Vi 6 Wi $= 0, i = 1, 2. Since and Rkπ W2 Rkπ = −W2 ,

Rkπ W1 Rkπ = W1

(5.54)

W would properly intersect both V1 and V2 , contradicting their irreducibility; therefore (5.53) is the unique decomposition. 5.4.3.7 Cubic decompositions Let ea be any cubic basis; we choose the unit vectors i , j , k along the three fourfold axes, in such a way that P (ea ) = Cijk – see §5.1. According to ˆ a ) is 1-dimensional, being given by the tensors Table 5.1, the space C(e   V = αV1 ,

V1 =

1 √ 3

1

0 0

0

0

1

0.

0

1

(5.55)

ˆ a )⊥ there are two mutually orthogonal i.i. subspaces: a 2-dimensional In C(e one, V2 = V2 , V3 , and a 3-dimensional one, V3 = V4 , V5 , V6 , where the Vr are as follows:     V2 =

1 √ 6

1

0 0

 V4 =

1 √ 2

0 

V6 =

1 √ 2

0

0

0

1

0

0

−2

0

0

,

1,

0

1

0

0

1

0

0

1 √ 2



0

1

V3 =



0

0.

0

0

1

−1

0

0

 V5 =

1 √ 2

0

0

0

0 1

0

0

0,

(5.56)

0 1



0

0,

0

0

(5.57)

The reduced symmetry group P on V2 has order six, and – recall (5.31)1 – is generated by π 4π 2π 3 f ≈ Rk2 and r( ) ≈ Ri +j (5.58) +k ; 3 the action of Cijk on the typical element of V2 produces six orthorhombic ‘mixed axes’ variants. Corresponding to the centerings P, F, I of the cubic basis ea , the orthorhombic centering, given by Table 5.1, is C, I, F. As in the two previous sections, the subspace of V2 of equation x = 0 is more symmetric, indeed is tetragonal with holohedry Tk and centering P, I, I. Two

164

EXPLICIT VARIANT STRUCTURES

more tetragonal subspaces of the same centering exist in V2 , of equation √ y = ±x 3/3 and holohedries given in Table 5.2. The reduced group P on V3 is a group of orthogonal matrices whose order is twentyfour and whose generators are π

π

− R1 ( ) ≈ Rk2 2

and

π

π

− R2 ( ) ≈ Rj2 ;

(5.59)

2

here Ri ( π2 ) denotes the rotation by π2 about the i-th axis in R3 . This group is actually the full symmetry group, in the crystal class ¯43m, of a regular tetrahedron whose threefold axes are along the body diagonals of the unit cube in R3 . The action of Cijk on the typical element of V3 produces twenty-four triclinic variants. Again, V3 contains subspaces of symmetry higher than triclinic. It is not diﬃcult to see that the only nontrivial subspaces left invariant by some subgroup of tetrahedral symmetries are the four body diagonals, on each one of which the holohedry is one of the four rhombohedral ones given in Fig. 5.1, and the three lines joining the midpoints of opposing tetrahedral edges, for each one of which the holohedry is Oijk , giving orthorhombic ‘cubic edges’ variants, also given in that ﬁgure. Accordingly, the centering of the latter is P, F, I. ˆ a ) there are no cubic i.i. subspaces Lemma 5.3 guarantees that in C(e other that the ones listed above.

CHAPTER 6

Energetics As anticipated in the Introduction, in recent years there has been a successful eﬀort to use nonlinear elasticity theory, based on molecular considerations, for modelling the behavior of crystals in the range of ﬁnite deformations, like the ones encountered in mechanical twinning or in symmetrybreaking diﬀusionless phase transformations. These typically lead to the formation of coherent microstructures in which the phase variants mix in a great variety of conﬁgurations, and these eﬀects are important for understanding the macroscopic behavior of materials; the latter can be very interesting also from the point of view of the applications. For instance, certain alloys exhibit shape-memory properties due to the ability of the material to form an array of self-accommodating equilibrium phase mixtures, involving periodic (patches of) twinned microstructures, in response to the imposed boundary conditions such as given loads or deformations. The nonlinear elastic model for phase transitions and twinning in crystals presented here originated from the work of Ericksen (1977), (1980b), (1984), (1987), (1989); based on molecular theories and the Cauchy-Born hypothesis, he obtained new invariance properties for the constitutive functions of nonlinear elasticity, described by a group conjugate to GL(3, Z). In this chapter we introduce the constitutive equations that are at the basis of the elastic approach to mechanical twinning and phase transitions in crystals, and study their general properties. In addition to the aforementioned references, this approach is presented, among others, by Ball and James (1987), (1992), Fonseca (1987), Kinderlehrer (1987), (1988), (1992), Chipot and Kinderlehrer (1988). We do not consider restrictions coming from imposed displacements on the boundary, and only in chapter 10 mention the corresponding Dirichlet boundary-value problem. Also applied loads are neglected (again, the dead-load boundary-value problem being mentioned in §10.1), with the incidental exception of a pressure load in §6.6.1; pressure could be added in general, and here is neglected for simplicity. Harder, and more interesting, is the case of a nonhydrostatic stress imposed on the boundary; the reader may see, for instance, Parry (1982a), (1984b), James (1984a), (1986a), Budiansky and Truskinovsky (1993), Simha and Truskinovsky (1996). Aiming at formulating a phenomenological nonlinearly elastic model of crystal mechanics, we postulate the existence of a potential function for a crystalline lattice which, consistently with the indications of molecular 165

166

ENERGETICS

theories, can be calculated explicitly as the potential energy φ per unit cell associated with the interatomic forces. Following the early works of Cauchy (1828a,b), (1829) and, later, Poincar´e (1892), Voigt (1910), Born (1915), the classical molecular theories of elasticity, based on a number of assumptions regarding the atomic force laws and the lattice deformations, construct the internal (or stored) energy or the stress in a simple lattice as a function of the positions of the lattice points. In addition, since the energy is unaﬀected by any permutation of the physically indistinguishable lattice points, and since the set of all lattice positions is uniquely determined by a current lattice basis ea up to an inessential translation, the energy φ depends on the lattice conﬁguration only through the basis itself. More recent examples of such derivations can be found in Love (1927), Stakgold (1950), Born and Huang (1954), Eftis et al. (1971), Hill (1975), Ericksen (1977). We add the assumption that φ also depends on a thermal variable, which can be either the absolute temperature or the entropy per unit cell. Intuitively, these variables should take into account the fact that the lattice points are actually average, or most probable, positions about which the lattice units vibrate; this oscillatory motion too having an associated energy. One may proceed more formally by statistical arguments, assuming the lattice points to be distributed in space according to a triply periodic probability density. Then, the entropy per unit cell is the integral over the cell of the negative logarithm of the probability density, while the absolute temperature appears as a parameter if the probability distribution is the canonical one. With this as a motivation only, we take both internal energy and entropy into account by considering the (Helmholtz) free energy φ per unit particle as the thermodynamic potential for simple lattices. The molecular calculations are thus consistent with the hypothesis that the free energy density φ of the lattice, which at zero absolute temperature coincides with the internal energy, be given in general as a function of the lattice vectors and the temperature, and this will be our basic assumption in §6.1. Our subsequent analysis will be based on as few a priori requirements as possible on the form of the constitutive function for φ; the development of the model will only rely on: 1. the analysis of the fundamental invariance properties of the lattice constitutive equations, which will be inferred from the global symmetry of lattices studied in chapter 3 (§6.1); 2. a general hypothesis for bridging the molecular and the continuum descriptions: lattice vectors behave as material vectors (§§6.2, 6.3); and 3. some basic assumptions on the minimizers (potential wells) of φ (§6.4). The continuum theory based on a free energy function having the invariance so obtained is ﬂexible enough to deal with large deformation phenomena in multiphase elastic crystals. On the other hand, this invariance is quite diﬀerent from the one following from the common view that the material symmetry of a crystalline solid is given by the point group describing the

6.1 INVARIANCE OF SIMPLE-LATTICE ENERGIES

167

geometric symmetry of the reference crystal lattice. And, indeed, there is no compelling reason for this assumption, as the additional symmetries, which are not accounted for in the classical theories, are essential for describing phase transformations and twinning. Perhaps the assumption may be acceptable for suitably small deformations, as linear elasticity has been successfully used to model the behavior of crystalline solids in small strain. The two positions can be reconciled (§§6.5.1, 6.5.2) by localizing the present theory, that is, by suitably restricting the domain of the constitutive functions. This is done by means of Proposition 4.1, so as to avoid the large shears connected with plasticity phenomena, and by using the results in §§4.3, 4.4; the ensuing energy well structure is analyzed in §§6.5, 6.6, and some of the many implications will be studied in the next chapters. In this approach the invariance of the constitutive functions depends on the choice of a (reference) conﬁguration and on the way the functions’ domains are restricted about that conﬁguration. This construction points out explicitly the subdomain of the energy function which is relevant in many interesting cases of symmetry-breaking solid-state phase transformations, and of connected phenomena like mechanical twinning. It can also rationalize the aforementioned practice of choosing the invariance group of the energy to be the point group of the reference lattice. Under suitable hypotheses, this method also leads to the classical linear theory of elasticity of crystals, which is universally adopted to establish the elastic moduli of the diﬀerent substances, as recalled in §6.7. We stress that in our framework energies satisfying the invariance dictated by molecular considerations will be constructed, for instance in polynomial form as in chapter 7, rather than calculated from molecular models. 6.1 The energy function of a simple lattice and its global invariance Until chapter 11 we take for granted that the structure of crystals can be described by means of a simple lattice in all their allowed conﬁgurations.1 Therefore, as anticipated above, we assume the free energy φ per unit cell of a deformable crystalline body whose current conﬁguration is a simple lattice L(ea ) to be given by a smooth enough function φ¯ of the current lattice basis ea and of the absolute temperature θ: ¯ a , θ). φ = φ(e (6.1) 1

While, as mentioned in Remark 3.1, several elements crystallize exactly as simple lattice structures, this is not true in general. However, as was mentioned in the introduction to chapter 3, depending on the phenomena of interest and on how detailed their description is, it is sometimes possible to regard as simple lattices also crystals whose structure is essentially a multilattice (chapter 11), by disregarding the internal variables giving the positions of any atoms in the unit cell. This simpliﬁcation is often successfully adopted in the literature even for metallic alloys, such as some shape-memory alloys, which are not true crystals because the alloying element may be randomly substitutional in the lattice.

168

ENERGETICS

In principle, the domain of φ¯ is the space B of all bases2 together with a suitable interval I of admissible temperatures. We suppose φ to be a Galilean invariant quantity, hence the function φ¯ is invariant under superimposed orientation-preserving isometries, following the more conservative point of view mentioned in footnote 8, chapter 2: + ¯ a , θ) = φ(Qe ¯ φ(e a , θ) for any ea ∈ B, Q ∈ O , θ ∈ I.

(6.2)

We also admit φ to depend on the positions of the points of L(ea ), hence the value of φ¯ in (6.1) must be the same at any basis generating the same lattice L(ea ). By Proposition 3.1 the lattice vectors ea for L(ea ) are determined up to transformations in GL(3, Z); this result and (6.2) imply that the energy function φ¯ of a simple lattice exhibits the following basic invariance: ¯ a , θ) = φ(m ¯ b eb , θ) = φ(m ¯ b Qeb , θ), φ(e a a

(6.3)

for all ea ∈ B, Q ∈ O , m ∈ GL(3, Z), and θ ∈ I. The GL(3, Z)-invariance of φ¯ reﬂects the 3-dimensional periodicity of lattices. The fact that GL(3, Z) contains the central inversion −1 implies that (6.3) must actually hold for all Q ∈ O. Equivalently, φ actually depends on the vectors ea only through their scalar products, that is, it only depends on the current lattice metric C – see (3.2): +

¯ a , θ) = φ(C, ˆ φ = φ(e θ) ,

Cab := ea · eb ;

(6.4)

the domain of φˆ is the 6-dimensional space Q> 3 of all simple lattice metrics times the interval I. Then (6.3) is equivalent to: ˆ ˆ t Cm, θ), φ(C, θ) = φ(m

(6.5)

for all C ∈ Q> 3 , all m ∈ GL(3, Z), and all θ ∈ I. This fundamental invariance of the energy functions φ¯ and φˆ of simple lattices was remarked by Parry (1976) and Ericksen (1977), (1979), (1980a,b), (1989); based on relation (6.5), Ericksen proposed GL(3, Z) as the global invariance group for the constitutive function of any simple lattice.3 In §6.3 we will restrict the energy domain to the wt-nbhds in B or Q> 3 (Proposition 4.1), and will then study how the energy invariance is aﬀected, showing that it actually reduces to the crystallographic invariance assumed in the classical theories of crystal mechanics. If the crystal structure of the material is more complex than a simple lattice (a multilattice or n-lattice as in chapter 11), the function φ¯ in (6.1) 2 3

Pitteri (1990) shows how and under which conditions the invariance properties found below can hold also for an energy function with a smaller domain. The constitutive functions for the other thermodynamic potentials in §2.5.1 exhibit the same invariance. Ericksen (1977) also remarks that (6.4)– (6.5) hold for the internal energy function of a simple lattice if, following Cauchy (1828a,b), (1829), the lattice points are assumed to interact through central forces whose magnitude depends only on the distance between the interacting atoms, that is, if the force derives from a pair potential depending only on their mutual distance. See §6.6.2 for some details on the well known Cauchy relations that are a consequence of these hypotheses.

6.2 THE CAUCHY-BORN HYPOTHESIS

169

depends also on a number of internal variables pi , i = 1, . . . , n − 1, called shifts, which give the current positions of the extra atoms (the motif ) in the unit skeletal cell of the crystal – see chapter 11, in particular (11.88): ¯ a , pi , θ) φ = φ(e

(6.6)

(Ericksen (1970), (1997), Parry (1978), (1981), (1984c), (1987)). Throughout this volume we assume that the 1-lattices or the n-lattices remain such in all their allowed conﬁgurations; in other words, the given number n of their constituting simple lattices does not change during the physical processes the multilattice may undergo.4 Also for multilattices one can obtain a function φE (ea , θ) of the form in (6.1) by minimizing out the internal variables pi from the energy function (6.6). The procedure, outlined in §11.7.1, is an interpretation (Ericksen (1980a), (1997), (2001b), Parry (1981), (1982b), (1984a), (1987), James (1987)) of one suggested by Born and generally accepted, and gives the energy of a multilattice as a possibly multivalued function of the lattice vectors and of the temperature. In fact, in order to obtain a theory that resembles elasticity as much as possible, the shift vectors of a multilattice are treated as molecular counterparts of internal variables in an extended elastic model (Maugin (1990)). The energy is then expressed as a function of an adequately deﬁned deformation gradient, related to the selected lattice vectors ea by (6.7) below. The analysis also shows the kind of problems that may arise when crystals are modelled as simple lattices glossing over the details of the motif. 6.2 Passage to the continuum description: the Cauchy-Born hypothesis For a long time linearly elastic theories have been successfully used to model various aspects of the behavior of crystalline solids, and the molecular theories of elasticity provide a way to ascertain the properties of the energy density of crystals considered as continua. In order to take advantage of the methods of continuum mechanics, and obtain a phenomenological model for crystals viewed as thermoelastic continua (§2.5) we must establish a con4

Consequently, we take n to be the smallest possible, and use the corresponding descriptors, which are called essential. Based on the symmetry properties of multilattices, to be obtained in chapter 11, we will describe the domain and invariance of the energy functions that are appropriate for crystals, away from the submanifolds on which the multilattice descriptors become nonessential (§11.7). As remarked in §11.2.3, physically this procedure is not always right, because crystals are known, for instance, to undergo phase changes from f.c.c. to h.c.p. conﬁgurations; the ﬁrst being simple lattices (§3.4) and the second 2-lattices (§11.5.2). In fact, Parry (1981) considers a particular phase change from simple lattice to 2-lattice. Certain aspects of an energetic model based on nonlinear elasticity theory for such phenomena require further thinking, and we do not consider them here.

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nection between the molecular and continuum descriptions of materials.5 This is done by linking the macroscopic deformation of the crystal to the change of atomic positions in the lattice. The earliest attempts by Cauchy (1828a,b), (1829), to obtain macroscopic elastic stress and energy from atomic-level calculations, were based on the hypothesis that the atomic motion simply agrees with the gross deformation. Born (1915) (also in Born and Huang (1954)), elaborating on earlier ideas of Thomson (Lord Kelvin) (1878), appreciated that this is not always the case, and modiﬁed Cauchy’s assumption by making the physically reasonable hypothesis that only the skeletal structure of the crystalline lattice is embedded in the macroscopic deformation, while the microstructural motif is free to adjust so as to reach equilibrium. This has become the standard assumption, used quite universally albeit often tacitly, for relating the deformation of the continuum to that of the lattice in the molecular theories of elasticity, not only linear. It is sometimes referred to as the Cauchy-Born hypothesis or, brieﬂy, the Born rule (Ericksen (1977), (1984), (1997), Zanzotto (1992), (1996a)); a critical presentation and historical remarks are also given by Love (1927), Stakgold (1950), Hill (1975). 6.2.1 The Born rule In order to state the rule more precisely, assume that a crystal, viewed as a deformable continuum, occupies a homogeneous reference conﬁguration R as in §2.5. Let us also assume the molecular analysis of the crystal in R to reveal that, possibly away from the boundary, its atoms or molecules be arranged in a periodic structure describable, if extended to the whole space, by a reference multilattice M0 constituted by a simple skeletal lattice L0 := L(ea0 ), and possibly a motif given by variables pi0 , (i = 1, . . . , n − 1) (see §11.1.1). The ea0 are called reference lattice vectors or reference basis. Now, suppose that the crystal in the reference conﬁguration R with lattice L0 experiences a homogeneous deformation whose gradient is F . The Born rule states that the vectors ea obtained from the reference lattice vectors ea0 through the macroscopic deformation gradient: ea = Fea0 ,

(6.7)

constitute a skeletal basis for the crystal in the macroscopically deformed conﬁguration. In other words, the lattice vectors of L0 are assumed to behave as material vectors (§2.5), that is, L0 is embedded in the continuum and participates in the macroscopic deformation.6 5

6

These two descriptions, involving diﬀerent observation lengthscales and diﬀerent experimental methods, are often used for the same material samples. The ﬁrst usually takes advantage of X-ray diﬀraction or electron microscopy, and analyzes the periodic arrangement of the atoms or molecules in the crystal lattice. The second is based on the various procedures that are available for the analysis of the macroscopic deformation of solids. The rule does not assume that the microstructural motif participates of the gross

6.2 THE CAUCHY-BORN HYPOTHESIS

171

The Born rule is of widespread use not only in theoretical considerations, but also in the experimental practice. For instance, it is implicitly used for connecting the measurements of compressibility or thermal expansion of a material, performed directly on the crystalline lattice with microscopic techniques, to the data obtained through macroscopic experimental methods (with exceptions, see next subsection). 6.2.2 Failures of the Born rule Although a quite natural hypothesis of very widespread use, the Born rule does not always hold, and is indeed observed to fail in various circumstances. Indeed, in real materials the deformation of the skeletal lattice is not always related in a straightforward way to the one of the macroscopic body. Here we only brieﬂy mention some of the related questions. First of all, in various cases one must choose in a suitable way the reference skeletal basis ea0 in order for (6.7) to hold; the latter formula may be false for unsuitable choices of the skeletal basis (Ericksen (1980a,b). Secondly, there are literal violations, of various degree. The simplest violations of the rule occur, rather unexpectedly, in the ‘small’ deformations involved in the homogeneous thermal expansion of some materials. For instance, Balzer and Sigvaldson (1979) report a disagreement in the results of simultaneous measurements of the change in the lattice parameters and in the macroscopic dimensions of zinc single crystals during free thermal expansion; the authors assume that atomic vacancies in the lattice are responsible for the discrepancies observed between the microscopic and macroscopic observations. Furthermore, in certain deformations experimentally observed, particularly the twinning shears that will be studied in §8.2, the Born rule does not apply to the skeletal lattice, but only to a sublattice, which in general depends on the twinning mode considered. This has forced students of the phenomenon of twinning to formulate the so-called Mineralogists’ Assumption as a twinning condition (§8.2). For several materials these twinning deformations have the property that no choice of the reference vectors ea0 in (6.7), not even for a sublattice, produces the correct lattice vectors for describing all the twinned lattices in the deformed conﬁgurations. We will return to these violations of the Born rule in §8.4.2, where we comment on the implications of twinning for the rule and for the material symmetry of crystals. deformation as does the skeletal lattice: the shifts pi of the lattice in the deformed conﬁguration and the reference shifts p0i are not required to obey a kinematical rule like (6.7). Rather, as already mentioned, Born assumes that the shifts pi evolve so as to guarantee immediate internal equilibration of the motif in the deformed multilattice, and thus are determined, for each skeletal basis ea , by conditions (11.97) or (11.99) for absolute or local minimization of the free energy at given ea . Other evolution laws are also possible, such as a gradient ﬂow – see footnote 40 in chapter 11. The Born rule (6.7) is indeed independent of any evolution assumptions made for the internal variables pi .

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These considerations suggest that the application of the Cauchy-Born hypothesis should be carefully checked in each case, and may not always be taken for granted. This is an important problem for several reasons; for instance, in lack of an hypothesis like Born’s, it is unclear how to describe the energy of the crystal regarded as a deformable continuum, and how to establish from molecular arguments any invariance properties of its constitutive function or the structure of its minimizers. In general it may even be diﬃcult to decide whether and how any continuum theory such as nonlinear elasticity can describe the behavior of a material. We refer to Zanzotto (1988), (1992), (1996a), for a more detailed discussion of these problems and some of their consequences, and to Ericksen (1997), (1999), (2000a,b), (2001c), (2002b,c) for a description of material behavior when the rule need not be called upon. 6.3 Thermoelastic constitutive equations for crystals The main goal here is to use continuum methods for modelling crystal behavior, which requires qualifying the constitutive functions of crystalline materials within nonlinear thermoelasticity theory. To do this we only need to consider homogeneous deformations – see §2.5. By means of the Born rule (6.7) we obtain, for a crystalline material, a constitutive function that depends on the gradient F = Dχ of the macroscopic deformation χ of the body, deﬁned in (2.47). Let R be a reference conﬁguration for a crystalline body whose structure is described by a reference simple lattice L(ea0 ), and also assume, for simplicity, that the unit cell of L(ea0 ) has unit volume (for otherwise there is a constant factor in front of the function φ˜ deﬁned in (6.8) below). From the free energy function φ¯ of the lattice in (6.1) the following deﬁnition produces a macroscopic constitutive function φ˜ that gives the free energy per unit reference volume of the crystal considered as a continuous medium: 0 ˜ , θ) := φ(Fe ¯ ¯ a , θ), φ(F , θ) = φ(e (6.8) a

where the reference lattice vectors ea0 are understood as ﬁxed. The energy function φ˜ depends on the current macroscopic deformation gradient F ∈ Aut, and ﬁts the usual format of nonlinear thermoelasticity (§2.5.1). By means of (6.7), from any function g¯(ea , . . .) of the current lattice vectors in the molecular description one can deﬁne, in the continuum description, a function g˜ given by g˜(F , . . .) = g¯(Fea0 , . . .).

(6.9)

In particular, also for crystals whose structure is described by a multilattice M0 as in chapter 11 the Born rule allows one to obtain a thermoelastic constitutive function φ˜ for the continuum free energy from the lattice energy φ¯ in (6.6). By analogy with (6.8) this is, in obvious notation, 0 ˜ , i , θ) = φ(Fe ¯ ¯ a , pi , θ), φ(F , F (p 0 + i ), θ) = φ(e (6.10) a

i

6.3 THERMOELASTIC CONSTITUTIVE EQUATIONS FOR CRYSTALS

173

where, again, the reference lattice vectors ea0 and shifts pi0 are held ﬁxed. The constitutive equation in (6.10) gives the macroscopic free energy of the continuum as a function of the current deformation gradient F and of the internal variables i , ﬁtting an extended format of elasticity theory (see, for instance, Parry (1981), (1987), Capriz (1989), Maugin (1990)). By the results in §11.7.1 we can also apply the analogue of (6.8) to the function φ¯E (ea , θ) in (11.98), obtaining a free energy function φ˜E (F , θ) for a complex crystal in which the internal variables are internally equilibrated: φ˜E (F , θ) = φ¯E (Fea0 , θ).

(6.11)

In this function the internal variables do not appear, but φ˜E (F , θ) may be multivalued because this may hold for φ¯E (ea , θ), as discussed in §11.7.1.7 ˜ , θ) introduced in (6.8) one By means of the constitutive function φ(F deﬁnes the free energy functional of the unloaded body in the reference conﬁguration R, in agreement with (2.75):

˜ Φ[χ; θ] := φ(Dχ(x ), θ) dx . (6.12) R

˜ , i , θ) in Of course in the modelling also the functions φ˜E (F , θ) or φ(F (6.11) or (6.10) may be used, the latter together with some evolution law for the i , as for instance in Bhattacharya et al. (1993). As mentioned in §2.5.2, from the mathematical point of view a general goal is understanding the minimizers of (6.12) at a given temperature θ, possibly subject to suitable boundary conditions. In this book we address only the most elementary cases of this problem, in which no boundary conditions of displacement or loads are imposed, and in which the diﬀerent solid phases are mixed in the simplest geometries. The main ingredients in these phenomenological models of transition phenomena are appropriate hypotheses regarding the minimizers of the integrand in (6.12), whose invariance, coming from (6.3) and (6.8), is not the invariance of the energy functions classically considered in thermoelasticity. We discuss these matters in more detail in the following sections of this chapter. The literature quoted in the Introduction and in §2.5.2 adds details on the treatment of many variational problems based on the functional (6.12), which arise in the phenomenological modelling of crystalline materials undergoing martensitic phase transformations. 6.3.1 Invariance of the response functions of elastic crystals The invariance of the constitutive function φ˜ in the continuum theory is a consequence of its deﬁnition (6.8) and of the invariance of the lattice energy function φ¯ in (6.3). In §3.1 we have deﬁned the global symmetry group 7

According to the remarks in §6.2.2, for some multilattices there are no bases e0a to which (6.7) can be applied to obtain a macroscopic energy function as in (6.11).

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G(ea ) of any simple lattice L(ea ); recall that G(ea ) is the representation of GL(3, Z) – see (3.9) – consisting of the unimodular tensors H represented by an integral matrix in the lattice basis ea : Hea = mba eb

or H = mba eb ⊗ e a , (mba ) ∈ GL(3, Z).

(6.13)

Formulae (3.9), (6.3), (6.8), and (6.13) imply that the invariance of φ˜ is: ˜ , θ) = φ(QFH ˜ φ(F , θ) for any F ∈ Aut, H ∈ G(ea0 ), θ ∈ I, Q ∈ O, (6.14) ˜ which summarizes both Galilean invariance and material symmetry of φ. By comparison with (2.58) and (2.60), in this model the material symmetry of a crystalline body regarded as a thermoelastic continuum is given by the global symmetry group G(ea0 ), where ea0 is a basis of the reference crystal lattice. This constitutes the proposal of Ericksen (1970), (1977), (1979), (1980a,b), (1987) for the material symmetry of elastic crystals.8 The above point of view transfers into the continuum theory the basic fact that a crystalline material has a periodic structure, rather than tying the invariance of its continuum energy to the point group symmetry of any one of its equilibrium conﬁgurations, as is implicitly done in the classical theories (Truesdell and Noll (1965) or Truesdell (1977)). We have seen in §6.1 that the invariance of the lattice energies is given by the group GL(3, Z) and is the same for all crystals; likewise, in the continuum theory the material symmetry group, which is a representation of GL(3, Z), is also abstractly the same for all crystals, and always contains nonorthogonal transformations. According to Ericksen (1977), (1987), the role of the latter is of main importance for a continuum theory of mechanical twinning (§8.2). As in §2.4.2, the orthogonal invariance in (6.14) implies that the continuum free energy depends on F only through the right stretch tensor U appearing in its polar decomposition F = RU – see (2.20) – or, equivalently, on the Cauchy–Green tensor C = F t F = U 2 – see (2.59): ˜ , θ) = φ(U ˜ , θ) =: φ(C ˇ , θ). φ(F

(6.15)

Here, as in (3.79), C is represented in the reference basis ea0 by the lattice metric C in (6.4); therefore ˇ , θ) = φ(e ˆ 0 · Ce 0 , θ), φ(C a b

or

ˆ ˇ ab e a ⊗ e b , θ). φ(C, θ) = φ(C 0 0

(6.16)

Clearly, (6.15) or (6.5) imply that the invariance of φˇ is given by: 8

Also for a multilattice the Born rule can be used to construct a macroscopic consti˜ ˜ i , θ) as in (6.10). In this case φ(F, i , θ) inherits its invariance tutive function φ(F, from the one of the multilattice energy (6.6), which is given in §11.7. The same considerations as the ones leading to (6.14) apply to the invariance of the free energy functions φ˜E (F, θ) and φ¯E (ea , θ) of an internally equilibrated multilattice ˜ θ) – see (6.11) and (11.98) – so that φ˜E (F, θ) exhibits the same invariance as φ(F, in (6.14). This gives, for a complex crystalline material whose motif is disregarded, a thermoelastic response function with properties analogous to the ones of simple lattices, except perhaps for smoothness.

6.4 ENERGY MINIMIZERS AND THEIR GENERAL PROPERTIES

ˇ , θ) = φ(H ˇ t CH , θ), φ(C

175

for any C ∈ Sym > , H ∈ G(ea0 ), θ ∈ I. (6.17)

Notice that any tensor H ∈ G(ea0 ) such that H t CH = C is represented in the reference basis by an element of the lattice group of L(Ue 0a ). A ﬁnal remark: the constitutive functions for the continuum and the material symmetry groups introduced above all depend on the reference conﬁguration R chosen for the body. If the reference state is changed as in §2.5.1, and the reference lattice vectors are also correspondingly changed according to the Born rule, the functions and their invariance groups change as expected in elasticity theory, that is, as in (2.57)–(2.61). 6.4 Energy minimizers and general properties of the constitutive functions of elastic crystals In the preceding sections we have clariﬁed the basic invariance properties of the constitutive functions of elastic crystalline solids. The next most important step in the formulation of a thermoelastic model for multiphase crystals is making suitable assumptions regarding the minimizers of the constitutive functions.9 Such minimizers describe the homogeneous deformations that are (meta)stable equilibria of the crystal in the absence of external loads or assigned boundary displacements, that is, the energetically ‘most favorable’ lattice structures of that crystal at any given temperature. In the rest of this chapter we discuss these points, and a domain restriction for the response functions.10 6.4.1 Multiplicity of the symmetry-related minimizers An important consequence of the invariance properties (6.3) regards the ¯ a , θ) of structure of the sets of minimizers for the free energy function φ(e ˆ ˜ ˇ a crystalline material, or for the functions φ(C, θ), φ(F , θ), φ(C , θ). As in §6.2.1, let ea0 be the reference lattice basis for the crystal, and assume the ¯ a , θ). The invariance properties (6.3) basis ¯ ea ∈ B to be a minimizer of φ(e ¯ of φ immediately imply that in B there are inﬁnitely many other minimizers 9

10

Unless otherwise stated, here ‘minimizer’ refers to a local minimizer of a given constitutive function for ﬁxed θ, and we assume that, except otherwise dictated by symmetry requirements and except at isolated temperatures, such local minimizers can be recognized through the analysis of the second derivatives at constant θ. For instance, ˆ in the case of φ(C, θ) we assume minimizers to be such that the matrix of the second ˆ be positive deﬁnite. We cannot make derivatives with respect to Cab (the hessian of φ) ¯ a , θ) related to the corresponding assumption for the minimizers of the function φ(e ˆ φ(C, θ) through (6.4), as Galilean invariance implies the existence of null vectors in ¯ Analogously for φˇ and φ. ˜ the space B for the second diﬀerential of φ. Although we will usually be more explicit regarding the properties of the lattice constitutive functions, it will be understood that the properties of the latter translate, through the deﬁnitions in §6.3, into analogous properties of the response functions for the thermoelastic continuum.

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¯ which are collected in inﬁnitely many disjoint copies of the set O of of φ, ¯ (6.3) says that φ¯ is also orthogonal tensors. Indeed, when ¯ ea minimizes φ, minimized by all the bases in B belonging to the O-orbits ¯¯ Omba¯ eb = {mba Q¯ eb : Q ∈ O} = {Q H ea : Q ∈ O},

(6.18)

¯ ∈ G(¯ for m ∈ GL(3, Z) or, equivalently, for H ea ), the group given by the analogue of (3.9) for ¯ ea . Recall that the bases in the O-orbits above are all the elements of B that are symmetry-related to the minimizer ¯ ea (see (3.35)). These bases generate, up to orthogonal transformations and changes of lattice vectors, one and the same stable stress-free lattice structure L(¯ ea ) of the crystalline substance at the temperature θ. An O-orbit ¯ of minimizers of φ¯ is often called a well of φ. A well O¯ ea ⊂ B corresponds to a single lattice metric C¯ ∈ Q> 3 which ˆ minimizes the function φ(C, θ); likewise, any other well Omba¯ eb in (6.18) corresponds to a metric ¯ mt Cm for m ∈ GL(3, Z).

(6.19)

These metrics – see Fig. 4.1 – constitute the GL(3, Z)-orbit of C¯ in Q> 3, ˆ and they all minimize φ due to the invariance properties (6.5). The orbits (6.18) and (6.19) give the basic structure of the minimizers of the crystal’s energies φ¯ and φˆ in (6.1) or (6.4), for each ﬁxed θ in the interval I of admissible temperatures. As θ varies in I, the basis ¯ ea and the metric C¯ in general also vary and, accordingly, so do the elements in the ¯ eb and mt Cm. orbits Omba¯ We can thus consider the curves of minimizers ¯ ea (θ) ∈ B, and C¯ = C(θ) ∈ Q> ¯ ea = ¯ 3 , whose temperature dependence describes the thermal expansion of the crystal (§6.6.1); the analogue holds ˇ for the minimizers of φ˜ and φ. As is detailed in chapter 7, for smooth, generic constitutive functions the curves of minimizers have piecewise constant lattice groups; thus, as θ varies, they are piecewise conﬁned to given ﬁxed sets in B or Q> 3 , respectively. This indeed happens to the (meta)stable conﬁgurations of real ¯ materials. From now on, when considering a curve C(θ) of minimizers of ˆ φ, we always refer, unless otherwise stated, to each portion that belongs ¯ to a given proper ﬁxed set (§3.5), so that the lattice group along C(θ) is constant. Here, a set of θ-dependent, symmetry-related minimizers of φ¯ or φˆ as in (6.18) or (6.19) with constant lattice group in some temperature interval is what is loosely referred to as a phase of the crystal. 6.4.2 Multiphase crystals: minimizers that are not symmetry-related The phenomenological model for phase transformations in multiphase crystals is based on the assumption that there exist minimizers of the free energy function of the material which are not symmetry-related, that is, minimizers not all included in (6.18) or (6.19).

6.4 ENERGY MINIMIZERS AND THEIR GENERAL PROPERTIES

177

For the case of a crystal with two distinct phases, the basic hypothesis is that above a certain temperature θM ∈ I the absolute minimizers of φ¯ are given by a θ-dependent set of wells as in (6.18) with ¯ ea = ¯ ea (θ), while below θM the absolute minimizers of φ¯ are given by another set of wells ˆ ∈ G(ˆ ea (θ); for m ∈ GL(3, Z) or H ea (θ)) these are based on ˆ ea = ˆ ˆˆ Ombaˆ eb = {mba Qˆ eb : Q ∈ O} = {Q H ea : Q ∈ O}.

(6.20)

Here ˆ ea (θ) is a curve of minimizers in B with constant lattice group. Except ea (θ) in (6.18) and (6.20) for some isolated values of θ, the curves ¯ ea (θ) and ˆ are not symmetry-related; they usually have inequivalent lattice groups and belong to arithmetically inequivalent ﬁxed sets, so that the lattices they generate are of diﬀerent Bravais type in B. At the temperature θM the two sets of wells (6.18) and (6.20) all give ab¯ and at that temperature they exchange their roles solute minimizers of φ, as absolute minimizers. This temperature is denoted by θM to recall, as discussed in §6.5.3, that it indicates to the so-called Maxwell temperature, which need not be the temperature at which a bifurcation producing the phase change occurs. In some temperature range near θM the two sets of ¯ to model the possible wells above may both give relative minimizers of φ, existence, for the same crystalline material, of a stable and a metastable lattice structure, with diﬀerent symmetry, coexisting near the transformation temperature.11 ˆ Equivalently, one assumes that the absolute minimizers of the function φ, > t¯ deﬁned on Q3 × I, are the elements of the GL(3, Z)-orbit m Cm in (6.19) for θ > θM , while for θ < θM they are the elements of a diﬀerent orbit ˆ ˆ ˆa (θ) ∈ B mt Cm, m ∈ GL(3, Z), where Cˆ = C(θ) ∈ Q> 3 is the metric of e appearing in (6.20). Since eˆa (θ) and ¯ ea (θ) have inequivalent lattice groups, the curves of minimizers ¯ ˆ C = C(θ) and C = C(θ) are not symmetry-related, and belong to inequivalent ﬁxed sets in corresponding two GL(3, Z)-orbits of minimizers, ¯ : m ∈ GL(3, Z)} {mt Cm

ˆ : m ∈ GL(3, Z)}, and {mt Cm

(6.21) Q> 3;

the

(6.22)

are therefore distinct. At θM the curves in (6.21) exchange their role as ˆ but in some temperature range near θM both C(θ) ¯ absolute minimizers of φ, ˆ ˆ and C(θ) may give relative minimizers of φ (see Fig. 7.3 for an example). In §6.5 we better detail the structure of the minimizers of the free energy when the two wells containing ¯ ea (θ) and ˆ ea (θ), or the two curves in (6.21), belong to a wt-nbhd in the spaces B or Q> 3 , respectively. Then, in chapter 7 we study the conditions under which such curves of minimizers or of critical points can meet, examples being given in §§7.6.2 and 7.6.4. This is a basis for modelling phase transformations with spontaneous symmetry breaking. 11

See §§7.6.1–7.6.2 for examples of energy functions with the properties stated here.

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6.4.3 Lack of convexity and symmetry-induced instabilities The multiplicity of minimizers of the constitutive functions of elastic crystals has an important consequence: the existence of symmetry-induced instabilities that are typical of crystalline substances – see Ericksen (1977). Indeed, the existence of unstable states for a crystal is a consequence of the periodicity of the lattice and the ensuing invariance properties (6.3)–(6.5) ˜ As a consequence of this invariance, cerof the free energy functions φ¯ or φ. tain necessary conditions for the existence of minimizers of the functional (6.12) fail. This causes various diﬃculties in the variational problems based on (6.12). References are given in the Introduction, in §2.4.3 and in chap˜ which cannot be ter 10. Here we show that, due to (6.14), the function φ, convex by orthogonal invariance, fails to be quasiconvex or, actually, also ¯ is an integral rank-1 convex. Quasiconvexity of φ˜ : R9 → R at a given F property (Morrey (1966)), roughly stating that, for any bounded open set ¯ x minimizes the free energy A ⊂ R, the homogeneous deformation y = F among all suﬃciently smooth deformations y = χ(x ) which are deﬁned in A, have almost everywhere a deformation gradient belonging to the domain ˜ and satisfy the linear displacement conditions y = F ¯ x for x on the of φ, boundary ∂A of A; in formulae

˜ ˜F ¯ , θ). φ(Dχ(x , θ) dx ≥ Vol A φ( (6.23) A

¯ in its domain. The function φ˜ is quasiconvex if it is quasiconvex at all F As deﬁned above, quasiconvexity states that, under a linear displacement imposed on the boundary, the corresponding homogeneous deformation minimizes the energy. Failure of this condition implies that the body may spontaneously move from a homogeneous conﬁguration to a inhomogeneous ˇ one to lower its energy (internal buckling, see Silhav´ y (1997), p.268 and references therein). This same condition is used to prove the existence of minimizers of (6.12) in the direct method of the calculus of variations. ¯ if it satisﬁes the inequality The function φ˜ is (F -) rank-1 convex at F ˜F ˜F ˜F ¯ (1 + υa ⊗ n), θ) ≤ (1 − υ)φ( ¯ , θ) + υ φ( ¯ (1 + a ⊗ n), θ) φ(

(6.24)

for any vectors a, n and 0 ≤ υ ≤ 1. Notice that, given the directions of a and n, this is equivalent to the convexity of the real-valued function ˜F ¯ (1 + υa ⊗ n), θ), υ → φ(

−∞ < υ < ∞.

(6.25)

¯ implies rank-1 convexity there (see for instance Since quasiconvexity at F ˇ Silhav´ y (1997), p.277), failure of the latter implies failure of the former. To prove failure of rank-1 convexity, let 1 + a ⊗ n be any lattice-invariant shear – see (3.14) – that is, any simple shear represented by an element of GL(3, Z) in the reference basis ea0 ; notice that there are inﬁnitely many of ˜ these. The function (6.25) is periodic of period 1 by the invariance of φ, hence it is convex if and only if it is constant. An argument based on the

6.5 CONSTITUTIVE FUNCTIONS FOR WEAK PHASE TRANSITIONS

179

ˇ density of the rationals in the real numbers (for instance in Silhav´ y (1997), ˜ ˜ ¯ ¯ p.292) implies that φ(F , θ) = φ(F S , θ) for any simple shear S , and this is not acceptable for solids. Therefore there are values of υ, a, n for which condition (6.24) fails, as anticipated. Correspondingly, if the function φ˜ is of class C 2 , the Legendre-Hadamard condition (6.46) below also fails (this condition is necessary for rank-1 conˇ vexity to hold – see for instance Silhav´ y (1997), p. 278). An argument which supports this last failure is given by Ericksen (1977), and is similar to the one given above for rank-1 convexity. All these failures contradict traditional assumptions for the energy of (thermo)elastic solids. Notice that along the 1-dimensional family of deformations (6.25), for which υ is a measure of strain, the derivative of the energy function with respect to υ, which is a measure of stress in agreement with (2.55)1 , is also a periodic function. This contrasts various assumptions of monotonicity for the stress-strain relations, which are considered in the classical theories of elastic materials. Thus we have for thermoelastic crystals a far-reaching generalization of the classical constitutive equations of the van der Waals type for ﬂuids. A 1-dimensional example is shown in §7.6.1 below. 6.5 Constitutive functions for elastic crystals undergoing weak phase transformations So far we have assumed the domains12 of the constitutive functions introduced in §§6.1–6.3 to be a priori the entire conﬁguration spaces of simple lattices. However, this may not be realistic nor necessary in many cases; and, indeed, for many purposes it is best to cut down the domain of the constitutive functions to suitable subdomains. Proposition 4.1 provides a rational way for such a restriction, which is analyzed here in some detail together with its implications. Among other things, we will see that reducing the domain also reduces the global invariance (6.3)–(6.5) of the constitutive functions to a suitable lattice-group invariance. This makes the model that we are developing locally compatible with the classical linear and nonlinear theories of crystal elasticity – see (6.27), (6.29) and Remark 6.1 below. This approach clearly limits the scope of the model, restricting it to phenomena involving lattice distortions that are ﬁnite but not too large (as in §4.2). As anticipated in the introduction to this chapter, in many crystalline materials the observed stable structures can be ¯ ˆ described by temperature-dependent metrics C(θ) and C(θ) that minimize φˆ as in (6.21) and are suitably close to each other in Q> . The description 3 of these phase changes and the related twinning phenomena only requires considering lattice strains that indeed are not too large. Furthermore, as 12

Since the interval I of temperatures does not aﬀect our discussion, for brevity hereafter we refer to the set where the variable ea [or C] vary as to the ‘domain’ of the ˆ analogously for φ˜ or φ. ˇ constitutive functions φ¯ [or of φ];

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also anticipated, in general it may not be useful to consider too large lattice deformations in a thermoelastic model for crystalline solids, as such strains can lead to plastic phenomena for which elasticity theory may not be an appropriate framework. The restrictions considered in this section are therefore not too severe as long as one’s goal is an elastic theory. 6.5.1 Weak and symmetry-breaking transformations As in the introduction to chapter 4, we call weak a solid-state phase transformation in a crystal whose temperature-dependent lattice metrics are on ¯ ˆ curves C(θ) and C(θ) that are not too far apart (at ﬁxed theta) in the space of metrics; explicitly, they can be taken to belong to a suitable wtnbhd in Q> 3 . As a consequence of the analysis in §4.1, the lattice groups of ¯ ˆ the minimizers C(θ) and C(θ) in a weak transformation are subgroups of a common ﬁnite supergroup, which is the lattice group of the center of the wt-nbhd. In particular, when there is a direct group-subgroup relation: ˆ ¯ L(C(θ)) ≤ L(C(θ)),

(6.26)

the weak phase transformation is called symmetry-breaking; in this case any ¯ of the metrics C(θ) can be taken to be the center of the wt-nbhd wherein the minimizing curves lay. We sometimes use the nomenclature common in the metallurgical literature, and call austenite (or parent phase ) the high-symmetry phase of the crystal, and martensite (or product phase ) the low-symmetry phase.13 As mentioned in the Introduction, symmetry-breaking transformations are very common in crystalline materials, for instance in shape memory alloys, and have great interest from both the theoretical and applicative points of view. We will focus on these transitions and related phenomena in the following chapters of this book.14 We will see that the relevant energy wells for the analysis of these phase changes are the (ﬁnitely many) wells describing the variants of the low-symmetry phase in the vicinity of a well describing the high-symmetry phase. This fact considerably simpli13

14

In many cases the high-symmetry austenite and the low-symmetry martensite are observed to be stable at the higher and lower temperatures, respectively. Also in the prototypical α-γ (that is, b.c.c.-to-f.c.c.) transformation in iron (§3.6.3) the stable phase at the higher [lower] temperatures is called austenite [martensite], although in this case the lattice groups are not in a group-subgroup relation, and it is not possible to qualify one symmetry to be higher than the other. The case of weak but not symmetry breaking transformations is also of interest, and can be treated in an analogous way. We recall that crystalline solids do also undergo nonweak phase changes, in which the lattice groups are not in a group-subgroup relation nor are they subgroups of a common lattice supergroup; a relevant example is the α-γ phase transformation in iron mentioned in footnote 13. In this case the ¯ ˆ two curves of minimizers C(θ) and C(θ) in (6.21) are not close in the space Q> 3 . In principle, also the latter transformations can be studied within the general framework for lattice symmetry and energy invariance described so far; however, a number of diﬃculties arise. See for instance the restricted analysis of Rosakis and Tsai (1994), or better Conti and Zanzotto (2002).

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181

ﬁes the elastostatic variational calculations based on the functional (6.12), which allow for the analysis of twinning and microstructure formation to be presented in chapters 8–10. 6.5.2 Domain restrictions for the constitutive functions: reconciliation of the global and local symmetries of crystals For any given lattice basis ¯ ea or metric C¯ consider the restriction of the constitutive function φ¯ to a wt-nbhd N¯ea , or of φˆ to a wt-nbhd NC¯ of C¯ as in Proposition 4.1. For simplicity we denote the restricted constitutive functions by the same symbols used so far for the unrestricted ones. Due to the properties of the wt-nbhds, the restricted constitutive functions are no longer invariant under all the elements of GL(3, Z) as in (6.3) and (6.5), but only under those operations mapping the neighborhood to itself; these, by Proposition 4.1, are the ones leaving invariant the center of the restricted domain. Explicitly, the restricted constitutive functions have the following invariance properties: ¯ a , θ) = φ(m ¯ b Qeb , θ) for any ea ∈ N¯e , Q ∈ O, m ∈ L(¯ φ(e ea ), θ ∈ I. (6.27) a

a

Equivalently we have ˆ ˆ t Cm, θ) for any C ∈ NC¯ , m ∈ L(¯ φ(C, θ) = φ(m ea ), θ ∈ I.

(6.28)

Suppose now R is the reference conﬁguration of a crystalline body, with ¯ and consider the restriction of the reference lattice L(¯ ea ) and metric C; ˜ continuum constitutive function φ to the neighborhood N1 derived from ˇ the wt-nbhd N¯ea as in Proposition 4.1 (with an analogous restriction for φ). Due to the properties of N1 indicated in the ﬁrst part of that Proposition, the function φ˜ restricted to N1 is no longer invariant under G(¯ ea ) as in (6.14); rather, for C = F t F , ˜ , θ) = φ(QFH ˜ ˇ , θ) = φ(H ˇ t CH , θ), φ(F , θ) or φ(C (6.29) for any F ∈ N1 , Q ∈ O, H ∈ P (¯ ea ), and θ ∈ I. Remark 6.1 Formulae (6.27)–(6.29) show that reducing the energy domain greatly simpliﬁes the constitutive framework, because then the invariance is given by the ﬁnite crystallographic point group P (¯ ea ). In particular, for ﬁnite but not too large strains from the reference conﬁguration ¯ ea , the constitutive theory for elastic crystals presented here is in agreement with, and gives a rationale for, the approach generally adopted in linear and nonlinear elasticity. According to the latter, the material symmetry of a crystal is described by a ﬁnite crystallographic point group, tacitly obtained from some equilibrium state of the crystal – see Coleman and Noll (1964), Truesdell and Noll (1965); see also footnote 9 in chapter 2. This follows from the above procedure when the wt-nbhd consists of appropriately small deformations from an isolated strict minimizer, as is explained below in more detail. In this way one also recovers the classical

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results on the invariance of the energy functions of crystalline solids in the linear theory – see for instance Love (1927), Landau and Lifˇsits (1959), Truesdell and Toupin (1960), Jaunzemis (1967), Gurtin (1972), Wang and Truesdell (1973), Huo and Del Piero (1991), Forte and Vianello (1996) – where the free energy function φˇ is approximated by a quadratic form in terms of the strain tensor E – see (2.53) – exhibiting suitable symmetries in its coeﬃcients (the elastic moduli, detailed in §§6.6.2 and 6.8). This compatibility is important because linear elasticity successfully describes the mechanical behavior of anisotropic solids under small deformations from a natural state. Notice, however, that in the classical theories the constitutive invariance is ﬁxed once and for all; here, on the contrary, the material symmetry (6.27)–(6.29) has a local character, and changes with the reference basis ¯ ea : for instance, depending on the latter, near 1 the symmetry of φˇ is dictated by the holohedry of ¯ ea , and this is diﬀerent if the reference is taken in the martensite rather than in the austenite. This adds ﬂexibility to the theory, and makes it suitable to model phase transitions, twinning and other related phenomena, as we will see in the next chapters.15 6.5.3 Energy wells in the wt-nbhds In the preceding section we have indicated how to reduce the domain and symmetry of the constitutive function for a lattice or for its continuum counterpart; for the latter, no special properties were required from the selected reference basis ¯ e at the center of the wt-nbhd. However, depending on each physical circumstance, there may be neighborhoods that are most relevant, and the choice of reference conﬁguration should reﬂect this fact. This happens, for instance, when one considers a weak phase change. As a prototypical case, we analyze a symmetry-breaking phase transformation for a two-phase crystalline material. Recall that for a two-phase crystal the minimizers of φ¯ and φˆ are as in (6.18), (6.20) and (6.21). As (6.26) holds, the results of chapters 4 and 5 show that the minimizers within the wt-nbhds N¯ea (θM ) or NC(θ ¯ M ) have a special arrangement. This is because, in the space of metrics Q> 3 , the orbit ˆ of the less symmetric minimizers ‘winds around’ each element in the mt Cm ¯ of the more symmetric minimizers, in such a way that each orbit mt Cm high-symmetry minimizer is surrounded by a set of ‘nearby’ variants (a local orbit) of low-symmetry minimizers – see Fig. 4.1. Similarly for the wells of φ¯ indicated in (6.18) and (6.20). Thus, by restricting the energy domain to a wt-nbhd we only need to consider a ﬁnite number of energy wells. In this section we analyze the structure of the minimizers in the wt-nbhds, and make some hypotheses suitable for modelling symmetry-breaking phase transitions (see also Ball and James (1992), Luskin (1996b)). Examples of 15

The analogue can be done for the energy of multilattices, based on Proposition 11.9.

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183

energy functions whose minimizers satisfy the conditions described hereafter are given in §§7.6.2, 7.6.4. As in §6.5.1, we say that a two-phase crystal undergoes a symmetrybreaking phase transition when there are two piecewise smooth curves ¯ ea (θ) ¯ ˆ and C(θ) in and ˆ ea (θ) such that the corresponding curves of metrics C(θ) > ˆ (6.21) (relatively) minimize φ in Q3 , are contained in some wt-nbhd in this same space, and have lattice groups satisfying the inclusion relation (6.26). The basis ¯ ea (θ) represents the undistorted high-symmetry austenitic phase, and ˆ ea (θ) the undistorted low-symmetry martensitic phase of the crystal. We stress the following assumptions on the curves above: ¯ (i) There exists a temperature θM at which ¯ ea (θ) and ˆ ea (θ) (or C(θ) and ˆ C(θ)) have the same energy. In this case one often chooses the reference conﬁguration at the stress-free equilibrium state of the high-symmetry austenitic crystal given by the lattice basis ¯ ea (θM ). (ii) The curves ¯ ea (θ) and ˆ ea (θ) can be chosen to belong to a neighborea (θM ) to which the energy domain is restricted. hood N¯ea (θM ) ⊂ B of ¯ ¯ ˆ Correspondingly, the curves C(θ) and C(θ) are in a neighborhood > ˆ Analo¯ NC(θ ¯ M ) ⊂ Q3 of C(θM ) which is the restricted domain of φ. gously for the constitutive functions φ˜ and φˇ of the continuum, when the Born rule holds. (iii) For all temperatures in a suitable interval around θM , the above curves of minimizers of φ¯ and φˆ remain in their own proper ﬁxed sets in B and Q> 3 , respectively; equivalently, they have constant lattice groups. This property is actually a consequence of suitable regularity hypotheses16 ˆ see §7.2. on φ, (iv) For each temperature θ > θM the absolute minimizer of φˆ in NC(θ ¯ M) ˆ ¯ is given by C(θ); for θ < θM the absolute minimizers of φ in NC(θ ¯ M) ˆ ¯ M )-variants (see (6.5)) are given by C(θ) and its C(θ ˆ mt C(θ)m,

¯ M )). m ∈ L(C(θ

(6.30)

Correspondingly, the absolute minimizers of φ¯ in N¯ea (θM ) are, for each θ > θM , in the orbit O¯ ea (θM ) (austenitic well ), while, for θ < θM , consist of eˆa (θ) and all its symmetry-related bases; as dictated by the invariance (6.27), these belong to the O-orbits ¯ M ))} {Oˆ ea(1) (θ), . . . , Oˆ ea(N ) (θ)} = {Ombaˆ eb (θ) : m ∈ L(C(θ

(6.31)

(martensitic wells or martensitic variants ). 16

If φˆ is smooth enough, any curve C(θ) of critical points of φˆ is smooth and isolated, by the implicit-function theorem, wherever the matrix of second derivatives of φˆ has maximal rank. At values of θ where that rank is not maximal two or more θ-dependent curves of critical points may meet and loose some regularity. Except for such values of θ, the curves of minimizers necessarily have constant lattice groups, so that each belongs to a suitable proper ﬁxed set in Q> 3 .

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At the temperature θM the high-symmetry orbit O¯ ea (θ) and the low(1) (N ) symmetry orbits Oˆ ea (θ), . . . , Oˆ ea (θ) all have the same energy, which can be set conventionally to zero, and exchange their roles as ¯ absolute minimizers of φ. Remark 6.2 The orbit O¯ ea (θ) ceases to minimize absolutely φ¯ for θ < θM , but may still give, in some temperature interval θ¯ ≤ θ ≤ θM , relative minimizers of φ¯ which represent metastable states of the austenitic phase ˇ the at low temperatures. Likewise, in a suitable interval θM ≤ θ ≤ θ, (1) (N ) ea (θ) may give relative minimizers low-symmetry orbits Oˆ ea (θ), . . . , Oˆ ¯ that is, metastable conﬁgurations of the low-symmetry martensitic of φ, phase variants at higher temperatures. At the temperature θM switch in the absolute stability of the austenitic and martensitic phases occurs; on the interpretation of it as the transition temperature, see Remark 6.5. We can summarize the above hypotheses on φ¯ as follows: there are functions ¯ ea (θ) and eˆa (θ) such that, except for what is dictated by the invariance (6.27), in N¯ea (θM ) the minimizers of φ¯ satisfy the conditions if θ > θM if θ = θM if θ < θM

¯ ea (θ), θ); ¯ a , θ) > φ(¯ φ(e ¯ a , θM ) > φ(¯ ¯ ea (θM ), θM ) = φ(ˆ ¯ ea (θM ), θM ); φ(e ¯ a , θ) > φ(ˆ ¯ ea (θ), θ), φ(e

(6.32) (6.33) (6.34)

for all bases ea $= ¯ ea (θ) in N¯ea (θM ) . The corresponding hypotheses on 17 ˆ the minimizers of φ in NC(θ Fig. 7.3 in §7.6.1 ¯ M ) can be derived easily. represents these conditions schematically in a speciﬁc 1-dimensional case. Constitutive functions satisfying the hypotheses above under various assumptions of symmetry and smoothness are presented by Ericksen (1980b), (1986a,b), (1987), (1988), (1992), (1993), (1996b), M¨ uller and Wilmanski (1980), Parry (1981), (1984a), Chan (1988), Collins and Luskin (1989), (1991), Fu et al. (1993), Raniecki and Lexcellent (1994), Simha and Truskinovsky (1996). See also Ball and James (1992), Luskin (1996b), Truskinovsky and Zanzotto (2002). Remark 6.3 Because of Galilean invariance it is convenient to normalize the orthogonal transformation out of both ¯ ea (θ) and ˆ ea (θ), by choosing ¯ (θ)¯ ¯ (θ) a stretch. By (ii) above this stretch is ¯ ea (θ) = U ea (θM ), with U symmetry-preserving for ¯ ea (θM ), and gives the free thermal expansion of the austenitic phase of the crystal. One also introduces the transformation stretch UM such that ˆ ea (θM ) = UM ¯ ea (θM ), along with the symmetryˆ (θ)ˆ ˆ (θ) for ˆ ea (θ) = U ea (θM ). The preserving stretch U ea (θM ), deﬁned by ˆ ˆ stretch U (θ) gives the free thermal expansion of a variant of the martensitic phase. Thermal expansion is thus represented by the dependence of the 17

By means of the Born rule, these requirements on the functions φ¯ or φˆ can be phrased in terms of the minimizers of the constitutive functions φ˜ and φˇ of the continuum, deﬁned in (6.8) and (6.15).

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185

energy minimizers on temperature. The latter displaces the potential wells in a way that is characteristic of each phase while maintaining them within their own proper ﬁxed sets; see also §6.6.1. Remark 6.4 By item (i) above, the transformation stretch UM is symmetry breaking for the reference austenitic basis ¯ ea (θM ); therefore its variants ea (θM ), . . . , OUN ¯ ea (θM ), generated by the point group P (¯ ea (θM )) OU1¯ acting on U1 = UM as in (4.40), deﬁne one of the variant structures described in chapter 5.18 For this reason hereafter we transfer to the energy wells the nomenclature about variants introduced there. Remark 6.5 Recall that, as θ varies, the curves of minimizers ¯ ea (θ) and ¯ ˆ ˆ ea (θ) in B (or the curves C(θ) and C(θ) in Q> ) may meet, or the critical 3 points may loose stability or cease to exist, or curves of unstable critical points may become stable, etc., giving rise to the bifurcation patterns for simple-lattice energies which are considered in detail in the next chapter. In the above description of a two-phase elastic crystal a bifurcation occurs at the temperature θM only if the curves ¯ ea (θ) and ˆ ea (θ) actually meet at θM . If so, the transformation stretch is trivial, UM = 1, and the phase change is called of second order ; moreover, θM is the temperature at which the bifurcation occurs, and coincides with the limits of metastability θˇ and θ¯ in Remark 6.2. This situation is compatible with reduced problems (2), (3), (5) in chapter 7, and is illustrated by the energy landscape in Fig. 7.2. In most cases, though, the transformation stretch is nontrivial, UM = 1, and the phase change is called of ﬁrst order. For instance, this necessarily happens in reduced problem (4), §7.4.4, because all the bifurcating branches are generically unstable. A simple model of the phase change is the following, and is sketched in Fig. 7.3: a curve ea (θ) of unstable martensitic equilibria ¯ and connects to the bifurcates from the austenitic branch at temperature θ, ˇ The bifurcation tempermetastable martensitic branch at temperature θ. ¯ the unstable bifurcating branches are restabilized by a turning ature is θ; ˇ and the Maxwell temperature θM is the point (§7.4.1) at temperature θ; one at which the two phases have the same energy, and exchange their role as absolute minimizers of the energy. Given the prejudice that for any value of θ the equilibrium conﬁguration of the crystal be an absolute energy minimizer at that temperature, one would be forced to regard θM as the temperature at which the phase change occurs. But energy barriers may prevent the immediate attainment of the absolute minimum, and the crystal may remain for a while in a metastable conﬁguration, in the austenitic phase upon cooling and in the martensitic one upon heating. This is similar to overheating and supercooling in the van der Waals theory of ﬂuids, and implies that the transition shows hysteresis. And, indeed, hysteretic be18

The variants are detailed in §5.1 or §5.2 when the reference basis ¯ ea (θM ) is of maximal (cubic or hexagonal) symmetry; otherwise the variant structure is obtained as described in §5.3. This is useful for the analysis of phase transitions and related phenomena given in the following chapters.

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havior is very common in phase changes of shape memory alloys; more, a certain temperature interval is necessary for the new phase to nucleate and progress throughout the body, both in heating and in cooling. In the literature four temperatures are usually given: Ms , the temperature at which martensite starts nucleating in austenite, Mf < Ms , the temperature at which the transformation of austenite into martensite is complete, and the reverse analogues As and Af > As . A branch of martensitic equilibria which, like ea (θ), exists above θ¯ near the bifurcation point, that is, in a temperature interval where the austenitic phase is (meta)stable, is usually called subcritical, and supercritical if it ¯ Also, the bifurcation leading to a ﬁrst-order phase change exists below θ. like the one in Remark 6.5 (and Fig. 7.3) is called a subcritical bifurcation. Remark 6.6 The above hypotheses on φ¯ do not rule out the possibility that a third phase may be metastable (a local minimizer) in all or part of the interval I of temperatures, possibly becoming a global minimizer at some transition temperature outside I. For the time being we exclude this possibility and consider the simplest case of two phases, although threeand four-phase materials, with progressively reduced symmetries, do occur in nature and have very interesting properties. Models of multiphase materials are based on energies whose minimizers are on three or four variant structures that suitably exchange their roles as absolute and relative energy minimizers in a given temperature interval. These more complex models are just beginning to be investigated (see for instance Ericksen (1996b), Truskinovsky and Zanzotto (2002)). An example is sketched in §7.6.4. 6.6 In the vicinity of an energy well In order to retrieve from our framework some classical notions of the linear elastic theory of crystals, we consider now the restriction of the constitutive functions to a wt-nbhd containing only one energy well, namely its center. The action of the global symmetry group GL(3, Z) will give us, in the vicinity of that well, a description of material symmetry which agrees with the classical one involving the crystallographic point groups; the latter have proven successful and satisfactory in a wide variety of circumstances. The ˆ ˇ )], when restricted to such a neighborhood constitutive function φ(C) [φ(C ¯ ¯ ], is convex at the center,19 and its invariance is directly of a minimizer C [C ¯ ]. The linked to the geometric symmetry of (the unique minimizer) C¯ [C constitutive function is approximated by a quadratic polynomial based on the classical anisotropic fourth-order tensors of elastic moduli, belonging to the well known symmetry classes (see the references in Remark 6.1). In this section we give some details on this approximation. 19

Due to the orthogonal invariance (6.2) and (6.14), the constitutive functions φ¯ and φ˜ are not convex, even when their domain is reduced to a wt-nbhd of one energy well.

6.6 IN THE VICINITY OF AN ENERGY WELL

187

6.6.1 Thermal expansion and compressibility of a crystal Before introducing the elastic moduli, it is worth giving more details on free thermal expansion of crystals, which has already been mentioned in §6.5.3. ¯ (θ) and There we have considered the temperature-dependent stretches U ˆ (θ) which, by hypothesis, are symmetry-preserving for the minimizers U ¯ ea (θM ) and eˆa (θM ) of φ¯ belonging to the wt-nbhd N¯ea (θM ) . ˜¯e (θ ) ⊂ N¯e (θ ) which only We focus our attention on a neighborhood N a M a M ¯ ¯ (θ) which, contains the austenitic well O¯ ea (θM ) of φ, and on the function U ¯ due to the choice of ¯ ea (θM ) as reference basis, satisﬁes U (θM ) = 1.20 If the Born rule holds, the one-parameter family of homogeneous deforma¯ (θ) describes, up to tions given by the symmetry-preserving stretches U rigid-body rotations, how the stress-free equilibrium conﬁguration of the crystal changes with θ, away from phase transitions. This is the thermal ¯ (θ) makes the austenitic well O¯ expansion of the austenitic phase: U e (θM ) drift slightly with temperature in the neighborhood N1 . It is useful to also consider the case in which the environment can exert a nonnegligible pressure p, which in §2.5.2 has been denoted by p0 and then altogether set to zero for simplicity. As we have seen there, for p > 0 the ˜ thermoelastic potential to be minimized is the Gibbs free energy γ˜ = φ+Jp ˜ in (2.73) rather than φ, the latter being the continuum analogue of the function φ¯ considered so far.21 Then in formulae (6.32)–(6.34) all quantities depend on both environmental variables p and θ, and so does the above ¯ (p, θ). The presence of p family of symmetry-preserving stretches: U = U ¯ in U (p, θ) describes the homogeneous stable equilibria of the crystal under varying environmental conditions p and θ, accounting also for the diﬀerent compressibility exibited by the material at various temperatures. Let us consider, as an explicit example, a hexagonal crystal. Various materials, such as quartz, magnesium and titanium, in suitable ranges of p and θ have an hexagonal skeletal lattice for which the sixfold axis is along a certain vector k and the basis ¯ ea (θ) is given by (3.50). In this case the invariance of the constitutive function γ˜ restricted to N1 is γ˜ (U , p, θ) = γ˜ (Q t UQ, p, θ)

(6.35)

for all p, θ in suitable ranges, all U ∈ N1 and Q in the hexagonal holohedry P (¯ ea (pM , θM )) = Hk listed in Table 5.9. Also, due to the choice of N1 , for ¯ (p, θ) such that all p, θ there is a unique stretch U = U ¯ (p, θ), p, θ) for all U $= U ¯ (p, θ) in N1 . γ˜ (U , p, θ) > γ˜ (U

(6.36)

¯ (p, θ) as a minimizer in N1 forces Then, by (6.35), the uniqueness of U ¯ (p, θ)Q = U ¯ (p, θ), Q tU 20

21

Q ∈ Hk ,

(6.37)

A similar discussion holds if we take another neighborhood in N¯ea (θM ) containing only one of the low-symmetry martensitic wells, for instance Oˆ ea (θM ). As noted in §2.5.2, φ˜ and γ ˜ have the same invariance.

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for all admissible values of p and θ; the analogue being true for the cor¯ (p, θ) = U ¯ 2 (p, θ). By a proposition of responding Cauchy-Green tensor C Coleman and Noll (1964) exploiting Corollary 2.11, equality (6.37) in turn ¯ (p, θ) must have a special form detailed in Table 6.1; for implies that U Q ∈ Hk this is ¯ (p, θ) = ;(p, θ)1 + δ(p, θ)k ⊗ k . U (6.38) Here ; and δ are scalar-valued functions such that ;(pM , θM ) = 1,

δ(pM , θM ) = 0,

(6.39)

because of the choice of the reference basis. The functions δ and ;, which depend on the physical characteristics of each (hexagonal) crystal, give the details of how the material responds macroscopically to the varying environmental parameters and changes its shape, away from phase transitions. The anisotropic compressibility and thermal expansion of these crystals along their sixfold symmetry axis is accounted for by the function δ(p, θ).22 This discussion refers to the case in which ¯ ea (pM , θM ) has hexagonal symmetry. For minimizing bases in the other crystal systems, the family of symmetry-preserving stretches may involve more (p, θ)-dependent functions than the ; and η appearing in (6.38); their number depends on the crystal system, and can be obtained from Table 6.1 but also from the tables given in chapter 5 or from the representations in §5.4.3.23 In any case, when φ˜ is continuously diﬀerentiable, the ﬁrst-derivatives condition for Gibbs free energy minimization is (see also (2.68)2 or (2.72)) ∂ φ˜ ¯ (p, θ); = −p J F −t at U = U (6.40) ∂F equivalently, the Cauchy stress T in the crystal reduces to the uniform environmental pressure p. Since the above treatment of thermal expansion and compressibility for the continuum is based on the invariance of the lattice energy and on the Born rule (6.7), the functions δ and ; also describe the macroscopic changes in shape of the hexagonal lattice as p and θ vary, because the vectors ¯ (p, θ)¯ ¯ ea (p, θ) = U ea (pM , θM )

(6.41)

give the basis of the hexagonal lattice cell in the deformed conﬁguration at In this case we have proved what was assumed in §6.4.1, that is, that the basis e¯a (p, θ) ¯ θ) belong to a given ﬁxed set in B or Q> (or, equivalently, that and its metric C(p, 3 ¯ U(p, θ) is symmetry-preserving for the hexagonal basis e¯a (pM , θM ) – compare also with U1 in Table 5.9). We will return to the aforementioned assumption in §7.2. 23 Experimental work on the phenomena of thermal expansion and compressibility of crystals is very abundant in the literature. See, for instance, Thurston (1974) and references therein. From such data one can estimate the (p, θ)-dependence of the lattice parameters of crystalline materials (however, see remarks in §6.2.2), for instance establishing explicitly in the case of hexagonal crystals the two constitutive functions (p, θ) and δ(p, θ); this amounts to giving an explicit form to the (local) minimizers of the constitutive function φ˜ of a material. For an explicit example, see Zanzotto (1989), (1990) and the references cited therein. 22

6.6 IN THE VICINITY OF AN ENERGY WELL

189

Table 6.1 Restrictions on the right Cauchy-Green tensor of the deformation between undistorted states of anisotropic solids Crystal system

Restriction on C

Triclinic Monoclinic Orthorhombic Tetragonal Rhombohedral Hexagonal Transverse isotropy Cubic

no restriction k is an eigenvector of C i, j, k are eigenvectors of C C = 1 + δk ⊗ k C = 1

given p and θ. Nevertheless, there are cases of thermal expansion in metals in which the Born rule does not hold experimentally; see remarks in §6.2.2. 6.6.2 The elasticity tensor Let us now set again p = 0, and consider the one environmental parameter ˇ , θ) near its minimizers θ and the features of the energy function φ = φ(C 24 ¯ C (θ) mentioned above. We concentrate on the continuum function φˇ for the sake of comparison with the classical theories of crystal elasticity, which are always concerned with the description of continua. We have seen that, for a crystal whose reference conﬁguration includes the lattice generated by a basis ea0 , the invariance of φˇ in its entire domain Sym> is given by (6.17). ¯ (θ), generates inﬁnitely Any (relative) minimizer of φˇ for given θ, say C > ¯ (θ)H for H ∈ many other (relative) minimizers in Sym , given by H t C 0 G(ea ). These minimizers correspond to a GL(3, Z)-orbit of the metric ¯ (θ)eb0 C¯ab (θ) = ea0 · C

(6.42)

and, due to the discreteness of GL(3, Z), any such minimizer is strict and isolated. We assume25 this to remain true also in case φˇ admits another ˆ (θ) (and also H t C ˆ (θ)H ) in Sym> , not symmetrycurve of minimizers C ¯ related to C (θ), as in §6.4.2. We brieﬂy discuss the properties of φˇ near one of these minimizing points. For deﬁniteness, let us take θ = θM (see ¯ (θM ). As usual, for the purpose of §6.5.3), and consider the minimizer C ˇ ¯ analyzing φ near C (θM ) we actually change the reference basis from ea0 to ¯ (θM )ea0 , ¯ ea (θM ) = U 24

25

where

¯ 2 (θM ) = C ¯ (θM ), U

(6.43)

When the Born rule (6.7) holds, all the properties and quantities that we will discuss starting from φˇ can easily be translated into corresponding ones derived from φˆ by ˇ means of the relation (6.16) between φˆ and φ. This is necessarily true if φˇ is twice continuously diﬀerentiable and its hessian matrix ¯ at C(θ) is positive deﬁnite, as we assume later.

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using, if necessary, formula (2.57) to deﬁne a new constitutive function; this ˇ , θ) for simplicity. Now, by hypothesis, is denoted by the same symbol φ(C ˇ φ(C , θ) has a minimum in (1, θM ); and we restrict the domain of φˇ to a neighborhood N1 ⊂ Sym > which is related by (6.42) to a wt-nbhd of ¯ M ). In the rest of this chapter we understand φˇ to denote the free C(θ energy function restricted to N1 . So, φˇ has only one minimizer, that is, 1: ˇ , θM ) > φ(1, ˇ θM ) for all C ∈ N1 \{1}. φ(C

(6.44)

Remark 6.7 Any deformation y = χ(x ) which is continuously diﬀerentiable and minimizes the energy functional (6.12) for the restricted density φˇ has a stretch which is 1 everywhere. The rotation appearing in the polar decomposition remains arbitrary, and may well depend on x . In this case natural states have a deformation gradient of the form F (x ) = R(x ), with R(x ) orthogonal for any x . A well known theorem of Euler states that any deformation26 whose gradient is everywhere orthogonal is a rigid displacement; equivalently, the deformation gradient is orthogonal and constant. A proof under the smoothness assumed above (Euler’s requires second derivatives) is given by Gurtin (1981), and the result still holds if smoothness is suitably weakened (Proposition 10.3). We conclude that the only energy minimizers for the restricted φˇ are rigid displacements of the whole body. ˇ , θ) is supposed to admit continuous second derivatives with reIf φ(C spect to C , one can introduce the classical fourth-order elasticity tensor, also called ‘the tensor of the elasticities’: ∂ 2 φˇ C= . (6.45) ∂C ∂C The tensor ﬁeld C can be viewed as a (C , θ)-dependent symmetric map from the 6-dimensional space Sym to itself. A classical assumption guaranteeing φˇ to have a (local) minimum at C is that the elasticity tensor at C be positive deﬁnite. This also implies that φˇ is strictly convex at, and near, C . We cannot make an analogous convexity ˜ as this is a Galilean invariant function assumption on the energy density φ, of F . A standard necessary condition for φ˜ to have a local minimum at F is the Legendre-Hadamard inequality: for any vectors a and n Cijhk ai ah nj nk ≥ 0,

Cijhk :=

∂ 2 φ˜ (F , θ). ∂Fij ∂Fhk

(6.46)

As indicated in §6.4.3, this condition cannot hold everywhere in the domain of the energy function φ˜ of elastic crystals. The tensor of elastic moduli used in linear elasticity, to be denoted by L, is deﬁned as four times27 the tensor of the elasticities at the minimizer 1: L = 4C(1, θM ). 26 27

(6.47)

Remember that, in our deﬁnition, χ is invertible and has a connected domain. This is due to the 1/2 in the deﬁnition (2.53) of the strain tensor E used in (6.48).

6.6 IN THE VICINITY OF AN ENERGY WELL

191

Near 1, that is, for (small) stretches in N1 , the free energy function φˇ is approximated by the quadratic form 1

φ = E · L[E ],

(6.48)

2

where E = (C − 1)/2 is the nonlinear strain28 tensor – see (2.52). Notice that, for C ∈ N1 , E belongs to a neighborhood N0 of 0 in Sym. The second-derivatives test for a minimum of φˇ at (1, θM ) requires that L be positive semi-deﬁnite: E · L[E ] ≥ 0 for all E ∈ Sym.

(6.49)

Also, the invariance (6.17) of φˇ implies that the tensor L of elastic moduli for a crystal with reference lattice L(¯ ea ) satisﬁes the following condition: Q t L[E ]Q = L[Q t EQ] for all E ∈ Sym and Q ∈ P (¯ ea ).

(6.50)

¯ of φˇ and of the elasticity tensor on The dependence of the minimizer C temperature is described in §§6.6.1 and 6.6.3, respectively. Remark 6.8 Given any orthonormal basis in R3 and the induced coordinate system, one has, for the map L : Sym → Sym: (L[E ])ij = Lijhk E hk ,

Lijhk =

∂ 2 φˇ (0, θM ), ∂Eij ∂Ehk

(6.51)

ˇ ij , θ) being the explicit expression, in that coordinate system, of φ(C ˇ , θ) φ(E for C = 1 + 2E – see (2.52). Thus, by deﬁnition, L enjoys the so-called minor and major symmetries, that is, the invariance under the exchange of the indices in ﬁrst or in the second pair, and under the exchange of the ﬁrst and the second pairs of indices, respectively: Lijhk = Ljihk = Lijkh = Lhkij .

(6.52)

Thus the elasticity tensor has at most 21 independent entries. This same conclusion also follows for the entries of the tensor C of the elasticities. Given an orthonormal basis (i , j , k ) one has in a canonical way the following basis U1 , . . . , U6 of Sym:29       1

U1 =  0 0



0

0,

0

0

0

0

U4 =  0 0 28

29

0

0

0

U2 =  0 0

0



0

1  , U5

1

0



0

0

1

0  , U3

0

0

0

= 0 1

0

0

= 0 0

1



0

0,

0

0



0

0

0

0,

0

1

0

1

U6 =  1 0

0

(6.53)



0

0.

0

0

In linear elasticity one actually approximates φˇ near 1 by means of a quadratic form whose variables are the components of the linearized strain tensor (2.53). The tensor corresponding to this quadratic form coincides with L in (6.48). For a coordinate-free deﬁnition: U1 = i ⊗ i, . . . , U4 = j ⊗ k + k ⊗ j, . . . , etc.

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ENERGETICS

Notice the ordering of the UA , A = 1, . . . , 6, in formula (6.53), which comes from the usual Voigt notation for 3 by 3 symmetric matrices when represented as sextuples of numbers by means of the following index convention: 11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, 12 → 6.

(6.54)

The six independent entries in the symmetric matrix representing any tensor W in Sym with respect to the basis (i , j , k ) give, through the convention (6.54), also the six components of that tensor expressed as a linear combination W A UA of the basis vectors UA in (6.53); one can also represent W as a linear combination WA U A of the dual vectors 1

U A = UA for A = 1, . . . , 3,

U A = UA for A = 4, . . . , 6.

(6.55)

2

The traditional choice is to represent (symmetric) stress tensors (see (2.55)) in the basis UA , so that σ11 → σ 1 , σ23 → σ 4 , etc., and the (symmetric) strain tensors in the dual basis U A , so that E11 → E1 , E23 → 12 E4 , etc. Then, given the symmetries (6.52), one can represent the fourth-order tensor L in (6.47) as a 6 by 6 symmetric matrix (LAB ) of the elastic moduli: (L[E ])A = LAB EB ;

(6.56)

here, in agreement with the above conventions, L12 = L1122 , L14 = L1123 , L45 = L2331 ,

etc.,

(6.57)

and, equivalently, in this notation, the quadratic form (6.48) has the expression 1 φ = LAB EA EB . (6.58) 2

In §6.8 we recall the classical forms of this matrix for the various classes of anisotropic linearly elastic solids. Remark 6.9 As mentioned in Remark 6.8, the independent entries of any tensor C of the elasticities are at most 21. Cauchy (1828a,b), in his model of elasticity for a simple lattice whose points interact through a central binary force, derived a set of extra symmetries for L, which are known as the Cauchy relations:30 Lijhk = Lihjk . (6.61) 30

By the assumptions, the interaction of any pair of lattice points derives from a potential energy V = V (r2 ), r being their mutual distance, and dependence being set on r2 rather than r for convenience. We attribute half of the energy to each point in the pair, and consider the total energy of a point in the lattice by adding its interaction energies with all the other lattice points. Here and below we assume the function V to decrease suﬃciently fast with the reciprocal distance for inﬁnite sums to converge and to commute with diﬀerentiation. Trivially, all lattice points have the same total energy, so we compute the energy φ of the lattice point at the origin: φ=

1 2

∞

n=1

2 V (r(n) ),

2 r(n) = y(n) · y(n) = x(n) · C x(n) ,

(6.59)

C being the Cauchy-Green deformation tensor from the reference lattice conﬁguration (under the Cauchy-Born assumption), and x(n) the position vector of the nth point

6.6 IN THE VICINITY OF AN ENERGY WELL

193

These symmetries imply, together with (6.52), that L is symmetric with respect to the exchange of any two indices, and reduce the number of independent entries in the elasticity tensor of crystals to a maximum of 15. While materials like beryl and rock-salt do indeed satisfy (6.61), at least approximately, it is well known that experimentally the Cauchy relations in general do not hold for L. This prompted various authors interested in deriving the expression for the energy of crystalline solids from molecular calculations – notably Poisson (1842), Thomson (Lord Kelvin) (1890), Poincar´e (1892), Voigt (1910), Born (1915) – to revise and generalize Cauchy’s model and assumptions, so as to avoid (6.61) being a consequence of a molecular theory. Some of these proposals, for instance the one by Born (1915), also presented by Born and Huang (1954), entailed replacing the simple lattices of Cauchy by the more realistic multilattices, whose geometry and kinematics is described in chapter 11. For a more detailed discussion of this point of view we address the reader to Love (1927), Stakgold (1950), Ericksen (1977), (1984). More recent proposals are based on statistical mechanics or ‘ab initio’ calculations; see for instance Penrose (1979), Weiner (1983), Salje (1990), or Parr and Yang (1989), Sutton (1993). There is an analogue of the Cauchy relations for third-order elastic constants and 3-body interactions; using the Ricci tensor ;ijk we write it ;ikm Lijklmn = 0, or L112233 + 2L122331 = L112323 + L223131 + L331212 . This was ﬁrst derived by Zucker and Paik (1979). See also Parry (1979). 6.6.3 Temperature-dependence of the elastic moduli In §6.6.2 we have chosen for deﬁniteness the value θ = θM and the cor¯ (θM ) = 1. Clearly, one can responding basis ¯ ea as a reference, so that C deﬁne the fourth-order tensor of elasticities as in (6.45) for each θ and ¯ (θ). This tensor is then a function of θ, and the elastic each minimizer C moduli of a crystal do indeed depend on temperature. However, the tensor components in (6.45) are not the moduli that are experimentally measured as functions of temperature and are reported in the literature for various materials (see for instance Love (1927), Thurston (1974)). Indeed, the standard experimental practice is based on procedures that give the moduli at any given θ = θ¯ when one takes as reference conﬁguration the natural state ¯ with the basis ¯ of the crystal at θ, ea (θ) obtained from ¯ ea (θM ) through the ¯ ¯ ¯ (θ). This is repeated for each value θ. stretch U To compare elasticities at diﬀerent conﬁgurations we return to formula (2.57) for the transformation properties of the energy function, neglecting the change of observer and changing slightly the notation to accommodate in it. By diﬀerentiating twice we obtain, for the tensor of elasticities C in (6.45), 2Cijhk =

∞

n=1

2 V (r(n) ) xi

(n) (n) (n) (n) xj xh xk ,

(6.60)

the prime denoting diﬀerentiation; therefore C is symmetric in all its indices, and so is in particular L.

194

ENERGETICS

various possibilities. So, let us denote by F0 the gradient of the deformation mapping the conﬁguration R into R . These can respectively represent austenite [martensite] at temperatures θM and θ, or austenite and martensite at θM , or other choices. By diﬀerentiating twice (2.57) with respect to F , and then setting F = 1, we obtain Lijhk (θ) = (det F0 )−1 (F0 )iL (F0 )jM (F0 )hR (F0 )kS CLM RS (F0t F0 , θ),

(6.62)

where L and C are the tensors of elastic moduli and of the elasticities for ˜ respectively. So, if R and R represent austenite the functions φ˜ and φ, at temperatures θ and θM , formula (6.62) details how the moduli (L) at temperature θ diﬀer from the elasticities (C) evaluated at the minimizer at that temperature. Assume now R and R to represent martensite and austenite at temperature θM , the reference conﬁguration to be the austenite at that temperature, and the free energy to be given by a polynomial in the stretch components. Formula (6.62) can help determining the coeﬃcients of the polynomial by means of experimental data. Indeed, while the moduli of austenite at θM only detail the quadratic part of the polynomial, the moduli (L) of martensite at θM give information also on the higher-order terms, which are essential for the transition (chapter 7). An equivalent form of (6.62), actually expressing C in terms of L, is given in (7.6)2 below. 6.7 Standard forms for the elasticities of anisotropic solids In this section we give the standard forms assumed, in suitable coordinates, by the fourth-order elastic tensor deﬁned in (6.56). These reduced forms, due to Voigt (1910), can be found for instance in Love (1927), Landau and Lifˇsits (1959), Jaunzemis (1967), Gurtin (1972). The standard forms below are representatives of the symmetry classes of all the fourth-order elasticity tensors under the following equivalence relation: two such tensors L and L are in the same class if they satisfy (6.50)1 for Q in groups G and G , respectively, that are (contained in and) conjugate in O (or, equivalently, in O+ because (6.50)1 is unaﬀected by replacing Q by −Q). Forte and Vianello (1996) show that there are exactly eight such classes: seven are determined by the requirement that (6.50) be satisﬁed by the holohedral point groups in each of the seven crystal systems; the eighth class refers to (6.50) being satisﬁed for all Q in O (isotropic elastic tensors). Below we list the classical reduced forms of the 6 by 6 matrices L in (6.58) for all these eight classes. A diﬀerent classiﬁcation criterion has been proposed by Huo and Del Piero (1991). For any group P of rotations they consider the whole space L(P ) of elastic tensors that are invariant (at least) under P according to the analogue of (6.50). Then two such groups P and P are elastically equivalent if there is a rotation R such that L ∈ L(P ) ⇔ Lijhk = Rip Rjq Rhr Rks Lpqrs , L ∈ L(P ). This necessarily holds if P and P are or-

6.7 ANISOTROPIC ELASTICITY

195

thogonally conjugate, but not only. For elasticity tensors with the major and minor symmetries, as here, Huo and Del Piero (1991) show that the elastic equivalence classes are exactly 10, the 9 crystallographic determined by Voigt (1910) plus isotropy (see also Love (1927)). This larger number is reasonable if one notices that here one must match two whole subspaces of elastic tensors, and not only two of their respective elements, as in the previous criterion; see Forte and Vianello (1996) for more comments. In §5.4.3 we have deﬁned a speciﬁc orthonormal basis (i , j , k ) for each crystal system. The related canonical basis U 1 , . . . , U 6 of Sym is given by (6.55). It is in the latter basis of Sym that the matrix LAB representing the fourth-order elastic tensor L as in (6.56) takes the standard form.31 Triclinic symmetry The elasticity tensors that are invariant as in (6.50) under the triclinic group {1} are given by generic 6 by 6 symmetric matrices, with twenty-one distinct entries. Monoclinic symmetry For the monoclinic system we choose the basis (i , j , k ) as in §5.1.4, that is, the holohedry is {1, Rjπ }. The elasticity tensors with this invariance have thirteen independent entries, as follows:   11 L L12 L13 0 L15 0 0 L25 0   L12 L22 L23   13 23 33 L L 0 L35 0  L (6.63) .  0 0 L44 0 L46   0   15 L L25 L35 0 L55 0 46 0 0 0 L 0 L66 One can refer to the discussion in §5.4.3.2 on the monoclinic decompositions of Sym to see how such a standard form arises. In the 4-dimensional space ˆ 0 ) any four orthonormal vectors can be chosen to generate irreducible C(e a invariant subspaces. To each one of these we assign a diﬀerent eigenvalue. ˆ 0 )⊥ . Then, for The analogue holds in the 2-dimensional complement C(e a α, β, γ = 1, 2, a, b, c = 1, . . . , 4, Rab , Rαβ orthogonal matrices, and VA the unit vector of the canonical basis vector UA of Sym associated with the chosen basis (i , j , k ), L = a,b,c λa Rab Rac Vb ⊗ Vc + α,β,γ λα Rαβ Rαγ Vβ ⊗ Vγ . (6.64) It is a matter of bookkeeping to obtain the representation (6.63) of L in the basis (i , j , k ), the elements LAB being expressed as functions of the 6 + 1 angles in Rab , Rαβ and of the six eigenvalues. One can see an explicit similar construction for the tetragonal symmetry, below. 31

The standard form is not the ‘simplest’ form an elastic tensor can be given. For instance, the matrix LAB , being symmetric, can always be diagonalized. However the diagonal form, in which at most six independent parameters (the eigenvalues) appear, conceals other independent parameters related to the choice of the eigenvectors.

196

ENERGETICS

Orthorhombic symmetry For the orthorhombic system the basis (i , j , k ) is chosen as in §5.4.3.3, and the holohedry is Oijk . Elasticity tensors with this invariance have nine independent entries:  11  L L12 L13 0 0 0 12 22 23 L L 0 0 0  L  13  L23 L33 0 0 0  L (6.65)  . 0 0  0 0 L44  0   0 0 0 0 L55 0 66 0 0 0 0 0 L With R a 3 by 3 orthogonal matrix; the UA canonical basis vectors of Sym associated with the chosen basis (i , j , k ), and the VA given in (5.24), this form results from representing in the basis (i , j , k ) above the tensor 3 3 L = a,b,c=1 λa Rab Rac Ub ⊗ Uc + α=1 λα Vα ⊗ Vα . (6.66) Rhombohedral symmetry For an easier comparison with standard references, we consider the orthonormal basis (c1 , c2 , c3 ) introduced in §5.4.3.4, in terms of which the 2π/3 rhombohedral holohedry is generated by Rc3 and Rcπ1 . The elasticity ten32 sors with this invariance have six independent entries and the following form in the chosen orthonormal basis:  11  L L12 L13 L14 0 0 0 0  L12 L11 L13 −L14   13  L13 L33 0 0 0 L  (6.67)  14 . −L14 0 L44 0 0 L    0 0 0 0 L44 L14 1 14 11 0 0 0 0 L − L12 ) 2 (L If c2 rather than c1 is along a twofold axis, as in Love (1927) and Landau ¯ has the form (6.67) except and Lifˇsits (1959), then the new tensor, say L, 14 24 15 24 56 ¯ ¯ ¯ ¯ ¯ that L = 0 = L while L = −L = −L need not vanish. With R a 2 by 2 orthogonal matrix; V1 , V2 as in (5.28); W3 , W4 the vectors given by (5.38) for x = 1, y = 0 and x = 0, y = 1, respectively, and W5 , W6 their analogues for σ ¯ replaced by σ ¯ + π2 , the form (6.67) results 32

2π/3

An elasticity tensor with nonholohedral Rc3 -symmetry and corresponding i.i. ˆ 0 ) and four to eigenspaces has seven moduli (Love (1927)), three related to C(e a ˆ 0 )⊥ . It turns out that an extra twofold axis of symmetry, of unit vector u say, necC(e a essarily exists in the c1 , c2 -plane, as was already clear to Landau and Lifˇsits (1959). Therefore this elasticity tensor is actually invariant under a rhombohedral holohedry. The latter, though, as the vector u, depends on the elastic tensor, among those enjoy2π/3 ing Rc3 -symmetry, and no single rotation matches all such tensors to a holohedral one. Therefore this symmetry is diﬀerent from the one associated with the rhombohedral holohedry if one uses the criterion of Huo and Del Piero (1991).

6.7 ANISOTROPIC ELASTICITY

197

from representing the tensor 2 4 6 a,b,c=1 λa Rab Rac Vb ⊗ Vc + λ3 i=3 Wi ⊗ Wi + λ4 i=5 Wi ⊗ Wi (6.68) in the basis (c1 , c2 , c3 ). Hexagonal symmetry (and transverse isotropy) Consider the basis (i , j , k ) in §5.4.3.6. Elasticity tensors with hexagonal33 invariance have ﬁve independent entries as follows:  11  L12 L13 0 0 0 L 12 11 13 L L 0 0 0 L   13  L13 L33 0 0 0 L  (6.69)  . 44 0 0 L 0 0  0    44 0 0 0 0 L 0 1 11 (L − L12 ) 0 0 0 0 0 2 With R a 2 by 2 orthogonal matrix; Vr , r = 1, 2 given by (5.50), and ¯ r, r = Vr , r = 3, . . . , 6 represented in the basis (i , j , k ) by the matrices of V 3, . . . , 6 in (5.40)-(5.41), this form results from representing in the basis (i , j , k ) the tensor 2 4 6 a,b,c=1 λa Rab Rac Vb ⊗ Vc + λ3 b=3 Vb ⊗ Vb + λ4 b=5 Vb ⊗ Vb . (6.70) Tetragonal symmetry Elasticity tensors with this invariance, when represented in the basis (i , j , k ) of §5.4.3.5, have six independent entries as follows:34 

 L11 L12 L13 0 0 0 12 11 13 L L 0 0 0  L  13  L13 L33 0 0 0  L (6.71)  . 0 0 L44 0 0   0   0 0 0 0 L44 0 66 0 0 0 0 0 L We give here a few details which will be used in §7.5. In the notation of §5.4.3.5 and in terms of an angle ϕ, the spectral representation of a tetragonally symmetric L is: L = µ1 (cos ϕV1 − sin ϕV2 ) ⊗ (cos ϕV1 − sin ϕV2 )+ µ2 (sin ϕV1 + cos ϕV2 ) ⊗ (sin ϕV1 + cos ϕV2 ) +

(6.72)

µ3 V3 ⊗ V3 + µ6 V6 ⊗ V6 + µ4 (V4 ⊗ V4 + V5 ⊗ V5 ). 33

34

We obtain this same matrix if we impose transverse isotropy about the axis k, that is, invariance for all rotations with axis k. π/2 Elasticity tensors with nonholohedral Rk -symmetry and corresponding i.i. ˆ 0 ), three with eigenspaces have seven moduli (Love (1927)), three related to C(e a W1 W2 and one with W3 . Also in this case, as in footnote 32, an extra twofold axis of symmetry, depending on the elasticity tensor, necessarily exists in the i, j plane. Remarks analogous to the ones in footnote 32 can be made.

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ENERGETICS

In the basis (i , j , k ) the nonvanishing components of L are 1

1

1

3

6

2

L11 = L22 = A + B + 2C + µ3 , 1

1

1

3

6

2

L12 = A + B + 2C − µ3 , 1

1

3

3

L13 = L23 = A − B − C, 1

2

3

3

L33 = A + B − 4C, L66 = 2µ6 ,

L44 = L55 = 2µ4 ,

(6.73) (6.74) (6.75) (6.76) (6.77)

for A = µ1 cos2 ϕ + µ2 sin2 ϕ, B = µ1 sin2 ϕ + µ2 cos2 ϕ, √

C=

2

6

Cubic symmetry Elasticity tensors with this (i , j , k ) of §5.4.3.7, have three  11 L L12 12 L11 L  12 L12 L  0  0  0 0 0 0

(µ2 − µ1 ) sin ϕ cos ϕ.

(6.78) (6.79)

invariance, when represented in the basis independent entries as follows:  L12 0 0 0 L12 0 0 0   L11 0 0 0  (6.80) . 0 L44 0 0   0 0 L44 0 0 0 0 L44

Also here we give a few details which will be used in §7.5. In the notation of §5.4.3.7 the spectral representation of a cubically symmetric L is: L = λ1 V1 ⊗ V1 + λ2 3a=2 Va ⊗ Va + λ3 6a=4 Va ⊗ Va . (6.81) Therefore in the basis (i , j , k ) the nonzero components of L are 1

L11 = L22 = L33 = (λ1 + 2λ2 ),

(6.82)

3

1

L12 = L23 = L13 = (λ1 − λ2 ),

(6.83)

3

44

55

66

L = L = L = 2λ3 , (6.84) as is well known – see, for instance, Thomson (Lord Kelvin) (1878) or Mehrabadi and Cowin (1990). Isotropy The elastic tensors having this invariance have two independent entries, and are represented in any orthonormal basis by the matrix in (6.80) for L44 = 12 (L11 − L12 ). Concerning the standard isotropic Lam´e coeﬃcients λ and µ, we point out that these coincide with L12 and L44 , respectively, and thus L11 = λ + 2µ. The coeﬃcient µ is the shear modulus, while the modulus of compression is λ + 23 µ.

CHAPTER 7

Bifurcation patterns 7.1 Introduction In general, for a crystalline solid immersed in a heat- and pressure-bath, the location of the energy wells depends on the environmental temperature and pressure, regarded as control parameters. Bifurcation theory studies how the equilibria – here the critical points of the free energy density – change in number and stability character as the control parameters are varied continuously. An analysis in which both environmental temperature and pressure vary is given by Ericksen (1996a). For simplicity, here we assume pressure to be zero, temperature remaining the only control, and show how the energetics in chapter 6 can be used to describe a family of static bifurcations (phase transitions) in simple lattices, included in the larger class of martensitic phase transformations. It is diﬃcult to give a general deﬁnition of phase,1 its characterization being clear in speciﬁc cases. One of the generally accepted features of the transformations considered here is that a change of symmetry is realized by a cooperative, diﬀusionless, temperature-driven deformation of the crystalline lattice: the lattice points change their positions in a continuous way while the holohedry, or the lattice group, changes abruptly. In terms of the analysis of multiphase crystals in §6.4.2, this happens when the θ-dependent branches of minimizers of the free energy function φˆ meet. Hereafter, following Ericksen (1989), (1991a), (1992), (1993), (1996b), we study the typical bifurcation problem that arises when branches of critical points meet at suitable bifurcation points where the symmetry of the equilibrium states may change. In agreement with the treatment of symmetry in the previous chapters, by change of symmetry we mean change of Bravais lattice type, or its strengthened version for weak transformations, that is, change of variant structure – see §§4.3 and 5.3. We adopt here suitable assumptions of smoothness and genericity for the constitutive functions. In more generality, ideas of genericity, stability, 1

In his fundamental memoir, Gibbs (1878) introduces the concept of phase to refer solely to the composition and thermodynamic state of any homogeneous body which can be formed out of any set of component substances, without regard to its quantity or form. He calls such bodies as diﬀer in composition or state diﬀerent phases of the matter considered, and regards all bodies which diﬀer only in quantity and form as diﬀerent examples of the same phase. Finally, he calls coexistent phases which can exist together, divided by a planar interface, in an equilibrium which does not depend on passive resistances to change. 199

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classiﬁcation of critical points, etc., have their home in the theory of singularities of diﬀerentiable maps (Golubitsky and Guillemin (1974), Arnold et al. (1986)), to be called below singularity theory for short, and in Catastrophe Theory – see for instance Arnold (1984) – which is the part of singularity theory more widely known and applied in the natural and social sciences. We address the reader to the references above for details; on smoothness assumptions, we recall that Ericksen (1980b) analyzes a special bifurcation problem, of interest in the theory of solid-state phase transitions, based on a free energy that need not be more than twice continuously diﬀerentiable. Among other contributions to the theory of martensitic phase transformations, we mention the one by Khachaturyan and his school – see for instance Roitburd (1978), Khachaturyan (1983). That theory has been qualiﬁed as geometrically linear because there the free energy is taken to be quadratic near each one the energy wells. We address the reader to Bhattacharya (1993) for a detailed comparison of Khachaturyan’s approach and the geometrically nonlinear one we adopt here. 7.1.1 The Landau theory Since the thermodynamic state, which here coincides with the lattice conﬁguration, changes continuously near and at the transition, what is being described is a second-order phase transformation (sometimes also called of the second kind). If, instead, the positions of the lattice points are allowed to jump at the transition, the latter is called ﬁrst-order (or also of the ﬁrst kind). These deﬁnitions are introduced in the seminal, fundamental work of Landau (1965), pp. 193 and 216 – also in Landau et al. (1980) – who started an approach to phase transformations in crystalline solids based on minimizing a suitably symmetric analytic free energy function under certain assumptions of genericity. This approach has been subsequently applied to weak ﬁrst-order solid-state phase transformations2 and to a variety of other kinds of transformations in which a ﬁnite number of order parameters governing the transition can be identiﬁed. Also these versions of the original approach by Landau go under the name of ‘Landau’ or ‘Landau-like’ theories. A summary of such extensions for crystalline solids and reference to the vast related literature, in particular to the work of Yu.M. Gufan and his coworkers, is given by Tol´edano and Tol´edano (1987) and Tol´edano and Dmitriev (1996). Moreover, the latter book also analyzes reconstructive 2

In certain cases this is necessary because the bifurcating branches are all unstable, and need to be suitably re-stabilized. This leads to a so-called subcritical ﬁrst-order (weak) bifurcation; an example is given in §7.4.4. In principle, one can distinguish a second-order from a weak ﬁrst-order transition by looking at the behavior of certain quantities (speciﬁc heats, succeptibilities, etc.) near the transition. Details are given by Tol´ edano and Dmitriev (1996), for instance. In practice this may be diﬃcult, being related to the actual magnitude of the experimental errors, which may blur the diﬀerence in the aforementioned behavior altogether.

7.1 INTRODUCTION

201

phase transformations; these, unlike the ones we consider here, cannot be described by restricting the domain of the constitutive equations to a weaktransformation neighborhood. An important example is the α-γ transition in iron – see §3.6.3. We also mention Salje (1990) for applications of the Landau theory to many solids of interest in mineralogy and materials science, including extensive references, and Stanley (1971) for an introductory analysis of critical phenomena in ﬂuids and magnetics which goes beyond the Landau theory. Indeed, for such materials, it was soon realized that the theory, in spite of its beauty and successes, failed to give the correct dependence on temperature of certain physical quantities related to derivatives of the thermodynamic potential: the power-laws governing such a dependence in the Landau theory have exponents, called critical indices or exponents, which do not agree with the experimentally determined ones. Roughly, this is due to the fact that, near the phase transition, ﬂuctuations become important and should be considered somehow in the thermodynamic potential, for instance by allowing the order parameters to vary in space. Various statistical mechanical models and molecular calculations have been used to obtain realistic potentials – see for instance Salje (1990). The problem of critical indices has produced much research on critical phenomena, of both theoretical and experimental character, part of which is mentioned in the review article by Stanley (1999). Since a number of solid-state phase transformations associated to elastic instabilities seem to be fairly well described by the Landau theory away from domain walls or dynamic phenomena – see Salje (1990), p. 12 for the ferroelastic and co-elastic transitions – we disregard ﬂuctuations and present the classiﬁcation of generic symmetry-breaking transitions in simple lattices within the Landau theory, with some preliminary comments. Remark 7.1 The ﬁrst comment concerns the regularity of the thermodynamic potential. This nontrivial issue does not seem to have been suﬃciently explored. Ericksen (1980b) gives an example of how regularity may aﬀect a phase transformation: if the potential is at least three times continuously diﬀerentiable, a second-order cubic-to-tetragonal transition in simple lattices is generically not possible because the lower-symmetry branches are unstable by the second derivatives test; if the potential is only twice continuously diﬀerentiable, then such a transition is possible by a version – see Ericksen (1980c) – of a theorem by Poincar´e on the exchange of stabilities. Landau’s original idea is that, near the transition, the potential be analytic in the order parameters, which are the variables driving the transition itself. Later researchers – for instance Salje (1990), Tol´edano and Dmitriev (1996) – regard Landau’s polynomial expansion not as an approximation but as the actual potential, except perhaps near the transition, where extra nonsmooth terms may become important to describe ﬂuctuations. Here we assume the potential to be smooth enough to have a certain Taylor expansion near the transition. By excessively restricting a priori the

202

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smoothness of the free energy density we shake the very foundations of the Landau theory; we have to face the problems of deﬁning what it means for a bifurcation to be generic, and then of classifying the diﬀerent possible bifurcation patterns. Remark 7.2 The second comment concerns the various points of view on the choice of reference conﬁguration, to which the free energy density is related as in chapter 6. The ﬁrst is to take right at the beginning a ﬁxed energy minimizer at some given temperature, as we will initially do in §7.2. When describing bifurcations, two other choices are usually made. One is to choose as reference the conﬁguration of the crystalline lattice at the bifurcation point, as does, for instance, Ericksen (1989), (1993) and we will also do from §7.2.1 on. The point of view of Landau (1965) and his followers – see Landau et al. (1980) – seems diﬀerent: the thermodynamic potential is actually an excess function, that is, it consists of the diﬀerence between the potentials at the less symmetric lattice conﬁguration (martensite) and at the more symmetric one (austenite) at the same temperature, the latter then being a running reference. This assumption is not always clear in the presentations of the Landau theory, but is very explicitly stated by Salje (1990), for instance. He also points out that one has to extrapolate the behavior of the austenitic phase, assumed to be stable above the transition temperature for deﬁniteness, down below that temperature, where this phase is unstable, hence not observable. In many cases, particularly in minerals, one cannot have suﬃcient data for the extrapolation, for instance because melting or chemical reactions occur not far from the transition temperature. It is then common to turn to approximations, one being to altogether neglect the temperature dependence of the parameters of the austenite near the transition, thus taking their available experimental values nearest to the transition itself. The systematic error involved is acceptable for most ferroelastic and co-elastic minerals, which have large transformation stretches; it makes the approximation unreliable when the transformation stretch is small and comparable with the thermal expansion of the austenite. Another method of obtaining the parameters of unobservable austenite below the transition temperature is by suitably averaging the parameters of the observable martensitic phase, in all possible orientations. Now no guessed austenitic data are necessary, and indeed this method is very good in many circumstances. It fails when there are volume anomalies in the austenitic phase near the transition. Thus neither approximation exempts the researcher from actually studying the lattice parameters of the phases; see Salje (1990), ch. 4, for details. In §7.7 we show that the choice of reference conﬁguration proposed here and the subsequent theory are fully consistent with Landau’s ideas. Roughly, the ‘running reference’ is automatically realized when one constructs the so-called reduced or Landau potential: the zeroth-order term in this potential is not a constant but rather a function of environmental

7.1 INTRODUCTION

203

temperature (and also environmental pressure when this is present as an additional control parameter), and represents the free energy of the highsymmetry phase for the given values of the controls. As we have seen in chapter 6, natural states are the simplest minimizers, that is, the homogeneous ones, of the potential

Φ[χ; θ] = φ˜ Dχ(x ), θ dx R

– see (6.12) – at a given temperature θ. The general invariance of the free energy density φ˜ for simple lattices dictates the location of all the symmetry-related energy wells once one of them is known. To cope with phase transformations, though, we must allow for the existence of wells that are not symmetry-related: these represent the phases with diﬀerent symmetries which may co-exist in a neighborhood of the transition temperature. For our purposes it is suﬃcient to assume the existence of a ﬁnite number of families of symmetry-related wells; in particular, in §6.4.2 we considered two phases, and also here we restrict the attention to this case. A more general situation is considered by Ericksen (1996a), and Truskinovsky and Zanzotto (2002) present a model for the tetragonal-orthorhombic-monoclinic phase changes of zirconia (ZrO2 ), including the analysis of the triple point. A general treatment of multiphase crystals can be found in Tol´edano and Dmitriev (1996). Our bifurcation analysis will be local, in the sense of §4.3. Transitions for which all phases belong to some neighborhood NC satisfying the conditions in Proposition 4.1 are studied in detail by Ericksen (1989), who calls them weak; a notation we also adopt here. For their description it is reasonable to restrict the free energy density to NC , disregarding arbitrarily large lattice-invariant shears; these are usually associated with plasticity, and would make the analysis much harder. For weak martensitic phase transitions in the neighborhood NC , the lattice group of any phase is a subgroup of the lattice group of the center C. Second-order transitions are necessarily weak, the center C representing the conﬁguration at the transition, which has the highest (local) symmetry. In this case the analysis in §4.3 allows us to describe symmetry changes in terms of holohedries and stretches from the conﬁguration at the transition, taken as a reference, rather than in terms of lattice groups and stretches from an arbitrary reference. This result considerably simpliﬁes the analysis because one and the same treatment works for all transitions for which the reference conﬁguration has a given holohedry, irrespective of its centering. We stress that there are interesting and technologically important transitions that are not weak, as, for instance, the ones along which the crystal system remains the same while the Bravais type changes: in the cubic system we have the so-called α-γ transformation in iron, of utmost importance in metallurgy, where the lattice type changes from b.c.c. to f.c.c. Other ex-

204

BIFURCATION PATTERNS

amples are the f.c.c.-to-h.c.p. transition, for instance in Al, Cu, Ag, Au,..., or the b.c.c.-to-h.c.p. transition, for instance in Li, Na, Fe,...; in these cases a simple lattice structure transforms into a multilattice, and vice versa. A systematic analysis of such phase transitions and of bifurcations in more complex crystal structures, along the lines for the weak transitions presented below, remains to be done and is of both theoretical and applicative interest for understanding the behavior of crystalline materials. Partial results in these directions are given by Parry (1981), (1982a), (1984c), (1987), Tol´edano and Tol´edano (1987), Salje (1990), Tol´edano and Dmitriev (1996), Ericksen (2001b), Conti and Zanzotto (2002). A weak transition can only take place if certain kinematic and energetic requirements are satisﬁed. In §5.3 we have discussed some of the kinematic restrictions resting on the sole symmetry of the phases, while in §7.4 below we present the generic weak bifurcations allowed by a smooth elastic free energy, following Ericksen (1993). A comparison of kinematic and energetic restrictions is summarized in §7.4.7, and the correlation with the ‘running reference’ approach of the Landau theory is given in §7.7. For the scheme to be meaningful, the free energy density is required to have a minimum number of continuous derivatives, depending on the energy invariance in the selected wt-nbhd; typically 6 derivatives are suﬃcient. 7.2 Branches of isolated critical points, and bifurcation points Denoting by ea0 the reference lattice basis from which the deformations are ˇ , θ) calculated, we base the bifurcation analysis on the energy function3 φ(C 0 deﬁned in (6.15), whose invariance is given by the group G(ea ) in (3.9). If the function φˇ is twice continuously diﬀerentiable, the ﬁrst-derivatives test for a minimum at ﬁxed θ provides the equilibrium conditions ∂ φˇ = 0, (7.1) ∂C while, as a necessary condition, the second-derivatives test requires the elasticity tensor C in (6.45) to be positive semi-deﬁnite at any equilibrium (C , θ) satisfying (7.1): E · C(C , θ)[E ] ≥ 0 for all E ∈ Sym.

(7.2)

As in §6.6.2, here we assume the reference basis ea0 to give a minimizing conﬁguration at some temperature θ¯ (see Remark 7.2), so that 1 is ˇ , θ) ¯ and (7.2) holds for C = 1, thus becoming idena minimizer of φ(C tical to (6.49). We then consider a neighborhood N1 of 1 in Sym such 3

ˆ The analysis can be also done for the function φ(C, θ) in (6.4), but then it must be repeated when we change the centering of the basis e0a within a given system, because the ﬁxed sets and the lattice groups do change. Instead, based on Remark 4.6, only ˇ one analysis in terms of φ(C, θ) is needed if the diﬀerently centered bases e0a in a system have the same holohedry P , of course in that system; indeed, the function φˇ has local invariance given by P , irrespective of the centering of L(ea ).

7.2 ISOLATED CRITICAL POINTS AND BIFURCATION POINTS

205

that Ne0a = N1 ea0 is a wt-nbhd of ea0 as in Proposition 4.1, and restrict ˇ1 of 1 whose elements are squares the domain of φˇ to the neighborhood N ˇ1 may well contain other of the stretches in N1 . The restricted domain N ˇ (relative) minimizers or critical points of φ. By (6.5), the invariance of φˇ ˇ1 is dictated by the holohedral point group P (ea0 ), as in restricted to N (6.17). From §7.2.1 on we will simplify the description of bifurcations by choosing θ¯ and ea0 to be the temperature and lattice conﬁguration at which bifurcation can occur: there, (6.49) holds as an equality for some nonzero strain tensor E . ¯ and the linear map E → L[E ] be invertible. Let 1 solve (7.1) for θ = θ, ˇ1 × I of Then, by the implicit function theorem, in a neighborhood N ⊆ N ¯ (1, θ) the equilibrium equations (7.1) have exactly one continuous solution ¯ (θ). This function gives an isolated branch of critical points of φˇ C =C in N . The uniqueness of this curve of critical points also implies that the ¯ (θ)ea0 is independent of θ: symmetry of the basis U ¯ (θ)ea0 ) = P (ea0 ), P (U

¯ 2 (θ) = C ¯ (θ). for U

(7.3)

¯ (θ). Otherwise, by symmetry, there would be additional equilibria not on C When the eigenvalues of L are all positive, this branch consists of stable equilibria, and describes thermal expansion of the crystal, in agreement with the treatment in §6.6.1 when θ is the only control parameter. The analysis above implies that two or more curves of equilibria can meet ¯ or a change of symmetry can occur at (1, θ) ¯ along any such curve, at (1, θ), only if at least one of the eigenvalues of L vanishes; that is, ker L $= {0}.

(7.4)

¯ is called a bifurcation point. In either the case of multiple If so, (1, θ) branches or of a symmetry change along one, all the lattice groups of ¯ are subgroups of L(e 0 ) by continuthe branches passing through (1, θ) a ity. Moreover, when (7.4) holds, there is always a branch passing through ¯ whose points all have lattice group L(e 0 ) – see §7.4 below. We will (1, θ) a see that, generically, such a high-symmetry branch is divided by the bifurcation point into two parts, one of which is constituted of minimizers, while the other is made of unstable equilibria. For deﬁniteness, we will assume ¯4 the stable part to be the one corresponding to θ ≥ θ. By (7.4) at least one eigenvalue of the elasticity tensor L goes to zero ¯ along any one of the branches meeting there. Since as we approach (1, θ) the eigenvalues can be expressed in terms of the elastic moduli, this means that a certain combination of moduli gradually vanishes while the body comes near the bifurcation point. The elasticity tensor also determines the acoustic tensor, which governs the propagation of acoustic waves, and the 4

For the other possibility, the bifurcating branches, together with their stability character, can be obtained from the ones detailed here by consistently replacing θ − θ¯ with its negative in the energy and in the computations based on it.

206

BIFURCATION PATTERNS

approach to bifurcation inﬂuences the speed of these waves, some of which may tend to zero. This is in agreement with experiments, which sometimes reveal the occurrence of a symmetry-breaking transition by means of a modulus softening, connected to the vanishing of certain wave speeds. 7.2.1 Neighborhoods of bifurcation points For an arbitrary choice of reference basis ˆ ea , a basis ea0 = R0 U0ˆ ea and a ¯ temperature θ are a bifurcation point if the function φ(C , θ) for the free energy density related to the reference basis ˆ ea satisﬁes the equalities 2 ∂φ ¯ = 0 and ker C $= {0}, C = ∂ φ (C0 , θ), ¯ C0 = U 2 . (7.5) (C0 , θ) 0 ∂C ∂C ∂C In this case, as anticipated, we greatly simplify the bifurcation analysis by choosing ea0 as a reference basis, and we do so henceforth.5 Thus the ¯ is such ¯ (θ), deﬁned for θ near θ, continuous equilibrium branch C = C ¯ ¯ that C (θ) = 1 and (7.5) becomes

∂ φˇ ¯ = 0 and (1, θ) ∂C

ker L $= {0}.

(7.10)

¯ we assume By restricting, if necessary, the range of temperature near θ, ¯ ¯ ker C(C (θ), θ) to be nontrivial only for θ = θ, and any branch to be contained in a neighborhood N like the one introduced above. As above, the ¯ (θ), described by the holohedry P (U ¯ (θ)ea0 ) – see (7.3)2 – symmetry of C ¯ remains constant for θ near θ and, by continuity, is a holohedral subgroup of P (ea0 ) which, in general, depends on the branch itself. The analysis below, which rests on the study of variant structures and of i.i. subspaces presented in §5.4, allows one to describe all the possible equilibrium branches, their symmetry and their stability. All these qualitative features only depend on the choice of the holohedry P (ea0 ) and of the P (ea0 )-i.i. subspace of Sym to be chosen as the kernel of L. 5

Clearly, all the results that we obtain below with this choice of reference basis can be expressed in terms of the arbitrary reference ˆ ea . Setting F0 = R0 U0 , the equalities (F0 )−1 P (e0a )F0 = H < G(ˆ ea ), ¯= (see (6.62) ), for d0 = det U0 , V

¯ ] = d0 (F0 )−1 L[V ](F0 )−t C[V

(F0 )t V F0 ,

¯ ] ¯ ]H−t = C[Ht VH H−1 C[V

(7.6)

and the transformation rule (see (6.50))

¯ ∈ Sym, for any H ∈ H, V

(7.7)

allow us to perform the aforementioned translation. Notice that the orthogonality of H-invariant subspaces is correctly deﬁned by the following scalar product [ , ]: ¯ V ¯ ] := W ¯ · C−1 V ¯ C−1 . [W, 0 0

(7.8)

With the deﬁnition B0 = R0 C0 Rt0 , we have L[V ] = λV ¯ ] = λV ¯ C[V

⇔

¯ C−1 , ¯ ] = λC−1 V C[V 0 0

⇔

L[V ] = λB0 VB0 .

or (7.9)

¯ in (7.6). In particular, the elements of the kernels of L and C are related as V and V

7.2 ISOLATED CRITICAL POINTS AND BIFURCATION POINTS

207

7.2.2 Genericity The analysis of bifurcation patterns for simple lattice energies rests on an assumption of genericity for the constitutive function φˇ restricted to N . This hypothesis allows one to determine only the bifurcations that are not ‘very unlikely’ to occur. Let S be a ﬁnite set of equations involving a ﬁnite number of derivaˇ S has to be compatible with tives of the free energy density function φ. ˇ implies certhe invariance (6.17), which, by successive diﬀerentiation of φ, tain identities among derivatives. All relevant such identities have to be included in S, which should be an altogether compatible set of conditions. We avoid degenerate situations by assuming S not to contain more independent scalar equations, compatible with invariance, than the (seven) ˇ Any extra conditions may only be satisﬁed by independent variables in φ. ˇ special choices of φ, and can be violated under arbitrarily small perturbations, even in the class of the symmetry-preserving ones. That a property or behavior holds generically under the conditions in S means that it holds for all functions φˇ satisfying S without any additional constraints, possibly involving derivatives. In particular, φˇ itself is a generic function subject to S if it does not satisfy any additional constraints besides those in S. It turns out that such genericity can be characterized by a ﬁnite set of strict inequalities for φˇ and its derivatives, and thus the above property or behavior persists if we ‘slightly perturb’ φˇ while preserving its invariance and the imposed conditions S. As in other contexts, one can introduce a topology in terms of which such generic free energy functions constitute an open and dense subset of the space of all admissible free energies. This is the point of view of singularity theory, which also analyzes the sensitivity of a bifurcation to imperfections through the construction of the so-called universal unfolding of the singularity; this includes in parametric form – the number of essential parameters being called the codimension of the singularity – all small perturbations of the given bifurcation. We will not be so general and formal here, and will restrict our attention to the simplest singular critical points, the ones of codimension 0 in the language of that theory. We address the reader to Golubitsky and Schaeﬀer (1985), Golubitsky et al. (1988) or Tol´edano and Dmitriev (1996) for more details on these issues, and to Sattinger (1983) for a compact introduction. As a ﬁrst example, assume that a unique branch of stable equilibria ¯ in this case S consists of (7.10)1 , and the inequalities passes through (1, θ); characterizing the generic φˇ express the positivity of the eigenvalues of L. ¯ is a bifurcation point, so that (7.10) Consider now the case that (1, θ) holds. Genericity requires the rank of L to be lowered as little as possible. In the absence of symmetry the condition would be that only one of the eigenvalues vanishes. The presence of (local) symmetry forces certain eigenvalues to be equal, and thus more than one may have to vanish at

208

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the bifurcation point. Indeed, recall that the elasticity tensor obeys the transformation rule (6.50): Q t L[E ]Q = L[Q t EQ] for all E ∈ Sym and Q ∈ P (ea0 ). Therefore, if E is an element of the eigenspace Sλ of L corresponding to the eigenvalue λ, so is also Q t EQ for any Q ∈ P (ea0 ); and the subspace of Sym, Sλ say, generated by these tensors is invariant – see §2.2.2 – under the action of P (ea0 ). If there is some element of Sλ which does not belong to Sλ , we can repeat the construction and conclude that Sλ is an invariant, reducible subspace of Sym. Equivalently, certain eigenvalues of L are equal – that is, certain equalities hold among second derivatives of φˇ – without this being imposed by invariance. Genericity then requires all eigenspaces of L to be irreducible invariant subspaces of Sym under the action of P (ea0 ). This must hold, in particular, for ker L. ¯ to be a bifurcation point, the number of indepenAltogether, for (1, θ) dent equilibrium equations and the conditions on the eigenvalues of the elasticity tensor L imposed by the invariance of φˇ and by the bifurcation condition (7.4) should not exceed the number, seven, of independent variables (C , θ) at our disposal. This is best seen if one moves along an equilibrium branch of the same (maximal) symmetry as the bifurcation point, which always exists: along it all the eigenvalues that need not be equal by invariance remain distinct while varying with θ in order to satisfy the equilibrium equations; and the only degree of freedom left is used to have ¯ The situation is more compliexactly one of the eigenvalues go to zero at θ. cated along a branch of lesser symmetry than the bifurcation point: here, the number of generically distinct eigenvalues is larger, but there are also more independent equilibrium conditions; moreover, to comply with the higher symmetry of the bifurcation point, more than one eigenvalue may approach zero, and some may approach the same nonzero value, as θ goes ¯ Some details are given below, particularly in §7.5. to θ. 7.3 Reduced bifurcation problems; order parameters We are now interested in classifying the solutions of the generic bifurcation problem in Sym> stated in the previous section. This problem depends on the symmetry group P (ea0 ) of the bifurcation point and on the choice of the i.i. kernel of the elasticity tensor L (that is, of the i.i. eigenspace whose eigenvalue is zero). Based on the results of §5.4.3, here we show that such a general bifurcation problem reduces to one of six lower-dimensional (that is, 1-, 2- or 3-dimensional) reduced problems, where the appropriate space of order parameters, that is, of relevant coordinates, is introduced, together with the reduced action of P (ea0 ) on that subspace. This analysis is an example of the so-called Liapunov–Schmidt reduction – see §7.8. We also show that stability in Sym> is equivalent to stability in the order parameter space.

7.3 REDUCED BIFURCATION PROBLEMS; ORDER PARAMETERS

209

¯ be a bifurcation point satisfying (7.10), with symmetry P (e 0 ). Let (1, θ) a The latter holohedry produces one of the decompositions of Sym into P (ea0 )-i.i. subspaces analyzed in §5.4.3. Let the tensors VA , A = 1, . . . , 6, be a basis of Sym induced by such a decomposition, as is speciﬁed in that section for each crystal system, and represent the typical element of Sym in terms of this basis: V = A yA VA .6 Since the kernel of L must be one of the i.i. subspaces in the decomposition, we assume for deﬁniteness that the ﬁrst N ≤ 3 among the VA generate ker L. Any tensor C ∈ Sym > in a neighborhood of 1 diﬀers from 1 by a tensor V ∈ Sym belonging to a suitable neighborhood of 0. The equilibrium equations (7.1) take the form ∂ φˇ ˇ 1 , . . . , y6 , θ) := φ(1 ˇ + 6 yA VA , θ). (7.11) = 0, A = 1, . . . , 6, φ(y A=1 ∂yA ¯ and all By hypothesis, (7.11)1 holds at the bifurcation point (0, . . . 0, θ), the eigenvalues of L are positive except those pertaining to the subspace generated by V1 , . . . , VN . Then the last 6 − N of the equations (7.11)1 can be uniquely solved for the corresponding yr , r = N + 1, . . . , 6 near yr = 0, ¯ producing smooth functions θ = θ, yr = fr (y1 , . . . , yN , θ), or yr = fr (yi , θ), i = 1, . . . , N, (7.12) ¯ = 0. The variables y1 , . . . , yN are called the order such that fr (0, . . . , 0, θ) parameters of the bifurcation. The functions (7.12) give the equilibrium values of all the other variables in terms of the order parameters. Some of the yr may well vanish at the bifurcation point, as the order parameters, in which case they are often called secondary order parameters; unlike the order parameters, they do not aﬀect the basic qualities of the bifurcation (see for instance Tol´edano and Dmitriev (1996)). An example of trivial coupling (cubic-to-tetragonal transition) is given in §7.6.2, where also a nontrivial coupling (tetragonal-to-monoclinic) is presented. Replacing the variables yr by the functions fr in φˇ gives the reduced (or Landau) potential ˇ i , fr (yi , θ), θ). Φ(yi , θ) := φ(y (7.13) The remaining equilibrium equations (7.11)1 for A = 1, . . . , N, are equivalent to the critical-point conditions for Φ: ∂Φ (y , θ) = 0, i = 1, . . . , N, y = (y1 , . . . , yN ), (7.14) ∂yi ¯ The assumption that the subspace which in particular hold at y = 0, θ = θ. of Sym generated by (y1 , . . . , yN ) be the kernel of L is now equivalent to ∂2Φ ¯ = 0 i, j, = 1, . . . , N. (0, θ) ∂yi ∂yj 6

(7.15)

Throughout this chapter indices denoting components in R6 are written as subscripts for notational convenience, while superscripts indicate exponents of powers.

210

BIFURCATION PATTERNS

Notice that the existence of a stable equilibrium branch for all suﬃciently small θ > θ¯ requires all the eigenvalues of L to be nonnegative. Genericity forces only one of the distinct eigenvalues to vanish, hence all the remaining ones are strictly positive and, for some positive real number κ, ∂ 2 φˇ ¯ θ)z ¯ s zs ≥ κz2 , z ∈ R6−N . (0, fr (0, θ), (7.16) ∂ys ∂ys Therefore, by (7.11)–(7.12) and by the continuity of the second derivatives appearing in (7.16), we have ˇ i , yr , θ) ≥ φ(y ˇ i , fr (yi , θ), θ) = Φ(yi , θ) φ(y (7.17) ¯ ¯ for suﬃciently small (y1 , . . . , y6 ) and |θ−θ|. So, near (0, θ), an equilibrium branch is stable if and only if the corresponding curve yi = yi (θ) is a curve of local minima for the reduced potential; stability can thus be judged through the derivatives of Φ alone, which plays the role of a subpotential. The reduced potential inherits some invariance properties from those of the free energy. We have seen in §5.4.2 that any element of P (¯ ea ) induces an orthogonal transformation ρ in the six-dimensional space of parameters (y1 , . . . , y6 ) which is made of orthogonal blocks – see (5.17). This implies that any sextuple (yi , yr ) solving (7.11) is transformed by ρ into another solution of (7.11). By the uniqueness of the functions in (7.12) we have fr (ρij yj , θ) = ρrs fs (yi , θ) ,

(7.18)

hence, for any block submatrix ρ = (ρij ) in the reduced group P on ker L Φ(ρij yj , θ) = Φ(yi , θ).

(7.19)

A priori Φ could have additional symmetries, which would then imply identities among derivatives of the original potential φˇ not required by its invariance. Since this cannot hold generically, we only consider henceforth the case that the symmetry group of the reduced potential Φ is exactly the reduced symmetry group on the irreducible invariant kernel of L. Notice that all the equilibria given by y = 0 correspond to lattice bases whose symmetry is P (ea0 ). Diﬀerentiation of (7.19) implies the identity ∂Φ(yi , θ) ∂Φ(ρij yj , θ) = ρhk . ∂yh ∂yk

(7.20)

For all the reduced groups except P = {1} equality (7.20) holds at y = 0 and any given θ if and only if, there, ∂Φ/∂yh = 0, h = 1, . . . , N ; thus the branch y = 0 is always an equilibrium branch near the tempera¯ Since in the case P = {1} the equilibrium branches have P (e 0 )ture θ. a symmetry (§7.4.1), we conclude that in all cases an equilibrium branch of the same symmetry as the bifurcation point passes through the point itself, as anticipated.7 Also, condition (7.18) provides restrictions on the 7

The location of the critical points of any P-invariant reduced potential can be ob-

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

211

admissible values of the yr . For instance, if ker L is contained in the ﬁxed ˆ 0 ), hence P = {1}, a reasoning like the one above based on (7.20) set C(e a ˆ 0 )⊥ , fr (yi , θ) = 0. Thereimplies that, for all the indices r related to C(e a 0 ˆ fore the equilibrium branch belongs to C(ea ), a result we will recover in a diﬀerent way in §7.4.1. Remark 7.3 As was mentioned in Remark 5.8, by changing the basis VA in Sym we obtain a generally diﬀerent reduced group and reduced potential. The Liapunov–Schmidt reductions turn out to be equivalent, and to produce equivalent bifurcation patterns (that is, with the same qualitative behavior) – see Golubitsky and Schaeﬀer (1985), Golubitsky et al. (1988). For this reason we restrict the attention to the choices of bases proposed in §5.4.3 for the various crystal systems. 7.4 Analysis of the reduced bifurcation problems The discussion in the preceding section shows that the characteristics of the bifurcation pattern near the bifurcation point in (7.10) for a generic simple-lattice energy are completely determined by the features of the corresponding bifurcation for the reduced potential in the reduced space of order parameters (ker L) under the action of the corresponding reduced symmetry group P. In this section we see that all the possibilities are covered by six distinct reduced problems. Then, depending on how any such problem arises in one of the invariant subspaces in one of the decompositions of Sym detailed in §5.4.3, we can trace back from the reduced problem the full 6-dimensional original bifurcation pattern in Sym.8 This is summarized in the second and third tables of Table 7.1. There and in the text below we classify the possible cases by the dimension N of ker L, which gives the number of independent order parameters, and, for each N , consider all the possibilities for the reduced symmetry group P. The two simplest cases, (1) and (2) in Table 7.1, which exhaust the possibilities for a 1-dimensional order-parameter space, give the standard analysis of the turning (or limit) point and of the pitchfork bifurcation. These are treated in most references, for instance Ioos and Joseph (1980), Golubitsky and Schaeﬀer (1985), or Tol´edano and Dmitriev (1996) pp. 60– 64 and pp.51–60, respectively, and are included below for completeness.

8

tained by a theorem of Michel (1971): the critical orbits are exactly those which are isolated components of their strata; a stratum is a subset of the reduced space whose points have conjugate stabilizers – see §2.2.2 – under the action of P. An explicit description of the critical orbits is also given by Michel (1993), (1996). This last step is important because, as is visible in the tables, one and the same reduced problem may correspond to many actual transitions, whose details cannot be obtained from the reduced problem alone. Even more important is a full kinematic description for the transitions in complex crystals, where one usually focusses on the kinematic variables that are relevant, thus addressing directly the reduced problem. This procedure allows one to avoid the complexities of starting from the full kinematics; but only the latter can describe all the possibilities, some of which may be otherwise overlooked.

212

BIFURCATION PATTERNS

Table 7.1 The reduced bifurcation problems, listed by the dimension of the orderparameter space and the generators of the reduced group P. The corresponding general (G) and θ-controlled (θ) – see §7.4.7 – symmetry-breaking continuous transitions in simple lattices are indicated in the second and third table, without explicit mention for 1-dimensional order-parameter spaces (problems (1) and (2)), where they coincide. Wherever necessary, the centerings of the high-symmetry phase and the corresponding centering of the low-symmetry one, in the same order, are indicated. A comparison of G- or θ-symmetry breaking with the kinematically possible continuous paths of §5.3 is sketched in §7.4.7. The overall notation is taken from §5.4.3, which contains the details about the i.i. subspaces and the related centerings. Also, ‘tp’ means turning point, a change of stability but not of symmetry – see §7.4.1 Reduced problem

(1)

(2)

(3)

(4)

(5)

Dimension Generators

1

1

2

2

2

3

1

−1

f, r( π 2)

f, r( 23π )

f, r( π 3)

π −R1 ( π 2 ), −R2 ( 2 )

(1)

(6)

Triclinic

Monoclinic

Orthorhombic

Rhombohedral

(§5.4.3.1)

(§5.4.3.2)

(§5.4.3.3)

(§5.4.3.4)

tp

tp

tp

PC → triclinic

PFI → monoclinic PCC C → monoclinic P or C (details in §5.4.3.3)

tp

(2)

G: triclinic θ: monoclinic ‘face diagonals’ C

(4)

Tetragonal

Hexagonal

Cubic

(§5.4.3.5)

(§5.4.3.6)

(§5.4.3.7)

(1)

tp

tp

tp

(2)

(a) PII → orthorhombic ‘cubic edges’ PFI (b) PII → orthorhombic ‘mixed axes’ CIF

(3)

G: triclinic θ:(a) PI → monoclinic ‘cubic edges’ PC (b) PI → monoclinic ‘face diagonals’ CC

(4)

G: ‘optic axis’ monoclinic P θ: orthorhombic C

G: PFI → orthorhombic ‘mixed axes’ CIF θ: PFI → tetragonal PII

(5)

G: triclinic θ:(a) ‘basal diagonals’ monoclinic C (b) ‘basal side-axes’ monoclinic C

(6)

G: triclinic θ:(a) rhombohedral (b) PFI → orthorhombic ‘cubic edges’ PFI

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

213

Also the other cases have been treated in the literature, to which we refer in the appropriate places, but apparently have not been related to an organized treatment of symmetry breaking in simple lattices. For this reason here we give a direct, elementary description of these bifurcations, based on the discussion of Ericksen (1993),9 keeping details to a minimum. For instance, below we use in an informal way invariant theory, which has already been used by Smith and Rivlin (1958) to describe the energy functions of anisotropic elastic materials. The polynomial invariants we introduce can be shown to be an integrity (or Hilbert) basis – see Hilbert (1890), (1893): any invariant polynomial function is a polynomial in the elements of the basis. The analogue holds for any smooth invariant function. We address the reader to Tol´edano and Tol´edano (1987) §4.5, Golubitsky et al. (1988), Tol´edano and Dmitriev (1996) p. 30, or Olver (1999), and references cited therein, for more details.10 For convenience, hereafter subscripts on Φ will denote partial derivatives. 7.4.1 Reduced problem (1) In this case the reduced potential is a function Φ(y, θ) of one order param¯ it must be eter, with no symmetry. For a bifurcation at (0, θ) Φy = 0 = Φyy

¯ ; at (0, θ)

(7.21)

¯ In particular it by genericity no other conditions can be satisﬁed at (0, θ). ¯ $= 0, so that the equilibrium equation must be Φyθ (0, θ) Φy (y, θ) = 0

(7.22)

can be solved by the implicit function theorem, producing a smooth function θ = f (y) which is deﬁned near y = 0 and satisﬁes ¯ f (y) = − f (0) = θ, f (y) = −

Φyy (y, f (y)), f (0) = 0, Φyθ

and

(7.23)

(Φyyy + 2Φyyθ f )Φyθ − Φyy Φyθθ f . (Φyθ )2

(7.24)

¯ $= 0, so that, near Again by genericity, this means that f (0) = − Φyyy (0,θ) yθ y = 0, f (y) − θ¯ is either positive or negative and diﬀers by o(y 2 ) from a ¯ $= parabola; o(·) denoting an inﬁnitesimal of higher order. Since Φyyy (0, θ) ¯ 0 and Φyy (0, θ) = 0, the function Φyy (y, f (y)) changes sign at y = 0, where the stability thus changes. Also, since P = {1}, the symmetry of the equilibria on this branch is the same as the one of the bifurcation point. We thus have a turning (or limit) point: a bifurcation with change of stability but no change of symmetry. This is called an isostructural transition by Φ

9 10

Case (3) is corrected here. The analysis of symmetry through invariant theory is important also in other areas of physics, see for instance Sartori (1991).

214

BIFURCATION PATTERNS y

y

θ − θ¯

(a)

θ − θ¯

(b)

¯ Solid lines represent the stable branches Figure 7.1 Turning point for θ ≥ θ.

Tol´edano and Dmitriev (1996), §I.4.5. As we assume a stable branch of the same symmetry as the bifurcation point to exist for θ ≥ θ¯ (but keep footnote 4 in mind), we have the two possibilities shown in Fig. 7.1. 7.4.2 Reduced problem (2) Also in this case Φ = Φ(y, θ), but Φ is an even function of y: Φ(−y, θ) = Φ(y, θ). Then the equilibrium condition (7.21)1 necessarily holds at (0, θ) ¯ giving a branch of equilibria y = 0, passing through the for all θ near θ, bifurcation point and with its same symmetry. We introduce the new variable z = y 2 and think of the reduced potential ˜ θ), z ≥ 0, subject to no symmetry requirements. It is Φ as a function Φ(z, not diﬃcult to check that a function θ = θ(y) solves (7.21)1 for y $= 0 if ˜ and only if θ = θ(z) solves ˜ z (z, θ) = 0 Φ (7.25) 11 ¯ near z = 0, θ = θ. Now, if Φ is regular enough, we can calculate the ¯ = 1 Φyyθ (0, θ), ¯ which is non zero by genericity; thus ˜ zθ (0, θ) quantity Φ 2 (7.25) can be solved by the implicit function theorem, producing a function ˜ θ = θ(z),

˜ = θ, ¯ θ(0)

Φyyyy ¯ $= 0 , θ˜ (0) = − (0, θ) 6Φyyθ

(7.26)

˜ 2 ) is, up to the last inequality again holding by genericity. The function θ(y higher order terms, a parabola with vertex at the bifurcation point. This follows in particular if one takes as (approximation of the) potential the ﬁrst three terms of the polynomial F (y, θ) = a(θ) + b(θ)y 2 + c(θ)y 4 + d(θ)y 6 , (7.27) ¯ = c(θ) ¯ and Φyyθ (0, θ) ¯ = b (θ). ¯ The sixth-order in which case Φyyyy (0, θ) term in (7.27) is introduced now for convenience; it will be used in §7.6.1. 11

¯ and take into account that all For instance write a Taylor formula for Φ near (0, θ), the derivatives of Φ containing an odd number of diﬀerentiations with respect to y vanish at (0, θ) by symmetry.

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

215

¯ > 0 to have the branch y = 0 stable for θ > θ. ¯ We assume Φyyθ (0, θ) 2 ¯ ¯ ˜ If Φyyyy (0, θ) is positive, then, by (7.26), θ ≤ θ and 0 ≥ 2θ y = θ y. ¯ By Thus for z near 0 the branch of minimizers (7.26) exists only for θ ≤ θ. diﬀerentiating the identity Φy (y, θ(y)) = 0 and writing the Taylor formula for Φyθ (y, θ(y)) we have Φyy (y, θ(y)) = −θ (y)Φyθ (y, θ(y)) (7.28) ¯ θ (0) + Φyyθ (0, θ)] ¯ y + o(|y|); = −θ (y)[Φyθθ (0, θ) so in this case the branch of minimizers (7.26) is stable – see Fig. 7.2(a). ¯ negative, which In an analogous way we can discuss the case of Φyyyy (0, θ) is sketched in Fig. 7.2(b). Since the order of the reduced group is 2 (hence its orbit consists of two points) we have indeed two symmetry-related branches of critical points with y $= 0, constituting the parabola (7.26), and the order of the symmetry group along each one of these branches is half the order of the symmetry group on the branch y = 0. When their points are stable, these symmetryhalving bifurcations are the likely ones associated with second-order phase transitions according to Landau et al. (1980). 7.4.3 Reduced problem (3) This problem is also analyzed by Golubitsky et al. (1988) in ch. 17, §1(a) (see Fig. 1.2 there) and §7, and by Tol´edano and Dmitriev (1996), pp.64– 70. Also, this is one of the reduced problems in the introductory example of Tol´edano and Dmitriev (1996), which deals with a structural transition; there, the variants turn out to have C4v symmetry. In this case the reduced potential is a function Φ(y1 , y2 , θ) with invariance Φ(y1 , y2 , θ) = Φ(−y1 , y2 , θ) = Φ(y1 , −y2 , θ) = Φ(y2 , y1 , θ).

(7.29)

¯ and there the conditions The bifurcation point is (0, 0, θ), Φy1 = 0 = Φy2 ,

Φyr ys = 0, r, s = 1, 2,

(7.30)

hold. The invariance (7.29) implies that Φy1 (0, y2 , θ) = 0 = Φy2 (y1 , 0, θ)

(7.31)

¯ in identically, so that the equilibrium branch y1 = 0 = y2 exists near θ, agreement with the statement following (7.20). Other equilibrium branches exist12 because, if we choose y1 = 0, the equilibrium equations reduce to Φy2 (0, y2 , θ) = 0, as (7.31)1 identically holds by symmetry. The function Φ(0, y2 , θ) is even in y2 , hence we have the pitchfork bifurcation discussed in §7.4.2, where y is identiﬁed with y2 : y1 = 0 = y2 12

and y1 = 0, θ = f (y22 ).

(7.32)

This result and similar later ones also follow from the general Proposition 7.1 below.

216

BIFURCATION PATTERNS y

θ

Φ θ − θ¯

Φyyθ > 0 Φyyyy > 0

(a)

θM = θ¯

y

y θ − θ¯

(b)

(c)

Φyyθ > 0 Φyyyy < 0

¯ The derivaFigure 7.2 (a), (b) Pitchforks for the (y = 0)-branch stable above θ. tives of Φ are taken at the transition, and the stable branches are represented by solid lines. (c) Energy proﬁle at various temperatures for case (a). The analogue for case (b) can be obtained from Fig. 7.3 by restricting the attention to neighborhoods of the origin that only contain the unstable branches (dotted lines there)

Transforming this pitchfork by means of the reduced group P we get another pitchfork in the plane y2 = 0: y1 = 0 = y2

and y2 = 0, θ = f (y12 );

(7.33)

no more solutions exist in these two planes. Each equilibrium with y1 = 0 = y2 is mapped to itself by P, hence has the same symmetry as the bifurcation point; each equilibrium on any one of the four pitchfork branches in (7.32) or (7.33) has three additional equilibria on its P-orbit, each on one of the other branches, so its symmetry is given by a subgroup of index 4 of the symmetry group of the bifurcation point. According to the discussion in §5.4.3.5, in the symmetry breaking of simple lattices this bifurcation point has tetragonal symmetry, and any four P-related equilibria on the pitchfork branches above belong to a monoclinic ‘cubic edges’ variant structure whose centering is also described in §5.4.3.5. We observe that the orthogonal change of parameters √ √ 2ξ = y + y 1 2 √ √2y1 = ξ − η , or (7.34) 2η = y2 − y1 2y2 = ξ + η

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

217

produces a new reduced potential ˜ η, θ) := Φ(y1 (ξ, η), y2 (ξ, η), θ) Φ(ξ,

(7.35)

with the same invariance P as Φ; that is, (7.34) is a change of coordinates that leaves P invariant. By the same reasoning as above13 we infer the existence of pitchforks of equilibria of the form ξ = 0 = η,

˜ 2 ), ξ = 0, θ = θ(η

˜ 2 ). and η = 0, θ = θ(ξ

(7.36)

The ﬁrst curve is the same branch as (7.32)1 . Each point on one of the pitchfork branches has three additional P-related points on the remaining branches, so its symmetry is given by a subgroup of index 4 of the symmetry group of the bifurcation point. According to the discussion in §5.4.3.5, where also the centerings are described, in the symmetry breaking for simple lattices all of the P-related equilibria on the pitchfork branches belong to a monoclinic ‘face diagonals’ variant structure. The previous example poses the question whether or not there are other orthogonal (or linear) changes of coordinates that conjugate P to itself, because, as (7.34), any such change would produce other generic equilibrium branches. It is possible to see by a direct calculation that there is none; another argument rests on the direct proof, to be given below, that there ¯ in the generic are no additional equilibrium branches meeting at (0, 0, θ) case. Preliminarly, let us introduce the polynomial P-invariant potential ¯ 1 , y2 , θ) = (a + bX + cY + dX 2 ) X=X(y ,y ) , Φ(y (7.37) 1 2 Y =Y (y1 ,y2 )

where a, b, c and d are smooth functions of θ, and the polynomials X = y12 + y22

and Y = y12 y22

(7.38)

are the invariants forming an integrity basis for the polynomial functions (or the smooth functions) that are invariant under the reduced group P. For such a potential the pitchforks can be computed easily and explicitly, together with their stability character, and we only summarize the results. In the (y1 , y2 ) plane, the stability of the branch y1 = 0 = y2 for θ > θ¯ ¯ = 0, b (θ) ¯ > 0, the prime denoting derivative; the requires generically b(θ) θ-dependent branches of equilibria (7.32)3,4 , (7.33)3,4 have now the form 0, ± −b/(2d) and ± −b/(2d), 0 , (7.39) while the ones in (7.36)3−6 become ± −b/(c + 4d), ± −b(c + 4d) ,

(7.40)

with all combinations of signs. Notice that, by suitably choosing c and d, the two pairs of pitchforks can exist above and below θ¯ in all possible ¯ both above, and one above and one below. combinations: both below θ, 13

The existence of these pitchforks can be obtained by directly applying the implicit function theorem in the planes y1 = −y2 and y1 = y2 , respectively.

218

BIFURCATION PATTERNS

The branches (7.39) belong to a single P-orbit, and thus have the same stability character by symmetry. The same is true for those in (7.40). The ¯ where b ≤ 0; this branches (7.39) are stable only if they exist for θ ≤ θ, ¯ requires generically d > 0 at (and near) θ in order to have a real square root ¯ Then, on these curves, Φ ¯ y y is positive for θ < θ¯ if and only if for θ ≤ θ. 1 1 ¯ c > 0 at (and near) θ, while the hessian determinant is then automatically positive. Analogously, the branches (7.40) can be stable only if they exist ¯ which requires c + 4d to be positive at θ. ¯ This necessarily holds for θ ≤ θ, ¯y y if the branches (7.39) are stable. But then, along the ones in (7.40), Φ 1 1 is positive because so is d, while the positivity of c forces the hessian to be negative; thus these branches are unstable. The role of the two sets of branches can be reversed, and if at θ¯ we have c < 0, d > 0, c + 4d > 0, the branches (7.40) exist for θ ≤ θ¯ and are stable, while the ones in (7.39) necessarily exist for θ ≤ θ¯ but are unstable. Also, if both pairs of pitchforks ¯ then one of them is stable and the other unstable. exist for θ ≤ θ, Since any smooth P-invariant reduced potential Φ diﬀers from the poly¯ in (7.37) by inﬁnitesimals of (higher) order o([y 2 +y 2 ]2 ), the exact nomial Φ 1 2 bifurcating curves diﬀer from the (approximate) ones constructed by means ¯ The hessian of Φ ¯ by inﬁnitesimals of order o(|θ − θ|). ¯ evaluated along of Φ the approximate bifurcation curves also diﬀers from the exact hessian by ¯ and thus the approximate potential Φ ¯ captures the qualitative o(|θ − θ|), bifurcation landscape near the bifurcation point.14 We now go back to the existence of other equilibria, besides the ones in (7.32)–(7.36). These pitchforks are exactly the equilibria contained in the planes of the (y1 , y2 , θ)-space where the jacobian ∂(X, Y ) = 4y1 y2 [y12 − y22 ] ∂(y1 , y2 )

(7.41)

vanishes. Any other equilibrium branch must have, in any neighborhood of ¯ points on which the jacobian is diﬀerent from zero. On each one (0, 0, θ), of these points the equilibrium equations are equivalent to FX = 0 = FY , F X(y1 , y2 ), Y (y1 , y2 ), θ := Φ(y1 , y2 , θ), (7.42) ¯ too. By using the approxso, by continuity, (7.42)1,2 must hold at (0, 0, θ) ¯ imate potential Φ in (7.37) we see that (7.42)1 is the bifurcation condition (7.30)3,4 , whereas (7.42)2 is an additional condition on the fourth-order derivatives of Φ, which cannot hold generically. Therefore the four pitchforks above are the only generic equilibrium branches.

14

In this way the essential features of the bifurcation are determined from the Taylor expansion of the potential up to fourth-order terms, or third-order terms in the equilibrium equations. In the deﬁnition of Golubitsky and Schaeﬀer (1985) – see also Field and Richardson (1992a), (1992b) – this bifurcation is 3-determined.

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

219

7.4.4 Reduced problem (4) Ball and Shaeﬀer (1983) come to this reduced problem starting from the equilibrium of an elastic dead-loaded body. A detailed analysis, which includes the universal unfolding, can also be found in Sattinger (1983), and other references are Golubitsky et al. (1988), ch. 15 §4, and Tol´edano and Dmitriev (1996), pp.71–76. Also in this case the reduced potential is Φ(y1 , y2 , θ), and invariance pro¯ in accordance with the duces the equilibrium y1 = 0 = y2 for θ near θ, statement below (7.20). Indeed, Φ(−y1 , y2 , θ) = Φ(y1 , y2 , θ) by the invariance under f in (5.31)1 , hence, at any point (0, y2 , θ), Φy1 = 0,

Φy1 θ = 0 = Φy1 y2 ,

while the invariance under Φy2 = 0 = Φy2 θ ,

r( 2π 3 )

Φy1 θθ = 0 = Φy1 y2 θ = Φy1 y2 y2 ,

(7.43)

implies that, at any point (0, 0, θ),

Φy1 y1 = Φy2 y2 ,

Φy1 y1 y2 = Φy2 y2 y2 .

(7.44)

¯ the derivatives in (7.44)3 In particular, at the bifurcation point (0, 0, θ) vanish, while those in (7.44)4 do not, as well as Φy2 y2 θ , by genericity. Let us ﬁrst look for the existence of other equilibrium branches in the plane y1 = 0, where the only equilibrium condition that needs to be imposed is Φy2 (0, y2 , θ) = 0. By the condition Φy2 y2 θ $= 0, the equilibrium equation can be uniquely solved by applying the implicit function theorem ¯ 2 −1 Φy (0, y2 , θ) with initial point (0, 0, θ); ¯ this produces the to (θ − θ)y 2 unique equilibrium branch y1 = 0, y2 = f (θ),

¯ = 0 $= f (θ) ¯ = −2 f (θ)

Φy 2 y2 θ ¯ (0, 0, θ). Φy 2 y2 y 2

(7.45)

Invariance under P implies that there are the two additional branches y1 = ±

√

3 2

f (θ), y2 = − 12 f (θ).

(7.46)

As in §7.4.3 we can quickly discuss the possible existence of further branches of equilibria and the stability properties of all the solutions by introducing ¯ of Φ: the fourth-degree polynomial approximation Φ ¯ y =y (X,Y ) = F (X, Y, θ) = a(θ) + b(θ)X + c(θ)Y + d(θ)X 2 , Φ (7.47) 1 1 y2 =y2 (X,Y )

where X = y12 + y22 , Y = y23 − 3y12 y2 ,

∂(X, Y ) = 6y1 [3y22 − y12 ], ∂(y1 , y2 )

(7.48)

X and Y being P-invariant polynomials that constitute an integrity ba¯ > 0, so that the sis for P-invariant smooth functions. We assume b (θ) ¯ equilibrium branch (0, 0, θ) is stable for θ > θ. Again, the existence of an equilibrium branch not fully contained in one of the planes in which the jacobian (7.48)3 vanishes forces the vanishing at ¯ of both b – and this is already true because (0, 0, θ) ¯ is a bifurcation (0, 0, θ)

220

BIFURCATION PATTERNS

¯ which cannot be point – and c; the latter is a third-order derivative of Φ, ¯ zero at (0, 0, θ) by genericity. So, only the equilibria (7.45)–(7.46), with ¯ = − 2b (θ), ¯ exist generically. f (θ) 3c To check the stability of the branch (7.45) we calculate along it  ¯ − 6c(θ)f ¯ (θ)](θ ¯ ¯ = 6b (θ)(θ ¯ − θ) ¯ − θ)  Φy1 y1 = [2b (θ) ¯ ¯ ¯ ¯ ¯ ¯ , (7.49) Φ = [2b (θ) + 6c(θ)f (θ)](θ − θ) = −2b (θ)(θ − θ)  y2 y2 Φy 1 y 2 = 0 ¯ so this solution is always unstable.15 By symmetry, the up to o(|θ − θ|), same conclusion of instability holds for the other two branches (7.46). Notice that the qualitative properties of the equilibrium branches near ¯ mentioned above are fully determined by the quadratic and cubic (0, 0, θ) terms in the Taylor expansion of Φ. In the latter, for deﬁniteness, we have ¯ > 0, so that the high-symmetry phase be stable at high assumed b (θ) temperature. Here we show how the quartic term can be used to re-stabilize the branch (7.45), and in §7.6.2 how to construct an energy function that produces this transition, generalizing a proposal of Ericksen (1986a). In either case we set a(θ) = 0 for simplicity. By using the quartic term in (7.47) we can introduce on the branch (7.45) ¯ after which the branch becomes stable. In a turning point at some θˇ > θ, ¯ in (7.47) are given by detail, for y1 = 0 the nonzero critical branches of Φ √ 9c2 − 32bd −3c y2 = y˜2 ± , y˜2 = . (7.50) 8d 8d One of the signs produces the analogue of the branch f (θ) in (7.45), while ¯ the other corresponds to a branch passing through the point (0, 2˜ y2 , θ). ¯ This point can be seen to be stable if and only if d(θ) > 0; in this case at ¯ we have F = − 27c43 (θ) ¯ < 0. The two branches above can meet (0, 2˜ y2 , θ) 256d if, for instance, we assume c and d to be constant and b a linear function ¯ for which ¯ indeed they meet at y˜2 when θ takes the value θˇ (> θ) of θ − θ: 9c2 b = 32d > 0. The branch corresponding to y2 between y˜2 (excluded) and 27c4 ˇ 2˜ y2 (included) is stable, and at y˜2 we have F = 4096d 3 (θ) > 0. Along this branch the potential F vanishes (hence the corresponding conﬁguration has the same energy as the high-symmetry conﬁguration y1 = 0 = y2 at the same temperature) at the (Maxwell) temperature θM , θ¯ < θM < θˇ such c2 that b = 4d . The corresponding value of y2 is −c 2d . Altogether, in this type of symmetry breaking we have a typical subcritical bifurcation, which seems to be the simplest way16 of having a transition 15

16

This also follows from a general result of Golubitsky et al. (1988), p. 90. Also, in the spirit of Landau et al. (1980), generic instability is due the fact that invariance does not force the cubic term in the potential to vanish. Actually, this subcritical bifurcation turns out to be included in the universal unfold¯ = 0) and to provide a full qualitative ing of the simplest degenerate bifurcation (c(θ) ¯ > 0 and small; if instead c(θ) ¯ < 0, then other secondary bifurcadescription for c(θ) tions typically occur – see for instance Golubitsky et al. (1988) or Sattinger (1983).

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

221

from a stable high-temperature phase, for instance, cubic, to a stable lowtemperature phase, in this case tetragonal. 7.4.5 Reduced problem (5) This case is considered by Golubitsky et al. (1988), ch.13, §5(b), as well as Tol´edano and Dmitriev (1996) pp. 80–83. We proceed as in the previous two cases, by introducing the sixth-degree ¯ to the reduced potential Φ, which is the lowest polynomial approximation Φ compatible with invariance under P and not under the whole of O(2); this is given by (7.47)1 for F = F (X, Y˜ ) = a + bX + cX 2 + dX 3 + eY˜ ;

(7.51)

here a, b, c, d and e are functions of θ, and – compare with (7.48) X = y12 + y22 , ∂(X, Y˜ ) ∂(y1 , y2 )

Y˜ = Y 2 = [y23 − 3y12 y2 ]2 , = 12Y y1 [3y22 − y12 ].

(7.52)

¯ = 0, b (θ) ¯ > 0, so that the high-symmetry phase is stable We assume b(θ) above θ¯ and unstable below. The same arguments as in the two previous reduced problems tell us that there cannot be generic solutions outside the locus where the jacobian vanishes. Now, besides the same 3 planes √ y1 = 0 and y1 = ± 3y2 (7.53) as in §7.4.4, the jacobian vanishes also on 3 other planes where Y = 0: √ y2 = 0 and y2 = ± 3y1 . (7.54) The sets of planes (7.53) and (7.54) are each an independent orbit under P. Therefore any solutions in each set exist together and have the same stability character. It is enough to restrict our attention to the equilibria in the two planes y1 = 0 and y2 = 0, where the potential becomes, respectively, F = a + by + cy 2 + (d + e)y 3 , y = y22 , and F = a + bx + cx2 + dx3 , x = y12 . (7.55) The bifurcation diagrams are the respective pitchforks −c + c 1 − 3bc−2 (d + e) 2 y1 = 0, y2 = f (θ) = , (7.56) 3(d + e) √ −c + c 1 − 3bc−2 d ¯ b ¯ 2 ¯ (7.57) , f (θ) = − (θ) = g (θ). y2 = 0, y1 = g(θ) = 3d 2c Thus the equilibrium branches for this reduced problem consist of two Notice that a subcritical bifurcation like the one above results from the snapsprings model studied by Truskinovsky and Zanzotto (1995), (1996).

222

BIFURCATION PATTERNS

¯ > 0 or for triples of pitchforks which, by (7.57)4,5 , all exist for θ < θ¯ if c(θ) ¯ ¯ θ > θ if c(θ) < 0. The stability analysis is analogous to the one of the reduced problem (3), so here we only outline the results. First, only the branches existing ¯ > 0. Furthermore, on the for θ ≤ θ¯ can be stable, which requires c(θ) branches in (7.56) [(7.56)] the hessian matrix is diagonal, and the second derivative Φy2 y2 [Φy1 y1 ] is automatically positive. To analyze the sign of the remaining second derivative one must approximate the square root in (7.56) ¯ >0 [(7.57)] up to quadratic terms. Then (7.56) [(7.57)] are stable if e(θ) ¯ ¯ [e(θ) < 0]. We conclude that if one set of pitchforks exists for θ ≤ θ, then also the other does, and exactly one of the two is stable. 7.4.6 Reduced problem (6) For this case see also Michel (1980) and Tol´edano and Dmitriev (1996), pp. 89–91. We proceed as before. The integrity basis I = y12 + y22 + y32 , J = y1 y2 y3 , K = y14 + y24 + y34 ,

(7.58)

for smooth P-invariant functions has jacobian ∆=

∂(I, J, K) = 8(y12 − y22 )(y22 − y32 )(y32 − y12 ), ∂(y1 , y2 , y3 )

(7.59)

and we can base the analysis on the quartic reduced potential ¯ J, K) = bI + cJ + dI 2 + eK, Φ(I,

(7.60)

¯ = 0, b (θ) ¯ > 0, so that the high-symmetry phase is stable above with b(θ) ¯ As in the previous cases, there are no generic equilibria outside the locus θ. where ∆ in (7.59) vanishes; this consists of the 2-dimensional subspace y 2 = y3

(7.61)

and on its six transforms under the tetrahedral group P. We analyze the equilibria on this subspace, as all the other branches are obtained by the action of P and have the same stability. On (7.61) the potential becomes F (X, Y, θ) = b(X 2 + 2Y ) + cXY + d(X 2 + 2Y )2 + e(X 4 + 2Y 2 ), (7.62) ¯ = 0 as bifurcation condition, and with b(θ) X = y1 ,

Y = y22 ,

∂(X, Y ) = 2y2 . ∂(y1 , y2 )

(7.63)

For y2 $= 0 the jacobian (7.63)3 does not vanish, and the equilibrium equations are equivalent to the vanishing of the ﬁrst derivatives of F . Since ¯ is a critical point of F where the hessian (0, 0, θ) ∂ 2 F¯ ∂ 2 F¯ ∂X 2 ∂X∂Y ¯ det = −c2 (θ) (7.64) ∂ 2 F¯ ∂X∂Y

∂ 2 F¯ ∂Y 2

7.4 ANALYSIS OF THE REDUCED BIFURCATION PROBLEMS

223

does not vanish by genericity, the implicit function theorem provides one local equilibrium solution (X(θ), Y (θ)). Existence and uniqueness also hold on each one of the six other subspaces; the transform of (X(θ), Y (θ)) by an element of P is an equilibrium, hence is the solution on the corresponding subspace. Since diﬀerent pairs among y1 , y2 , y3 are equal or opposite on different subspaces, necessarily Y (θ) = X 2 (θ). This condition, when inserted ∂F into the equilibrium equation ∂Y = 0, yields ¯ or X(θ) = − 2b (θ)(θ ¯ − θ) ¯ + o(|θ − θ|). ¯ ¯ = − 2b (θ), X (θ) c c Altogether, we have the two branches

y1 = X(θ), y2 = y3 = y1 ,

and y1 = X(θ), y2 = y3 = −y1 ,

(7.65)

(7.66)

the points of each one of which lie along one of the body diagonals of the unit cube centered at the origin. It is not diﬃcult to see that the tetrahedral reduced group P maps each point of a branch to three other points, each on one of the other main diagonals of the unit cube, so that each point, whose P-orbit consists of four points, has rhombohedral symmetry. For the stability of these solutions we compute the hessian matrix. The ¯ while the determinant diagonal terms are equal and behave like 2b near θ, 3 behaves like −32b , hence these branches are always unstable. We can reach the same conclusion by the method mentioned in footnote 15. Lastly, we consider the equilibria in the subspace y2 = 0 = y3 , where the jacobian in (7.63) vanishes. Here the potential becomes K(y1 , θ) = by12 + (d + e)y14 ,

(7.67)

producing a pitchfork bifurcation given by ¯ = 0, f (θ) ¯ =− y2 = 0 = y3 , y12 = f (θ), f (θ)

2b ¯ (θ). (d + e)

(7.68)

The P-orbit of each equilibrium for θ $= θ¯ consists of six points, each one of which has orthorhombic ‘mixed axes’ symmetry – see §5.4.3.7. In particular, k and i ± j are the axes for the branches in (7.68). The latter are stable ¯ that is, when d + e > 0. In addition, along only if they exist for θ ≤ θ, them Φy1 y1 , Φy2 y2 and (Φy2 y2 )2 − (Φy2 y3 )2 must be all positive. The ﬁrst ¯ <0 condition is automatically satisﬁed, while the other two require e(θ) ¯ and c(θ) = 0, respectively. Therefore the branches in (7.68) are generically unstable, and so are the other two pitchforks in their P-orbit. 7.4.7 Comparison with the kinematic transitions of §5.3 The results in §§7.4.3–7.4.6 show that in all spaces of order parameters of dimension N > 1 the generic bifurcations occur along special 1-dimensional subspaces, whose points are actually ﬁxed under maximal subgroups of the reduced group P. This is related to Proposition 7.1 below, and has been the

224

BIFURCATION PATTERNS

basis for the so-called Maximal Isotropy Subgroup Conjecture (MISC) – see footnote 23. The relevant subspaces to which these bifurcating θ-controlled branches, or θ-branches for short, belong have already been identiﬁed geometrically in §§5.4.3.4–5.4.3.7, and are schematically presented in Table 7.1 under the heading θ. In the rest of this subsection we do not discuss stability issues for brevity, understanding that, at least when all the bifurcating branches are unstable as in §7.4.4, suitable re-stabilization occurs as there. On the other hand, in §5.3 we have examined the kinematically possible weak symmetry-breaking continuous paths, which can be extracted from the trees of groups in Figs. 5.1 and 5.2. Any lattice conﬁguration whose symmetry group G is in one of the trees can be transformed into one whose symmetry is a subgroup of G. For instance, a cubic lattice can become triclinic. Only a selected set of these kinematically possible transitions can be driven by changing temperature, as is evident from Table 7.1. For instance, a cubic lattice can θ-transform to become tetragonal (problem (4)), rhombohedral or orthorhombic ‘cubic axes’ (problem (6)), but not, for instance, monoclinic. There are two elements responsible for this selection: the requirement that the eigenspaces of the elasticity tensor L at the transition be irreducible invariant, and the fact that there is one control parameter, the temperature θ. The latter condition was adopted for simplicity, and for each reduced problem one may obtain bifurcating branches of lower symmetry if more control parameters are available: at least the pressure p,17 and perhaps some shear stress component. This point of view is adopted, for instance, by Tol´edano and Dmitriev (1996), to which we refer for details. Here we only point out that, with suﬃciently many controls, we can start bifurcating in an arbitrary direction within the chosen kernel of L; for kernels of dimension N > 1 this gives new symmetry-breaking transitions, to be called general for deﬁniteness. The lowest symmetry possible for these branches for each reduced problem and crystal system is summarized in Table 7.1 under the heading G (general). As an example, in problem (4) for a cubic high-symmetry phase the θ-branches are tetragonal, while the general ones are orthorhombic ‘mixed axes’.18 Analogously, controlling (at least) p and θ, in the reduced problem (6) one can explore suitable 2-dimensional subspaces of monoclinic ‘face diagonals’ variants (containing the 1-dimensional subspaces of rhombohedrals) and of monoclinic ‘cubic edges’ variants (containing the 1-dimensional subspaces of orthorhombics 17 18

As is known, the arrangement of the (stable) phases in the (p, θ) control plane is the phase diagram of the material. Details on the cubic-to-tetragonal and tetragonal-to-orthorhombic transition lines, and the triple point, are given by Tol´edano and Dmitriev (1996), §3.3.3(b); there the situation is more complex because the tetragonals are divided into two phases, corresponding to the height of the tetragon being bigger or smaller than the edge of the square basis, and a further (anti-isostructural) transition between them at the cubic subspace is considered. So, at the triple point, which is rather called a Landau point, 4 phases coexist, and that point is an isolated point of second-order transition ending two curves of ﬁrst-order transition from cubic to either long or short tetragons.

7.5 BEHAVIOR OF THE MODULI ALONG THE TRANSITIONS

225

‘cubic edges’), while (at least) a third independent parameter is necessary to explore triclinic domains. This example shows how by using suﬃciently many controls, one can recover in a thermoelastic model all the kinematically possible transitions, by either second-order or subcritical ﬁrst-order bifurcations. The same holds for the other bifurcation problems above, with the exception of the orthorhombic system. There, θ- and general bifurcations coincide, and the least symmetric variants obtainable are monoclinic. If one is looking for triclinic variants, then a second control, say p, has to be used to force a second eigenvalue of L to vanish at the transition, so that the kernel is a reducible invariant subspace under the orthorhombic holohedry. This possibility, based upon controlling p and θ, has been used by Ericksen (1996a) to describe the meeting at a tetragonal [orthorhombic] conﬁguration, of two branches of orthorhombics [monoclinics] of diﬀerent Bravais type, thus justifying by thermodynamics certain kinematically possible continuous deformations pointed out by Pitteri and Zanzotto (1996a) in an analysis of the classiﬁcation criterion of lattices proposed by Bravais (1850). The possibility of having more than one eigenvalue vanishing, and thus working with reducible invariant subspaces of order parameters, seems to be relevant for other transitions, in particular for the reconstructive ones (see Tol´edano and Dmitriev (1996), §4.4 and references therein). A coupling of the kind proposed there, of two i.i. subspaces into a reducible one, has been used by Ericksen (2001b) to describe certain phases of quartz. Following the point of view of Landau, in the above treatment we have used polynomial potentials as approximations to suitably smooth potentials near the bifurcation point. As mentioned, in the literature the polynomials are often regarded as the true potentials, and the question arises of which is the simplest polynomial that produces all the bifurcations that we have called general, for any given reduced problem. We refrain from addressing this nontrivial issue here, and refer to Tol´edano and Dmitriev (1996) for detailed information. 7.5 Behavior of the moduli along the transitions In the previous sections we have described the bifurcation patterns in terms of eigenvalues and of irreducible, P (ea0 )-invariant subspaces of the elasticity tensor L at the transition. The assumption of genericity requires any two eigenvalues to be equal if and only if the corresponding eigenvectors belong to the same invariant subspace, and exactly one among the diﬀerent eigenvalues to vanish. The introduction of the reduced energy, reduced group, etc., allowed us to treat bifurcations in the most compact, eﬃcient way. On the other hand, experimentalists and material scientists usually describe the phase transitions above in terms of the behavior of the elastic moduli of the austenitic and martensitic phases. This description can be recovered from the one given above, and examples of energies producing

226

BIFURCATION PATTERNS

the desired transitions are given in §7.6, where, for convenience, we only give the example of the cubic-to-tetragonal and tetragonal-to-monoclinic transitions; the other cases can be treated similarly. Take the cubic [tetragonal] symmetry axes as in §5.4.3.5 [§5.4.3.7], and consider the corresponding elasticity tensor described in detail in §6.7. For easy comparison, the orthonormal basis (i , j , k ) used to represent nonlinear strains and elasticities has been chosen to be the same in the two cases. Let L be an elasticity tensor of cubic symmetry as described by (6.80)(6.84). As we move along the cubic equilibrium branch, the components of L with respect to the cubic reference axes all remain zero except those satisfying (6.82)–(6.84), with eigenvalues λ1 , λ2 , λ3 that vary with θ and stay generically diﬀerent. Then, at a generic cubic-to-tetragonal transition, the only condition to be met is that the eigenvalue λ2 vanishes, while the other two remain positive and diﬀerent; equivalently, in terms of moduli, the diﬀerence L11 −L12 vanishes by (6.82)–(6.84), with no other restrictions. Let us now consider the reference conﬁguration to be the limit of tetragonal conﬁgurations whose symmetry is described in §5.4.3.5. Notice that in the cubic case the three eigenspaces of L are generated by V1 , V2 to V3 , and V4 to V6 , respectively, while in the tetragonal case they are generated by V1 to V2 , V3 , V6 , and V4 to V5 , respectively. At the transition the moduli in (6.73)–(6.76) must all be equal, and so do L44 and L55 . The latter condition is equivalent to requiring the eigenvalues µ6 and µ4 to become equal (and equal to λ3 of the cubic phase), and the corresponding 1and 2-dimensional eigenspaces to ‘merge’ and generate the 3-dimensional cubic eigenspace. The ﬁrst condition is equivalent to requiring ϕ, µ2 and µ3 to vanish. This means that the 1-dimensional tetragonal eigenspace generated by V1 becomes the 1-dimensional eigenspace of the cubic phase (and λ1 = µ1 ), while the 1-dimensional eigenspaces generated by V2 and V3 correspond to eigenvalues that both go to zero, and merge to generate the 2-dimensional kernel of the cubic elasticity tensor at the transition. As the example shows, the behavior of the elastic moduli at a generic transition has the simplest description on the equilibrium branch of higher symmetry. Along it exactly one combination of moduli (L11 − L12 in the cubic case) goes to zero (modulus softening) while the eigenspace structure remains constant. The situation is more complex if we approach the transition along a branch of lower symmetry. In this case more than one combination of moduli may vanish, other relations among them may hold, and the eigenspace structure may change so as to comply with the higher symmetry of the conﬁguration at which the transition occurs. 7.6 Examples of energy functions for simple lattices So far we have assumed the invariance of the constitutive functions to be based on the lattice description of the crystal, and to be independent of ˇ so will also be our considany speciﬁc form of the energy functions φ˜ or φ;

7.6 EXAMPLES OF ENERGY FUNCTIONS FOR SIMPLE LATTICES

227

erations on twinning and microstructure formation in chapters 8–10. However, it is useful to mention some of the polynomial energy functions that have been proposed in the literature to model symmetry-breaking phase transitions in crystalline solids, for the purpose of both theoretical and numerical investigations. In this section we give some examples of energy functions for elastic simple lattices which exhibit minimizers with properties discussed in §6.5.3. Such constitutive functions of the Cauchy-Green tensor C and the temperature θ are usually obtained directly as polynomials in the entries of C , or from suitable manipulation of polynomials in some of the entries (order parameters), with θ-dependent coeﬃcients, following the practice established in the Landau theory (see Wayman (1964), Landau et al. (1980), Khachaturyan (1983)). More details on the constitutive functions that enjoy these and related properties, under various assumptions of symmetry and smoothness, can be found in Ericksen (1980b), (1989), (1992), (1996a,b), (1997), Chan (1988), Collins and Luskin (1989), Budiansky and Truskinovsky (1993), Luskin (1996b), Simha and Truskinovsky (1996), Truskinovsky and Zanzotto (2002). 7.6.1 A schematic 1-dimensional example Before giving energies deﬁned on the spaces B or Sym> , whose symmetry properties are as in §6.5.2, we ﬁrst brieﬂy discuss a schematic 1-dimensional example which shows the main properties of minimizers needed in phase transformation models. As we have seen in §6.5.3, the main idea is that, locally, in suitable neighborhoods in its domain of deﬁnition, the energy function of a crystal should have minimizers which depend on temperature in such a way that for high θ, say, there is only one minimizer (austenite), while for low θ there are several (in this 1-dimensional case, two) symmetryrelated minimizers (variants of martensite) distinct from the austenitic one. We describe the energy landscape and the bifurcation diagram in terms of a parameter y which measures the distance of the martensitic lattice from the austenitic one at the same temperature, in the spirit of the Landau theory and in agreement with the above description of reduced potentials. If the transformation stretch is the identity, hence the transformation is of the second order, then the analysis in §7.4.2 describes well what happens near the transition. It is suﬃcient to consider the second and third terms in the polynomial potential (7.27), in which for simplicity in the rest of ¯ or b to be this subsection the coeﬃcient b is assumed to be linear in θ − θ, constant, c and d to be constant, and a to vanish. Then the transition is generated by the quartic potential included in (7.27) for b > 0, c > 0. The bifurcation diagram and the energy landscape are shown in Fig. 7.2(a),(c). If the transformation stretch is nontrivial, then the transition is of the ﬁrst order, and can be modelled by a subcritical bifurcation based on the full sextic potential in (7.27) for b > 0, c < 0, d > 0. Analogously to what was done in §7.4.4, the highest order term in the polynomial is used to

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BIFURCATION PATTERNS

θ

Φ

σ = Φ

θˇ

θM θ¯ y

y

Figure 7.3 A 1-dimensional example based on the sextic potential in (7.27). Both energy and its derivative, which measures stress, are plotted versus the order parameter y, which is a measure of strain, for various values of θ near the transition ¯ At the shown temperature between θ¯ and θM the dotted lines plot temperature θ. energy and stress versus strain in the absence of the restabilizing sextic term

restabilize the unstable subcritical branches. The energy landscape and the stress-strain relation are shown in Fig. 7.3. In detail, by analyzing the bifurcation diagram in terms of the variable z = y 2 ≥ 0, the nonzero critical branches of F in (7.27) are given by √ c2 − 3bd c z = z˜ ± , z˜ = − > 0. (7.69) 3d 3d The minus sign produces the analogue of the branch z(θ) included in (7.26), while the plus sign corresponds to a branch passing through the point ¯ This point can be seen to be stable if and only if d > 0, as indeed (2˜ z , θ). ¯ we have F = 4c32 < 0, hence the we have assumed; in this case at (2˜ z , θ) 27d energy of the martensite is lower than the one of the austenite. The two c2 ¯ branches above meet at z˜ when θ takes the value θˇ = θ¯ + 3db > θ. The branch corresponding to z between z˜ (excluded) and 2˜ z (included) is stable, 8c3 and at z˜ we have F = − 27d2 > 0. Along this branch the martensite and the austenite at the same temperature have the same energy at the (Maxwell) c2 ¯ ˇ temperature θM = θ¯ + 4db , so that θ < θM < θ. The corresponding value c of z is − 2d . Of course, we must in the end recall that z = y 2 to represent energy or stress as a function of y, as in Fig. 7.3. 7.6.2 Energies for cubic-to-tetragonal and for tetragonal-to-monoclinic transitions We now consider a crystalline material in a cubic reference conﬁguration, with primitive cubic lattice vectors as in (3.52), and construct an energy

7.6 EXAMPLES OF ENERGY FUNCTIONS FOR SIMPLE LATTICES

229

ˇ , θ) that is suitable for the description19 of the reversible cubicfunction φ(C to-tetragonal transition undergone by InTl crystals somewhere between 25◦ C (In-23at%Tl) and 105◦ C (In-18.5at%Tl), depending on composition. For this transition, based on the polynomial reduced potential in (7.47), one can construct a polynomial free energy function which agrees with the one in the constrained theory of Ericksen (1986a), (1988), and with its extension proposed by James (1988a) and used by Collins and Luskin (1989), (1991), and Luskin (1996b), among others, for modelling numerically the cubic-totetragonal transition in InTl alloys. Indeed, consider the following version of the cubic decomposition in §5.4.3.7, related to the basis vectors Vi introduced there: V = y3 V1 + y1 V2 + y2 V3 + y4 V4 + y5 V5 + y6 V6 .

(7.70)

Also, take as free energy function the expression ˇ 1 , . . . , y6 , θ) = F (X, Y, θ) + 3f (y3 − g(θ))2 + 1 e(y 2 + y 2 + y 2 ), (7.71) φ(y 4 5 6 2

with e, f temperature-dependent positive coeﬃcients, g(θ) a suitable func¯ = 0, and F given by (7.47) for a(θ) = 0. It is straighttion such that g(θ) forward to show that F (X, Y, θ) is the reduced potential, and that the function g(θ) describes the thermal expansion of the cubic austenite. By (7.70), (7.11), (5.55)–(5.57), the following relations hold: √ √ √ trC − 3 y3 = √ , y4 = 2C12 , y5 = 2C23 , y6 = 2C13 , and (7.72) 3 √ X = λ21 + λ22 + λ21 , Y = 3 6 λ1 λ2 λ3 , for (7.73) 1 λr = Crr − trC , r = 1, 2, 3, λ1 + λ2 + λ3 = 0. (7.74) 3 Therefore √ F (X, Y, θ) = b(λ21 + λ22 + λ23 ) + 3 6 c λ1 λ2 λ3 + d(λ21 + λ22 + λ23 )2 . (7.75) To within renaming the coeﬃcients in the last formula,20 the potential φˇ in (7.71) for y3 ≡ 0 (that is, trC ≡ 3) coincides with Ericksen’s constrained potential. The extension of the latter given by φˇ is slightly diﬀerent from the one presented by Luskin (1996b), for instance, but it provides essentially the same equilibria. The related stability analysis has been already given in §7.4.4, and the coeﬃcients in (7.71) can be determined in terms of the experimental moduli by a procedure proposed by James (1988a). Similarly we can reconstruct from the same reduced potential (7.47) the quartic polynomial energy used by Ericksen (1996b) to describe rhombohedral-to-monoclinic transitions. 19 20

Various materials undergo the same symmetry-breaking transition, and can be treated in the same way with appropriate changes in the coeﬃcients appearing in (7.71). c d b → 6b , c → √ , d → 36 . The invariants J, K used by Ericksen are given by 6 6 √ X = 6J, Y = 6 6K.

230

BIFURCATION PATTERNS

One more remark on the energy (7.71): since φˇ is quadratic on the eigenspaces orthogonal to ker L, the variation of the order parameters y1 , y2 does not inﬂuence the other strain variables: the functions fr in (7.12) reduce to functions of θ at most. This is indeed the simplest way to construct an energy function with the right properties, but there are in principle more complex possibilities; and in certain cases this is forced by the experimental data, which show that also the order parameters, besides temperature, inﬂuence the other strain components. An example is the energy for a tetragonal-to-monoclinic transition in Zirconia (ZrO2 ) considered by Simha (1994) and Simha and Truskinovsky (1996). In this case the re˜ 1, V ˜ 2 , V3 , . . . , V6 duced problem is (3) (§7.4.3) and, in terms of the basis V in footnote 16 of chapter 5, (5.44) and (5.45), and of the order parameters y4 , y5 , the simplest guess for the energy, analogous to (7.71) for g(θ) ≡ 0 for simplicity, would be the following: ˇ 1 , . . . , y6 , θ) = (αy 2 + 2γy1 y2 + βy 2 + δy 2 + ;y 2 )/2 + F (X, ¯ Y¯ , θ); (7.76) φ(y 1 2 3 6 here α, γ, β, δ, ; are functions of θ; the corresponding quadratic polynomial is positive deﬁnite; the function F is given by (7.37), (7.42)2 , and the ¯ Y¯ are given by the analogue of (7.38) when the parameters invariants X, y1 , y2 in it are replaced by the present y4 , y5 . In this way we reproduce the quadratic and quartic terms in the potential given by Simha (1994), and no coupling between y4 , y5 and the other yr is present. We stress that the sextic contribution in the potential of Simha (1994) is related to producing a ﬁrst-order subcritical bifurcation, and restabilizes the unstable bifurcating branches, analogously to the example in §7.6.1. The same can be obtained by adding an X 3 term in (7.37). Therefore we now concentrate on the cubic terms proposed by Simha (1994) to describe the inﬂuence of y4 , y5 on the other yr . We refer to (7.12) and (7.18), recalling that the latter restrict the functions fr . Assuming enough smoothness, the simplest choice is to have the fr depend on θ at most, as was done for the cubic-to-tetragonal case; the simplest instance is (7.76), whose corresponding reduced potential is easily seen to be F . The next simplest choice is to add a linear dependence of y1 , y2 , y3 , y6 on y4 , y5 . Since each one of the former variables describes a 1-dimensional i.i. subspace of Sym, and therefore either is left ﬁxed or mapped to its negative by the tetragonal symmetries, it follows from (7.18) that the linear dependence is necessarily trivial. One tries next a quadratic dependence; since one of the matrices (ρij ) in (7.18) exchanges y4 and y5 , each one among y1 , y2 , y3 , y6 must be a quadratic function of both y4 ¯ which is the only and y5 , and the independent ones are: y42 + y52 = X, ¯ quadratic invariant under the reduced group P, and then y42 − y52 =: X ¯ . The action of the tetragonal holohedry on the i.i. subspaces and y4 y5 =: X then forces the dependence ¯ y1 = Γ1 X,

¯ y2 = Γ2 X,

¯ , y3 = Γ3 X

¯ , y6 = Γ6 X

(7.77)

on the quadratic parts of (7.12), with the Γr arbitrary functions of θ. The

7.6 EXAMPLES OF ENERGY FUNCTIONS FOR SIMPLE LATTICES

corresponding equilibrium equations are ¯ = 0, φˇy1 = αy1 + γy2 + Λ1 X ¯ = 0, φˇy2 = γy1 + βy2 + p3 X ¯ = 0, φˇy = δy3 + Λ2 X 3

√ √

231

2Λ1 =: p1 + p2 ,

(7.78)

2Λ2 =: p1 − p2 ,

(7.79)

¯ = 0, φˇy6 = ;y6 + ω X where α to ; are the functions of θ appearing in (7.76), and Λ1 , Λ2 , p3 , ω can be easily related to the Γr , for instance Γ3 = −Λ2 /δ, etc. Moreover, for convenient comparison with the energy of Simha (1994), Λ1 , Λ2 are given in terms of the parameters p1 , p2 which, together with p3 , ω, appear in the cubic terms of that energy. Since the additional terms in the energy will be functions of y4 , y5 and θ, (7.78)-(7.79) are indeed the ﬁrst four equilibrium equations for the full energy, and produce all the cubic and quadratic terms, ¯ in the term F of (7.76). Notice that, by adding the latter term, except bX we already obtain a reduced potential of the form (7.37), in which all the parameters in (7.78)-(7.79) only appear through two suitable combinations ¯ 2 in F , respectively. The positivity which form the coeﬃcients c, d of Y¯ , X of the nonvanishing eigenvalues of L then produces the inequality c + 4d < 0, which forces the bifurcating branches in the planes y4 = ±y5 to be subcritical and unstable – see §7.4.3. If this condition is too restrictive, one ¯ 2 or Y¯ , to can introduce an independent quartic term, either of the form X uncouple c and d, and thus obtain all the possibilities detailed in §7.4.3. Remark 7.4 Neither the energy for the cubic-to tetragonal nor the one for the tetragonal-to-monoclinic transitions was obtained from the procedure outlined above. Rather, as already mentioned, both Ericksen (1986a) and Simha (1994) started from an energy which, in agreement with the Hilbert basis theorem (see before §7.4.1), is a polynomial of a suitable degree in the basic strain invariants for the respective holohedries, detailed by Smith and Rivlin (1958), for instance. This simpler procedure requires more eﬀort in the subsequent discussion of the bifurcation diagram, which is instead already known from the analysis of the reduced problems presented above. 7.6.3 Orientation relationships and lattice correspondence As the analysis in §§5.3, 5.4.3 and 7.4.7 shows, kinematics and thermoelasticity impose deﬁnite patterns on weak phase transformations. Kinematics describes the possible point groups and centerings along progressive symmetry breaking from any given high-symmetry lattice (see Tables 5.1, 5.2, 5.3). Generic temperature-controlled (θ) thermoelastic weak transitions are a deﬁnite subset of the kinematically possible ones. For instance, in Table 5.3 we see that three tetragonally conjugate classes of monoclinic point groups can be reached by breaking the tetragonal symmetry; and, unless the tetragonal lattice is primitive, the resulting monoclinic lattices are not distinguished by their centering. To select one of these classes from the

232

BIFURCATION PATTERNS

others more detail is needed about the direction of the monoclinic axis, at least. Such an information is often present in the metallurgical literature in the form of orientation relationships, which compactly describe the relative orientation of the high- and low-symmetry lattices. Since both phases must be present in a suﬃciently homogeneous specimen in order to determine the orientation relationships, these are sometimes diﬃcult to obtain or not accurate enough. This information, however, may be crucial in the modelling of the transition. For instance, a direct tetragonal-to-monoclinic transition driven by temperature generically prevents the monoclinic axis from coinciding with the former tetragonal axis (k in Table 5.2 and §5.4.3.5), and one should have this type of information if the model is to be eﬀective. As an example, consider zirconia (ZrO2 ), which is one of the main toughening agents for ceramics. An approach to modelling the well known tetragonal-to-monoclinic transformation in this material is proposed by Simha (1994) and Simha and Truskinovsky (1996). Their energy function (§7.6.2) indeed produces monoclinic variants whose axis is either i or j , but also i ± j if a suﬃciently high tensile stress is applied. If, experimentally, the monoclinic axis was observed to be k , or if another zirconia phase with orthorhombic symmetry could also be present, the model should be reconsidered. Truskinovsky and Zanzotto (2002), to which we address for details, have constructed a polynomial free energy for an elastic crystal which undergoes tetragonal-orthorhombic-monoclinic (t-o-m) weak phase transformations (with ﬁrst-order character), and which exhibits a triple point in its pressure-temperature phase diagram. This framework ﬁts the modelling of zirconia behavior, as ZrO2 has a phase diagram of this kind (see Fig. 1.6). Based on the analysis summarized in Fig. 5.3, Truskinovsky and Zanzotto (2002) examine the possible paths of t-o-m symmetry breaking, and check them against the experimental evidence. The latter is not conclusive as, for instance, the orientation relationships between the phases are not established with complete certitude. The analysis, however, suggests that the symmetry-breaking path be Tk → Oijk → Mk (unlike the Tk → {Mi , Mj } path considered by Simha (1994) and Simha and Truskinovsky (1996)). Given this kinematics, the authors take advantage of the decomposition presented in §5.4.3.5, and recognize that the qualitative features of the phase diagram of zirconia can be obtained with a suitable choice of a 2dimensional manifold of order parameters in strain space, in which two successive subcritical bifurcations occur; ﬁrst from Tk to Oijk , then from Oijk to Mk . The analysis of bifurcation and stability is based on a sixthdegree polynomial energy which we do not detail here. We only reproduce in Fig. 7.4 the phase diagram and the energy landscape at a point inside the gray area, where (meta)stable equilibria of all three symmetries coexist. The bifurcation diagram along the line t-s in Fig. 7.4(a) is illustrated in Fig. 7.5, and Fadda et al. (2002) give a quantitative analysis for ZrO2 . For weak phase transformations the kinematics presented above, based

7.6 EXAMPLES OF ENERGY FUNCTIONS FOR SIMPLE LATTICES

233

Figure 7.4 (a) Phase diagram in the temperature-pressure plane; T3 is our Tk , etc. The thin lines represent stability boundaries, and the thick lines the Maxwell lines, concurring into the triple point. The darker gray area represents states in which all three phases coexist as (meta)stable equilibria. The bifurcation diagram along the straight line t-s is shown in Fig. 7.5. (b) Energy landscape illustrating the variety of wells existing in the typical point of the three-phase coexistence region highlighted in darker gray in (a); y3 and y6 are the order parameters From Truskinovsky and Zanzotto (2002), courtesy of Elsevier.

s

Tk

δ

1

δ δ

γ β

0

Oijk

Mk

Oijk Mk

δ

α

Mk β

Oijk

t

Tk

−1

γ

Oijk

Mk

t

t t

−1

0

1

Figure 7.5 Bifurcation pattern along the s-t line shown in Fig. 7.4. The points marked here correspond to the ones having the same name in that ﬁgure. The dotted lines indicate the unstable portions of the equilibrium branches From Truskinovsky and Zanzotto (2002), courtesy of Elsevier.

234

BIFURCATION PATTERNS

on symmetry-breaking stretches, automatically produces a lattice correspondence between the reference and the stretched simple lattices, and one does not have to select in any speciﬁc way the basis for the reference lattice. This does not hold in general and, for a given phase transformation, one has to select a speciﬁc lattice correspondence, that is, he has to decide which structural unit of the parent lattice is deforming into a unit of the product lattice. As commented by Wayman (1964), this is in fact a matter of educated guesswork. A ﬁrst example is the Bain correspondence for the α-γ transition in iron, detailed in §3.6.3. Another explicit example is given in §9.2.4.2, in terms of both lattice bases and symbols usually found in the metallurgical literature. Sometimes the choice of correspondence is helped by the transformation stretch being small, as in the case of InTl alloys. It also helps if reliable orientation relationships are available, as the ones of Kurdjumov and Sachs (1930)21 for ferrous martensites; these support the lattice correspondence involved in the Bain model (§3.6.3). Other correspondences are conceivable for the α-γ transition in iron, but these produce comparatively larger stretches; also, Jaswon and Wheeler (1948) have shown that the Bain correspondence involves the smallest possible atomic displacements, an idea that is often used in other transitions as well. For the β-α transition in UCr alloys, Lomer (1956) (mentioned by Wayman (1964)) has systematically studied about 1600 lattice correspondences, only one of which was found to give a transformation stretch with eigenvalues smaller than 0.1. An example of how a more accurate determination of the orientation relationships can aﬀect the lattice correspondence is provided by the description of long period stacking order structures of martensites in certain β-phase alloys, for instance in CuZnAl. These shape memory alloys have a martensitic phase that was initially thought to be orthorhombic, while the basically b.c.c. austenite was described as a B2 or DO3 structure according to the scheme in the Strukturberichte (1915-1940). The symmetry of martensite was obtained by choosing as structural unit in the austenite a suitable stacking of 9 atomic planes for the B2 structure, and equivalently 18 planes for the DO3 one. As detailed by Otsuka et al. (1993), when subsequent analysis revealed beyond doubt that the martensite was not far from orthorhombic but actually monoclinic, a smaller unit was preferred, which better ﬁts some of the observations; see also Pitteri and Zanzotto (1998b), Hane (1999), James and Hane (2000) and the references therein. 7.7 Relation with the Landau theory The bifurcation theory presented in §§7.2–7.4 is indeed the Landau theory for simple lattices, in spite of the apparently diﬀerent starting point: here 21

¯11] ¯ b is approximately In this case: (011)b is approximately parallel to (111)f , and [1 parallel to (¯ 101)f , the indices b and f denoting b.c.c. and f.c.c., respectively.

7.8 GENERAL REFERENCES

235

we choose from the beginning a convenient, ﬁxed reference conﬁguration, the one of the more symmetric phase at the transition temperature, and compute energies and strains from it. Apparently, Landau and his followers have in mind a temperature-dependent reference, which is the conﬁguration of the more symmetric phase at that temperature, and consider the excess free energy, that is, the diﬀerence between the free energy at an arbitrary conﬁguration and temperature and the free energy at the corresponding reference. We now argue that there is essentially no diﬀerence if one considers the reduced potential Φ rather than the free energy density ˇ while the latter is referred to a ﬁxed reference, the very construction of φ: the former implicitly introduces the ‘running reference’. Let us remember that, except for the reduced problem (1) (ﬁxed set), the curve yi (θ) = 0, i = 1, . . . , N represents parametrically the unique equilibrium branch of the austenitic phase near the transition temperature ¯ the function Φ(0, . . . , 0, θ) is the (§7.3). Therefore in all these cases, near θ, free energy density of the austenitic equilibrium at the temperature θ, by deﬁnition. This coincides with the zeroth-order (in the order parameters) term a(θ) appearing in all the polynomial approximations to Φ introduced above, so that the remaining polynomial terms do constitute the excess free energy of the Landau theory. For the reduced problem (1) the choice of reference in the ‘excess energy’ description is not so obvious, because all the equilibria near θ¯ have ¯ the meaning of the same symmetry. But, due to the construction of Φ, Φ(0, . . . , 0, θ) is again clear: it is the free energy of the conﬁguration which, for the value 0 of the order parameter (which describes a 1-dimensional ˆ 0 )) and θ of the temperature, equilibrates subspace of the ﬁxed set C(e a the other coordinates in the ﬁxed set – that is, makes the free energy stationary with respect to their variations. Remember that in this case the equilibrium values of the coordinates describing the orthogonal compleˆ 0 )⊥ are necessarily zero (see above Remark 7.3). As an example, ment C(e a assume the (reference) lattice L(ea0 ) at the transition to have hexagonal symmetry – see (3.50) – and lattice parameters a0 , c0 ; and the kernel of L to consist of uniaxial strains along the optic axis, so that the order parameter is the increment of the axial size c. Then, for any choice of (c, θ) ¯ there is a unique partial equilibrium (that is, equilibrium for near (c0 , θ), the remaining 5 parameters), consisting of hexagonal lattices whose basal parameter a is a function a(c, θ). The ‘running reference lattice’, on which the free energy takes the value Φ(0, . . . , 0, θ) and from which the excess is considered, is the hexagonal lattice with lattice parameters a(c0 , θ), c0 ; this ¯ is not an equilibrium except for θ = θ. 7.8 General references A general treatment of both static and Hopf bifurcation theory is given, for instance, by Golubitsky and Schaeﬀer (1985) when no symmetry is present.

236

BIFURCATION PATTERNS

In their analysis the equilibrium equations have the form f (x , λ) = 0,

f : R n × R → Rn ,

(7.80)

where x is the state variable and λ is the bifurcation parameter; the lefthand side need not be the gradient of a potential. A large class of problems exhibiting multiple equilibria can be reduced to analyzing the bifurcation diagram of a single (that is, for n = 1) equilibrium equation of the form (7.80). This is accomplished by a procedure called Liapunov–Schmidt reduction (and called elimination of the passive coordinates by Thompson and Hunt (1973)). The methods of singularity theory are best applied to a bifurcation problem after Liapunov–Schmidt reduction has been performed. Sattinger (1983), p. 27, applies this reduction also to inﬁnite-dimensional problems, attributing it to Michel and Radicati (1973). Golubitsky and Schaeﬀer (1985) present in detail Liapunov–Schmidt reduction for equilibrium equations that are equivariant – see (7.81) – under a compact Lie group. They show how to construct invariant complements, how symmetry is inherited by the reduced equations, and how to analyze stability in the ﬁnite-dimensional case. Moreover, they address the issues of equivalence of two bifurcation problems (which means, roughly, that their sets of equilibria have the same qualitative dependence on λ); of ddeterminacy (which means that the qualitative behavior of a bifurcation problem (7.80) is dictated by the Taylor expansion of the function f up to order d); and study perturbations of singularities by unfolding theory, which intends to model the eﬀect of imperfections on the set of equilibria by suitably embedding the original problem into a multiparameter one. All these matters are systematically extended to systems with symmetry by Golubitsky et al. (1988), a short account of some related issues being given by Sattinger (1983). One of the important applications is spontaneous symmetry breaking: for a certain value of the bifurcation parameter λ the equilibrium equations can acquire solutions with lower symmetry than the equations themselves, in addition to one which has the full symmetry. The analysis is more complicated than in the nonsymmetric case because symmetry forces multiple eigenvalues in the gradient of the function f in (7.80). In the equilibrium case, Golubitsky et al. (1988) base bifurcation in the presence of symmetry on the following equivariant branching lemma.22 Proposition 7.1 Consider a group G ≤ O(n) acting on Rn , and any subgroup G ≤ G such that the set of points stabilized by G be 1-dimensional, and assume the steady-state problem (7.80) to be equivariant: for all (x , λ) and all g ∈ G, f (gx , λ) = gf (x , λ). (7.81) 22

This was ﬁrst proved by Michel (1972) for Higgs–Landau theories, in which the vector ﬁeld f in (7.80) is the gradient of a potential. The theory was then extended to general vector ﬁelds by Vanderbauwhede (1980) and Cicogna (1981), who covered what is called the real case. Extensions to the complex and quaternionic cases were given by Golubitsky and Stewart (1985) and Gaeta (1987), (1990), respectively.

7.8 GENERAL REFERENCES

237

Then we can associate with G a unique branch of solutions for (7.80).23 They also obtain stability for the branches of solutions in the generic case; that is, in agreement with §7.2.2, when no additional constraints are satisﬁed by derivatives of f besides the ones that are dictated by symmetry and allow for the existence of a bifurcation point. Also, the authors apply equivariant unfolding theory to model the eﬀect of imperfections that are still compatible with the symmetry of the original problem. In §5.4.1-5.4.2 we directly introduced some of the necessary concepts and results, for instance irreducible invariant subspaces and reduced symmetry groups. Other issues, except unfolding theory, have been addressed in this chapter directly, keeping the mathematical formalism to a minimum.

23

For our purposes the assumption that the ﬁxed set of G be 1-dimensional is suﬃcient; it is shown to hold for the Landau theory when the number of order parameters is less than four (see Michel and Mozrzymas (1978), also Tol´edano and Dmitriev (1996), pp.35–37), and is extended by Michel (1980) to what Golubitsky et al. (1988) call the Maximal Isotropy Subgroup Conjecture (MISC). The class of bifurcation problems satisfying the conjecture (which includes the cases discussed here) was then guessed to exhaust, at least in a generic sense, all bifurcation problems with symmetry. This conjecture turns out to be too restrictive for certain applications, for instance in theoretical physics. Indeed, Tol´edano and Dmitriev (1996), p.36, give reference to counterexamples, but stress that the conjecture still covers most of the relevant cases. Progress in the direction of clarifying the status of the MISC has been made by Field and Richardson (1992a), (1992b), and is also summarized by Field (1996). The authors provide a large number of ‘generic’ counterexamples to MISC.

CHAPTER 8

Mechanical twinning As we discussed in chapter 1, one of the main ways in which crystalline materials deform under an external action or along a displacive phase transformation is by mechanical twinning, to which we will mostly refer henceforth when using the word twinning.1 In this chapter we assume crystals to be stress free, under neither loads nor displacement boundary conditions, and deﬁne mechanical twinning deformations as special pairwise homogeneous natural states (that is, stable, stress-free equilibria) for a crystal involving any pair of symmetry-related minimizers of its free energy (see Ericksen (1977), (1981a), (1987), Parry (1980), James (1981), (1984a), Gurtin (1983), Pitteri (1985b); also Ball and James (1987) and Bhattacharya (1991)). Within the elastic model considered in chapter 6, twins are the simplest nonhomogeneous minimizers of the functional (6.12), for no load or boundary displacement imposed. As we will see below and in the next chapter, when the energy density has the invariance properties described in §6.3.1 nonlinear elasticity becomes ﬂexible enough to describe twins as well as more complex formations, to be called microstructures (chapter 10), which are produced to accommodate external actions. In agreement with experience, we assume the interfaces between twin individuals created by loads or in displacive transitions to be coherent ;2 this means that the displacement is continuous across each interface, while its gradient suﬀers a jump discontinuity. Also, the energy associated with such surfaces is usually regarded to be low, and is thus neglected in the present framework, as anticipated in §1.2. This simpliﬁcation gives some basic insight in the phenomenon of twinning. However, in complicated arrangements as in Fig. 1.5, or other ﬁne structures in which one or more lengthscales are present, interfacial energy cannot be neglected; see for instance Parry (1989), Kohn and M¨ uller (1992), (1994), Dolzmann and M¨ uller (1995), M¨ uller (1998), (1999), Conti (2000). As shown in Figs 1.4 and 1.5, usually numerous twins, arranged in a variety of microstructures, may form in a body as a consequence, for instance, 1

2

We mostly analyze twinning in simple lattices. For multilattices the theory is not as well developed. In §11.9 some examples indicate the main phenomena involved, such as shuﬄe twinning, and the need of using multiple cells. In fact, in metallurgy one is also forced to deal with interfaces that are only partially coherent or worse (greasy). These require more modelling eﬀorts than the coherent ones; see for instance Truskinovsky (1983a,b), Grinfeld (1991), Cermelli and Gurtin (1994), Leo and Hu (1995), Sutton and Balluﬃ (1996). 239

240

MECHANICAL TWINNING

of a phase transformation. Some of these complex phenomena are addressed in chapter 10. Here we concentrate on a single twin, and thus on a perhaps small region of the crystal containing a regular portion of the interface separating two twin individuals, like the one in the inset of Fig. 1.4(b). As anticipated in §1.2, in that region the two adjacent individuals of a twin are homogeneously deformed and related by both a simple shear, the twinning shear,3 and an orthogonal transformation, the twinning operation, which in the great majority of the mechanical twins observed experimentally is an operation of period two as in §2.3.1. These physical characteristics4 of mechanical twinning deformations are described for instance by Cahn (1954), Hall (1954), Klassen-Nekliudova (1964) or Kelly and Groves (1970). The problem of kinematic compatibility in general is discussed in §8.1, following Ericksen (1991b). From §8.2 on, special attention is given to the case of compatibility for symmetry-related energy wells. The existence of rank-1 connections for the latter correponds to the existence of mechanical twins for the crystalline material (Ericksen (1987)). The assumption that the minima of the potential be symmetry-related and the compatibility conditions for the coherent deformation lead to the twinning equation, which characterizes mechanical twins in general. If, in addition, the wells belong to a weak-transformation neighborhood, the rank-1 connections among them describe the transformation twins for the crystal (chapter 9). More complex connections among the energy wells emerge in the microstructures that form along phase transitions (chapter 10). In order to relate our description of twinning to the one more common in the metallurgical and crystallographic literature, in §8.2 we use the Born rule to obtain a ‘molecular’ version of the twinning equation and some classical results regarding the rationality of the twinning shear elements. We list here for convenience a number of sources which give the crystallographic data for twinning modes in a great number of minerals and metals: Mathewson (1928) p. 563, 565-6, Tertsch (1949) p. 56, Cahn (1954) p. 411, Hall (1954) p. 56-7, Reed-Hill et al. (1963) p. 979, Klassen-Nekliudova (1964) p. 166-76, Barrett and Massalsky (1966) p. 4156, Kelly and Groves (1970) p. 303-4, Zoltai and Stout (1984) p. 65. 8.1 Coherence and rank-1 connections Let a crystalline body be in a reference conﬁguration R with reference lattice basis ea0 . We consider for the crystal a continuous, pairwise homogeneous deformation χ whose gradient takes constant values F1 and F2 in 3 4

Cahn (1954) always observed a shear, except for Dauphin´e and other twins, which are of an essentially distinct nature. Some of these were discussed by the german mineralists of the early 1900’s, like M¨ ugge (1883), Johnsen (1914), Niggli (1924), which are credited by KlassenNekliudova (1964), p. 10-12 for ﬁrst pointing out the problem of the existence of structural shuﬄes – see the Mineralogists’ Assumption in §8.2.

8.1 COHERENCE AND RANK-1 CONNECTIONS

241

two complementary subregions R1 and R2 of R, which meet along a smooth interface: F1 x + c1 for x ∈ R1 y= , (8.1) F2 x + c2 for x ∈ R2 where c1 and c2 are constant vectors. Later on we will also require the gradients F1 and F2 to be such that χ is a minimizer of the functional (6.12), but in this section we only concentrate on the consequences of the kinematic requirement that χ be continuous on the surface where its gradient undergoes a jump. Hadamard’s compatibility conditions (§2.4.1) require the tensors F1 and F2 to be rank-1 connected, that is, to diﬀer by a tensor of rank 1: F2 = F1 + a ⊗ n ∗ = F1 (1 + a ∗ ⊗ n ∗ ) = (1 + a ⊗ n)F1 , a ∗ := (F1 )−1 a, n := (F1 )−t n ∗ ;

(8.2)

here a and n ∗ (hence a ∗ and n) are constant vectors, and at any point of the reference surface separating R1 and R2 the tangent plane is orthogonal to n ∗ .5 Thus the interface is a plane orthogonal to n ∗ , whose equation is x · n ∗ = c with c a constant given by ca = c1 − c2 . This interface, which in R is a coherent boundary for the crystal, in the deformed conﬁguration is orthogonal to n and is called the composition plane. Notice that the vector a ∗ and the (co)vector n ∗ are not uniquely deﬁned: given any pair (a ∗ , n ∗ ), all the equivalent pairs have the form (ka ∗ , k1 n ∗ ) for some real number k $= 0. Usually this indeterminateness is eliminated by selecting n ∗ of unit norm, but we will not do so unless explicitly stated. Bilby and Christian (1956), also credited by Wayman (1964) p.88, show that a homogeneous deformation has an invariant plane (here the composition plane) if and only if the corresponding stretch has ordered eigenvalues 0 < λ1 < λ2 = 1 < λ3 . Acton et al. (1970), extending work by Bevis and Crocker (1968) on the twinning equation (§8.2), reduce an equation equivalent to (8.2) to three quadratic scalar equations containing as parameters the ratio of the determinants of F2 and F1 and either the unit vector of a or the unit vector of n (see also Crocker (1982)). Necessary and suﬃcient conditions for (8.2) have been proved by Ball and James (1987) and Ericksen (1991b); here we follow the latter. The proof of the former, in R3 , is ˇ y (1999), (2001). extended to Rn by Forclaz (1999) and Silhav´ For given F1 and F2 consider the corresponding Cauchy–Green tensors C1 and C2 . From (8.2) we have C2 − C1 = nr∗ ⊗ n ∗ + n ∗ ⊗ nr∗ ,

nr∗ := C1 a ∗ +

a ∗ ·C1 a ∗

n ∗.

(8.3)

2

Notice that the indeterminateness in (a ∗ , n ∗ ) implies the analogue for (nr∗ , n ∗ ): all the equivalent pairs are of the form (knr∗ , k1 n ∗ ). The following proposition is the n-dimensional analogue of one by Ericksen (1991b). 5

As in §2.4.1, we use asterisks to indicate vectors and covectors in the reference conﬁguration.

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MECHANICAL TWINNING

Proposition 8.1 Necessary and suﬃcient condition for (8.3), or (8.2), to hold in Rn is that there be a basis ai , i = 1, . . . , n, and two scalars µ1 and µ2 such that (2.21) holds for C = C1 and 0 < λ1 = (1 −

µ21 2

) ≤ 1 = λ2 = . . . = λn−1 ≤ λn = (1 +

µ22

).

(8.4)

2

According to (2.22), condition (8.4) can be phrased in terms of eigenvalues of U1−1 C2 U −1 (Ball and James (1987)). Proof. By Proposition 2.4, relations (2.21) hold with all the λr s positive because C2 is positive deﬁnite; and, as is always possible, we order these positive numbers increasingly. Also (see §2.3), the rank rk of C2 − C1 is the number of the λr s diﬀerent from 1 and, by (8.3), this is at most two. The rank rk is zero if and only if (8.4) holds with all the λr equal to 1, that is, C2 = C1 . In this case (8.3) implies that, for some real number k, C1 a ∗ = kn ∗ ,

k(2 + a ∗ · n ∗ ) = 0.

(8.5)

Besides the trivial solution F1 = F2 of (8.2), which corresponds to k = 0, we have the inﬁnitely many solutions with a ∗ · n ∗ = −2, all of which produce determinants of F1 and F2 of opposite sign. These nontrivial solutions are usually discarded in the analysis of mechanical twins, for which F1 and F2 can always be chosen with a determinant of the same sign (see footnote 7). To discuss the two other cases, and ﬁrst the double implication between (8.3) and (2.21), (8.4), we introduce the deﬁnitions b 14 a := nr∗ · C1−1 nr∗ , b := n ∗ · C1−1 n ∗ , κ := nr∗ · C1−1 n ∗ , χ := ± , (8.6) a

in terms of which the positive deﬁniteness of C2 will be proved below to be equivalent to √ ab < 1 + κ. (8.7) The rank of C2 − C1 is one if and only if either λ1 or λn is the only eigenvalue diﬀerent from 1. In (8.3) this rank condition requires nr∗ and n ∗ to be nonzero and parallel. Then √ κ n∗ n∗ √ , and ab = |κ|. (8.8) C2 − C1 = 2κv ∗ ⊗ v ∗ for v ∗ = √r = |κ| b a If κ > 0, (8.8)1 is equivalent to (2.21) for v ∗ = an , 1 < λn = 1 + 2κ, and (8.7) holds trivially, as does the positive deﬁniteness of C2 . If κ < 0, (8.8)1 is equivalent to (2.21) for v ∗ = a1 , λ1 = 1 + 2κ < 1, and the positive deﬁniteness of C2 is equivalent to 1 + 2κ > 0, which is (8.7) by (8.8)4 . Vice versa, (2.21) implies (8.8)1 ; this in turn can be put in the form (8.3) by choosing an arbitrary positive a and deﬁning b, nr∗ , n ∗ by (8.8)4,2,3 . If C2 − C1 is of rank 2, then nr∗ and n ∗ in (8.3) are linearly independent and, remembering (8.6), we introduce the positive reals µ1 , µ2 and the vectors a1 , an as follows: √ √ µ21 = 2( ab − κ), µ22 = 2( ab + κ), (8.9)

8.1 COHERENCE AND RANK-1 CONNECTIONS

χnr∗ − χ−1 n ∗ = µ1 a1 ,

χnr∗ + χ−1 n ∗ = µ2 an .

243

(8.10)

It is now a matter of computation to check that, with these choices, (2.21) and (8.4) hold, the latter with strict inequalities; and that the positive definiteness of C2 , that is, λ1 > 0, is equivalent to (8.7). Conversely, starting from (2.21) and (8.4), use the latter to construct µ1 , µ2 , then (8.9) to con√ satisfy (8.7). By arbitrarily ﬁxing the positive struct ab and κ, which then √ number a we obtain b from ab, χ from (8.6)4 , and nr∗ , n ∗ from (8.10). Again, one can check that (8.3) holds for these choices of nr∗ , n ∗ , and the numbers a, b, κ are related to nr∗ , n ∗ through (8.6)1−3 . So far, we have shown (2.21) and (8.4) to be necessary and suﬃcient for (8.3)1 , and (8.3) to be necessary for (8.2). To show that (8.3) is also suﬃcient for (8.2), we ﬁrst identify the vectors n ∗ in these formulae, as has been done so far, and use (8.3)2 to obtain a ∗ , so proving the desired suﬃciency: a ∗ = C1−1 (nr∗ − k1 n ∗ ), for k1 b = 1 + κ − (1 + κ)2 − ab. (8.11) Since in (8.3) we can exchange the roles of nr∗ and n ∗ , we can take nr∗ rather than n ∗ as the normal to the interface, and obtain another gradient ¯ 2 which is rank-1 connected to F1 : F ¯ 2 = F1 + ar ⊗ nr∗ = F1 (1 + ar∗ ⊗ nr∗ ) = (1 + ar ⊗ nr )F1 , F

(8.12)

nr = (F1 )−t nr∗ , ar = F1 ar∗ , ar∗ = C −1 (n ∗ −k2 nr∗ ), k2 = k1 b/a. (8.13) ¯ 2 , which have the same Cauchy–Green tensor C2 , are Thus both F2 and F rank-1 connected to F1 ; they correspond to the two aforementioned choices of the normal to the interface, and are called reciprocal or conjugate. If either µ1 or µ2 vanish, that is, if (8.8) holds, then ar∗ = (sgnκ b/a)a ∗ , ¯ 2 = F2 , and the two reciprocal solutions coincide. If µ1 and µ2 are both F diﬀerent from zero, the reciprocal solutions are distinct, and are the only solutions to (8.2) sharing their Cauchy–Green tensor. Notice that the actual amplitudes and normals a, ar and n, nr are all coplanar; indeed, by (8.11) and (8.13), a = nr − k1 n,

ar = n − k2 nr .

(8.14)

In general the analogue does not hold for the reference amplitudes and normals. The two deformations 1 + a ⊗ n and 1 + ar ⊗ nr exhibit an interesting geometric relation in the case of twinning (see §8.3.3). The formulae above can also be used to ﬁnd all the tensors that are rank-1 connected to any given F1 , regardless of their having a prescribed Cauchy–Green tensor. This can be done as follows (Ericksen (1991b)): pick any two vectors nr∗ and n ∗ satisfying the inequality (8.7) and deﬁne C2 by (8.3)1 ; then C2 is positive deﬁnite by (8.7), and is the Cauchy–Green ¯ 2 satisfying (8.12). tensor of a deformation gradient F2 satisfying (8.2), or F Remark 8.1 If det C1 = det C2 , then the determinant of U1−1 C2 U1−1 , or

244

MECHANICAL TWINNING

the product of the λr in (8.4), is 1, hence λ1 is the reciprocal of λn . Thus in this case either none or two rank-1 connections exist. Formulae giving the vectors n ∗ , nr∗ , a ∗ , ar∗ , in terms of the eigenvectors and eigenvalues of U1−1 C2 U1−1 (the λs and cs in (2.22)) are given by Ball and James (1987)6 (also Bhattacharya (1991), Forclaz (1999)). Here these vectors are obtained from the solutions of the eigenvalue problem (2.23), where extracting the square root of C1 is not needed. Similarly for the ˇ formulae of Silhav´ y (1999), (2001). In particular, a corollary of the following ˇ result (Silhav´ y (1999)) will be used to compute n ∗ and nr∗ in chapter 9. To state it, remember that the symmetric tensors U1−1 C2 U1−1 −1 and C2 −C1 have the same number of positive and negative eigenvalues by Sylvester’s inertia theorem (§2.3), hence C2 − C1 has exactly one positive and one negative eigenvalue if its rank is 2 and F1 and F2 are rank-1 connected. Proposition 8.2 Assume F1 and F2 to be rank-1 connected, and the symmetric tensor T = C2 − C1 to have ordered eigenvalues (ξ1 , 0, ξ3 ), ξ1 < 0, ξ3 > 0, and corresponding orthonormal eigenvectors v1 , v2 , v3 . Then, up to the usual indeterminateness, the reference normals are: √ √ √ √ ξ3 v3 + −ξ1 v1 ξ3 v3 − −ξ1 v1 ∗ ∗ √ √ n = , nr = . (8.15) 2 2 Indeed, compare the spectral representation of T and (8.3)1 . As an alternative to the expressions (8.11), (8.13) for a ∗ , ar∗ , the followˇ ing formula (Silhav´ y (1999)) gives a ∗ in terms of C1 , C2 and n ∗ : a ∗ = b−1 (r−1 C1−1 − C2−1 )n ∗ , r = det U2 / det U1 = (1 + a ∗ · n ∗ ); (8.16) here b is given by (8.6)2 , and the analogue of (8.16) holds when a ∗ , n ∗ are everywhere replaced by ar∗ , nr∗ (so, b becomes a). We prove (8.16), without actually losing generality, by assuming F1 in (8.2) to be a stretch U1 , and write the polar decomposition F2 = RU2 . Of course U12 = C1 and U22 = C2 . Then, by taking the inverse transpose of (8.2), and by extracting R from (8.2), we obtain, respectively, RU2−1 = U1−1 (1 − r−1 n ∗ ⊗ a ∗ ) and R = U1 (1 + a ∗ ⊗ n ∗ )U2−1 . (8.17) By applying (8.17)1 to n ∗ we have RU2−1 n ∗ = r−1 U1−1 n ∗ ,

(8.18)

which implies (8.16) by substituting (8.17)2 for R and then extracting a ∗ . 8.2 The twinning equation We now look for the continuous, pairwise homogeneous stress-free stable equilibria of a crystal at a given temperature θ. As in §6.2.1, we assume 6

The formulae for amplitudes and normals obtained above reduce, for C1 = 1, to the ones of Ball and James (1987), in whose analysis the conﬁguration of the crystal in the subdomain R1 is chosen as the reference, as is always possible.

8.2 THE TWINNING EQUATION

245

that in the reference conﬁguration R the crystalline lattice is L(ea0 ), with ˇ ) reference lattice vectors ea0 , and consider a free energy function φ = φ(C which satisﬁes the invariance (6.17) and has an isolated minimum at C1 . Then any homogeneous conﬁguration whose deformation gradient F1 is such that F1t F1 = C1 is a natural state. The same is true for any homogeneous conﬁguration with deformation gradient F2 such that the corresponding Cauchy–Green tensor is C2 = H t C1 H , for H ∈ G(ea0 ): this is a minimizer which is symmetry-related to C1 , and diﬀers from C1 if and only if m ∈ / L(F1 ea0 ), m representing H in the basis ea0 . In the absence of boundary conditions, the continuous deformation χ given by (8.1)–(8.2) is a minimizer of the energy functional (6.12) if and ˜ that is, if they only if both F1 and F2 minimize the energy function φ, belong to the energy wells. Of particular importance are the deformations χ whose gradients F1 and F2 belong to wells of φ˜ that are symmetryrelated, because so are all the energy wells away from phase transitions. In this case we have F2 = RF 1 H ,

for some R ∈ O and H ∈ G(ea0 ).

(8.19)

While in some growth twins orientation-reversing operations are involved, so that the determinants of F2 and F1 can have diﬀerent signs, this possibility seems not to be of interest for mechanical twins:7 there is always a way of choosing the lattice vectors in the subregion R2 in such a way that the determinants of F1 and F2 have the same sign, and therefore so do the determinants of R and H in (8.19). We take this point of view henceforth, postponing to chapter 9 the additional assumption that H be orthogonal. Relations (8.2), (8.19) and the above assumption on the determinants hold together only if F1 satisﬁes the following twinning equation for suitable vectors a ∗ and n ∗ : RF 1 H = F1 (1 + a ∗ ⊗ n ∗ ) = SF1 , a ∗ · n ∗ = 0, S = 1 + a ⊗ n , a · n = 0, a = F1 a ∗ , n = F1−t n ∗ .

(8.20) (8.21)

If so, we say that the symmetry-related wells OF1 and OF2 = OF1 H of ˜ , θ) are rank-1 connected. A schematic representathe energy function φ(F tion of such a connection (actually a pair, see §8.3.3) is given in Fig. 8.1. By Proposition 8.1 and the fact that C1 and C2 have the same determinant, the above wells, if distinct, are rank-1 connected if and only if λ2 = 1; equivalently, C2 − C1 has a nontrivial kernel, or det(C2 − C1 ) = 0 (Ericksen (1985)). Indeed, the product of λ1 and λ3 is one, which makes λ1 < 1, λ3 > 1. If we can use the Born rule (6.7) to deﬁne vectors ea = F1 ea0 , 7

(8.22)

See for instance Ericksen (1997) p. 198, (2000b), (2002c); cases in which this assumption is not made are considered by Ericksen (2001a,c).

246

MECHANICAL TWINNING

and m ∈ GL(3, Z) represents H in the reference basis ea0 , then (8.20) and (8.19) hold if and only if the orbits Oea = OF1 ea0 and Omba eb = OF2 ea0 ¯ are rank-1-connected wells of the lattice energy φ. A pairwise homogeneous deformation χ satisfying (8.19) and (8.20) is called a mechanical twin if the lattices L(Rea ) and L(ea ) on the two sides of the composition plane do not coincide; otherwise it may be called a pseudotwin. The aforementioned two lattices coincide if and only if in (8.20) R ∈ P (ea ).

(8.23)

One can check that (8.23) holds in (8.20) if and only if S is a (possibly trivial) lattice-invariant shear in G(ea ). We will henceforth understand these cases to be excluded in twinning,8 in particular the trivial one9 S = 1 by requiring m ∈ GL(3, Z)\L(ea ); this guarantees that S $= 1 and that the two wells are distinct: Omba eb $= Oea . The orthogonal transformation R ∈ / P (ea ) appearing in (8.20) is called the twinning operation, and the simple shear S the twinning shear.10 If the homogeneous reference conﬁguration R is chosen to be the one of the subregion R1 , in the twinning equation (8.20) we can set F1 = 1, hence F2 coincides with the shear S . This is the common practice in the experimental literature on mechanical twinning, where the shear S is always observed and often explicitly measured for mechanical twins (Cahn (1954), Hall (1954), Klassen-Nekliudova (1964)). One can eliminate the obvious ambiguity in the choice of a and n, or a ∗ and n ∗ , by imposing n ∗ or n to be of unit length, but in general we will not do so. Remark 8.2 In general, for twins, the tensor H in (8.20) belongs to the symmetry group G(ea0 ) of the energy function, but is not orthogonal; for instance, if the homogeneous reference conﬁguration is the one of the subregion R1 , then in (8.20) F1 = 1 and the material symmetry operation H cannot be orthogonal. This remark of Ericksen (1977), (1987), was one of the main reasons for rethinking the classical ideas on the material symmetry of crystals in the context of nonlinear elasticity (presented for instance by Truesdell and Noll (1965)), and for proposing the invariance discussed in §6.3.1 for the energy functions of crystalline solids. Also, in general, the lattice groups and the holohedries of the reference conﬁguration and of the twin individuals are not in a group-subgroup relation. However, in chapter 9 we will simplify the analysis of transformation twins by choosing the reference conﬁguration in such a way that H is an orthogonal operation and belongs to the point group of the reference, of which the point groups of the twin individuals are subgroups. So, for instance, if the reference vectors ea0 8 9 10

The nontrivial lattice-invariant shears above are automatically excluded for rank-1 connections of symmetry-related wells in a wt-nbhd (§9.1). This is the only possibility in a theory of material symmetry like the one of Coleman and Noll (1964), where a reference is used for which F1 = 1 and H ∈ O. In the metallurgical literature the shear S is often called lattice-invariant shear, a name we have reserved for the special class of deformations in (3.14).

8.2 THE TWINNING EQUATION

247

have cubic symmetry, as in chapter 5, a symmetry operation transforming the tetragonal stretch U1 (§5.1.1) into the tetragonal U2 is orthogonal, while the same transformation is not so if the reference vectors are taken to be the tetragonal U1 ea0 . For any choice of reference the orthogonal symmetry operations in the twinning equation are just a very special instance, although an important one. It is interesting to see how a twinning deformation aﬀects the crystalline lattice. To do this we write a molecular version of the twinning equation by applying both sides of (8.20) to the vectors ea0 , and using (8.22): Sea = mba Reb

with R ∈ O, (mba ) ∈ GL(3, Z).

(8.24)

This equation says that the sets of vectors Rea and Sea generate the same lattice. The basis ea generates the crystal lattice in one of the individuals of the twin, and if we take ea as reference basis and consider the eﬀect of a twinning deformation, we see that the portion of the lattice L(ea ) which is on one side of the composition plane K1 = {y : y · n = 0} is left undeformed, while the shear S acts on the portion of L(ea ) on the other side of K1 in such a way that the twinned lattice generated by Se a does not coincide with the original unsheared lattice L(ea ), but is congruent to it, through the orthogonal transformation R. For instance, it is typical that the sheared lattice be restored in mirror symmetry across the plane K1 (Figs. 1.4(c), 1.7(a)), but other twinning operations are also possible (§8.3.5 and chapter 9). As is discussed in detail by Zanzotto (1988), (1992), condition (8.24) is satisﬁed by all mechanical twins experimentally observed in simple lattices. The case of multilattices is more complex, and does not permit such a straightforward description. In general the so-called Mineralogists’ Assumption holds (Cahn (1953), (1954), Crocker (1965), Bevis (1968)): the restoration in shear of a sublattice of L(ea ) suﬃces, so that in a mechanical twin a fraction of the points of L(ea ) usually undergoes inhomogeneous shuﬄing (Fig.1.7(a)). The Born rule is a guideline for choosing the lattice vectors ea in each case (Rapperport (1959), Reed-Hill et al. (1963), Crocker and Bevis (1970)), but this procedure causes serious problems for a continuum theory (Zanzotto (1992), also Ericksen (1997)). The twinning equation, in the form (8.20) or (8.24), has been extensively studied under various assumptions, for example by Crocker (1962), (1982), Bilby and Crocker (1965), Bevis (1968), Bevis and Crocker (1968), (1969), Acton et al. (1970), Ericksen (1981b), (1985), (1986b), (1987), (1991b), (1997), (2000b),11 James (1981), (1982), Gurtin (1983), Pitteri (1985b), (1986), (1987), Ball and James (1987), Zanzotto (1988), Fosdick and Her11

Ericksen (1997), (2000b) presents a new point of view, called the X-ray theory, which aims at avoiding problems related to the Born rule in crystal mechanics (see §6.2.2) by only using hypotheses based on X-ray observations without reference to macroscopic deformation.

248

MECHANICAL TWINNING

F2 OU1H1 U2

F1

SU1 ¯2 F

U1

OU1 R1U1

SrU1 O 1−υ 1

R1R2U3

F υ

OU1H2 U3

Figure 8.1 Schematic description of three martensitic, symmetry-related wells and one austenitic well, where the reference is set. The upper part shows reciprocal (as in §8.3.3) rank-1 connections between the wells OU1 and OU1 H1 when the data in the twinning equation are (F1 , H1 ) (see (8.20), (8.12)) and (U1 , H1 ) (see (8.25), (8.27)). The lower part contains the construction of a habit plane between austenite and a simple laminate of variants U1 and U3 of martensite; see the formulae in §10.4.1, which involve variant U2 instead of U3

ˇ tog (1990), Bhattacharya (1991), Ou (1996), Silhav´ y (1997) §2.5, (1999), (2001), Forclaz (1999), and Adeleke (2000). 8.3 Solutions of the twinning equation For classiﬁcation purposes it is useful to remark that a rigid-body rotation superposed to the twinning deformation χ in (8.1) gives another twinning deformation solving (8.20), with the same H , and the twinning operation and twinning shear changed by conjugacy. It is thus natural to disregard the rotation R1 in the polar decomposition of the tensor F1 = R1 U in (8.20), rewriting the twinning equation in the form RUH = SU ,

S = 1 + a ⊗ n , a · n = 0;

(8.25)

here U ∈ Sym and S is the twinning shear in (8.21). In this case the twinning deformation χ deﬁned on the reference conﬁguration R is given by the gradients (U , SU ) rather than the ones (F1 , F2 ) in (8.20). The situation is schematized in Fig. 8.1. For equation (8.25), in which a reference basis ea0 is understood, the data are usually a tensor H ∈ G(ea0 ) and a stretch U , which speciﬁes one individual of the twin; this is a minimizer of the crystal’s free energy >

8.3 SOLUTIONS OF THE TWINNING EQUATION

249

function. Analogously, the data for (8.24) are a matrix m ∈ GL(3, Z) and a basis ea = Uea0 . Then either (8.25) or (8.24) has solutions (R, S ), each one giving a rank-1 connection between the energy wells Oea = OUea0 and Omba eb = OH t UH ea0 (8.26) ¯ of the lattice energy φ. A class of elements H of G(ea0 ) for which (8.20) has a solution are the lattice-invariant shears (Ericksen (1984)), whose representative matrix m can be shown to be conjugate to – see (3.15)   1

m ¯ = 0 0

n

0

1

0,

0

1

n ∈ Z\{0}.

For data (U , H ), with H a given lattice-invariant shear and U an arbitrary stretch, (8.25) is solved by the pair (R, S ) with R = 1, S = UHU −1 . So, also this solution is included in the pseudotwins, while the reciprocal solution (see §8.3.3) in general is not. A more important class, the one of (conventional) Type-1 and Type-2 twins, will be analyzed in §8.3.5. The special status of lattice-invariant shears and conventional twins with respect to the twinning equation is detailed in Proposition 8.5 below. Apart from these two classes, due to the inﬁnite, discrete invariance imposed by GL(3, Z), an algorithm is wanting for a neat classiﬁcation of the solutions (R, S ) of (8.25) for given data (U , H ) (or of (8.24) for data (ea , m)), although Adeleke (2000) provides, for each m, restrictions on the possible ea and gives the corresponding R, a, n, or, alternatively, the possible solutions (ea , m) for data (n, R). As a very preliminary result, for any given H ∈ G(ea0 ), (8.25) holds for some U , R, S (Pitteri (1987)). A reduction in the number of cases to be tested is provided by Remark 8.4 below. 8.3.1 Diﬀerent descriptions of the same twin and cosets Based on the molecular twinning equation (8.24), we can inquire to which extent the lattice structure of one twin pair determines, or is determined by, the solutions of that equation. Many diﬀerent descriptions of the same crystalline conﬁguration are possible if, as usually happens in the X-ray inspection of twinned specimens, only the lattice structure of a twin pair is known and the Born rule holds, without any further information on the actual macroscopic deformation that possibly took place. In fact, R and S are determined as solutions of an equation of the type (8.20)2 only up to point group operations and lattice-invariant shears with same invariant plane, respectively. This leaves open a wide range of possibilities that are used in trying to match macroscopic changes of shape with X-ray observations; see for instance Hall (1954) or Nishiyama (1978). Often an hypothesis of smallest amount of shear is used to select one among various possible choices diﬀering by a lattice-invariant shear. The possibilities increase fur-

250

MECHANICAL TWINNING

ther if the Born rule does not hold, and one assumes that only a sublattice has to follow the shear deformation (Crocker (1965)). When a solution to (8.24) is known, it is easy to check that a new solution with the same shear is obtained if the matrix m and the tensor R ∈ O ¯ t respectively, with m are replaced by mm ¯ and RQ ¯ ∈ L(ea ) representing ¯ Q ∈ P (ea ) in the lattice basis ea . This means that the substitution of m and R in a solution of (8.24) by any other corresponding elements of the cosets L(ea )m and RP (ea ) does not change the twinning deformation χ with gradients (U , SU ): we obtain a diﬀerent description of the same twin, that is, the same rank-1 connection between the two given energy wells. The converse is also true: if, given a basis, two solutions of the twinning equation have the same shear, then they correspond to symmetries m that are in the same left coset of L(ea ), and have twinning operations in the same right coset of P (ea ). Another way to check this assertion is by recalling that all the elements of the left coset L(ea )m in GL(3, Z), when acting on the well ¯ produce one and the same well: Omb eb = Om ¯ ba eb if and only Oea of φ, a if m ¯ ∈ L(ea )m. Given Oea , any element of such a coset can be chosen to generate the well Omba eb and, if a rank-1 connection exists between the two wells, it is the same regardless of the chosen symmetry element in L(ea )m entering the twinning equation (8.24). Only the twinning operation is aﬀected by this choice, and the distinct twinning operations describing the same twin form themselves a right coset of P (ea ). These observations are summarized in the following Proposition 8.3 For any lattice basis ea let (R, S ) solve the twinning ˜ S ) is a soluequation (8.24) with data (ea , m), m ∈ GL(3, Z). Then (R, tion of (8.24) for data (ea , m) ˜ with the same shear, if and only if m ˜ = mm ¯ ˜ = RQ ¯ t , and m ¯ ∈ P (ea ) in the basis ea . for m ¯ ∈ L(ea ), R ¯ represents Q Analogous considerations apply to the choices of H and R in the twinning equation (8.25). Consequently, it is useful to take as data for (8.25) a stretch U together with an entire left coset of elements H . The solutions are then constituted by two reciprocal twinning shears (§8.3.3), each one of which has its own right coset of twinning operations, all describing the same rank-1 connection between OU and OH t UH . Notice that the twinning operations in a coset do not all have the same properties; for instance, the rotation axes and angles do change. In general, it is convenient to describe a twin by means of the simplest operations, the most relevant example of this being the conventional twins – see §8.3.5. 8.3.2 Crystallographically equivalent twins We have mentioned before (8.25) that the superposition of an orthogonal transformation Q to a twin gives another twin which, trivially, solves (8.24) with the same m and the vectors Qea as data. The latter belong to the same energy well as ea , however the lattices L(ea ) and L(Qea ) do not in general

8.3 SOLUTIONS OF THE TWINNING EQUATION

251

coincide. The case in which they do, that is, Q ∈ P (ea ), is particularly interesting: both twins can take place in the same body at one time, with distinct composition planes in L(ea ), and this is often observed. Such twins are called crystallographically equivalent; an important property is that the normals to their composition planes and the axes of their twinning operations are crystallographically equivalent in L(ea ), indeed related by Q. This means that the composition planes and the axes of the twinning operations have the same indices in Q-related lattice bases of L(ea ). 8.3.3 Reciprocal twins By (8.19) the Cauchy-Green tensors C1 and C2 have the same determinant. Thus, according to Remark 8.1, two distinct wells either admit two rank-1 connections or none. In the ﬁrst case the reciprocal solutions to (8.25) never coincide; the corresponding pairwise homogeneous deformations are called reciprocal or conjugate twins, and the same nomenclature is used for their shears. Precisely, the results in §8.1 imply that if there is a solution (R, S ) of (8.25) for data (U , H ), then there is a reciprocal solution (Rr , Sr ) of (8.25) with same data. The situation is schematized in Fig. 8.1. In general, the fact that the tensors C1 and C2 in (8.3) are symmetryrelated in twinning does not simplify much the computations necessary to ﬁnd the vectors n ∗ , nr∗ , a ∗ , ar∗ that solve the twinning equation;12 these can still be obtained from the data (U , H ) by means of the general formulae (8.3), (8.4), (8.6), (8.9)–(8.11) and (8.13). They can also be found in terms of the eigenvectors and eigenvalues of U −1 H t U 2 HU −1 by means of the formulae of Ball and James (1987) or Forclaz (1999), and also by ˇ the ones, (8.15), (8.16), of Silhav´ y (1999). The corresponding formulae for transformation twins will be presented in §9.1. Anyway, if the wells are symmetry-related we have n ∗ · a ∗ = nr∗ · ar∗ = 0, and the reciprocal twinning shears S and Sr are in a very interesting geometric relation, often mentioned in the literature on twinning. Once we show that the reciprocal shear Sr introduced here coincides with the ¯r , we can refer one so denoted in §2.3.2, which is temporarily denoted by S to that section for the detailed geometrical description. Indeed, the shear ¯r in §2.3.2 has the property of being the unique nontrivial solution of S ¯ a rotation which does not ¯r = RS ¯ (see (2.36)4 ), with R the equation S necessarily belong to the holohedry of the lattice. Consequently, with the ¯ ¯ r = RR, ¯ r, S ¯r ) solve deﬁnition R we can directly verify that (R, S ) and (R (8.25) with the same data (U , H ): ¯ r UH = S ¯r U . RUH = SU ⇔ R

(8.27)

¯ r = Rr and S ¯r = Sr , as we claimed. So, by uniqueness, R As was detailed in §2.3.2, the remarkable property of the two reciprocal 12

The computations become considerably simpler when H has period two (§8.3.5).

252

MECHANICAL TWINNING

shears solving the twinning equation is that K1 and η2 in one twinning shear coincide with K2 and η1 in the other, and vice versa, and the amounts of shear are equal. These properties can be also derived, for C1 = U 2 , from formulae (8.2), (8.3), (8.6), (8.11)–(8.13) specialized to the present case, in which the equalities n ∗ · a ∗ = nr∗ · ar∗ = 0 imply the following ones: k1 b = κ = k2 a, or, equivalently, (1 + κ)2 − ab = 1. (8.28) 8.3.4 Generic twins When a rank-1 connection exists between two symmetry-related wells, we ask what happens to it when, roughly, the wells are perturbed so as to preserve symmetry (and their symmetry-relation (8.19)). A rank-1 connection between two wells that exists regardless of any such symmetry-preserving perturbations will be called generic, as well as the related mechanical twin. More precisely, ea0 being the reference basis, consider a stretch U , the basis ea = Uea0 , and a matrix m ∈ GL(3, Z)\L(ea ) such that the two distinct symmetry-related energy wells Oea and Omba eb be rank-1 connected: (8.24) holds for ea and m. We call this rank-1 connection generic if (8.24) also holds for data (¯ ea , m), with ¯ ea any other basis in the proper ﬁxed set E(L(ea ))∗ of ea , for suitable R and S . Equivalently, we require that if (8.25) holds with data (U , H ), then it must also hold, with same H and ¯ and S ¯ , for any other U ¯ such that L(U ¯ ea0 ) = L(Ue 0a ). The appropriate R ¯U ¯ ea0 ) = L(SUea0 ); last condition can be replaced by the following: L(S 0 ¯ ¯ ¯ ¯ indeed, remember relations S U = RU H , H ∈ G(ea ), and (3.26)–(3.27). This formulation, which may seem odd here, will be used in Remark 8.4. The relevance of generic connections is clear: such a connection continues to exist regardless of any stress-free thermal expansion of the crystal, away from phase transitions. Also, a generic connection is one whose existence depends on the symmetry of the stretch U appearing in the twinning equation, and not on the particular choice of U compatible with that symmetry. 8.3.5 Type-1 and Type-2 (conventional) twins Given one energy well, one class of solutions to the twinning equation (8.25) is well understood: the one for which the coset of H (or m) generating the second well contains an element of period two: H 2 = 1; in this case the computation of the solutions (R, S ) is also much simpler. Indeed, by Proposition 8.4 below, when the data are (U , H ), with U an arbitrary stretch and H of period two, the twinning equation (8.25) necessarily has two reciprocal solutions, corresponding to what are referred to as Type-1 and Type-2 twins in the mineralogical and metallurgical literature. These twins, often called conventional, are the geometrically simplest and by far the most signiﬁcant twins from the experimental point of view. In agreement with Crocker (1962) p. 1902, Bilby and Crocker (1965) p. 241, Bevis (1968), in a Type-1 twin the two individuals can be related by a rotation of π about

8.3 SOLUTIONS OF THE TWINNING EQUATION

253

the normal n to the invariant plane K1 of the twinning shear (recall that K1 is the twin interface); this means that the period-two rotation in (2.32): Rnπ = 2n−2 n ⊗ n − 1 can be taken as the twinning operation R in equation (8.25). Since the central inversion −1 always belongs to the holohedries of simple lattices, one can alternatively describe a Type-1 twin by means of the operation −Rnπ , which is the mirror symmetry across the composition plane K1 . This is how the orientation relationship between the individuals in a Type-1 twin is commonly described in the experimental literature (see Fig. 1.4(c)).13 Analogously, in a Type-2 twin the two individuals can be related by a rotation of π about the shear amplitude vector a – see (2.32)2 : Raπ = 2a−2 a ⊗ a − 1. For the same reason as above, the orientation relationship between the two individuals of a Type-2 twin can equivalently be described as a mirror symmetry across the plane orthogonal to a; this is often given as the deﬁnition of this twin (see again Fig. 1.4(c), which actually shows a (compound) twin that is at once of Type 1 and of Type 2 (see §8.3.6.). The following result (Pitteri (1986)) shows that conventional twins originate exclusively from period-two matrices m or tensors H (which need not be orthogonal) in the twinning equation (8.24) or (8.25), respectively. Proposition 8.4 For any data (U , H ) [(ea , m)], with U an arbitrary stretch14 and H 2 = 1 [with ea an arbitrary basis and m2 = 1], there are always two solutions of the twinning equation (8.25) [(8.24)], corresponding to reciprocal twinning modes of Type 1 and 2, respectively. Vice versa, reciprocal Type-1 and Type-2 modes are produced, through (8.25) [(8.24)], by one and the same H [m] of period 2. Indeed, assume det H > 0, which is not restrictive; then (§2.3.1) H is of period two if and only if, for suitable vectors b ∗ and n ∗ , it has the form H = 2b ∗ ⊗ n ∗ − 1,

for b ∗ · n ∗ = 1.

(8.29)

∗

The vector b is the eigenvector of H associated to the eigenvalue 1, while n ∗ is orthogonal to the 2-dimensional eigenspace related to the double eigenvalue −1. Then, given any stretch U , one can introduce the shears S = 1 + a ⊗ n and Sr = 1 + ar ⊗ nr , with n ar n = U −1 n ∗ , a = 2( −ar ), nr = 2(n − ), ar = Ub ∗ . (8.30) 2 n ar 2 Then, considering the period-two rotations Rnπ and Raπr , we deduce from 13

14

π For complex crystals without a center of symmetry, the operations Rπ n and −Rn give distinct twinned conﬁgurations having the same skeletal lattice but diﬀerent motifs within the skeletal cell. The following holds, a fortiori, for all stretches preserving any given lattice group; therefore conventional twins are, in particular, generic.

254

MECHANICAL TWINNING

(2.33) and the analogue of (2.34) for Sr that Rnπ UH = SU

and Raπr UH = Sr U ,

(8.31)

so that the Type-1 twin (Rnπ , S ) and the Type-2 twin (Raπr , Sr ) are the reciprocal solutions of equation (8.25) with data (U , H ). Vice versa, given any vectors a and n such that a · n = 0, consider the twinning modes (Rnπ , S ) and (Raπr , Sr ) as above; for any stretch U , they solve the twinning equation (8.25) with data (U , H ), where H is the tensor (see (2.31)) H = η2 ⊗ K1 − 1,

H 2 = 1,

η2 =

2n − a, n2

K1 = n.

(8.32)

Therefore Type-1 and Type-2 twins always come in reciprocal pairs; corresponding to any observed Type-1 twin there exists a reciprocal Type-2 twin with same H , which is in principle kinematically active and is indeed observed in various materials. We stress that using crystallographic indices for writing (8.32) is only a device for utilizing the recorded data. Such lattice vectors are here on an equal footing with any other basis, and, of course, any basis could be used, if convenient, for writing η2 and K1 explicitly. Indeed, various diﬀerent systems of indices, based on diﬀerent sets of vectors ea0 , are often used in the literature for the same crystal. The reader may see, for instance, Otte and Crocker (1965). The otherwise trivial observation that this does not inﬂuence the expression of H is sometimes of help, since in some cases it is not very clear which unit cells are used in the experimental literature. Remark 8.3 By Proposition 8.4, H 2 = 1 ⇒ R2 = 1; the converse need not hold. Also, Ericksen (1986b) produces twins where R3 = 1 = H 4 . These examples also indicate that there are restrictions on the stretches U which can satisfy the twinning equation for such choices of R and H . The case of H orthogonal and of period two arises in transformation twinning problems, and will be detailed in §9.1. In that case all the formulae above hold for b ∗ = n ∗ . Finally, recall that the twinning equation (8.25) has a solution for any choice of U when in the data (U , H ) the tensor H has order 2 or is a lattice-invariant shear; the following converse holds (Pitteri (1987)): Proposition 8.5 If for a given H ∈ G(ea0 ) the twinning equation (8.25) has a solution for data (U , H ), with U an arbitrary stretch, then either H 2 = 1 or H is a lattice invariant shear in G(ea0 ). 8.3.6 Compound twins It often happens in crystals of rather high symmetry that the observed twins are geometrically simple, as the one shown schematically in Fig. 1.4(c), which is at once of Type 1 and of Type 2. Such twins are called compound.

8.3 SOLUTIONS OF THE TWINNING EQUATION

255

By Propositions 8.3 and 8.4, they must satisfy the conditions (m1 )ba Rnπ eb = Se a = (m2 )ba Raπ eb , (m1 )2 = 1 = (m2 )2 ,

(8.33)

for suitable m1 , m2 ∈ GL(3, Z). Their main properties can be derived from those of Type-1 and Type-2 twins. For instance, they exist for data (ea , m1 ) with ea arbitrary, so, in particular, they are generic (see footnote 14); and the four elements of their twinning shear are all crystallographic (see §8.3.7). The following results are useful in the analysis of compound twins: Lemma 8.6 For any two (distinct) symmetry-related energy wells Oea and Omba eb , with m ∈ GL(3, Z)\L(ea ), there are at most two elements of ¯ give Om ¯ ba eb = Omba eb . period two in the coset L(ea )m whose elements m This is again a consequence of Propositions 8.3 and 8.4. Indeed, we have seen above that, given ea , all the elements of the coset L(ea )m solve the twinning equation (8.24) with the same twinning shear S and twinning operations in a right coset of P (ea ). If there is an element of order 2 in L(ea )m, then the corresponding twinning operation is necessarily either Rnπ or Raπ , whence the conclusion. Proposition 8.7 Given any two (distinct) symmetry-related energy wells Oea and Omba eb , with m ∈ GL(3, Z)\L(ea ), the two wells are rank-1 connected by two (reciprocal) compound twins if and only if there are two elements of period two in the coset L(ea )m. An equivalent condition for compound transformation twins is given by Forclaz (1999). Here, as in the proof of Lemma 8.6, the two elements of period two satisfy equation (8.24) with the same shear, so that (8.33) holds. Vice versa, by Proposition 8.3, if (8.33) is satisﬁed, the two period-two elements m1 and m2 are necessarily in the same coset of L(ea ). Remark 8.4 If (R, S ) is a solution of (8.25) for data (U , H ), then, easily, (Rt , S −1 ) is a solution for data (SU , H −1 ). The new solution, which is trivially obtained from the old (the tensor U appearing in either one satisﬁes the identical twinning equation (8.25)), corresponds to exchanging the deformation gradients U and SU in the pairwise homogeneous deformation (U , SU ) constituting a twin; or, equivalently, explicitly shows that the relation of being rank-1 connected among symmetry-related wells is symmetric. This conclusion also follows from the straightforward fact that the condition of having a nontrivial kernel (see below (8.21)) holds for H t CH − C if and only if it does for C − H t CH . The above exchange maintains the character of a twin of being conventional, compound, and also generic by the ﬁnal formulation in §8.3.4. Also, the corresponding twinning shears, S and S −1 , have the same amplitudes and opposite normals or vice versa; hence the same amount of shear. This simpliﬁes the description of transformation twins in chapter 9.

256

MECHANICAL TWINNING

8.3.7 Conventional twins and rationality conditions The following corollary gives the well known crystallographic properties of Type-1 and Type-2 twins usually mentioned in the literature, and explains why the reported experimental data regarding (K1 , η2 ) in Type-1 twins and (K2 , η1 ) in Type-2 twins are always integral numbers. In general, in a Type-1 twin the plane K2 , which is the interface of the conjugate Type-2 twin, is neither crystallographic nor material. Corollary 8.8 In a Type-1 twin the shear elements K1 and η2 are a crystallographic plane and row, respectively, in the lattice L(ea ). The same is true for the elements K2 and η1 in a Type-2 twin. This follows from Proposition 8.4 and the fact that any H of period two admits eigenvectors with rational components in any basis in which H is represented by an integral matrix. Zanzotto (1988) shows that the converse to this corollary is generally false (see below), but partially holds by the following useful construction: Proposition 8.9 Given any crystallographic plane which is not a symmetry plane in a simple lattice, there is an inﬁnite number of Type-1 twins whose interface is on that plane. This result is based on the fact that, for any choice of a crystallographic plane in a lattice, it is always possible to choose a set of lattice vectors such that two of them are on that plane. While we introduced Type-1 and Type-2 twins by means of the operations (2.32), in the great majority of the classical references in mineralogy or metallurgy another deﬁnition is adopted, according to which Type-1 twins are modes in which the twinning shear has elements K1 and η2 that are rational, while K2 and η1 are irrational; the converse being true for Type-2 twins. In addition, compound twins are called those in which all the four shear elements are rational (see for instance Cahn (1954), Hall (1954), Klassen-Nekliudova (1964), Bilby and Crocker (1965), Barrett and Massalsky (1966), Kelly and Groves (1970)). We call these modes of rational Type 1, rational Type 2, and rational compound, respectively. The common understanding is that these deﬁnitions are equivalent to the ones given in §§8.3.5, 8.3.6, and indeed they are often used interchangeably. While we have shown that Type-1 and Type-2 twins are of rational Type 1 and rational Type 2 respectively, the converse does not hold. Indeed, Zanzotto (1988) proves by an example that, if the Born rule applies, that is, if all the lattice points must shear as dictated by a macroscopic twinning shear S , then the two deﬁnitions above are actually not equivalent: the ﬁnal reorientation of a rational compound twin can be diﬀerent from the conventional laws Rnπ , Raπr , and the period of the matrix m can be essentially diﬀerent from 2. However, as we mentioned in §8.2, in twinning shears the Born rule sometimes must only be applied to a suitable set of sublattice vectors (Mineralogists’ Assumption). Then, the failure of the Born rule

8.4 SHORT REMARKS

257

may partially explain the assumed equivalence of the two diﬀerent deﬁnitions of Type-1 and Type-2 twins above; a fact that is also based on the experimental observation that the properties of orientation and rationality seem always to go together. This fact may be supported by the following: Proposition 8.10 If the elements of a shear S satisfy the conventional rationality conditions in a lattice, then S always restores at least a sublattice in one of the orientations Rnπ , Raπr in (2.32). Proof. Assume K1 and η2 to be rational and, as is always possible (Proposition 3.11), take lattice vectors ea such that e1 and e2 generate K1 ; then, if one denotes by e a the dual vectors, n ∧ e 3 = 0. By the rationality assumptions, we can choose vectors ηa ∈ Z

¯ n = e 3 , η¯2 = η a ea ,

(8.34)

a

along n and η2 , respectively, with the η diﬀering by a common factor from relatively prime integers. Then, by (2.33) and the arbitrariness in the lengths of n and η2 in (2.33)2 , we have   1 3 Rnπ Sea

=

rba

eb ,

(rba )

=

−1

0

0

−1

0

0

2η /η

2η 2 /η 3

.

(8.35)

1

Therefore the sublattice generated by ¯ e1 = e 1 , ¯ e2 = e2 , ¯ e3 = η 3 e3 satisﬁes the molecular twinning equation (8.24). A similar reasoning can be used to treat the other case of rationality. By this proposition, when Born’s assumption does not hold and a part of the lattice points can undergo some structural shuﬄing, the conventional twin laws (2.32) seem to be the best candidates for giving the ﬁnal reorientation of the whole twinned lattice, because a sublattice is already reoriented that way. For the cases in which the Born rule applies strictly, we are left with two non-equivalent deﬁnitions of Type-1 and Type-2 twins, and we keep the one involving orientations. 8.4 Short remarks 8.4.1 Experimental data We present in Table 8.1 some of the twin data taken from the sources mentioned in the introduction to this chapter and from James and Hane (2000). 8.4.2 Mechanical twinning and the Born rule Here we give arguments of caution in the applicability of the continuum theory developed in chapter 6 and applied to twinning in this chapter. The questionable point is the Born rule, and already in §8.2 we anticipated that it seems safer to replace it by a weaker hypothesis, such as

258

MECHANICAL TWINNING

Table 8.1 Some commonly reported twin data. The ρs and σs are suitable composition-dependent real numbers. ‘I’ means irrational, with approximate values in double quotes crystal

structure

K1

η1

K2

η2

s

α-Fe, Cr, Mo, Na Cu, Ag, Al, Au, γ-Fe, Co, Ge Be, Cd, Mg, Ti, Zr, Zn Mg Zr In, BaTiO3 , NiMn, InTl, NiAl, FeNiC β-Sn Bi, As, Hg, Sb, TiNi, AuCd, TbDyFe2 Anhydrite (CaSO4 ) CuAlNi

b.c.c. f.c.c.

(112) (111)

[¯ 1¯ 11] [11¯ 2]

(¯ 1¯ 12) (11¯ 1)

[111] [112]

0.707 0.707

h.c.p.

b.c.t.

(01¯ 12) (10¯ 11) (11¯ 22) (10¯ 1)

[0¯ 111] [10¯ 1¯ 2] [11¯ 2¯ 3] [101]

(01¯ 1¯ 2) (10¯ 1¯ 3) (11¯ 2¯ 4) (101)

[01¯ 11] [30¯ 32] [22¯ 43] [10¯ 1]

dep. on c/a 0.137 0.225 In 0.150

b.c.t. rhombo.

(031) (100)

[0¯ 13] [0¯ 1¯ 1]

(0¯ 11) (0¯ 1¯ 1)

[011] [100]

0.119 dep. on α

ortho. ortho.

(101) (100) (10¯ 1) (130) (112) (100) (110) (100) (011)

[10¯ 1] [0¯ 10] [1ρ1] [3¯ 10] “[3¯ 72] [001] [00¯ 1] [0ρ1 σ1 ] [0ρ3 σ3 ]

(10¯ 1) (0¯ 10) (1σ1) (1¯ 10) “(1¯ 72) (10¯ 1) (00¯ 1) (0ρ2 σ2 ) (0ρ4 σ4 )

[101] [100] [10¯ 1] [110] [312] [101] [110] [100] [011]

0.228 dep. on compos. 0.299 I 0.228 0.228 dep. on composition

α-U

ortho. C

Jordanite (Pb4 As2 O7 )

monocl.

TiNi

monocl.

the Mineralogists’ Assumption. We only sketch the main ideas, and refer to Zanzotto (1988), (1992), (1996a) for details. Additional criticism of the Born rule, from diﬀerent perspectives, has been given by Davini (1986), (1988), Davini and Parry (1989), (1991), and Friesecke and Theil (2001). Let us summarize the way we proceeded. First, we postulated that the global invariance group GL(3, Z) for the description of simple lattice conˆ Secﬁgurations be the invariance group for the lattice energy function φ. ondly, we used the Born rule to bridge molecular and continuum descriptions of deformation and energy, thus obtaining a thermoelastic potential φˇ whose invariance is dictated by a suitable discrete group G = G(ea0 ) of unimodular tensors; precisely, the group of those unimodular tensors that are represented by elements of GL(3, Z) in the reference lattice basis ea0 . Thirdly, given one energy well, the energy invariance guarantees the existence of inﬁnitely many more symmetry-related wells; and, whenever two such wells are rank-1 connected, but not by pseudotwins, we have a pairwise homogeneous natural state, actually a pair of reciprocal twins. If the two wells are described by the stretches U and H t UH , this happens if and only if the twinning equation (8.25) has a solution (R, S ) (actually, two) for data (U , H ). Now, we can reverse the use of the twinning equation, by rejecting a priori prejudices about the group G to which H belongs, and indeed using the experimental data on observed twins in a crystalline material to obtain information about G. Any twinning mode can be used

8.4 SHORT REMARKS

259

to construct the corresponding H , and if more twins are known for the material, a whole subgroup of G, to be called a twinning subgroup, is generated, whose properties can be investigated. The validity of the Born rule requires this subgroup to be representable by integral matrices in a suitable reference basis ea0 . The investigation shows that in most cases the twinning subgroups are not consistent with the discrete groups we would expect, and conﬂict with some of the common ideas associated with crystalline behavior. For instance, nonisolated energy minimizers, as for transversely isotropic materials, turn out to be the rule rather than the exception. In some cases the subgroups turn out to be high-dimensional continuous groups, possibly forcing G to coincide with the whole unimodular group; this would imply for the material a ﬂuid-like behavior. Although the class of discrete twinning subgroups is thoretically very special, it turns out to be experimentally important because a number of remarkable materials show a nongeneric behavior. They are essentially either crystals whose structure is described by a simple lattice, or shape memory alloys. For all these crystals, as far as we know, the Born rule holds when suitable lattice vectors are chosen, and therefore the nonlinearly elastic model presented here safely applies. The Born rule fails, for instance, for the hexagonal metals; the coupling of molecular and continuum descriptions of their twinning modes cannot be done in terms of a unique (sub)lattice basis; diﬀerent sublattices are operative, depending on the mode, in agreement with the Mineralogists’ Assumption. For such materials, which fail to obey the Born rule, it is not obvious how to ﬁnd a substitute that allows one to reconcile molecular and continuum descriptions, and to decide a priori whether, and how, elasticity theory might be useful. Investigating the possibility of a phenomenological approach to crystal mechanics independently of the Born rule, Ericksen (1997) has proposed what he calls the X-ray theory for crystalline solids, which is based on the kind of data one can obtain from X-ray observations alone, without reference to macroscopic deformation. This new point of view has interesting applications (Ericksen (1999), (2000a), (2001a,c), (2002a-c) ). 8.4.3 Growth twins One of the basic hypotheses leading to the twinning equations (8.24), (8.25) for mechanical twins is the continuity, across the twin interface, of the displacement from the reference conﬁguration. This condition is not sound in general for twins originated during crystal growth, because there is no reference conﬁguration for a growing crystal. In fact, Zanzotto (1988) gives a clear example, using a growth twin in alum which is carefully described by Friedel (1933) and Shalkolsky and Shubnikov (1933): small cubic crystals sprinkled onto the horizontal face of a large octahedron immersed into

260

MECHANICAL TWINNING

a satured solution were found to adhere to the octahedral face either in parallel or twinned positions. The analysis shows that there is no homogeneous reference conﬁguration from which the octahedron and the cubes in twinned positions can be obtained by a continuous displacement. The other condition leading to (8.24), (8.25), namely that both twin individuals be energy minimizers, remains tenable but is too weak by itself, and must be supplemented by other requirements. One condition is that the twin interface must have low associated interfacial energy. As exempliﬁed in §1.2, this has led to various matching conditions, each with counterexamples; indeed, a precise statement about the relation existing between the misﬁt at a crystalline interface and the energy stored there is still missing. This complex situation is detailed by Zanzotto (1988), who points out that the by far most commonly and frequently observed growth twins are the formations whose individuals meet at an interface which has crystallographically equivalent indices on the two sides, and share the lattice points on it (Type-1 and compound twins, for instance, share this property). Another general assumption seems reasonable in most cases: growth twins can be modeled as polycrystals which can exist in stable stress-free equilibrium states (natural states) while in contact with an environment described by the two control parameters p (pressure) and θ (temperature) varying in a domain of R2 . The scalar quantity p expresses the intensity of the external hydrostatic load on the body while it grows in contact with a heatbath at temperature θ. Thus, growth twins are similar to the shearstress-free joints of Ericksen (1983) and James (1984b). The two assumptions above seem to produce the most favorable conditions for crystal growth; indeed, the structure of a polycrystal meeting the second assumption above is able to adjust to the environmental changes possibly taking place during crystallization, and the crystal maintains itself in those natural states best suited for a continuous growth process consistent with environmental changes. Furthermore, by the ﬁrst assumption, large stresses are not created at the joints, and interfacial energies are likely to be maintained low (ideally vanishing), so that fracture or damage has less chance to occur. These formations are therefore the ones whose growth and survival thereafter are most favored; they have the highest chances of being observed with regularity in a given mineral, diﬀerently from what happens with fortuitous, randomly occurring associations. The formations so selected are the most likely to feature the main characteristics attributed to genuine twins, in particular reproducibility (see §1.2). As is shown by Ericksen (1983) and James (1984b), the point of view adopted here implies some deﬁnite conditions that an orientation has to satisfy in order to be considered a true twinning operation. Take a reference conﬁguration in which two individual crystals are already joined (they may constitute a crystal seed, for instance). By applying a reasoning like the one of §8.2 to the stretches that the reference crystal must undergo to comply

8.4 SHORT REMARKS

261

with a changing environment, one obtains an equation similar to (8.25): R(p, θ)U (p, θ)R0t = U (p, θ)(1 + a ∗ (p, θ) ⊗ n ∗ ),

(8.36)

where R0 is the rotation relating the two joined reference crystallites, and n ∗ is the normal to the reference interface. Notice that the the twin operation R(p, θ) is in general sensitive to the environmental conditions, a fact which does not appear to be appreciated in the literature. As detailed by Zanzotto (1990), one ﬁnds growth twins for which (8.36) holds for all kinds of submanifolds of the (p, θ) plane: open subsets (for instance, this is the case for conventional twins), lines, isolated points, the empty set. For more complex twin aggregates, like the staurolite crosses in Fig. 1.1, more conditions of the form (8.36) need to be imposed, thus restricting the subset of the (p, θ) plane compatible with the existence of the aggregate. As an example, Zanzotto (1989), (1990) analyzes a fourfold twin in quartz15 whose existence, based on the above model, requires p and θ to satisfy an equation that can be explicitly written in terms of the known quartz data. A few more details are given in §10.3. This equation can be used as a geological barothermometer: the presence of these quartz crosses in a geological bed can give information about the pressure in it at the time of crystallization if one knows the temperature at that time, or vice versa. This quantitative information is by far better than any other previously available to geologists.

15

This has been preferred to the analogue, much more common, cross in staurolite because there are many more data on the constitutive equations of quartz than on those of staurolite.

CHAPTER 9

Transformation twins Based on Pitteri and Zanzotto (1996b),1 and on the results of chapters 5 and 8, here we give a description of all the transformation twins that are possible in simple lattices. To do so we check all the rank-1 connections between any pair of variants in the same variant structure in the wt-nbhds. As in chapter 5, we consider wt-nbhds of reference bases whose symmetry is maximal, that is, bases of any of the three cubic types, or of the hexagonal type. The cases involving a reference lattice with nonmaximal symmetry can be derived from these. We describe the twins in §§9.2 and 9.3, indicating the composition planes and shear amplitude vectors based on the results in §9.1. In §9.4 we discuss the so-called Mallard law in the light of the results in §§9.2 and 9.3. 9.1 General properties To discuss mechanical twinning in general we have considered the rank-1 connections between any two symmetry-related energy wells. Now, in order to study the transformation twins that arise when a lattice undergoes a symmetry-breaking transition, we conﬁne our attention to pairs of wells belonging to the same variant structure2 in a wt-nbhd Nea0 , of which the reference basis ea0 is the center. From the discussion in §4.4, the two distinct wells are necessarily given by OUea0 and OUH ea0 , where the symmetrybreaking stretch U and the element H of P (ea0 ) satisfy: P (ea ) < P (ea0 ), H ∈ P (ea0 )\P (ea ) ⇒ H t UH $= U , ea := Uea0 . (9.1) So, unlike in the general case, here the holohedries of the reference basis ea0 and of a twin individual ea are necessarily in a group-subgroup relation. Thus, in the analysis of transformation twins, the data in the twinning equation (8.25) have the form (U , H ), for U a symmetry-breaking stretch and H an element of a left coset of P (ea ) in P (ea0 ). Analogously, in the data (ea , m) for the twinning equation (8.24) the symmetry element m belongs to a left coset of L(ea ) in L(ea0 ) rather than to all of GL(3, Z). 1

2

Part of the following information, with additional data on habit planes and microstructures, can also be found in Hane and Shield (1998), (1999a,b), (2000a,b), Hane (1999), James and Hane (2000). As for habit planes between austenite and (twinned) martensite, also interfaces between diﬀerent martensitic variant structures in a wt-nbhd are of interest both in theory and applications, but we do not consider them here. 263

264

TRANSFORMATION TWINS

If we choose the ﬁrst well U1 in the same, otherwise arbitrary, way as in chapter 5, the tensors giving the second well OUH necessarily belong to one of the cosets listed there. Remember that we restrict ourselves to the rotational subgroup of any symmetry group, with no loss of generality: the structure of (8.25) or (8.24) implies that considering only proper orthogonal symmetry elements and twinning operations does not entail any loss of twins; it only halves the number of distinct symmetry elements and corresponding twinning operations that describe the same twin. We ﬁrst study some general consequences of the hypothesis that in the twinning equation (8.25) the tensor H be proper orthogonal. Ericksen (1985) has given, in the nontrivial case H t UH $= U , a necessary and suﬃcient condition for that equation to hold; this is equivalent to det(H t CH − C ) = 0.

(9.2)

Later, Ericksen (1991b) has proven the following useful criterion: Proposition 9.1 Let U ∈ Sym > , H ∈ O+ with H t UH $= U , and let v be the unit vector of the axis of H ; the twinning equation (8.25) holds for data (U , H ) if and only if at least one of the following conditions holds: (a) H is a twofold rotation: H 2 = 1; (b) v either is an eigenvector of C = U 2 or is orthogonal to an eigenvector of C , which can be taken to be Cv ∧ v . The last condition can be equivalently stated by simply requiring v to be orthogonal to an eigenvector of C . Also, since U and C have the same eigenvectors, in this Proposition we can replace C by U . We give a proof diﬀerent from the original, following Pitteri (2001), and ﬁrst notice that condition (b) can be conveniently put in the form 0 = [C (Cv ∧ v )] ∧ (Cv ∧ v ) = v [v · Cv ∧ C 2 v ] or v · Cv ∧ C 2 v = 0, (9.3) (9.3)2 being an identity. Also the following holds,3 for instance by a direct computation in an orthonormal basis whose third unit vector is v : ω det(H t CH − C ) = −8 sin ω sin2 v · Cv ∧ C 2 v ; (9.4) 2 here ω is the angle of rotation about v . This provides the required proof with assertion (b) in the form (9.3)3 . The identity (9.4) still holds when H t CH and C are everywhere replaced by H t UH and U , respectively. Forclaz (1999) provides an extension of Proposition 9.1 to n space dimensions. Fosdick and Hertog (1996) and Ou (1996) (see also Forclaz (1999)) give a diﬀerent characterization of the solutions, which is equivalent to the 3

The case H t C H = C is also included; the characterization of the related H and C – see for instance Ou (1996), Forclaz (1999) before Lemma 9 – can be obtained from (9.4), (9.5), but also from Proposition 2.8 and the standard description of eigenvalues and eigenvectors of the rotation H given in §2.3.

9.1 GENERAL PROPERTIES

265

one in Proposition 9.1 by the identity v · Uv ∧ U 2 v = v1 v2 v3 (λ2 − λ1 )(λ3 − λ2 )(λ3 − λ1 ),

(9.5)

where λ1 , λ2 , λ3 are the eigenvalues of U and v1 , v2 , v3 the components of v in a corresponding orthonormal basis of eigenvectors. Identity (9.5) remains true when U is replaced by C and λ1 , λ2 , λ3 by their squares, in which case it is again equivalent to condition (b) in Proposition 9.1. In order to determine the characteristic vectors in certain pairs of reciprocal twins, the following results provide more eﬃcient tools than the general formulae in chapter 8, and will be repeatedly used in §§9.2, 9.3. Of course, they follow, in the speciﬁc cases, from those general formulae, which will be replaced by the present more direct methods. Proposition 9.2 Let the data (U , H ) in the twinning equation (8.25) be such that H t UH $= U with a period-two H ∈ O+ whose axis is along the unit vector w ∗ ; then the equation is solved by reciprocal Type-1 and Type-2 twins generated, to within the usual indeterminateness, by n ar n = U −1 w ∗ , a = 2( − ar ), nr = 2(n − ), ar = Uw ∗ . (9.6) n2 ar 2 Equivalently, for C = U 2 , the reference normals and amplitudes are n ∗ = w ∗ = ar∗ , a ∗ = 2(

C −1 w ∗ Cw ∗ ∗ ∗ ∗ −w ), n = 2(w − ) ⊥ n ∗. r w ∗ ·C −1 w ∗ w ∗ ·Cw ∗ (9.7)

This result covers case (a) in Proposition 9.1, and follows from (8.30) once we take into account that, here, b ∗ = n ∗ because H ∈ O. The orthogonality relation in (9.7)4 can be checked directly, but also obtained from (8.3)1 and the fact that C2 and C1 have the same trace. Therefore, the composition planes of all the conventional reciprocal transformation twins are orthogonal in the reference conﬁguration: n ∗ · nr∗ = 0 (see also (9.11) below). Also, Proposition 9.2 is essentially geometric: it holds for any rotation H of order 2, not only for those belonging to G(ea0 ) as required by the energetic reasoning leading to (8.19). Formulae equivalent to (9.6) are given, among others, by Crocker (1962), (1982), Bilby and Crocker (1965), Bevis (1968), Bevis and Crocker (1968), (1969), Acton et al. (1970), Ericksen (1981a), Gurtin (1983), Ball and James (1987), Bhattacharya (1991). Formulae (9.6) allow one to obtain the relevant information regarding the conventional transformation twins when H ∈ G(ea0 ) is orthogonal and of period two. In particular, we stress that the composition plane for the Type-1 twin (Rnπ , S ) is on the material image, through the stretch U , of the reference plane orthogonal to w ∗ , which is a symmetry plane for the reference lattice by hypothesis. This is called a lost symmetry plane because such material image is no longer a symmetry plane for the lattice after the stretch U is applied, due to the hypothesis H t UH $= U . Moreover, as

266

TRANSFORMATION TWINS

always in simple lattices, the normal w ∗ to the aforementioned symmetry plane is itself a 180-degree symmetry axis for the reference lattice. Formula (9.6)4 says that the axis of the reciprocal Type-2 twin (Raπr , Sr ) is exactly on the material image of such a 180-degree symmetry axis, lost by the reference lattice after the stretch U is applied. For short, we say that the twinning operations in the two reciprocal conventional twins (Rnπ , S ) and (Raπr , Sr ) are on lost symmetry elements for the reference lattice L(ea0 ). Remark 9.1 As many examples show, in general it is not true that the axis of the twinning operation R is the material image of a symmetry axis for the reference lattice. This necessarily holds only for the conventional twins, which involve the 180-degree twinning operations Rnπ and Raπr . The following corollary of Proposition 9.2 is useful when there are two rotations of period two in the coset of tensors generating the second well. Proposition 9.3 Let the reference vectors ea0 and the energy well OUea0 be given, and let the coset P (ea )H generating another well OUH ea0 contain π π ∗ two period-two rotations, say Rw and wr∗ of unit length. ∗ and Rw ∗ , with w r Then the two compound twins connecting these wells are described, to within the usual indeterminateness, by the vectors n = U −1 w ∗ ,

a = λUwr∗ ,

nr = µU −1 wr∗ ,

ar = Uw ∗ ,

(9.8)

where, as usual, C = U 2 and w ∗ ·C −1 wr∗ w ∗ ·Cwr∗ = −2 ∗ . ∗ −1 ∗ wr ·C wr w ·Cw ∗ (9.9) Equivalently, the reference normals and amplitudes are λ=2

w ∗ ·C −1 wr∗ w ∗ ·Cwr∗ = −2 ∗ , ∗ −1 ∗ w ·C w wr ·Cwr∗

n ∗ = w ∗ = ar∗ ,

µ=2

a ∗ = λwr∗ ,

nr∗ = µwr∗ ⊥ w ∗ .

(9.10)

First of all, (9.10)1,2 coincide with (9.7)1,2 . Then, by the assumptions, U π π ω commutes with Rw ∗ Rw ∗ , which coincides with Rg ∗ for some angle ω (which r ∗ ∗ ∗ will be shown to be π) and g = w ∧ wr . By Proposition 2.8 g ∗ is an eigenvector of U , hence Cw ∗ and C −1 w ∗ are coplanar with w ∗ and wr∗ . This and (9.7)3,4 imply (9.10)3,4 for some λ, µ, and the orthogonality relation in (9.10)4 ; and thus the expressions (9.9)1,3 for λ and µ. Equalities (9.9)2,4 now follow from the identity w ∗ = Cw ∗ (w ∗ · C −1 w ∗ ) + Cwr∗ (wr∗ · C −1 w ∗ ) and its analogue for wr∗ . In this case the composition planes of both twins are lost symmetry planes in the deformed lattice. Also, we stress that this result too has a π π 0 geometric character: the tensors Rw ∗ and Rw ∗ giving the well OUH ea r need not belong to P (ea0 ) for the proposition to hold. The two propositions above allow for a rapid discussion of many of the transformation twins, because the cosets in §§9.2 and 9.3 generating all the Ur in a variant structure from the given U1 often contain one or two rotations with period two. However, there are also variants generated by cosets

9.1 GENERAL PROPERTIES

267

containing no 180-degree rotation; then statement (b) in Proposition 9.1 gives the relevant information regarding the existence and genericity of the related nonconventional twins. In all these cases the vectors n, a, nr and ar are certainly provided by the general procedure outlined in §§8.1 and 8.3.3, or the general formulae in Ball and James (1987), Forclaz (1999) or ˇ Silhav´ y (1999), (2001). Nevertheless the formulae in Propositions 9.2 and 9.3 can still be useful. First of all, the reference normals n ∗ , nr∗ are given ˇ by Silhav´ y’s general formulae (8.15) adapted to this case: 1 1 n ∗ = ξe ∗, nr∗ = ξf ∗, e ∗ = √ (v3 +v1 ), f ∗ = √ (v3 −v1 ) ⊥ e ∗ . (9.11) 2

2

Indeed, −ξ1 = ξ3 =: ξ because C2 and C1 have the same trace. Secondly, the equality of the quadratic and cubic invariants of C2 and C1 implies the following (Ou (1996), Forclaz (1999)): v2 · C1 w ∗ = 0

for either w ∗ = e ∗

or w ∗ = f ∗

Indeed, in the basis (v1 , v2 , v3 ), C1 and C2 have    a b r a−ξ C1 =  b c s  and C2 =  b r s d r

or both.

(9.12)

the representations  b r c s  (9.13) s d+ξ

for suitable a, b, c, d, r, s and for the same ξ as in (9.11) By equating the trace of the cofactors (second invariants) we obtain the condition ξ = a − d > 0. This, substituted in C2 , yields (a − d)(s2 − b2 ) for the diﬀerence of the determinants, which vanishes if and only if (9.12) holds. Proposition 9.4 (Ou (1996), Forclaz (1999)) Assume the wells OU1 ea0 and OU2 ea0 , for U2 = H t U1 H , H ∈ O, to be distinct and rank-1 conπ π ∗ nected. Then U2 = Rw ∗ U1 Rw ∗ for any one of the two unit vectors w that satisfy (9.12)1 . Moreover, the amplitudes and normals of the two reciprocal shears in the solutions (R, S ), (Rr , Sr ) of (8.25) for data (U1 , H ) are given by (9.6)–(9.7) if one choice of w ∗ satisﬁes (9.12)1 , and by (9.8)– (9.10) if both do; in the latter case e ∗ and f ∗ are the choices of w ∗ and wr∗ , or vice versa. The corresponding twinning operations R, Rr can be obtained from (8.25), and in general are neither Rnπ nor Raπr . One can check the ﬁrst assertion by a direct computation, using (9.12), and the statements about the amplitudes and normals follow from Propositions 9.2 and 9.3 by their geometric character. The ﬁrst assertion about the twinning operations is straightforward; an example of the fact that that they need not be the conventional ones is given by the (1, 4)-twins in π §9.2.4.1. Of course, if the data are (U1 , Rw ∗ ) the shears remain the same as for data (U1 , H ), but the rotations are necessarily Rnπ and Raπr . Remark 9.2 The ﬁrst part of Proposition 9.4 details a result of James and Ericksen (Ericksen (1985), (1991b), Fosdick and Hertog (1996)): if the twinning equation (8.25) holds with R, H in O such that R2 $= 1, H 2 $= 1,

268

TRANSFORMATION TWINS

˜ H ˜: then it also holds for suitable 180-degree rotations R, RUH = SF

⇔

˜ H ˜ = SF , R ˜2 = 1 = H ˜ 2. RU

(9.14)

The meaning of (9.14) is that there always exists a suitable lattice L , congruent to the given lattice L(Uea0 ), such that while the latter produces two reciprocal twins through the solutions (R, S ), (Rr , Sr ) of the twinning equation, L generates a Type-1 and a Type-2 twins by means of the same shears. However L(Uea0 ) and L in general do not coincide. We stress that this result does not mean that all twins are conventional ˜ ∈ ˜ Type-1 or Type-2 twins, because in general H / P (ea0 ), or, equivalently, H is not in the left coset P (ea )H . Examples are the nonconventional twins in ˜ ∈ P (ea )H necessarily the next sections. Remember that the condition H follows from the assumptions that the two rank-1 connected deformations are energy minimizers; that these have to be symmetry-related away from phase transitions; and that the energy invariance in a wt-nbhd is given, say, by (6.29). This fact points out that in a theory of mechanical twinning it is not suﬃcient to adopt the continuum point of view, for instance by considering only equation (8.25) with no reference to a crystalline lattice, or with too broad a material symmetry group. A purely kinematic theory in which the tensor H in (8.25) is only assumed to be orthogonal is too coarse because in it, by (9.14), all twins would be of Type 1 or 2. It is in fact important to consider also the reference lattice basis, or, equivalently, a ﬁnite crystallographic symmetry group attached to the reference conﬁguration, to which H must belong. This ensures that also the crystallographic nature of the phenomenon is properly taken into account. 9.1.1 Procedure to determine the transformation twins Let us number the variants as in chapter 5, keeping its notation and denominations, and denote by #h and #l any pair of variants OUh ea0 and OUl ea0 in the wt-nbhd Nea0 of the reference basis ea0 . Besides Uh = Hht U1 Hh and its analogue for Ul , we have Ul = Hit Uh Hi for any Hi in a suitable left coset of P (Uh ea0 ) in P (ea0 ). A rank-1 connection between variants #h and #l, to be called an (h, l)-connection or an (h, l)-twin, is given by a solution (R, S ) of the twinning equation (8.25) for data (Uh , Hi ) as above. An (h, l)-connection exists if and only if there is a corresponding rank-1 connection between variant #1, whose stretch is U1 , and some suitable variant #k, that is, if and only if there is a suitable (1, k)-twin: the latter ¯ S ¯ ) to (8.25) for data (U1 , Hk ), where Hk = Hh Hi H t , is the solution (R, h t ¯ = Hh RH and S ¯ = Hh SH t . For this reason the (1, k)-connection is R h h called comparable in the reference lattice to the original (h, l)-connection. This equivalence relation maintains the properties of a twin of being conventional, compound, or generic because either one of the conditions in Proposition 9.1 holds for a pair (h, l) if and only if it holds for any other comparable pair; and the twinning elements, such as the composition plane,

9.1 GENERAL PROPERTIES

269

etc., are crystallographically equivalent in the reference lattice. Thus we only need to test whether and under which conditions any of the (1, k)twins exist for each one of the variant structures. Here and below two pairs that are comparable in the reference lattice are simply called equivalent. Remark 9.3 We will beneﬁt from the description of the variant structures and their cosets given in chapter 5, and use once more equivalence to reduce the number of (1, k)-connections to be tested. In addition, by the straightforward relation between any (1, k)- and (k, 1)-twins, which is given in Remark 8.4 and preserves the twin characteristics of being compound, conventional or generic, once we have examined a (1, k)-twin we can avoid detailing the (k, 1)-twins or their equivalents, which can all be reconstructed from the (1, k)-twins. Notice that any conventional (k, 1)-twin is already equivalent to a (1, k)-twin, by any period-two element in the coset generating well #k. The conceptually diﬀerent operations of equivalence and of exchange of order in a pair can thus only be distinguished in nonconventional twins, and indeed they do; see §§9.2.4.1, 9.2.4.2, 9.3.2.3. By Propositions 9.2 and 9.3 the cosets containing one [two] 180-degree rotations give wells that are connected to well # 1 by conventional [compound] twins; these are always generic (footnote 14 in chapter 8). The twin planes of the Type-1 and the twin axes of the Type-2 twins, which are material and crystallographic, are lost symmetry planes and axes of the reference lattice. For compound twins either one can be chosen as Type-1. Concerning the wells generated by cosets lacking any rotation of period two, the rank-1 connections, if they exist, are necessarily nonconventional. For data (U1 , H ), equation (9.3), which is a cubic equation in the components of the tensor U1 , gives the necessary information on the existence and genericity of the twins between OU1 ea0 and OU1 Hea0 . If (9.3) is satisﬁed for all stretches preserving the lattice group L(U1 ea0 ), then the reciprocal nonconventional twins with data (U1 , H ) are generic by deﬁnition. Otherwise, the rank-1 connections between these variants are nongeneric: they exist only when the lattice parameters (or the transition stretch components) of the deformed lattice satisfy relation (9.3). This selects a submanifold of stretches U – equivalently, a submanifold of bases in the ﬁxed set E(L(U1 ea0 )) or of metrics in I(L(U1 ea0 )) – for which twinning between OUea0 and OUH ea0 is possible. When such a submanifold actually reduces to, or is contained in, some ﬁxed set included in the ﬁxed set I(L(U1 ea0 )), we say that the two wells OU1 ea0 and OU1 Hea0 are not rank-1 connected: indeed they are connected only if U1 ea0 has strictly higher symmetry than initially supposed. All these cases do indeed occur: there are generically and nongenerically rank-1 connected variants, and also variants that are not rank-1 connected. Remark 9.4 We do not list all the possible connections, but give at least one representative for each class of equivalent twins, and one for any two classes of twins diﬀering by the ordering in the pairs (see Remark 9.3). The

270

TRANSFORMATION TWINS

Table 9.1 Rank-1 connections in the tetragonal variant structure Twins

(1, 2)

(1, 3)

Type

compound

compound

n∗

1 √

2

2(d−a) (1, 0, 1) ∼ [101] 2(d+a)

a∗

√

n∗ r

√

a∗ r s2

1 √

(1, 0, −1) ∼ (10¯ 1)

2(d−a) (1, 0, 1) ∼ (101) 2(d+a) 1 (1, 0, −1) ∼ [10¯ √ 1] 2 (d−a)2

2

(0, 1, −1) ∼ (01¯ 1)

2(d−a) (0, 1, 1) ∼ [011] 2(d+a)

√

2(d−a) (0, 1, 1) ∼ (011) 2(d+a) 1 (0, 1, −1) ∼ [01¯ √ 1] 2 (d−a)2

√

ad

ad

number of pairs given below for each variant structure or equivalence class counts the ordered pairs. Also, for any variant structure, we give the components of the reference normals and amplitudes in the orthonormal basis (i , j , k ) introduced in chapter 5 for that structure. These provide indices of reference twin planes and twinning directions in the associated cubic lattice or, equivalently, indices of actual planes and directions in the lattice obtained from it by the stretch U1 in that structure. Whenever necessary, we indicate how to obtain the indices in a standard cell (Remark 3.6). For simplicity, the quantities in the tables are given in terms of the components of the Cauchy-Green tensor C1 rather than those of the stretch U1 ; for each variant structure the former are given in terms of the latter in the text. 9.2 Rk-1 connections in a cubic wt-nbhd For each variant structure we refer to the section where the wells and the related cosets are numbered and described, and list in table form the characteristics of the rank-1 connections. The cosets containing two 180-degree rotations produce compound, generic twins by Proposition 8.7, and the twin elements are determined by means of Proposition 9.3 for one of the two equivalent choices of the unit vectors w ∗ and wr∗ . The cosets containing one 180-degree rotation give conventional (not compound) and generic twins, and the elements of the Type-1 and Type-2 twins are determined by means of Proposition 9.2. The characteristics of all these conventional twins are reported with no additional comment. 9.2.1 Tetragonal variant structure (§5.1.1) π/2

The two tetragonal twins in Table 9.1 are equivalent (by means of Rk ) and compound, thus generic, hence so are all 6 possible twins. The twin planes are the cubic symmetry planes that are lost in any of the tetragonal variants, and the normals to those cubic planes are the axes of the two 180-degree rotations in the corresponding cosets. Indices in Table 9.1 are

9.2 RK-1 CONNECTIONS IN A CUBIC WT-NBHD

271

Table 9.2 Rank-1 connections in the rhombohedral variant structure Twins

(1, 2)

(1, 3)

(1, 4)

Type

compound

compound

compound

n∗

(1, 0, 0) ∼ (100)

∗

− 2b a +b

n∗ r

−2b a

a

(0, 1, 0) ∼ (010)

(0, 1, 1) ∼ [0¯ 1¯ 1]

−2b a +b

(0, 1, 1) ∼ (0¯ 1¯ 1)

− 2b a

(0, 0, 1) ∼ (001)

(1, 0, 1) ∼ [¯ 10¯ 1]

−2b a +b

(1, 1, 0) ∼ [¯ 1¯ 10]

(1, 0, 1) ∼ (¯ 10¯ 1)

−2b a

(1, 1, 0) ∼ (¯ 1¯ 10)

a∗ r

(1, 0, 0) ∼ [100]

(0, 1, 0) ∼ [010]

(0, 0, 1) ∼ [001]

s2

8b2 (a−b)(a+2b)

8b2 (a−b)(a+2b)

8b2 (a−b)(a+2b)

preceded by a ‘∼’ and are taken with respect to the standard tetragonal cell; moreover a = α2 , d = δ 2 . Since here and below the representative matrices of C1 and U1 have the same structure, we denote each element of C1 by the latin letter corresponding to the greek letter denoting the element in the same position in the matrix of U1 . Here, a, d are the elements of indices 11 and 33, respectively, in the representative matrix of C1 , the corresponding elements of U1 being denoted by α, δ. Similar notational conventions are used below without additional comments. Notice that twins become trivial (s = 0) only on the ﬁxed set of the cubic reference lattice (α = δ). 9.2.2 Rhombohedral variant structure (§5.1.2) The (1, 3) and (1, 4) twins in Table 9.2 are equivalent to (1, 2) (by Riπ−j and Riπ−k , respectively) and compound, hence so are all 12 possible twins. The twin planes are the cubic symmetry planes that are lost in any of the rhombohedral variants, and the normals to those cubic planes are the axes of the two 180-degree rotations in the corresponding cosets. Indices are taken with respect to the standard rhombohedral basis, and we set a = α2 + 2β 2 , b = β(2α + β). The twins become trivial only for b = 0, that is, when U1 is actually a cubic stretch. 9.2.3 Orthorhombic variant structures 9.2.3.1 Orthorhombic ‘cubic edges’ wells (§5.1.3.1) The 30 pairs in this structure are grouped in 4 equivalence classes, the ﬁrst of 12 elements represented by (1, 2) (to which (1, 3) is equivalent through any nontrivial element of Oijk ), and the others of 6 elements each, represented by (1, 4), (1, 5), (1, 6).4 All four are invariant under exchanging the order in the pairs. Indeed, the last 3 classes are all made of compound twins, representatives of which are shown in Table 9.3. There, indices are 4

Here and below, within any given variant structure diﬀerent equivalence classes are indicated by diﬀerent type styles.

272

TRANSFORMATION TWINS

Table 9.3 Rank-1 connections in the orthorhombic ‘cubic edges’ variant structure Twins

(1, 2) (1, 3)

(1, 4)

(1, 5)

(1, 6)

Type

none none

compound

compound

compound

n∗

1 √

2

(1, 1, 0) ∼ (110)

1 √

2

(1, 0, 1) ∼ (101)

1 √

2

(0, 1, 1) ∼ (011)

∗

k(1, −1, 0) ∼ [1¯ 10]

l(1, 0, −1) ∼ [10¯ 1]

m(0, 1, −1) ∼ [01¯ 1]

n∗ r

k(1, −1, 0) ∼ (1¯ 10)

l(1, 0, −1) ∼ (10¯ 1)

m(0, 1, −1) ∼ (01¯ 1)

a∗ r

1 (1, 1, 0) ∼ [110] √ 2 (c−a)2

1 (1, 0, 1) ∼ [101] √ 2 (d−a)2

ac

ad

a

s

2

1 √

2

(0, 1, 1) ∼ [011] (c−d)2 cd

in the standard orthorhombic cell and 2(d−a) 2(d−c) 2(c−a) , l= √ , m= √ , a=α2 , c=γ 2 , d=δ 2 . k= √ 2(c+a) 2(d+a) 2(d+c)

(9.15)

Also the ﬁrst class is invariant under exchanging the order in the pairs, 4π/3 because (3, 1) is equivalent to (1, 2) by means of Ri +j +k . Unfortunately, as was noticed by Ball and James (1992), no rank-1 connection exists in this class. The assertion can be veriﬁed immediately by taking the diﬀerence of the squares of the two stretches, but we use Proposition 9.1 as an exercise. 2π/3 In this case (U1 , Ri +j +k ) can be chosen as twin data – see Table 5.4. Thus equation (9.3) must be checked for U = U1 and v = i + j + k . Since v = (1, 1, 1), v · Uv ∧ U 2 v = (β − α)(α − γ)(β − γ),

(9.16)

the (1, 2)-connection exists only if (9.3) holds, that is, in the given tensor U1 two of the (diagonal) entries coincide. This makes it belong to one of the tetragonal ﬁxed sets contained in the orthorhombic ﬁxed set; in other words, the connection exists only if the symmetry is actually increased from orthorhombic to tetragonal. Consequently there are no (1, 2)-twins in this variant structure, the same being true for all the other pairs in the same equivalence class. 9.2.3.2 Orthorhombic ‘mixed axes’ wells (§5.1.3.2) The 30 pairs in this structure are grouped in two equivalence classes, both invariant under exchanging the order in the pairs. The ﬁrst has 6 elements, is represented by the pair (1, 2), and consists of compound twins. The second contains the remaining 24 pairs, is represented by (1 , 3 ) (to which (1 , 4 ), (1 , 5 ) and (1 , 6 ) are equivalent through Rkπ , Riπ−j and Riπ+j ), and is the ﬁrst instance of conventional and generic, but not compound, twins. The interfaces of the Type-1 twins are on lost symmetry planes, and the axes of the Type-2 twins are lost 180-degree symmetry axes, for the cubic reference lattice. The interfaces of the Type-2 twins are neither crystallographic nor material (see Proposition 9.2). In Table 9.4 indices are taken with respect

9.2 RK-1 CONNECTIONS IN A CUBIC WT-NBHD

273

Table 9.4 Representatives of equivalence classes of rank-1 connections in the orthorhombic ‘mixed axes’ variant structure Twins

(1, 2)

(1, 3)

Type

compound

Type 1 and 2

n∗

(1, 0, 0) ∼ (100)

∗

a

− 2ab (0, 1, 0) ∼ [0¯ 10]

n∗ r

− 2ab (0, 1, 0) ∼ (0¯ 10)

a∗ r

(1, 0, 0) ∼ [100]

s

1 √

√

√

4b 2 a2 −b2

2

(1, 0, −1) ∼ (10¯ 1)

2 2(b2 −a2 +ad) (1, r1 , 1) ∼ [1r1 1] 2 2 a −b +ad 2(d−a) (1, s1 , 1) ∼ (1s1 1) a +d 1 √ (1, 0, −1) ∼ [10¯ 1] 2 3 a −ab2 −2a2 d+3b2 d+ad2 d(a2 −b2 )

to the cubic reference basis (i , j , k ); we use the deﬁnitions a=α2 +β 2 , b=2αβ, d=δ 2 , r1 =

2bd , s1 = 2b , a−d a2 −b2 −ad

(9.17)

and only report the data for (1, 2)- and (1, 3)-twins, which are representatives of the equivalence classes in this structure; we also do the same in the other tables below. It is not diﬃcult to reconstruct the other twin data; for instance, the indices of the (1, 4)-, (1, 5)-, and (1, 6)-twins are, respectively, n ∗ ∼ (1, 0, 1), a ∗ ∼ [1, r1 , ¯ 1], nr∗ ∼ (1, s1 , ¯ 1), ar∗ ∼ [1, 0, 1], ∗ ∗ ∗ n ∼ (0, 1, ¯ 1), a ∼ [r1 , 1, 1], nr ∼ (s1 , 1, 1), ar∗ ∼ [0, 1, ¯1], n ∗ ∼ (0, 1, 1), a ∗ ∼ [r1 , 1, ¯ 1], nr∗ ∼ (s1 , 1, ¯ 1), ar∗ ∼ [0, 1, 1],

(9.18)

with the same shear magnitude as the (1, 3)-twins. Notice that, in general, r1 $= s1 , hence a ∗ is not parallel to nr∗ . Therefore the reference normals and amplitudes are not generally coplanar. To obtain the standard orthorhombic indices, one has to express reference vectors and covectors in the respective (dual of one another) bases 1

1

2

2

c1 = i + j , c2 = j − i , c3 = k = c 3 , c 1 = (i + j ), c 2 = (j − i ), (9.19) the cr being now parallel to the reference orthorhombic axes. As detailed in §9.2.4.2, these are examples of lattice correspondences. For instance, the orthorhombic indices of the twin interfaces in the (1, 2)- and (1, 3)-twins can be shown to be (1¯ 10), (¯ 1¯ 10) and (1¯ 1¯ 1), (s1 +1 s1 −1 1), respectively. 9.2.4 Monoclinic variant structures 9.2.4.1 Monoclinic ‘cubic edges’ wells (§5.1.4.1) In this structure indices are given in the standard monoclinic basis, and the following notation is used:   a

0

b

C1 = U12 =  0

d

0 ,

b

0

c

a=α2 +β 2 , b=β(α+γ), c=β 2 +γ 2 , d=δ 2 .

(9.20)

274

TRANSFORMATION TWINS

Table 9.5 Representatives of equivalence classes of compound twins in the monoclinic ‘cubic edges’ variant structure Twins

(1, 2)

n∗

(0, 0, 1) ∼ (001)

(1, 3)

a∗

−2b c (1, 0, 0)

∼ [¯ 100]

n∗ r

− 2b c (1, 0, 0)

∼ (¯ 100)

a∗ r s

(0, 0, 1) ∼ [001] 4b 2 ca−b2

2

1 √

(1, 0, −1) ∼ (10¯ 1)

2 2(c−a) (1, 0, 1) ∼ [101] c+a+2b

√ √

2(c−a) (1, 0, 1) c+a−2b

∼ (101)

1 (1, 0, −1) ∼ [10¯ √ 1] 2 (c−a)2 ca−b2

Table 9.6 Representatives of equivalence classes of Type 1 and 2 rank-1 connections in the monoclinic ‘cubic edges’ variant structure n∗ a∗ n∗ r a∗ r s2

(1, 11)

(1, 7)

Twins 1 √

√

1 √

(0, 1, −1) ∼ (01¯ 1)

2 2(b2 +a(d−c)) (r2 , 1, 1) ∼ [r2 11] b2 −a(d+c) √

2(c−d) (s2 , 1, 1) ∼ (s2 11) c +d 1 √ (0, 1, −1) ∼ [01¯ 1] 2 3db2 −b2 c+ad2 −2acd+c2 a d(ac−b2 )

√

(1, −1, 0) ∼ (1¯ 10)

2 2(b2 +c(d−a)) (1, 1, r3 ) ∼ [11r3 ] c(a+d)−b2 √

2(d−a) (1, 1, s3 ) ∼ (11s3 ) a +d 1 √ (1, −1, 0) ∼ [1¯ 10] 2 3db2 −ab2 +cd2 −2acd+a2 c d(ac−b2 )

The 132 pairs are divided in 7 equivalence classes. The ﬁrst two, represented by the pairs (1, 2) and (1, 3), consist of 12 compound twins each, representative data being given in Table 9.5. Two more classes, of 24 elements each, are made of conventional, noncompound twins, and are represented by the pairs (1, 7) and (1, 11) (equivalent to (1, 8) and (1, 12), respectively, through Rjπ ); representative data are given in Table 9.6, with r2 =

−2bd , s2 = 2b , r3 = 2 −2bd , s3 = −2b . c−d d−a b2 +a(d−c) b +c(d−a)

(9.21)

Pairs in the last 3 classes give nonconventional twins, if they satisfy (9.3): the cosets generating wells #4–#6, #9 and #10 do not contain any 180degree rotation. One class, of 12 elements, is represented by (1, 4) and is invariant under exchange of the order in the pairs. The second and third classes, of 24 elements each, are represented by (1, 5) and (1, 9) (equivalent to (1, 6) and (1, 10), respectively, by means of Rjπ ). Also, (1, 5) is 2π/3

equivalent to (9, 1) by means of Ri +j +k , hence by exchanging the order in the pairs of the second class we obtain the ones in the third class, and vice versa. In so related pairs, one satisﬁes (9.3) if and only if so does the other, and the elements of the twinning shears have equal normals and opposite amplitudes, or vice versa (see Remark 9.3). Therefore, below, we will only analyze the (1, 5)-twins. π/2 For the variant pair (1, 4) we can choose the twin data as (U1 , Rj ),

9.2 RK-1 CONNECTIONS IN A CUBIC WT-NBHD

275

π/2 Rj

π/2 Rj

with ∈ Mj ,4 – see Table 5.6. Since the axis j of is an eigenvector of U1 for any admissible values of its stretch components, equation (9.3) in this case is identically satisﬁed. Therefore the (1, 4)-connections do exist and give nonconventional generic twins. These were recently observed in LaNbO4 (Jian and James (1997)), and have also been predicted by Simha (1997), with explicit values referred to zirconia (ZrO2 ); indeed, energy wells and connections in this variant structure arise in the model for the tetragonal-to-monoclinic transition analyzed by Simha (1994) – see also Simha and Truskinovsky (1996) and Section 5 in Zanzotto (1996b). We analyze the (1, 4)-twins by means of Proposition 9.4. Under the (nonrestrictive, see below) assumption b $= 0 and the deﬁnitions ξ = (c − a)2 + 4b2 , A = a − c + ξ > 0, x = 2b + A, y = 2b − A, (9.22) 3π/2

the ordered eigenvalues of the tensor Rj the corresponding unit eigenvectors are v1 =

√

1 A2 +4b2

π/2

− C1 are −ξ, 0, ξ, and

1

(2b, 0, −A). (9.23)

C1 Rj

(A, 0, 2b), v2 = (0, 1, 0), v3 =

√

A2 +4b2

One checks easily that (9.12) holds for both e ∗ and f ∗ , hence a choice of reference amplitudes and normals is given by (9.10): n∗ = √

1 2(x2 +y 2 )

(x, 0, y) = ar∗ ,

1 a∗ n∗ = √ 2 2 (y, 0, −x) = r , λ 2(x +y ) µ

(9.24)

(a − c)xy + b(y 2 − x2 ) (a − c)xy + b(y 2 − x2 ) , µ = −2 . (9.25) cx2 − 2bxy + ay 2 ax2 + 2bxy + cy 2 Away from the subspaces given by b = 0 and by c = a (equivalently, β = 0 and γ = α), on which the symmetry increases to orthorhombic ‘cubic edges’ and ‘mixed axes’, respectively, the map (c, a) → A is surjective and smooth, hence the composition planes of the reciprocal (1, 4)-twins are neither crystallographic nor material: by (9.24), their reference indices vary with two independent combinations of the stretch components. If one suitably substitutes in U1 the values of the stretches for LaNbO4 , one obtains the twinning elements reported by Jian and James (1997), up to the usual indeterminateness. Also the twinning operations agree with the ones reported: they have both axis j 5 and angles π/2 ± 5.4◦ , respectively, and are thus very diﬀerent from the conventional ones. They give the reciprocal λ = −2

5

This is v2 , necessarily. Indeed, since (9.12) holds for both e∗ and f∗ , in (9.13) b = 0 = s, hence v2 is an eigenvector of both C1 and C2 for the same eigenvalue c. Also U1 √ and U2 have v2 as eigenvector, corresponding to the eigenvalue c. Incidentally, since U2 = Ht U1 H, with H (in this case Ri+j+k ) a rotation of period other than two (hence with 1 as its only real eigenvalue), necessarily v2 is along the axis of H (apply the last equality to v2 ). We now conclude the proof: since v2 is orthogonal to both n∗ and n∗r (in fact parallel to a ∧n or ar ∧ nr ), it is along the axis of both R and Rr by (8.17)2 and its analogue for the conjugate solution.

276

TRANSFORMATION TWINS

orientation relationships of the individual in well #4 with respect to the one in well #1. 4π/3

4π/3

For the pair (1, 5) we use (U1 , Ri +j +k ) as data, with Ri +j +k in the coset Mj ,5 . In (9.3) we have v = (1, 1, 1), v ∧ Uv · U 2 v = (γ − α)(β 2 + δγ + αδ − γα − δ 2 ), (9.26) hence that equation is equivalent to either of the following: γ=α

or β 2 + δγ + αδ − γα − δ 2 = 0.

(9.27)

Condition γ = α increases the symmetry of U1 to orthorhombic, hence is not acceptable. Therefore the (1, 5)-twins exist only when (9.27)2 holds, and are neither conventional nor generic. Indeed, an indecomposable quadratic equation such as (9.27)2 deﬁnes a submanifold of I(L(U1 ea0 )) that is not contained in the ﬁxed set of any lattice group larger than L(U1 ea0 ): any such ﬁxed set is deﬁned by linear equations, and so is the corresponding set of representative matrices of the symmetry-breaking stretches. The mixed product in (9.3) for a given pair is unaﬀected when passing to an equivalent pair, hence the same condition (9.27)2 characterizes the existence of nongeneric, nonconventional twins for all pairs in the same class of (1, 5), in particular for (1, 6); it also holds for (1, 9) and its equivalent pairs, among which is (1, 10): indeed it is unaﬀected by exchanging the order in a pair (Remark 8.4 and (9.4)), and (9, 1) is equivalent to (1, 5). To summarize, all the nongeneric twins in this variant structure exist if and only if condition (9.27)2 holds for the components of U1 . A phase transition that reduces the symmetry of a cubic lattice to this monoclinic set of variants and satisﬁes (9.27)2 can itself be called nongeneric; in this case the number of twins that may form is quite larger than for a generic transition. A visual comparison of generic and nongeneric twins is given in Fig. 9.1. The aforementioned increase in the number of twins may happen, for instance, in CuZnAl alloys, which have in their phase diagram a monoclinic martensite belonging to this variant structure. The situation is analyzed by Soligo et al. (1999), who propose a way to ﬁnd concentrations at which the alloy may show the extra twins. The experimental check is under way. Since for γ $= α the axis v in (9.26) is not orthogonal to any eigenvector of U1 , condition (9.27)2 and identity (9.5) imply that U1 must have two equal eigenvalues. By the positive-deﬁniteness of U1 the two eigenvectors orthogonal to the monoclinic axis (in this case j ) cannot correspond to equal eigenvalues, and thus one of these eigenvalues must be equal to α, that is, to the eigenvalue of the monoclinic stretch (in this variant structure) which corresponds to the unit vector of the monoclinic axis. To obtain the reference amplitudes and normals we compute the eigen2π/3 4π/3 values and nonunit eigenvectors of Ri +j +k C1 Ri +j +k − C1 to be ξ1 = −ξ, ξ2 = 0, ξ3 = ξ, u1 = (

x+ξ b

,−

b2 +x2 +xξ 2

x

x x2

, 1), u2 = ( , b

b2

, 1), (9.28)

9.2 RK-1 CONNECTIONS IN A CUBIC WT-NBHD U1

U12

U2

U12 U3

U11

U2 U3

U10

U5 U8

U1

U11

U4

U10

U9

277

U4

U5

U9 U6

U8

U6 U7

U7

(a)

(b)

Figure 9.1 The generic ((a)) and the nongeneric ((b)) twins in the monoclinic ‘cubic axes’ variant structure; the variants are indicated by dots and the rank-1 connections by lines. The full set of possible twins is the union of (a) and (b)

u3 = (

x−ξ b

,−

b2 +x2 −xξ x2

, 1),

ξ=

4

b2 + b 2 +x2 , x

x = d − c.

(9.29)

By a symbolic manipulator (we used Mathematica ) one can obtain the normalized eigenvectors v1 , v2 , v3 , and check that equality (9.12)1 holds for both e ∗ and f ∗ . Therefore, by Proposition 9.4, reference amplitudes and normals can be obtained from (9.8)–(9.10). These formulae can be handled by the symbolic manipulator, but the resulting expressions are fairly cumbersome, hence are not reported. Of course, if one assigns numerical values to the components of U1 , the aforementioned symbolic procedure quickly delivers all the twin data. For instance, for the hypothetical Cu-18.6Zn-14Al (at%) alloy, whose lattice parameters are interpolated by Soligo (1998) from the available data, the proposed stretch components are (for the so-called M 18R cell, see Soligo et al. (1999), Hane (1999), James and Hane (2000)) α = 0.9274,

β = 0.0722,

γ = 1.0006,

δ = 0.8837,

(9.30)

and satisfy condition (9.27)2 up to 10−4 . The amplitudes and normals are n = (0.4020, 0.5622, −0.8382), nr = (0.2704, 0.1114, 0.1414),

a = (0.2186, 0.0750, 0.1551), (9.31) ar = (0.2311, 0.4390, −0.7877),

in agreement with numerical calculations of Soligo (1998) based on the formulae of Ball and James (1987). Both twinning operations have axis a ∧ n (see footnote 5), and rotation angles of ±8.5◦ within the expected error of a few percent; as for the (1, 4)-twins, they diﬀer much from the conventional ones. By (9.28)–(9.29) the components of the reference amplitudes and normals

278

TRANSFORMATION TWINS

Table 9.7 Representatives of equivalence classes of conventional twins in the monoclinic ‘face diagonals’ variant structure. In the second column r, s denote for short the corresponding components of a∗ Twins

(1, 2)

(1, 3)

Type

compound

Type 1 and 2

n∗

1 √

2

(1, 1, 0) ∼ (110)

(1, 0, 0) ∼ (100)

n∗ r

√ − 2 2c (0, 0, 1) ∼ [00¯ 1] d √ −2 2c (0, 0, 1) ∼ (00¯ 1) a +b

a∗ r

1 (1, 1, 0) ∼ [110] √ 2

(1, 0, 0) ∼ [100]

8c 2 (a+b)d−2c2

4(6ae2 −a2 d+5b(bd−2e2 )) (a−b)(d(a+b)−2e2 )

a

s

∗

2

(0,

2(e2 −bd) ad−e2

(0,

,

2e(b−a) ad−e2

−2b − 2e a , a )

∼ (0

) ∼ [0 r s]

−2b −2e a a )

Table 9.8 More representatives of equivalence classes of Type-1 and Type-2 twins in the monoclinic ‘face diagonals’ variant structure (1, 7)

(1, 5)

Twins n∗

1 √

a∗

k1 (r1 , 1, 1) ∼ [r1 11]

n∗ r

2

√

2(d−a) 2(e−b) ( d−a , 1, 1) a+d−2e

a∗ r

1 √

(0, 1, −1) ∼ (01¯ 1)

2

k2 (r2 , 1, −1) ∼ [r2 1¯ 1]

e−b ∼ ( d−a 11)

1 (0, 1, −1) ∼ [01¯ √ 1] 2

s2

(0, 1, 1) ∼ (011)

√

2(d−a) 2(e+b) ( a−d , 1, −1) a+d+2e

∼(

2(e+b) a−d

1¯ 1)

1 (0, 1, 1) ∼ [011] √ 2

s1

s2

(a−b)(d(a+b)−2e2 )

(a−b)(d(a+b)−2e2 )

all depend on the two independent parameters b and x in U1 ; hence the twin interfaces are in general neither crystallographic nor material. 9.2.4.2 Monoclinic ‘face-diagonals’ wells (§5.1.4.2) In this variant structure the indices of reference vectors are given with respect to the cubic reference basis (i , j , k ), and are not the standard monoclinic ones. To obtain the latter, one has to express reference vectors and covectors in the (dual of one another) bases given in (9.19), c2 being now parallel to the reference monoclinic axis. These expressions are usually described in the metallurgical literature through lattice correspondences (see §7.6.3); in this case, using the notations B19 and B2 of Strukturberichte (1915-1940) for the monoclinic and the cubic structures, referred to the respective bases (c1 , c2 , c3 ) and (i , j , k ), (9.19) can be written as follows in terms of indices of directions and planes: [100]B19 → [110]B2 , [010]B19 → [¯ 110]B2 , [001]B19 → [001]B2 ,

(9.32)

1 1 ¯ B2 , (001)B19 → (001)B2 . (9.33) (100)B19 → (110)B2 , (010)B19 → (110) 2

2

9.2 RK-1 CONNECTIONS IN A CUBIC WT-NBHD

279

For instance, the normals n ∗ in Table 9.7, whose indices in the basis (i , j , k ) are (110) and (100), have standard monoclinic indices (100) and (1¯10) in the basis (c 1 , c 2 , c 3 ), the ones in Table 9.8 have monoclinic indices (11¯1) and (111), and so forth. Equivalent (with monoclinic axis j +k ) correspondence relations in this variant structure are given by Hane and Shield (1999b). The 132 pairs in this structure are divided into 6 equivalence classes. The ﬁrst, of 12 elements, is represented by (1, 2) and consists of compound twins; representative data are given in Table 9.7, with a = α2 + β 2 + ;2 , b = 2αβ + ;2 , e = (α + β + δ);, d = δ 2 + 2;2 , (9.34) α, β, ;, δ being the components of U1 in Table 5.7. The remaining 5 are made of 24 elements each, and represented by (1, 3), (1, 5), (1, 7), (1, 6), and (1, 8), to which (1, 4), (1, 9), (1, 11), (1, 10), and (1, 12), respectively, are equivalent through Ri −j . Also, (1, 6) is equivalent to (8, 1) by means of π/2 Ri . Of these 5 classes, the ﬁrst 3 are made of Type-1 and Type-2 twins, whose representative data are given in Tables 9.7 and 9.8, with √

k1 =

2(b2 −a2 +ad−e2 ) , a2 −(b+e)2 +a(d+2e)

r1 =

2(e(a+e)−b(d+e)) , b2 −a2 +ad−e2

s1 =a −2a d+a(d −b −2be+3e )+(b−e)(e(d−2e)+b(3d+2e)). √ 2(b2 −a2 +ad−e2 ) 2(e(a−e)+b(d−e)) k2 = 2 , , r2 = a −(b−e)2 +a(d−2e) a2 −b2 −ad+e2 3

2

2

2

2

(9.35) (9.36) (9.37)

(9.38) Also here the reference amplitudes and normals are not coplanar, in general. If in the tables one substitutes the values of the stretch components for the Ti-49.75Ni(at%) alloy, one obtains the twin data reported by Hane and Shield (1999b); there one also ﬁnds information about other microstructures that are not detailed here. For the last two classes we can proceed as in §9.2.4.1. The cosets of P (U1 ea0 ) = Mi −j describing wells #6 and #8 do not contain any 180degree rotation, so the nonconventional connections that possibly exist between each one of these wells and well #1 rest on checking equation (9.3) for it. On the other hand, by changing the order in the pairs of one class we obtain the pairs in the other; consequently, also in this variant structure all the nongeneric twins either exist together or do not exist at all, as it results from the analysis of, say, the (1, 6) connections. In this case we can choose e = i in (9.3), and make the compatibility condition explicit: s2 =a3 −2a2 d+a(d2 −b2 +2be+3e2 )−(b−e)(e(d+2e)+b(2e−3d)).

e · Ue ∧ U 2 e = ;(β 2 − αβ + βδ − ;2 ) = 0.

(9.39)

Once the case ; = 0 is eliminated because it increases the symmetry to orthorhombic, we conclude that the nonconventional twins are all nongeneric and all exist if and only if the condition β 2 − αβ + βδ − ;2 = 0

(9.40)

holds for the components of the stretch U1 . This is the equation of a submanifold of I(Mi −j ) which is not contained in the ﬁxed set of any lattice

280

TRANSFORMATION TWINS

group larger than Mi −j . Also in this variant structure there are many more twins when (9.40) holds than otherwise. Pitteri and Zanzotto (1998b) analyze the case of Ti-49.75Ni(at%), which has a martensitic phase of this monoclinic type. They conclude that the components of U1 do not satisfy the extra twin condition (9.40), not even in a certain approximate sense, and therefore the extra twins should not be present in this alloy. To obtain the reference amplitudes and normals we compute the ordered π/2 3π/2 eigenvalues and nonunit eigenvectors of Ri C1 Ri − C1 to be 2 2 −(b +e ) e−b 4 (−ξ, 0, ξ), ξ = 3b2 + e2 +4e2 , u2 = ( , , 1), (9.41) b

u1 = ( u3 = (

b(b+e)

b+e

−b +e +b (e−ξ)−be(e+ξ) b(b−e) −ξ(b −e +bξ) 3

3

2

b(b2 −e(e+2ξ))

2

,

2

2

b(b2 −e(e+2ξ))

−b3 +e3 +b2 (e+ξ)−be(e−ξ) b(b−e)2 +ξ(b2 −e2 −bξ) b(b2 −e(e−2ξ))

,

b(b2 −e(e−2ξ))

, 1),

(9.42)

, 1).

(9.43)

A symbolic manipulator provides the normalized eigenvectors v1 , v2 , v3 , and one checks that equality (9.12)1 holds for both e ∗ and f ∗ . Therefore, by Proposition 9.4, reference amplitudes and normals can be obtained from (9.8)–(9.10); the expressions are not reported because cumbersome. If one assigns numerical values to the components of U1 , the symbolic manipulator quickly delivers all the twin data. For instance, for a hypothetical Ti-46.65Ni (at%) alloy, whose lattice parameters are interpolated by Soligo (1998) from the available data, the stretch components are α = 1.0321,

β = 0.0815,

; = −0.0429,

δ = 0.9729,

(9.44)

and satisfy (9.27)2 up to 10−4 . The amplitudes and normals are n = (0.0604, −0.2522, 0.9730), nr = (0.2494, 0.1827, 0.0847),

a = (0.2494, 0.1978, 0.0358), (9.45) ar = (−0.0651, −0.3476, 0.9417).

As for the nonconventional twins in §9.2.4.1, both twinning operations have axis a ∧n and diﬀer much from the conventional ones. Within the expected error of a few percent, the rotation angles are about ±9.1◦ . By (9.41)–(9.43) the components of the reference amplitudes and normals all depend on the two independent parameters b and e in U1 ; hence the twin interfaces are in general neither crystallographic nor material. 9.2.5 Triclinic variant structure (§5.1.5) There are nine period-two rotations in the cubic group Cijk ; each one of them gives a triclinic well connected by a pair of conventional twins to well #1. Since the triclinic ﬁxed set is the whole space Q> 3 , Proposition 8.5 implies that only such conventional twins are generic in this variant structure. The nonconventional twins in this case do not all exist under the same condition for the transition stretch components, so that there are several types

9.3 RK-1 CONNECTIONS IN A HEXAGONAL WT-NBHD

281

Table 9.9 Representative of compound twins in the orthorhombic variant structure; indices in italics are standard orthorhombics Twins n∗ a∗ n∗ r a∗ r

√ √

1 (1, 2

√

(1, 2) √ 3, 0) ∼ (1 3 0) ∼ (130 )

√ 3(c−a) √ ¯ ¯ 3a+c ( 3, −1, 0) ∼ [ 3 10] ∼ [3 1 0 ]

√ 3(c−a) √ ¯0) ( 3, −1, 0) ∼ ( 3 ¯ 10) ∼ (1 1 a+3c √ √ 1 (1, 3, 0) ∼ [1 3 0] ∼ [110 ] 2 3(a−c)2 4ac

s2

of nongeneric transitions to the triclinic symmetry, each of which admits its own speciﬁc nongeneric twins. We omit the details. 9.3 Rk-1 connections in a hexagonal wt-nbhd Throughout this section components and indices are given with respect to the orthonormal basis used in §5.2. Indications on how to obtain indices with respect to standard cells, through lattice correspondences, are given for each variant structure, and otherwise comments are kept to a minimum. 9.3.1 Orthorhombic variant structure (§5.2.1) The pair (1, 3) is equivalent to (1, 2) through Riπ , and the coset Oijk ,2 of P (U1 ea0 ) = Oijk in the hexagonal holohedry P (ea0 ) = Hk contains two 180-degree rotations. Thus all 6 orthorhombic pairs give compound and generic twins, a representative of which is detailed in Table 9.9, for a = α2 , c = γ 2 , d = δ 2 – see U1 in Table 5.10. Standard orthorhombic indices are taken with respect to the (mutually dual) bases c1 = i = c 1 , c2 =

√

3j ,

c3 = k = c 3 , c 2 =

1 √ j. 3

(9.46)

A twinning mode with these characteristics is reported for α-U (Hall (1954), Barrett and Massalsky (1966)), and is analyzed in a theory for complex lattice structures by Ericksen (2001a). 9.3.2 Monoclinic variant structures 9.3.2.1 Monoclinic ‘basal diagonals’ wells (§5.2.2.1) Here we can choose the standard monoclinic bases to be √

c1 = −

3j ,

1

c2 = i = c 2 , c 3 = k = c 3 , c 1 = − √ j

(9.47)

3

(remember that the net in the i , j plane is made of centered rectangles).

282

TRANSFORMATION TWINS

Table 9.10 Representative of compound twins in the monoclinic ‘basal diagonals’ variant structure; indices in italics are standard monoclinics Twins

(1, 2)

n∗

¯ 00 ) (0, 1, 0) ∼ (010) ∼ (1

a∗ n∗ r

−2b d (0, 0, 1) −2b c (0, 0, 1)

¯] ∼ [00¯ 1] ∼ [00 1 ¯) ∼ (00¯ 1) ∼ (00 1

a∗ r

¯ 00 ] (0, 1, 0) ∼ [010] ∼ [1

s2

4b2 cd−b2

Table 9.11 Representatives of classes of Type-1 and Type-2 twins in the monoclinic ‘basal diagonals’ variant structure; indices are standard monoclinics Twins n∗

1 (1, 2

a∗ r s2

(1, 3) ¯ 10 ) 3, 0) ∼ (3

1( 2

√ √ k1 ( 3, −1, r1 ) ∼ [3 , 1 , 3 r1 ]

a∗ n∗ r

√

√

√ 3(c−a) √ ( 3, −1, s1 ) ∼ (1 , 1 , s1 / 3 ) a+3c √ 1 (1, 3, 0) ∼ [1 ¯ 10 ] 2 3(5ab2 −b2 c+a2 d−2acd+c2 d) 4a(cd−b2 ))

√

k2 (1, √

(1, 4) 3, −1, 0) ∼ (110 )

√

¯, 1, r ] 3, r2 ) ∼ [1 2

√ 3(c−a) ¯ 3a+c (1, 3, s2 ) ∼ (3 , 1 , s2 ) √ 1 ( 3, −1, 0) ∼ [130 ] 2 3c2 d−3cb2 −6acd+7ab2 +3a2 d 4a(cd−b2 )

The 30 pairs in this structure are divided into 3 equivalence classes. The ﬁrst, of 6 elements, is represented by (1, 2) and consists of compound twins, whose data are given in Table 9.10 for – see U1 in Table 5.11 a = α2 , b = β(γ + δ), c = β 2 + γ 2 , d = β 2 + δ 2 .

(9.48)

The other two classes, of 12 elements each, are represented by (1, 3) and (1, 4) (to which (1, 5) and (1, 6) are equivalent through Riπ ), and consist of Type-1 and Type-2 twins. Their data are represented in Table 9.11, with √ k1 =

√ √ 3(b2 +d(a−c)) 3(b2 +d(a−c)) 4b =−√3s . , k2 = , r1 = 2 4ab =− 3r2 , s1 = a−c 2 b2 −d(3a+c) 3b2 −d(a+3c) b +d(a−c)

(9.49) 9.3.2.2 Monoclinic ‘basal side-axes’ wells (§5.2.2.2) Here we can choose the standard monoclinic bases to be c1 = i = c 1 , c 2 =

√

3j ,

c3 = k = c 3 , c 2 =

1 √ j. 3

(9.50)

The 30 pairs in this structure are divided into 3 equivalence classes. The ﬁrst, of 6 elements, is represented by (1, 2) and consists of compound twins, whose data are given in Table 9.12 for – see U1 in Table 5.12 a = α2 + β 2 , b = β(α + δ), c = γ 2 , d = β 2 + δ 2 .

(9.51)

9.3 RK-1 CONNECTIONS IN A HEXAGONAL WT-NBHD

283

Table 9.12 Representative of compound twins in the monoclinic ‘basal side-axes’ variant structure; indices in italics are standard monoclinics Twins

(1, 2)

n∗

(1, 0, 0) ∼ (100) ∼ (100 )

a∗ n∗ r

−2b d (0, 0, 1) −2b a (0, 0, 1)

¯] ∼ [00¯ 1] ∼ [00 1 ¯) ∼ (00¯ 1) ∼ (00 1

a∗ r

(1, 0, 0) ∼ [100] ∼ [100 ]

s2

4b2 ad−b2

Table 9.13 Representatives of classes of Type-1 and Type-2 twins in the monoclinic ‘basal side-axes’ variant structure; indices are standard monoclinics Twins n∗ a∗ n∗ r a∗ r s2

1 (1, 2

√

(1, 3) 1( 2

3, 0) ∼ (130 )

√ √ ¯ 3r ] k1 ( 3, −1, r1 ) ∼ [3 1 1 √ √ √ 3(a−c) ¯ (− 3, 1, s1 ) ∼ (1 1 s1 / 3 ) a+3c √ 1 (1, 3, 0) ∼ [110 ] 2 3a2 d−3ab2 −6acd+7cb2 +3c2 d 4c(ad−b2 )

√

√

¯1 r ] 3, r2 ) ∼ [1 2 √ 3(c−a) ¯ 3a+c (1, − 3, s2 ) ∼ (1 3 s2 ) √ 1 ( 3, 1, 0) ∼ [310 ] 2 k2 (−1,

√

(1, 4) 3, 1, 0) ∼ (110 )

3(5cb2 −b2 a+c2 d−2acd+a2 d) 4c(ad−b2 )

The other two classes, of 12 elements each, are represented by (1, 3) and (1, 4) (to which (1, 6) and (1, 5) are equivalent through Rjπ ), and consist of Type-1 and Type-2 twins. Their data are represented in Table 9.13, with √

k1 =

3(b2 +d(c−a)) , k2 = −3b2 +d(3a+3)

√

√ 3(b2 +d(c−a)) −4b =−√3s . , r2 = 2 4bc =− 3r1 , s2 = c−a 1 b2 −d(3a+c) b +d(c−a)

(9.52) 9.3.2.3 Monoclinic ‘optic axis’ wells (§5.2.2.3) Here U1 is given in Table 5.13, and we choose the monoclinic bases 1

c1 = −i , c2 = k = c 2 , c3 = i + 2

√

3

2

1

j , c 1 = −i + √ j , c 3 = 3

2 √ j. 3

(9.53)

These 30 pairs can be divided into 5 equivalence classes, of 6 elements each, represented by (1, 2), (1, 3), (1, 6), (1, 4) and (1, 5). The ﬁrst three consist of compound twins, whose data are given in Table 9.14, with a=α2 +β 2 , b=β(α+γ), c=β 2 +γ 2 , d=δ 2 , √ √ √ √ 3(c−a)−2b 3(c−a)−2b 3(c−a)+2b 3(c−a)+2b √ √ √ √ l1 = , m1 = , l2 = , m2 = . 3a−2 3b+c a+2 3b+c 3a+2 3b+c a−2 3b+3c π/3 The (1, 4) pair is equivalent to (5, 1) by means of Rk , hence

(9.54) (9.55)

these two classes have the same character and satisfy the same condition for rank-1 connectivity, which characterizes the existence of all these nonconventional

284

TRANSFORMATION TWINS

Table 9.14 Representatives of classes of compound twins in the monoclinic ‘optic axis’ variant structure; indices are standard monoclinics Twins

(1, 2)

n∗

¯ 01 ) (1, 0, 0) ∼ (2

a∗

− 2b c (0, 1, 0)

∼ [102 ]

n∗ r

−2b a (0, 1, 0)

∼ (001 )

a∗ r s

(1, 0, 0) ∼ [¯ 100] 4b2 ac−b2

2

1 (1, 2

√

(1, 3)

√

1 (1, − 2

¯ 02 ) 3, 0) ∼ (1

(1, 6) ¯01 ¯) 3, 0) ∼ (1

√ ¯01 ¯] l1 ( 3, −1, 0) ∼ [2

√ ¯ 01 ] l2 ( 3, 1, 0) ∼ [1

√ ¯ 00 ) m1 ( 3, −1, 0) ∼ (1 √ 1 (1, 3, 0) ∼ [001 ] 2

√ ¯ 01 ) m2 ( 3, 1, 0) ∼ (1 √ 1 (1, − 3, 0) ∼ [1 ¯01 ¯] 2

√

3(a2 +c2 )+4 3b(a−c)+4b2 −6ac 4(ac−b2 )

√

3(a2 +c2 )−4 3b(a−c)+4b2 −6ac 4(ac−b2 )

twins. We analyze in detail the (1, 4)-twins: since the coset of well #4 only contains rotations whose axis is k , and this is an eigenvector of the tensors U1 , (9.3) is trivially satisﬁed for all the admissible values of its variables α, β, γ, δ. Therefore all these nonconventional twins are generic. We give the data for the (1, 4)-twins, using again Proposition 9.4. With the deﬁnitions √ √ 3 ξ= (a − c)2 + 4b2 , A = 3(a − c) + 2 3b + 4ξ, 2 √ B = 3(a − c) − 6b, x = A + B, y = A − B, (9.56) π/3

the ordered eigenvalues of the tensor Rk the corresponding unit eigenvectors are v1 =

√

1 A2 +B 2

5π/3

C1 Rk

(−A, B, 0), v2 = (0, 0, 1), v3 =

√

− C1 are −ξ, 0, ξ, and 1

A2 +B 2

(B, A, 0).

(9.57)

One checks easily that (9.12) holds for both e ∗ and f ∗ , hence a choice of reference amplitudes and normals is given by (9.10): 1 1 a∗ n∗ n ∗ = √ 2 2 (−y, x, 0) = ar∗ , = √ 2 2 (x, y, 0) = r , (9.58) 2(x +y ) λ 2(x +y ) µ (c − a)xy + b(x2 − y 2 ) (c − a)xy + b(x2 − y 2 ) , µ = −2 . (9.59) ax2 + 2bxy + cy 2 ax2 − 2bxy + cy 2 Since e ∗ and f ∗ depend on the two independent parameters a − c and b, the twin composition planes are neither crystallographic nor material. According to footnote 5, k is the axis of the twinning operations R and Rr , which diﬀer from the conventional ones. λ = −2

9.3.3 Triclinic variant structure (§5.2.3) Each one of the seven period-two rotations in the hexagonal group Hk produces variants connected by two conventional twins to well #1; by Proposition 8.5, only these conventional twins are generic. Also in this triclinic structure the nonconventional twins do not all exist under the same condition for the transition stretch components. We omit the details.

9.4 THE MALLARD LAW

285

9.4 The Mallard law: material interfaces and twin genericity The ‘Mallard law’ is a conjecture, or rather, a rule of thumb, used to establish the transformation twins that arise generically in a symmetry-breaking transition (see for instance Bassett (1981), p. 226). According to this conjecture, the generic twins that are possible for the low-symmetry structure are all conventional, and the composition planes of the Type-1 twins coincide with symmetry planes that are lost in the lowering of symmetry; it appears understood that also the reciprocal Type-2 twins would be active. This version of the law seems to originate from the geometric treatment of the so-called twinning by pseudo-merohedry proposed in the 1800s by Mallard, a modern statement of which can be the following: ‘empirically, all twins of this type can be accurately described in terms of a twin plane which is almost a symmetry plane of the lattice, or in terms of a twin axis parallel to a lattice row which is almost a symmetry axis of the crystal structure’ (Friedel (1926), Cahn (1954) p. 371). The relative smallness of the stretches usually observed in symmetry-breaking transitions may have led one to consider the symmetry elements that are lost when the symmetry of a lattice is lowered, as its approximate symmetry elements in Mallard’s sense, and thus to interpret (at least some of) them as twin elements. Our analysis shows that all the lost symmetry planes, which are necessarily lattice planes, are interfaces of generic, conventional Type-1 twins. However, not all generic twins are conventional: for the monoclinic symmetries nonconventional but generic transformation twins do exist (§§9.2.4.1 and 9.3.2.3). In these, the twin axis can indeed be chosen to be a lost symmetry axis, as required by Cahn’s version of the law. Nevetheless, this is not a general property of the twinning operations; according to Remark 9.1, not all the lost symmetry axes act as twin axes in the deformed lattice. This suggests that the analysis in §§9.2 and 9.3 cannot be condensed in a statement reminiscent of the Mallard law reported above. Still, as was mentioned, that statement correctly summarizes most observations of transformation twins in symmetry-breaking transitions: for mechanical twinning, the experimental reports give clear evidence only about Type-1 and Type-2 twins, with the noticeable exception of a nonconventional twin found by Jian and James (1997). It is thus interesting to understand why this happens, phenomenologically, and to see if the Mallard law may be retrieved in some sense. Two tentative observations seem worth making. First, since the generic twins in a variant structure of given symmetry exist regardless of the particular stretch value compatible with that symmetry, they give rise to structures whose joints are stress-free in the sense of Ericksen (1983) and James (1984b). However, the generic twins do not always have a material composition plane: this is the case of the interface for any of the generic Type-2 twins (not compound), besides any of the nonconventional twins found above. In all these cases the components of the interface normal with respect to the reference lattice vectors depend on

286

TRANSFORMATION TWINS

the stretch components. If the stretch is perturbed, so is the interface in the reference conﬁguration, which then must sweep across material points in the crystal, with a transfer of matter from one individual to the other, to keep the twin stress-free. This possibility was considered unlikely by Ericksen (1983), who did not allow for it in the stress-free joints; notice however that the composition planes in twins, unlike those in glued joints, in many cases can be observed to move parallel to themselves in the material. So, there are, on the one hand, good physical reasons suggesting that twins with a material interface should be more likely to occur than those with a nonmaterial interface. And indeed, Type-1 twins, whose interface is always rational and material, are quite more commonly observed than Type-2 twins, whose interface is generally neither rational nor material. On the other hand, it is certainly too restrictive to impose that the contact plane be material in all twins, because Type-2 twins do occur, in spite of the fact that the interfacial energy of their noncrystallographic and nonmaterial composition planes must be comparatively high. Consequently, the Mallard law holds if in it we intend that (at least) one of the twins in a reciprocal pair be required to have a material composition plane. However, such an assumption is rather ad hoc and not very satisfactory; a better grasp of the role of interfacial energies in crystals may help our understanding of the phenomenon of twin formation. The second observation about the law regards the notion of twin genericity we considered so far. This notion seems rather natural for the investigation of twins in a given variant structure, because it picks out the twins that exist regardless of the stretch value, a property that certainly makes them the most important for that structure. Any time symmetry is broken in a given way the generic twins can be present, and they are stable against any symmetry-preserving additional stretch. However, the actual process of twin formation at the onset of a symmetrybreaking transition may well entail small and yet signiﬁcant stretch disturbances in which the symmetry of the new phase variants is not necessarily preserved. This would make it more likely for the material to form only those twins whose ‘seeds’ remain stress-free during any such dynamic process. In this sense, genericity of a rank-1 connection between two variants could be intended to mean existence of the connection regardless of any stretch disturbance, no matter whether symmetry-preserving or not. Proposition 8.5 then implies that, in this sense, the only generic twins are the conventional ones. Thus, were this stronger notion of twin genericity to be used, the Mallard law would again be true. In order to clearly justify this point of view further investigation is needed, most likely regarding the actual mechanisms and the dynamics of twin formation. It should be also remarked that the common occurrence of various types of defects in the crystalline lattice and their stability properties may play a role in selecting the observed twins in each material.

CHAPTER 10

Microstructures The assumed global invariance of the constitutive functions of simple lattices produces a certain lack of convexity and the possibility of a wide range of equilibrium conﬁgurations. This provides a good model of physics because crystals often prefer complicated morphologies, perhaps at a very ﬁne scale – called microstructures1 – in which various individuals or conﬁgurations are present. Microstructures frequently appear in metals and alloys undergoing martensitic phase transitions, the equilibrium arrangements being aﬀected by external loads or changes in temperature. These phenomena are at the basis of such material properties as shape memory, magnetostriction and ferroelectricity, and thus the development of new tecnologies based on these properties rests upon understanding this ﬁne-scale material behavior. In this chapter we limit ourselves to sketching some examples, keeping the technical aspects to a minimum and giving references for the interested reader. In particular, we neglect energies associated with the various interfaces mentioned below; this produces a theory without an intrinsic lengthscale: the ﬁne-scale morphologies should in principle become inﬁnitely ﬁne to minimize energy. Since this does not happen in nature, some lengthscale should be added, perhaps by penalizing interfaces. This is the object of active research, outlines of which are given, for instance, by M¨ uller (1998) and DeSimone et al. (2000). 10.1 Piecewise homogeneous equilibria The ﬁrst problem we consider is the one of minimizing the energy functional in the absence of any prescribed external loads or displacements of the boundary. According to the terminology of §2.5.2, such minimizers are called natural states. Any natural state must minimize the free energy density φ˜ almost everywhere in R. The simplest natural states are the ones with a constant ˜ For convenience we choose the deformation gradient F that minimizes φ. minimum value to be zero, hence the function φ˜ is nonnegative. 1

The word microstructure has a diﬀerent meaning in other contexts (see for instance Capriz (1989)): it denotes the presence of internal degrees of freedom in the structure of the material itself, which in the end inﬂuences the overall elastic behavior of it. Additional parameters describing the internal structure of crystals naturally arise in multilattices (chapter 11); see in particular §11.7.1. 287

288

MICROSTRUCTURES

Remember that in classical linear elasticity there is one natural state, up to trivial rigid-body rotations, that is, the reference conﬁguration. In the theory of nonlinear elasticity for crystals presented here there are other natural states that we can construct ‘by hand’, using the global invariance of the free energy. For instance, for any minimizer F1 of φ˜ we can always select another, say F2 = F1 + a ⊗ n ∗ , which is rank-1 connected to F1 , and construct a pairwise homogeneous conﬁguration with a planar interface p orthogonal to n ∗ , as in §8.1.1. From this point of view, in chapter 9 we have classiﬁed all the possible pairwise homogeneous conﬁgurations such that F1 and F2 are symmetry-related variants belonging to one of the wt-nbhds, that is, pairwise homogeneous transformation twins. Instead of dividing the reference conﬁguration R into two parts as above, we can think of dividing it into a ﬁnite number of layers by means of a ﬁnite number of planes parallel to p, assigning to each layer deformation gradient F1 or F2 , alternatively. This conﬁguration, called a simple laminate, is an example of a piecewise homogeneous natural state, and it is quite clear that we have a variety of these. A relevant characteristic of a simple laminate is the volume fraction, say υ, occupied by points whose deformation gradient is F2 , say. Simple laminates are also a basic ingredient in the construction of more general equilibria, for which υ plays an important role – see §10.1.2. The second question we are going to analyze concerns the loads and the displacements of the boundary compatible with equilibrium. The simplest boundary-value problems for our crystalline body with reference conﬁguration R and free energy functional (2.75) are the Dirichlet problem, in which the displacement of the boundary ∂R of R is prescribed, and the Neumann problem, in which an external body force in R and a dead load on ∂R are assigned. Unless the contrary is stated, we assume ∂R to be piecewise smooth. For deﬁniteness (and actually without loss of generality, see Bhattacharya et al. (1994)) we select the space S of deformations χ to be the space W 1,∞ (R; R3 ) of Lipschitz functions, that is, functions from R to R3 with essentially bounded gradient; and choose suitable functions χ0 and f in S, and g in its analog W 1,∞ (∂R; R3 ). Then the Dirichlet problem consists in minimizing the functional (2.75) over the set of admissible functions A(χ0 ) := {χ ∈ S : det Dχ $= 0 and χ = χ0 on ∂R},

(10.1)

while in the dead loading problem we must minimize the functional

˜ Φ[χ; θ] = φ˜ Dχ(x ), θ) dx − χ · f dx − χ · g dS (10.2) R

R

∂R

over the set of functions

B := χ ∈ S : det Dχ $= 0 and χdx = 0 .

(10.3)

R

The ﬁeld f represents an external body force per unit volume, and g is a dead traction. The integral condition in (10.3) normalizes the arbitrary

10.2 GENERALIZED SOLUTIONS

289

superimposed translation up to which, at least, the solutions of this problem are determined. In the crystal case the Neumann problem above need not be well posed. For instance, Fonseca (1987) shows that, in a large class of functional set˜ is bounded below if and only if f = 0 = g . Roughly, this is a tings, Φ consequence of the fact that in the global invariance of the free energy density there are arbitrarily large lattice-invariant shears, along which the loads can do an arbitrarily large amount of work, of either sign. We will not elaborate on this further, and will concentrate on the Dirichlet problem, for which a number of positive results have been obtained. 10.2 Generalized solutions In the absence of assigned boundary displacements the large invariance of the free energy density prevents us from using a naive format of the direct method in the calculus of variations to ﬁnd stable equilibria. Indeed, as we have seen in §6.4.3, the Legendre–Hadamard condition (6.46) cannot hold everywhere, and the functional (2.75) neither grows at inﬁnity nor is quasiconvex or lower semicontinuous (see Ball (1977a)). This remains true when the integrand has the restricted invariance associated with a wt-nbhd, with relative minimizers that are either the variants of a symmetry-breaking stretch or these together with the identity. In the ﬁrst case one looks for equilibria made of martensitic twins, in the second for these possibly coexisting with austenite. In both cases the free energy has more than one well and is not rank-1 convex, hence not quasiconvex either. So, except for special boundary data, the Dirichlet problem does not have a solution, but it may still admit minimizing sequences exhibiting ﬁne-scale oscillations between energy minimizers, which are needed to allow the energy to converge to its lowest possible value. Such sequences are a mathematical model for the physical microstructures mentioned early in this chapter. One important consequence of the above lack of convexity is the following: for general functions of the deformation gradient, and suitable kinds of convergence, the presence of ﬁner and ﬁner oscillations along a minimizing sequence does not allow the limiting value of the function to coincide with its value at the limit (or eﬀective) deformation gradient. The energy itself is an example: assume the minimizing sequence to be made – except on a thinner and thinner transition layer at the boundary, which is necessary for the boundary conditions to be met – of a ﬁner and ﬁner laminate involving the deformation gradients F1 and F2 with respective volume fractions 1−υ and υ. The deformation gradient tends, in the sense speciﬁed by (10.4) below, to the average, or eﬀective, gradient F = υF2 +(1−υ)F1 , at which the energy typically takes a positive value. On the other hand, by deﬁnition of minimizing sequence, the limit of the energy is zero. In spite of this general fact, there are special functions of the deformation gradient for which the limiting value is indeed the value at the limit (see §10.2.1).

290

MICROSTRUCTURES

Along a minimizing sequence the deformation gradient does not converge pointwise, but certain kinds of averages of it and of other physically relevant quantities related to the free energy density do often converge. This behavior is described in a classical problem in the calculus of variations proposed by Young (1969). He used as a paradigmatic example the minimization of travel time done by a sailor who has to sail downstream on a straight river, but against the wind that blows parallel to the river and upstream. Since the water speed is maximum in the middle and minimum along the banks, he takes best advantage of the river ﬂow by staying on the central streamline. But in this way he does not take advantage of the wind; to do so, he has to tack at the most eﬃcient angle, thus departing from the center of the river and reducing the drag by the ﬂow. If the cost of turning is negligible, in the best compromise he has to tack about the central streamlineline at the best angle, ideally inﬁnitely often, each time travelling an inﬁnitesimal distance. To describe similar situations, in which minimization forces oscillations, he introduced a concept of generalized solution, which in our context consists in a (possibly oscillatory) minimizing sequence of deformation gradient ﬁelds {Dχk } and an associated parametrized measure νx – now called the Young measure – which is a tool for taking the oscillatory properties of the minimizing sequence into account, and for computing the limit of certain averages. The Young measure νx gives the local probability distribution, near x , of gradients in the limit k → ∞. Generalized solutions of interest in nonlinear elasticity have been studied by means of Young measures (Murat (1978), Tartar (1984), Chipot and Kinderlehrer (1988), Fonseca (1988), Kinderlehrer and Pedregal (1991), Ball and James (1992)) and H-measures (Francfort and Murat (1986), Tartar (1990), Kohn (1991)). We only state here two related theorems; more information can be found in the references above. We also mention the review ˇ article of M¨ uller (1998) and the books of Silhav´ y (1997) and Pedregal (2000) for general analytical aspects, and the review article of Luskin (1996b) for a detailed description of numerical methods in the analysis of crystalline microstructures. All these references include an extensive bibliography. Theorem 10.1 (Existence of Young measures) Assume {Fk } ⊂ L∞ (R; Rs ) to be a sequence of (vector- or) tensor-valued functions on R and that, for some compact set M ⊂ Rs and any open U ⊃ M , the measure meas{x ∈ R : Fk (x ) ∈ / U } tend to zero as k tends to inﬁnity. Then there are: (i) a subsequence, still denoted by {Fk } for simplicity, and (ii) an associated family of probability measures νx on Rs , parametrized by x ∈ R, such that (1) νx is supported on M for almost all x in R. (2) For any continuous real-valued function ψ on Rs , ψ(Fk ) converges to the function x → Rs ψ(A)dνx (A) in the weak∗ topology of L∞ (R); this means that, for any function ϕ ∈ L1 (R),

10.2 GENERALIZED SOLUTIONS

291

ψ(Fk )(x )ϕ(x ) dx →

R

ϕ(x )

Rs

R

ψ(A)dνx (A) dx .

(10.4)

The following version of an average Young measure theorem (Kinderlehrer and Pedregal (1991)) implies that if boundary conditions are spatially homogeneous one can restrict the attention to homogeneous microstructures, that is, ones for which the Young measure is independent of x . Theorem 10.2 Assume R to be open, connected, bounded in R3 with Lipschitz continuous boundary ∂R. Assume the sequence {χk } in W 1,∞ (R; R3 ) to satisfy the boundary condition χk = F0 x on ∂R for some ﬁxed F0 and all k, and to converge weak∗ to χ, that is, {χk } converges uniformly to χ in R and {Dχk } converges weak∗ (the analogue of (10.4)) to Dχ in L∞ (R; R9 ). Let νx be the Young measure, with support on M , associated to the sequence of gradients by the previous theorem. Then there is a se¯ k }, such that χ ¯ k = F0 x on ∂R for any k, which converges weak∗ quence {χ 1,∞ ¯ := x → F0 x in W to χ (R; R3 ) and satisﬁes the following: ¯ is independent of x . (1) the Young measure ν¯ related to the sequence {Dχ} (2) ν¯ is supported on M .

(3) Vol R ψ(A)d¯ ν (A) = dx ψ(A)dνx (A) (10.5) R9

R

R9

for every continuous function ψ on R9 . A Young measure is trivial if it is a Dirac delta; the corresponding limit gradient is almost everywhere constant, and the deformation is essentially homogeneous. 10.2.1 The minors relations Both in theory and in practice we have a very limited control over the complexities of the microstructures that a crystalline body can form in a given environment. Indeed, actions are typically exerted on the boundary of the body, for instance by constraining it to displace itself in a given way (Dirichlet problem). We do not expect arbitrary boundary displacements to be compatible with the equilibrium of the crystal, once its constitutive equations, in particular the structure of its energy wells, are given. We would like to obtain as much information as possible on these admissible boundary displacements from the constitutive equations, and this is helped by the existence of suitable functions of the deformation gradient whose average only depends on the boundary data, and not on the details of any corresponding equilibrium microstructure. More formally, these are continuous functions ψ = ψ(F ) such that, for any suﬃciently smooth functions χ(x ), δχ(x ) on R, the latter with compact support,

ψ D(χ + δχ) dx = ψ(Dχ) dx . (10.6) R

R

292

MICROSTRUCTURES

Any such function ψ is called a null Lagrangian because the Euler-Lagrange equations for ψ, namely ∂(∂ψ/∂F rs )/∂xs = 0, r = 1, 2, 3, are identically satisﬁed, possibly in the sense of distributions, by all functions χ as above. Equivalently, the volume integral on the right-hand side of (10.6) is determined by the restriction of Dχ to the boundary ∂R of R. Based on a result of Ericksen (1962), extended further by Ball et al. (1981) to Lagrangians depending on higher-order gradients, Ball (1977a,b) characterizes the null Lagrangians to be the linear combinations ψ(F ) = Asr (F )rs + Brs (cofF )rs + C det F ,

(10.7)

where the matrix elements of the tensor cofF in an orthonormal basis are the cofactors of the matrix elements of F in the same basis. He also shows that a function χ → ψ(Dχ) is sequentially weak∗ continuous from W 1,∞ to L∞ (as speciﬁed in Theorem 10.2) if and only if it is a null Lagrangian. Consider now the homogeneous boundary condition of Theorem 10.2, and assume the Young measure ν¯ in it to be supported on a ﬁnite number of gradients, F1 , . . . , Fn , with respective volume fractions λ1 , . . . , λn . In this case, for any choice of χ satisfying the boundary conditions, the class of functions δχ in (10.6) includes F0 x − χ, hence the integrals there have the value Vol R ψ(F0 ). Then, by that theorem, by the weak∗ continuity of ψ, and by the hypotheses on the Young measure,

n Vol R ψ(F0 ) = ψ(Dχ) dx = Vol R ı=1 λi ψ(Fi ). (10.8) R

This equality, by the expression (10.7) of ψ and the arbitrariness of the coeﬃcients in it, imply the following minors relations: n n n F0 = ı=1 λi Fi , cof F0 = ı=1 λi cof Fi , det F0 = ı=1 λi det Fi . (10.9) More general related analytical results and references are given, for inˇ stance, by Dacorogna and Murat (1992), Silhav´ y (1997), M¨ uller (1998), Pedregal (2000). The minors relations are only necessary conditions, but require very little information about the microdeformations, and are the basic ingredient for obtaining relations between the microdeformations themselves and the corresponding macroscopic (homogeneous) deformation F0 . They have been used by Ball and James (1992), Bhattacharya (1992), Bhattacharya and Kohn (1997) (for polycrystals), James and Hane (2000), among others, to ﬁnd restrictions on structural parameters for the existence of various microstructures, some of which are mentioned in §10.4.3 below. 10.2.2 The N -well problem Going back to the Dirichlet problem, and recalling that the minimum of φ˜ has been set to zero, we can distinguish three diﬀerent cases. The ﬁrst is when the inﬁmum of the free energy functional over the set of admissible deformations is zero and is achieved, so that the minimizers are stress-free.

10.2 GENERALIZED SOLUTIONS

293

This is the case of the natural states, in which no boundary conditions are imposed and the minimum is achieved, among others, on homogeneous ˜ deformations minimizing the free energy density φ. The second case is when the inﬁmum is still zero but is not achieved. Then there is a minimizing sequence {χk } such that Dχk ∈ M (θ),

˜ , θ) = 0}, M (θ) := {F : φ(F

(10.10)

approximately, except on a subset of R the measure of which, as well as the approximation, tend to zero as k → ∞. For large k, R is partitioned into regions where Dχk is near M (θ), hence the stress is nearly zero, and transition layers, whose overall measure tends to zero, on which Dχk is not near M (θ) but is uniformly bounded. The third case is when the inﬁmum of the free energy functional is positive, hence the body is necessarily stressed, and the inﬁmum is attained or not depending on the free energy density and the boundary conditions. Explicit examples of nonattainment under nonzero stresses are provided by some phase–transforming polymeric systems called nematic elastomers, studied by DeSimone and Dolzmann (2002) and Conti et al. (2002). Besides homogeneous deformations, the natural states include all simple laminates. A nontrivial question is whether or not simple laminates describe, at least locally, any natural state. If deformations are Lipschitz functions, the problem of determining all natural states is equivalent to determining all deformations χ such that Dχ satisﬁes (10.10) almost everywhere in R. If we restrict the energy density φ˜ to one of the wt-nbhds and assume that it has only symmetry-related minima (as is reasonable away from phase transitions), then M (θ) = ∪N i=1 OUi ;

(10.11)

this can also cover the equilibria at the Maxwell temperature of a phase transformation (see §6.5.3) if we take the high-symmetry conﬁguration at that temperature as a reference, and enlarge the list of stretches Ui to include the identity. With this proviso, the problem of solving (10.10)– (10.11) is by now called the N-well problem, and is a reasonable restriction of the problem of ﬁnding all natural states for a free energy φ˜ having the general invariance group G(ea0 ). The importance of the set M (θ) is stressed by the following result of Ball and James (1992), which actually also applies to the second case mentioned above, thus extending the scope of the N well problem: for an energy which is inﬁnite outside the wt-nbhd, the Young measure of any minimizing sequence for the Dirichlet problem is supported on M (θ) - that is, (10.10)–(10.11) hold - if the inﬁmum of the free energy functional in the class of admissible deformations is zero. Consider ﬁrst the 1-well problem. The following proposition extends the result in Remark 6.7 and holds, again, not only for the natural states, but in general for solutions of the Dirichlet problem – see Kinderlehrer (1988) or Ball and James (1992), Theorem 4.3.

294

MICROSTRUCTURES

Proposition 10.3 The only microstructure possible in the 1-well problem is the trivial one: Dχ is essentially constant, say Dχ = F0 ∈ M (θ). The well mentioned may be either pure austenite or one of the variants of martensite. The admissible displacements of the boundary have the form χ(x ) = F0 x ,

F0 ∈ M (θ).

(10.12)

Therefore one must have an energy with a suitable set of wells, not only one, to construct laminates or more complex nonhomogeneous natural states, and in §10.3 we mention a few examples. The simplest nontrivial choice is the one of two wells, say OU1 and OU2 . In this case any homogeneous deformation with gradient in either well - that is, F0 ∈ OU1 ∪ OU2 - is a natural state; it also solves the Dirichlet problem for the corresponding linear boundary condition (10.12). If the two wells are rank-1 connected, that is, F2 − F1 = a ⊗ n ∗ for suitable Fr ∈ OUr , r = 1, 2, any simple laminate with reference normal n ∗ is also a natural state. Furthermore, the Dirichlet problem for linear boundary conditions (10.12) with F0 = υF2 + (1 − υ)F1 , 0 ≤ υ ≤ 1, is solved by a minimizing sequence made of laminates except on a small set of vanishing measure along the sequence. The scalar υ is the limit of the volume fraction of points whose deformation gradient is F2 . One can state this result compactly in terms of the Young measure associated to the minimizing sequence: νx = υδF2 + (1 − υ)δF1 ,

(10.13)

where, for r = 1, 2, δFr is the Dirac mass concentrated at Fr . In this sense F0 is an eﬀective deformation gradient. It is interesting to see to what extent these can be the only possible solutions for the Dirichlet problem with linear boundary conditions. If the two wells are not rank-1 connected and a certain technical condition holds ˇ ak (1993)), then any minimizing sequence converges to (Matos (1992), Sver´ an essentially homogeneous deformation, hence only the trivial microstructure exists; equivalently, the only Young measure supported on the two wells is trivial and independent of x – see also Bhattacharya et al. (1994). If there is one rank-1 connection (in our case this can only happen if the two wells are austenite and one of the variants of martensite) and Dχ is of bounded variation, then the natural state is necessarily a simple laminate (Fonseca (1989), Corollary 5.20). If there are two rank-1 connections (as is necessarily true if the two wells are two variants of martensite), then Ball and James (1992) characterize the linear boundary conditions (10.12) corresponding to which a minimizing sequence exists: take (8.2) into account, and deﬁne D := Lt F0t F0 L, c3 :=

a , c1 := a

L := U1−1 (1 − δc3 ⊗ c1 ),

F1−t n ∗ , F1−t n ∗

δ :=

1 a F1−t n ∗ . 2

(10.14) (10.15)

10.3 EXAMPLES OF MICROSTRUCTURES THAT ARE NOT LAMINATES

295

Then a minimizing sequence exists if and only if D has the representation   D11 0 D13 0 < D11 ≤ 1 + δ 2 ,  0  , 1 0 0 < D33 ≤ 1, , (10.16) 2 D13 0 D33 D11 D33 − D13 =1 in the orthonormal basis generated by c1 , c3 in (10.15). In this case the microstructure is not a simple laminate, in general, but rather a double (or rank-2) laminate: it is made of layers each one of which is itself a laminate. Rank-m laminates are described by Bhattacharya (1992), for instance, and M¨ uller (1998) introduces laminates as limits, in a suitable sense, of laminates of ﬁnite rank (or order). There, as well as in Dolzmann et al. (2000), one also ﬁnds analysis of the two-well problem, and of similar ones, in terms of properties of various types of convex hulls of the set M (θ), a topic we do not discuss here. The example leading to (10.13) may suggest that all microstructures be laminates, at least locally. This is not true, as is shown by an example ˇ ak (1992) in two dimensions, involving complex arrangements, and of Sver´ ˇ ak (1996), who notice that the set E := {x ∈ R : also by M¨ uller and Sver´ ˇ ak (1992) has inﬁnite Dχ(x ) ∈ OU1 } corresponding to the solution of Sver´ perimeter, this being a generalization of surface area to nonsmooth surfaces. If one imposes the additional condition that E has locally ﬁnite perimeter, then the natural state is locally a simple laminate – see Dolzmann and M¨ uller (1995). A generalization to the much harder case of three tetragonal wells in three dimensions is given by Kirchheim (1998). We quickly mention some results concerning the N -well problem when there are no rank-1 connections between the wells. The interested reader should consult Bhattacharya et al. (1994), or M¨ uller (1998) and Dolzmann et al. (2000) for more details and proper references. Consider the N -well problem in two space dimensions: if the wells do not admit rank-1 connections, then the only Young measure supported on the union of the wells ˇ ak (1993)), and thus the only natural states is trivial and constant (Sver´ are homogeneous deformations. The analogue need not hold for higher dimensions: there are nontrivial Young measures limits of gradients, hence nontrivial microstructures, supported on three (hence also more than three) pairwise incompatible wells in three space dimensions; a similar construction works also in dimension greater than three. So, the question remains of the existence of nontrivial Young measures supported on two incompatible wells in space dimensions greater than 2. As we mentioned above, there is none in the 3-dimensional case under certain conditions. 10.3 Examples of microstructures that are not laminates In some shape memory alloys it is rather common to observe structures, to be called crossing twins, which can be described as a twin plate zigzagging across a twin laminate. At any one of the fourfold corners four

296

MICROSTRUCTURES

planar surfaces meet at a line, and separate regions of homogeneous deformation. Such corners impose severe kinematic compatibility conditions, of the same kind as those guaranteeing stress-free joints (Ericksen (1983), James (1984b)); in obvious notation, we must have, for k = 4, Fi+1 − Fi = ai ⊗ ni∗ ,

i ∈ N mod k,

(10.17)

when the normals ni∗ all lie in one plane. Notice that (10.17) implies a1 ⊗ n1∗ + a2 ⊗ n2∗ + · · · + ak ⊗ nk∗ = 0 ;

(10.18)

for k = 3 this equality requires the normals to lie all in one plane, whereas this is not necessarily true for k > 3, and must be imposed as an additional condition. James and Kinderlehrer (1993) analyze crossing twins in the magnetostrictive material Terfenol, and propose a necessary and suﬃcient condition for the corresponding solution of (10.17); this is equivalent to the condition e1 · e2 = 0 in the following suﬃcient condition for the existence of a solution of (10.17) provided by Bhattacharya (1996): Proposition 10.4 For k = 4 (10.17) has a solution if there are orthogonal tensors Q1 , Q2 , Q3 and unit vectors e1 , e2 such that e1 · e2 = 0 and F2 = Q1 F1 Reπ1 , F3 = Q2 F2 Reπ2 , F4 = Q3 F1 Reπ2 .

(10.19)

There one also ﬁnds the amplitudes ai , the normals ni∗ and the orthogonal tensors Qi for a solution, which need not be unique. If exact compatibility is replaced by approximate compatibility in a minimizing sequence of laminated microstructures, then other interfaces are possible, for instance the so-called X-interface described in §10.4.2. Ericksen (1986b) provides other examples, not ﬁtting the hypotheses of the previous proposition, of piecewise homogeneous natural states that are not laminates, but rather cyclic twins, that is, axisymmetric orangelike arrangements of symmetry-related wedges. He is mostly interested in solutions of the twinning equation (8.20), that is, (10.17) for i = 1 and F2 = Q1 F1 H1 , Q1 ∈ O+ , H1 ∈ G+ (ea0 ), in the case of H1 of ﬁnite order greater than two. He characterizes a subclass of the rotation twins of Barrett and Massalsky (1966), namely the one for which H1 is linearly conjugate to Q1 : either a1 or the present normal n1 = F1−t n1∗ must be along the axis of the rotation Q1 , and all the possible solutions can be constructed starting from H1 . Of these, only the ones for which the axis of Q1 is parallel to the shear amplitude a1 can produce a cyclic twin. In the conceptually diﬀerent context of growth twinning, Zanzotto (1990) studies the stability of a fourfold cyclic twin in quartz under changes of the environmental temperature θ and pressure p. The deformations of the twin individuals from a reference state in which the cyclic twin is assumed to exist must again solve the compatibility conditions (10.17) for H1 conjugate to Q1 , hence the shear amplitude a1 must be along the axis of Q1 . The twinning law suggested by experience, on the contrary, requires a1 to be

10.4 HABIT PLANES IN MARTENSITE

297

orthogonal to the axis of Q1 , so a1 must vanish for the cyclic twin to survive under changing environmental conditions. This turns out to be a scalar equation on the environmental parameters p, θ which makes the cyclic twin a geological barothermometer, as anticipated in §8.4.3. 10.4 Habit planes in martensite In a martensitic phase transformation there is a range of temperature about the transition temperature in which the high-symmetry austenite coexists with one or more orientations of martensite, arranged in twin lamellae or more complicated twin morphologies. The general idea is that the material tries to minimize the elastic energy as much as possible, compatibly with avoiding fracture and complying with the boundary conditions. The twin microstructures are ways to accomplish this in the general case of absence of compatibility between austenite and a single variant of martensite. 10.4.1 Geometrically nonlinear theory A geometrically linear theory for the equilibria of martensitic crystals was developed by Eshelby (1961), Khachaturyan (1961), (1983), Khachaturyan and Shatalov (1969), and Roitburd (1969), (1978). This designation refers to the fact that the thermodynamic potential depends quadratically on the linear strain in a neighborhood of each minimum, there being at least two of these. Since the potential has local minima at multiple stress-free stretches, the theory is nevertheless nonlinear. More recently, Ball and James (1987) have proposed a fully nonlinear thermoelastic equilibrium theory for crystalline solids undergoing martensitic phase transformations. This theory, if suitably linearized, produces the previous one – see Kohn (1991) and Ball and James (1992). Also, the nonlinear theory extends and improves the earlier, very successful crystallographic theory of martensite – see Wechsler et al. (1953), Bowles and MacKenzie (1954), Christian (1975), Crocker (1982). Unlike in the crystallographic theory, the fact that results are obtained from energy minimization allows one to determine which changes should be made to take speciﬁc boundary conditions into account. Here we only recall some basic results, and address the reader to Bhattacharya (1993) for a comparison of the nonlinear and the geometrically linear theories, and to James and Hane (2000) for extensive references and crystallographic data for various alloys. If we ﬁx the reference conﬁguration in the austenite, then Proposition 8.1 for C1 = 1 requires one of the eigenvalues of the transformation stretch U2 to be 1. This condition is only observed in certain alloys at speciﬁc temperatures and compositions – see Bywater and Christian (1972). Thus austenite is generally not compatible with a single variant of martensite, and we must try to form an approximate interface between austenite and a twin lam-

298

MICROSTRUCTURES

inate of the martensite. The interface can be characterized by means of the following general result – see Ball and James (1987), James (1988b), James and Kinderlehrer (1989). Assume the following: {χk } is a minimizing sequence for the free energy functional, which converges weak∗ to χ in W 1,∞ (R; R3 ); the associated Young measure (§10.2) satisﬁes the condition (1 representing the austenite) supp νx ⊂ {F1 , F2 , 1},

F1 , F2 ∈ Aut ;

(10.20)

neither F1 nor F2 are rank-1 connected to 1, and both austenite and martensite are nontrivially mixed: there is (at least) a point x ∈ R such that, in any ball Bx ,r of center x and radius r, Dχ equals 1 in a subset whose measure is a fraction of the measure of Bx ,r , bounded away from both zero and one. Under the assumptions above there are vectors a, n ∗ , b, m ∗ , and a scalar υ, 0 < υ < 1, such that F2 − F 1 = a ⊗ n ∗

and F1 = 1 − υa ⊗ n ∗ + b ⊗ m ∗ ;

(10.21)

equivalent conditions are (10.21)1 and υF2 + (1 − υ)F1 − 1 = b ⊗ m ∗ . The latter is particularly signiﬁcant because is the compatibility condition between the stretch of the austenite, 1, and the eﬀective stretch, F := υF2 + (1 − υ)F1 , of the martensitic laminate. The vector m ∗ represents the reference normal to the habit plane between austenite and twinned martensite, and b is called the shape strain vector of the martensite. In a wt-nbhd of the austenite, the deformations F1 , F2 associated with the martensitic twins have the expression R1 U1 , R1 R2 U2 , respectively, for suitable symmetry-breaking stretches U1 , U2 and orthogonal R1 , R2 . In this case equations (10.21) become the following, which are equivalent to the ones of the crystallographic theory of martensite: R2 U2 − U1 = a ˆ ⊗ n ∗, a ˆ = R1t a, R1 υR2 U2 + (1 − υ)U1 = 1 + b ⊗ m ∗ .

(10.22)

Theorem 7 of Ball and James (1987) gives the general solution of (10.22) ¯ for ¯ t U1 R or (10.21). This can also be obtained as follows. Since U2 = R 0 ¯ some R ∈ P (ea ), (10.21)1 is the twinning equation (8.20), while (10.21)2 has again the form (10.21)1 when F2 is replaced by F1 +υa ⊗n ∗ and F1 by 1. In terms of given F1 , a, n ∗ satisfying (10.21)1 , Proposition 8.1 provides the condition for the existence of a habit plane: for C (υ) := C1 + υ(n ∗ ⊗ F1t a + F1t a ⊗ n ∗ ) + υ 2 a2 n ∗ ⊗ n ∗

(10.23)

the eigenvalue problem det(C (υ) − λ1) = 0 must have eigenvalues 0 ≤ λ1 ≤ λ2 = 1 ≤ λ3 . This turns out to restrict the principal stretches, that is, the eigenvalues of the stretch U1 , by suitable inequalities. Furthermore, the expressions of b and m ∗ in terms of the ordered eigenvalues of C (υ) and the corresponding eigenvectors can be obtained from (8.7)–(8.13). We address the reader to Ball and James (1987) for the details, as well as for the fully explicit example of the cubic-to-tetragonal habit planes (as, for

10.4 HABIT PLANES IN MARTENSITE

299

instance, in InTl alloys). A schematic description of this situation is given in Fig 8.1, where, for drawing convenience, the laminate is based on the variants labelled 1 and 3 instead of 1 and 2, respectively, as above. The aforementioned results allow for the analysis of the habit planes in the wt-nbhds; details are given by James and Hane (2000).

10.4.2 Self-accommodation in shape memory alloys Shape memory crystals undergo a martensitic phase transformation along which there is apparently no change of shape, and islands of martensite form in the bulk of austenite. This means that the martensite arranges itself in a microstructure which does not move its original boundary. The phenomenon is known as self-accommodation, and seems to be important for both the shape memory eﬀect and the reversibility of the transformation. Bhattacharya (1992) analyzes the conditions for self-accommodation, which turn out to depend on the crystal symmetry of the austenite. A necessary condition in all cases is that the phase transformation be volume preserving. This is also suﬃcient for cubic austenites, while for all the other holohedries additional equations on the components of the symmetry-breaking stretch must hold. For instance, in a wt-nbhd of monoclinic austenite there are two wells of triclinic martensite, and the conditions for self-accommodation can be read from two-well equations (10.14)–(10.16) for F0 = 1. These results may explain why austenite is cubic in all the shape memory materials that we know, and why for them the volume change at the phase transformation is very small. At the same time, it is reasonable to think that the volume change in some of these transitions is not exactly zero, so that the restrictions above are obeyed with some tolerance. The same presumably holds for the conditions guaranteeing the existence of wedges and other microstructures mentioned in §10.4.3. This rather vague idea is interpreted by Ruddock (1994) to express the fact that the observed equilibria do not necessarily minimize the free energy, but do so only approximately. He analyzes the so-called X-interface found in the cubic-totetragonal phase transition in InTl binary alloys: this X-shaped microstructure divides space into four regions occupied by homogeneous austenite adjacent to two laminates of martensite, both adjacent to homogeneous martensite. The theory of Ball and James (1987) presented above, when applied to this microstructure, shows that no energy minimizing sequence is compatible with this geometry. Furthermore, Ruddock (1994) shows that the geometrically linear theory of Eshelby (1961), Khachaturyan (1961), etc., mentioned above allows for the X-interface if and only if the transformation is volume-preserving up to higher order terms in the principal transformation stretch of multiplicity 2. This condition guarantees that the microstructure is energy-minimizing in the geometrically linear theory and is satisﬁed to a good approximation in InTl.

300

MICROSTRUCTURES

10.4.3 Wedges and other microstructures The results above on the existence of habit planes can also be used to determine conditions for the existence of wedge-like or spear-like microstructures – see Bhattacharya (1993). These, when they can form, are believed to be an important mechanism for transforming easily and reversibly austenite into martensite, starting at the boundary of the body or of a grain. Each one of them consists of two ﬁne laminates of martensitic twins coming together along a midrib and surrounded by austenite. Each one of the two interfaces between a martensitic laminate and austenite is a habit plane and must satisfy the aforementioned existence conditions. Correspondingly, the principal stretches of martensite satisfy certain inequalities. The additional compatibility conditions for the ﬁne laminates along the midrib, obtained by means of the minors relations (§10.2.1), imply that the shape strain vectors on the two habit planes must be parallel, which is satisﬁed on a submanifold of the space of the martensitic symmetry-breaking stretches. In the cubic-to-tetragonal transition the two principal stretches must satisfy a sextic algebraic equation, which turns out to be satisﬁed rather closely by the materials in which wedges are observed. Other special microstructures can be constructed by putting various wedges together. Some are self-accommodating and can exist surrounded by austenite; among these especially important are triangles (the edges of the triangle are habit planes for three wedges whose midribs concur in one internal point of the triangle) and diamonds (two back-to back wedges). Also for the existence of each one of these microstructures the lattice parameters of the martensite, or the transformation stretch components, must satisfy speciﬁc restrictions. We address the reader to James and Hane (2000), section 4, for details; in particular, one ﬁnds there an interesting critique of the identiﬁcation of the diamond microstructures with certain microstructures, called self-accommodating plate groups, which are very popular models for self-accommodation in the metallurgical literature.

CHAPTER 11

Kinematics of multilattices The simple lattices studied in chapter 3 cannot describe all the periodic structures that crystalline substances exhibit in nature. Indeed, more realistic models for the majority of crystals are given by the periodic arrangements of points called multilattices, which are ﬁnite unions of translates of a given simple lattice. Alternatively, multilattices can be thought of as one skeletal lattice (or skeleton) at whose points are placed congruent clusters of atoms: this gives the motif of the multilattice, which contributes in an essential way to characterizing the symmetry of the whole structure. There are various reasons that make it necessary to consider multilattice rather than simple-lattice models;1 for instance, only by means of the latter can we describe crystals which are not made of a single atomic species: simple lattices are in fact always monatomic, that is, constituted of physically indistinguishable points. Of course there are also very important cases of monatomic multilattices which are not simple lattices, one of the most common examples being the well known hexagonal close-packed structures that we encountered in §3.7.2 and will examine in §11.5.1. Another reason for considering multilattices is that only for multilattices the geometric symmetry, as observed in real substances, can be a proper subgroup of one of the holohedral groups detailed in Corollary 3.4; indeed, any one of the thirty-two kinds of crystallographic point groups mentioned in Theorem 3.3. This is impossible for simple-lattice structures. A third reason for considering multilattices is that, under the assumption of central binary interactions, the molecular theories for simple lattices necessarily lead to the Cauchy relations (Remark 6.9). These extra symmetries in the elasticity tensor are in fact not satisﬁed by most materials. As indicated by Born (1915) (also by Born and Huang (1954)), a way of avoiding this problem is to consider more complex structures, such as multilattices. Furthermore, the use of multilattices endows the model, in a natural way, with extra internal parameters and degrees of freedom that are essential for describing certain aspects of the behavior of crystalline solids, as for instance stress relaxation in §11.8.3. 1

However, as was mentioned in the introduction to chapter 3, depending on the phenomena of interest and on how detailed their description should be, it is sometimes possible to regard as simple lattices also crystals whose structure is essentially a multilattice. This simpliﬁcation is often successfully adopted in the literature even for metallic alloys, such as some shape-memory alloys, which are not true crystals because the alloying element may be randomly substitutional in the lattice. 301

302

KINEMATICS OF MULTILATTICES

The approach to the kinematics of multilattices taken in this chapter is most convenient for the development of an elastic model for phase transitions in complex crystalline structures because, as was the case earlier with simple lattices, it allows us to keep track in a natural way of the symmetry changes in multilattices that are deforming, and to obtain the invariance properties for their constitutive functions, to be sketched in §11.7. Our point of view on the symmetry and kinematics of multilattices is strictly analogous to the well investigated one of chapter 3, and is based on the works of Ericksen (1970), Parry (1978), Pitteri (1985a), (1990), (1998), Pitteri and Zanzotto (1998a). As for simple lattices, here the main goal is the description of the geometric symmetry of multilattices, given by the classical aﬃne space groups, and of their global and arithmetic symmetries. The latter are based on the action of suitable groups of integral matrices on the appropriate conﬁguration spaces of multilattices (§§11.1.2, 11.2.2, 11.4.1, 11.4.2); this is analogous to the action of GL(3, Z) and its ﬁnite subgroups on B or Q> 3 , which was discussed in chapter 3. Furthermore, also for multilattices the geometric symmetry is but a local version of the global one (§11.6), for they are mutually compatible in the range of ‘small but ﬁnite’ deformations. The readers familiar with the classical crystallographic description of multilattice symmetry, based on space groups, will ﬁnd a connection between the two points of view in §11.4.3. As for simple lattices, also here the arithmetic symmetry is ﬁner than the geometric aﬃne space-group symmetry usually considered in the literature. However, the notion of arithmetic symmetry of multilattices is not classical in crystallography, and only its very basic properties are known. Recent contributions to this knowledge have been given by Ericksen (1999), Fadda and Zanzotto (2000), (2001a), Adeleke (2001a), (2001b); and additional results would be beneﬁcial for our understanding of phase transitions in crystalline solids. Further investigation is also needed to develop a uniﬁed kinematics of simple lattices and multilattices; some remarks are given by Pitteri and Zanzotto (1998a). The natural setting for multilattices is the 3-dimensional aﬃne space A3 , and the Grassmann notation is often used. Recall (§2.4) that Aﬀ (3) is the group of invertible aﬃne maps of A3 , and E(3) (the Euclidean group) the subgroup of Aﬀ (3) consisting of the isometric aﬃne transformations. 11.1 Crystals as multilattices An ideal crystal can be directly deﬁned as an inﬁnite and discrete subset M ⊂ A3 of points2 admitting a simple lattice of translations, say L(ea ), mapping M to itself. This expresses the fact that M has 3-dimensional periodicity, and implies (Proposition 11.1) that M is a ﬁnite union of 2

As for simple lattices, each point of such a crystal represents either the actual, or the most probable, or the average position of an atom or of a representative point, like the center of mass, of a cluster of atoms whose structure we decide to ignore.

11.1 CRYSTALS AS MULTILATTICES

303

translates of some suitable aﬃne simple lattice, this being deﬁned by: L(P, ea ) := P + L(ea ) ⊂ A3 ,

with

P ∈ A3 .

(11.1)

A diﬀerent approach, suitable for both crystals and quasicrystals, is based on the deﬁnition of a discontinuum in A3 , that is, of an inﬁnite and discrete subset of A3 which is ‘spread uniformly’ over the whole space. The precise deﬁning conditions can be given by means of a set of axioms proposed by Sohncke (1874), Niggli (1918), and not recalled here – see Hilbert and CohnVossen (1973) or Engel (1986). If the discontinuum is also homogeneous or regular, that is, ‘looks the same’ when seen from any one of its points, then periodicity necessarily follows from a well known result in the theory of crystallographic groups, called the Sch¨ onﬂies–Bieberbach theorem (see for instance Engel (1986)). Periodicity in turn implies that a regular discontinuum necessarily coincides with a ﬁnite collection of translates of a suitable aﬃne simple lattice L(P, ea ) in A3 . Then an ideal crystal M is deﬁned in general as a ﬁnite union of regular discontinua with identical periodicity (see Engel (1986)), and this is equivalent to the deﬁnition recalled at the beginning of this section. A generalization of regular discontinua that leads to this same result is given by Dolbilin et al. (1998). It is also necessary to have a criterion for deciding when two points of M are occupied by physically indistinguishable atoms, or clusters of atoms. This criterion must have the formal properties of an equivalence relation, which then partitions the points of M into classes A1 , . . . , Ar , called the atomic species of M. We also assume that any two points of M which diﬀer by a translation in L(ea ) necessarily belong to the same atomic species. A crystal whose points all belong to one atomic species is called monatomic, and polyatomic otherwise. The simple lattices studied in chapter 3 were thus understood to be monatomic. The following Proposition summarizes the discussion above, and describes a crystal M as the union of a ﬁnite number of disjoint monatomic aﬃne simple lattices with the same periodicity. Proposition 11.1 Any monatomic or polyatomic ideal crystal M is a multilattice in the aﬃne space A3 ; that is, M is a subset of A3 such that

M = M(P0 , . . . Pn−1 , ea ) := M(Pi , ea ) := n−1 L(Pi , ea ), (11.2) i=0 where L(ea ) (see (11.1)) is a simple lattice in the translation space R3 of A3 . In (11.2) n is a suitable positive integer, and the points Pi ∈ M are all distinct and do not diﬀer by any vector in L(ea ). The points of each of the aﬃne simple lattices L(Pi , ea ) ⊂ A3 belong to the same atomic species. Here we take Proposition 11.1 as the deﬁnition of an ideal crystal, and use the words ‘crystal’ and ‘multilattice’ as synonyms henceforth.3 The 3

For brevity, sometimes the word ‘crystal’ will also be used for the ‘crystalline material’ or ‘substance’ whose conﬁgurations are given by multilattices as in (11.2). This should not generate confusion.

304

KINEMATICS OF MULTILATTICES

name ‘n-lattice’ is used when the number of simple lattices composing M has to be speciﬁed; in particular, 1-lattices (n = 1) are aﬃne simple lattices. 11.1.1 Descriptors and conﬁguration spaces for deformable multilattices By Proposition 11.1, the n lattice points P0 , . . . , Pn−1 and the lattice vectors ea uniquely determine an n-lattice M, hence can be used as descriptive parameters for M; a choice of these is called a description of M. One of our goals will be to establish the indeterminateness in these descriptions. One can obtain a more convenient set of descriptors by choosing a base point in M, for instance P0 , and introducing the n−1 shift vectors or shifts pi := P0 Pi

for i = 1, . . . , n − 1

(and p0 = 0);

(11.3)

the n-lattice M is then uniquely determined by P0 and the n + 2 vectors e1 , e2 , e3 , p1 , . . . , pn−1 :

M = M(P0 , ea , p1 , . . . , pn−1 ) = n−1 {P0 + pi + L(ea )}. (11.4) i=1 The lattice vectors and shifts satisfy the conditions e1 · e2 ∧ e3 $= 0,

a pi $= pj + lij ea ,

(11.5)

a for i, j = 0, . . . , n − 1, i $= j and lij any integers; this guarantees that the simple lattices L(Pi , ea ), i = 0, . . . , n − 1, included in M are 3-dimensional and mutually disjoint. It is often useful to collect the above vectors in one list εσ , σ = 1, . . . , n + 2:

εa := ea , a = 1, 2, 3 , and ε3+i := pi , i = 1, . . . , n − 1 ;

(11.6)

accordingly, the multilattice M in (11.4) is denoted by M(P0 , εσ ) .

(11.7)

The simple lattice L(ea ) ⊂ R appearing in (11.2) is called the skeletal lattice (or skeleton) of M(P0 , εσ ) in the given description, and its unit cell and lattice vectors are also called the unit cell and lattice vectors of M. As mentioned earlier, L(ea ) can be interpreted as a group of translational isometries that map M onto itself (see also below (11.38)). We can think of the multilattice M(P0 , εσ ) as a triply periodic distribution of congruent clusters of atoms, a representative of which is placed at each point of the base 1-lattice L(P0 , ea ). Such clusters create the microstructural motif, which contributes in an essential way to the symmetry of M; the shifts pi give the position of the atoms of the motif of M with respect to the representive atom. An equivalent interpretation is that the pi give the displacement from the base lattice L(P0 , ea ) to the remaining simple lattices included in M. For this reason the shift vectors pi are rather natural variables to be added to the lattice vectors in the kinematics and energetics of multilattices. The set of all (n + 2)-uples of vectors in R3 satisfying the conditions 3

11.1 CRYSTALS AS MULTILATTICES

305

m in (11.5) is denoted by Dn+2 , and is a natural conﬁguration space for de4 formable n-lattices. For aﬃne simple lattices (1-lattices) we have n = 1 , and the conﬁguration space is D3m = B, the set of all bases of R3 (§3.1). Let Qn+2 denote the space of all symmetric n + 2 by n + 2 real matrices; it is useful to extend to multilattices the notion of lattice metric in (2.9), and introduce the space Qm n+2 Qn+2 of the multilattice metrics K:

K = (Kστ ) ,

Kστ = εσ · ετ ,

σ, τ = 1, . . . , n + 2 ,

(11.8)

where the εσ satisfy the conditions (11.5) and (11.6). A matrix K ∈ Qm n+2 is in general only positive semi-deﬁnite because the vectors εσ are never linearly independent for n > 1; furthermore, the positive semi-deﬁnite matrices in Qn+2 do not all belong to Qm n+2 : by deﬁnition, the εσ in (11.8) must also satisfy (11.5) and (11.6). In general Qm n+2 is a conical submanifold > of strictly lower dimension than Qn+2 . For 1-lattices Qm 3 = Q3 (§3.1). It is not diﬃcult to check that, for any two sets of descriptors εσ and εσ as in (11.5)1 and (11.6), we have K = K

⇔

εσ = Qεσ for some Q ∈ O.

(11.9)

Due to (11.9) and to the Euclidean invariance of the elastic theory that will be sketched, also the space Qm n+2 is referred to as the ‘conﬁguration space’ of n-lattices. 11.1.2 Essential descriptions of multilattices A basic remark about the descriptive parameters (P0 , εσ ) of a multilattice M is that in (11.2) or (11.4) the number n is not determined by M. This ambiguity arises because, in general, the skeletal lattice L(ea ) in (11.2) may not describe the full translational invariance (periodicity) of M. Clearly, Proposition 11.1 implies that there is a maximal skeletal lattice, R say,5 that is, a maximal subgroup of translations mapping M onto itself. R contains all the translational isometries of M, hence it only depends on M itself, and not on any choice of its descriptors. The simple lattice R is generated by suitable sets of maximal lattice vectors for M, and when these are used in (11.2) or (11.4) the description of M is called essential. In 4

5

We can also describe M by means of n shifts π0 , . . . , πn−1 given by the analog of (11.3) when an origin O in A3 replaces P0 , and π0 := OP0 = 0. The descriptors (π0 , . . . , πn−1 ) treat the lattices forming M in a more symmetric way than (p0 , . . . , pn−1 ), and are chosen, for instance, by Adeleke (2001a), (2001b). Here we prefer the descriptors (P0 , εσ ) because multilattices diﬀering only by a translation, which have the same εσ , will be assumed to be energetically equivalent. Therefore the εσ are the natural independent variables in a translationally invariant multilattice energy function, while the base point P0 will not play any essential role when we will specify the multilattice conﬁgurations in an elastic model. In this perspective P0 is thus not included as a parameter in the conﬁguration space of multilattices. After §11.4 the point P0 will be altogether dropped from the notation, and the term ‘descriptors’ will be used for the vectors εσ alone. From the French word ‘r´eseau’ utilized for the maximal skeletal lattice of multilattices.

306

KINEMATICS OF MULTILATTICES

this case the number of shifts, as well as the volume of the elementary cells m of M, are minimal and no longer arbitrary, and the descriptors εσ ∈ Dn+2 themselves are called essential for M. In §11.2.3 we discuss some of the problems connected with the use of nonessential multilattice descriptors. Evidently, any description with the same number of shifts as an essential description is also essential. In any nonessential description of M the lattice vectors generate a sublattice of R – see (3.6) – and the number of shifts must be suitably increased to take all the points of M into account.6 For instance, consider the (essential) lattice vectors ea in (3.55) for a b.c.c. 1lattice R. The same R can be described by the three primitive cubic lattice vectors, ˜ ea say, as in (3.52), together with the shift (see Fig. 3.11) 1

˜ p = (˜ e1 + ˜ e2 + ˜ e3 ).

(11.10)

2

The skeletal simple lattice L(˜ ea ) in this case is clearly not maximal for R (its unit cell is twice the unit cell given by the ea ), and the descriptors (˜ ea , ˜ p ) are not essential for R. 11.2 The global symmetry of multilattices Here and in the next two sections we study the symmetries of multilattices in close analogy with the treatment of symmetries of simple lattices in chapter 3. We ﬁrst connect the global symmetry of multilattices to the general indeterminateness in the choice of the multilattice descriptors εσ introduced above; this is described by the action of a suitable inﬁnite and discrete group of integral matrices on the conﬁguration spaces (Proposition 11.2 and Corollary 11.3). This symmetry has important implications on the invariance of the response functions for crystalline substances (§11.7). Secondly, we study the local arithmetic symmetry of multilattices, which is described by the action of suitable groups of integral matrices, the ‘lattice groups’ of multilattices (§11.4). In Proposition 11.8 we show the relation with the space-group symmetry of multilattices classically studied and classiﬁed in crystallography, and in §11.6 we reconcile the local and global symmetries of multilattices, as was done in §4.1 for simple lattices. 11.2.1 Indeterminateness of the descriptors (P0 , . . . , Pn−1 , ea ) The classical question posed by crystallographers to describe the symmetry of a multilattice M = M(P0 , εσ ) in A3 is the following analogue of question (a) posed in §3.1 for simple lattices: (a ) to ﬁnd all the aﬃne isometries e ∈ E(3) which map7 M to itself. 6

7

Notice that when the number of shifts is not minimal there are descriptions for M with the same number of shifts (or same unit-cell volume) whose skeletal lattices are distinct (see an example in footnote 18). When the multilattice M is polyatomic, a map leaving M invariant is intended to do so also for each atomic species.

11.2 THE GLOBAL SYMMETRY OF MULTILATTICES

307

Once an origin O is chosen in A3 , an isometry in E(3) can always be given the form e = (t, Q) (see (2.43)–(2.44)). It is useful to choose the origin O coincident with the base point P0 in (11.4), and we do so henceforth unless otherwise speciﬁed. In this case an equivalent formulation of question (a ) ¯ 0, ε ¯σ ) of M of the form8 is: to ﬁnd all new sets of descriptors (P ¯ 0 := e(P0 ) = P0 + t, P

¯σ := Qεσ ε

(11.11)

for some t ∈ R3 , Q ∈ O, such that ¯ 0, ε ¯σ ) = M(P0 , εσ ) . M(P

(11.12)

Now, (a ) is a special instance of the following more general problem, similar to question (b) in §3.1 for simple lattices: ¯ 0, ε ¯σ ) generating the same multi(b ) to ﬁnd all new sets of descriptors (P lattice M as (P0 , εσ ), that is, such that (11.12) holds. Question (b ) also follows from the observation that a choice of descriptive parameters uniquely determines a crystal, while a crystal does not uniquely determine its descriptors given by Proposition 11.1. Indeed, a ﬁrst source of nonuniqueness in the representation (11.4)–(11.6) of a multilattice M comes from the fact that the number of shifts pi is not determined by M, and the descriptors may be nonessential. This and related diﬃculties make it necessary to introduce a restrictive hypothesis in question (b ), which we address ﬁrst. Indeed, we will suppose that in (b ) the two descriptors ¯ 0, ε ¯σ ) be both essential:9 both must preserve the maximal (P0 , εσ ) and (P translational invariance of M: R = L(ea ) = L(¯ ea );

(11.13)

henceforth R(ea ) will indicate the simple lattice generated by the vectors ea when this is the maximal skeletal lattice of M. The following Proposition answers question (b ) for essential descriptors (P0 , . . . , Pn−1 , ea ). Proposition 11.2 Let M be a polyatomic multilattice with essential descriptors10 (P0 , . . . , Pn−1 , ea ) as in (11.2), and let the set {0, . . . , n − 1} of indices be partitioned into subsets corresponding to simple lattices in M which belong to the same atomic species. Then, for any choice of a matrix m ∈ GL(3, Z), of integers nai and of a permutation f of {0, . . . , n − 1} 8

9

10

As in (2.44), the ﬁrst equality in (11.11) is a deﬁnition, hence it does not depend on the choice of the point O used for representing the isometry e through the pair (t, Q). On the contrary, the second equality in (11.11) holds if O = P0 ; if P0 = O we have ¯ 0 = P0 + t + (Q − 1)OP0 . The choice O = P0 thus allows for simpler representations P of the isometries of M. That the descriptions are essential is a mild imposition in view of the fact that the nonessential descriptors form ‘low-dimensional’ submanifolds in the conﬁguration m or Qm spaces Dn+2 n+2 , as will be discussed in §11.2.3. For nonessential descriptors see §11.2.3.

308

KINEMATICS OF MULTILATTICES

leaving the above subsets invariant, the formula ¯ ea = mba eb

and

¯ i = Pf (i) + nai ea , i = 0, . . . , n − 1 P

(11.14)

produces a (unique) new essential description of M; that is, ¯ 0, . . . , P ¯ n−1 , ¯ M(P0 , . . . , Pn−1 , ea ) = M(P ea ) .

(11.15)

¯ 0, . . . , P ¯ n¯ −1 , ¯ Conversely, given essential descriptors (P ea ) of M, necessarily n ¯ = n and there exist, unique and independent of one another, a matrix m ∈ GL(3, Z), a set of integers nai and a permutation f of {0, . . . , n − 1} leaving the above subsets of it invariant, such that (11.14) holds. We give the proof for a monatomic multilattice, in which case f is a permutation of {0, . . . , n − 1} without restrictions. It is immediate to extend the argument to the polyatomic case. Proof. Formula (11.14) implies (11.13), and also (11.15) because

¯ 0, . . . P ¯ n−1 , ¯ ¯ i, ¯ M(P ea ) = n−1 R(P ea ) = n−1 R(Pf (i) , ea ). (11.16) i=0 i=0 For the converse: the maximal skeletal lattice of M is independent of the essential descriptors used, that is, (11.13) holds; thus Proposition 3.1 implies that (11.14)1 necessarily holds for some element m of GL(3, Z). ¯ i is Furthermore, let us assume n ¯ ≥ n for deﬁniteness. Owing to (11.15), P then a point of M for any i = 0, . . . , n ¯ − 1; hence, by (11.2), there exist a map f : {0, . . . , n ¯ − 1} → {0, . . . , n − 1} and integers nai such that ¯ i = Pf (i) + nai ea ; P

(11.17)

this is (11.14)2 if we show that f is injective, hence (¯ n = n and) f is a permutation of {0, . . . , n − 1}. By contradiction, assume that f (i) = f (j) for some choice of i and j $= i. Then (11.14)1 and (11.17), together with the analogue of (11.3) for ¯ pi with ¯ p0 = 0, imply ¯ pi − ¯ pj = (nai − naj )ea = (nai − naj )(m−1 )ba ¯ eb ,

(11.18)

which contradicts (11.5) for ¯ ea and ¯ pi . Proposition 11.2, which should be compared with the analogous Proposition 3.1 for simple lattices, can be brieﬂy summarized as follows. If M = M(P0 , . . . , Pn−1 , ea ) is an essential and monatomic multilattice, any new ¯ 0, . . . , P ¯ n¯ −1 , ¯ essential parameters (P ea ) for M are obtained by choosing any set of lattice vectors for its maximal skeletal lattice R(ea ) and any permutation of its base points P0 , . . . , Pn−1 , up to the addition of elements of R(ea ) as in (11.17). If M is polyatomic,11 the permutation of the base 11

In a more general framework, it is possible to enlarge the list of descriptors of M by adding the atomic species, and parametrize the multilattice as follows: M = M(P0 , . . . , Pn−1 , ea , A0 , . . . , An−1 ),

(11.19)

where Ai is the atomic species of the 1-lattice R(Pi , ea ) included in M. For any new ¯ 0, . . . , P ¯ n−1 , e¯a , A ¯0 , . . . , A ¯n−1 ) for M, the geometric parameters set of descriptors (P still satisfy Proposition 11.2, to which we must add the following transformation rule

11.2 THE GLOBAL SYMMETRY OF MULTILATTICES

309

points is not arbitrary but must exchange among them only the ones of the same atomic species. 11.2.2 Indeterminateness of the descriptors (P0 , εσ ) The following Corollary is, for a monatomic M, a useful and more operative restatement of Proposition 11.2 in terms of the descriptors (P0 , εσ ) introduced in (11.4)-(11.6). To state it, we introduce the arithmetic (sub)group Γn+2 < GL(n + 2, Z),

(11.21)

constituted by the unimodular integral n + 2 by n + 2 matrices deﬁned in (11.22)–(11.23) below; Γn+2 plays for the monatomic n-lattices the role of global symmetry group that is played for simple lattices by the group GL(3, Z) in §3.1.12 By deﬁnition, the elements µ = (µτσ ) of Γn+2 have the following structure: for a, b = 1, 2, 3 and i, j = 1, . . . , n − 1,   b b b l 1 · · · l n−1

ma

µ ∈ Γn+2 ⇔

(µτσ )

 =  0 ..0 0 .

αji

0 0 0

 ,

(11.22)

where (mba ) is any matrix in GL(3, Z), lbi are arbitrary integers, and α = (αji ) is an n − 1 by n − 1 matrix belonging to the ﬁnite noncommutative unimodular group generated by the permutation matrices of the set {1, . . . , n − 1}13 and by the matrices of the following form, which are obtained from the identity by replacing one of its rows by a row of −1s:   1

0 ··· 0 · · · 0. · · · .. · · · −1. · · · .. 0 ··· 1 0 ··· 0

 0.. 1..  . .  −1. −1.  .. ..  0 0

0 0. .. −1 .. . 0 1

   .  

(11.23)

The set Γn+2 deﬁned in this way is indeed a subgroup of GL(n + 2, Z), as can be checked by a direct computation. Notice that the submatrix α of an element µ ∈ Γn+2 either is a permutation matrix or has a row of −1s. As for the atomic species:

12

13

¯i = Af (i) , i = 0, . . . , n − 1. A (11.20) In this way the general changes of descriptors also include the symmetry transformations belonging to the so-called color symmetry groups – see Shubnikov and Koptsik (1974) or Senechal (1990). For n = 1 the deﬁnitions give Γ3 = GL(3, Z). We will see that Γn+2 acts on the m conﬁguration spaces Dn+2 or Qm n+2 of n-lattices deﬁned in §11.1.2 in a way which is similar to the action of GL(3, Z) on the conﬁguration spaces B or Q> 3 of 1-lattices. The n − 1 by n − 1 permutation matrix α of a permutation f of {1, . . . , n − 1} is deﬁned as usual by αji vj = vf (i) for any numbers v1 , . . . , vn−1 ; so, the entries of the matrix α are all 0s, except for 1s in the f (i)-th row of the i-th column.

310

KINEMATICS OF MULTILATTICES

will be clear from the proof of Corollary 11.3 below, the group of matrices α is a representation of the permutations of n objects by means of n − 1 by n − 1 matrices. The explicit form of the elements of Γ4 and Γ5 is given in formulae (11.61) and (11.68) below. The structure of the matrices µ ∈ Γn+2 is justiﬁed by the following: Corollary 11.3 Let M(P0 , εσ ) be a monatomic n-lattice in an essential ¯ 0, ε ¯σ ) is a new set of essential descriptors for M, description.14 Then (P that is, (11.12) and (11.13) hold, if and only if there exist a set of integers na0 and a matrix µ ∈ Γn+2 that are (mutually independent and) such that ¯σ = µτσ ετ , µ ∈ Γn+2 , ε ¯ 0 = Pi(µ) + P

na0 ea

,

na0

and

(11.24)

∈ Z.

(11.25)

The integers and the matrix µ ∈ Γn+2 uniquely determine the descrip¯ 0, ε ¯σ ), and vice versa. tors (P The index i(µ) in (11.25) is determined by the submatrix α of µ in (11.22) as follows: if α is a permutation matrix, i(µ) = 0; if the r-th row of α is a row of −1s, then i(µ) = r. na0

For the reasons mentioned in footnote 4, in this and the following results the reader should mostly pay attention to the transformation rules for the vectors εσ , since the base point P0 plays a secondary role in the kinematics of multilattices in view of an elastic model. We stress that, by (11.22) and (11.24), the new lattice vectors and shifts are explicitly given by ¯ ea = mba eb ,

¯ pi = αji pj + lai ea ,

(11.26)

where (mba ) ∈ GL(3, Z), lai ∈ Z, and (αji ) is a permutation or a matrix (11.23) times a permutation, with a = 1, 2, 3, i, j = 1, . . . , n − 1. Corollary 11.3 states that any change of essential descriptors is obtained, in a unique way, through a matrix µ ∈ Γn+2 (and integers na0 ) as indicated in (11.24)–(11.25): the skeletal basis transforms to an equivalent basis, and ¯ 0 is chosen among the points in the (a) if i(µ) = 0 the new base point P 1-lattice R(P0 , ea ), and the new shifts are just a permutation of the old ones up to the addition of vectors in the maximal skeletal lattice R(ea ); ¯ 0 belongs to the 1-lattice R(Pi(µ) , ea ), (b) if i(µ) $= 0 the new base point P and, up to the addition of vectors in R(ea ), the new shifts are obtained by a matrix α as in (11.23), which ‘subtracts’ the shift pi(µ) from all the pr , r = 1, . . . , n − 1, possibly after they have been permuted. As anticipated earlier, this result shows that the global symmetry of any monatomic n-lattice in its essential description is given by the arithmetic group Γn+2 , and makes explicit the central role played by the matrices µ in the account of multilattice symmetry. Clearly, Γn+2 is independent of the (essential) descriptors used, and is only determined by the number n. For 14

See at the end of §11.2.3 for some comments on the case of nonessential descriptors.

11.2 THE GLOBAL SYMMETRY OF MULTILATTICES

311

polyatomic n-lattices a result analogous to Corollary 11.3 can be obtained in a rather straightforward way, showing that the global invariance group is reduced to a suitable subgroup of Γn+2 , whose generators have submatrices α leaving each atomic species invariant. In §11.5.1 we detail the cases of monatomic and diatomic 2-lattices. Proof of Corollary 11.3. We show that the necessary and suﬃcient conditions (11.14) are equivalent to (11.25) and (11.26). Since (11.14)1 is (11.26)1 and (11.25) coincides with (11.14)2 restricted to i = 0, we only have to show the equivalence of (11.26)2 and (11.14)2 for i = 1, . . . , n − 1. We write the latter – the right-hand side ﬁrst for convenience – in the form ¯0 + ¯ P0 + pf (i) + nai ea = P pi = P0 + pf (0) + na0 ea + ¯ pi ,

(11.27)

whence, for i = 1, . . . , n − 1 and a = 1, 2, 3, ¯ pi = pf (i) − pf (0) + lia ea ,

lia = nai − na0 .

(11.28)

If f (0) = 0, equation (11.28) has the form (11.26)2 in which α is the permutation matrix of f restricted to {1, . . . , n − 1}; if f (0) = r > 0, (11.28) has the form (11.26)2 when α = α1 α2 , when α1 is the matrix (11.23) in which the row of −1s is the r-th, and α2 the matrix of the permutation {1, . . . , r−1, 0, r+1, . . . , n−1} → {f (1), . . . , f (n−1)}. Conversely, (11.28)1 , (11.25) and nai deﬁned by (11.28)2 , imply (11.27), hence (11.14)2 for i $= 0. The constructive proof above details how the matrix µ is uniquely determined by m, by the permutation f , and by the integers nai , i = 0, . . . , n − 1, appearing in Proposition 11.2; it also shows that, by (11.28), the matrix α is uniquely determined by f , and vice versa. A change of descriptors as in (11.24) induces, in obvious notation, the following transformation of the multilattice metric K in (11.8): ¯ = µt Kµ , K → K

(11.29)

¯ $= K. This gives a natural action of Γn+2 on the where, in general, K conﬁguration space Qm n+2 which is analogous to the action (3.8) for simple lattices. By (11.9), for any choice of εσ the analogue of (3.24) holds for the multilattice metrics K ∈ Qm n+2 : µt Kµ = K ⇔ µτσ ετ = Qεσ for some Q ∈ O .

(11.30)

However, as noticed below (3.24) for 1-lattices, two multilattices M(P0 , εσ ) ¯ 0, ε ¯σ ) can be congruent without their descriptive parameters εσ and M(P ¯σ being related by an orthogonal transformation as in (11.30). and ε 11.2.3 Nonessential descriptors of multilattices As mentioned in §11.1.3, given a multilattice M, the selection of the number n in (11.2) is largely arbitrary, hence it is a matter of judgement to

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choose it appropriately and describe M as an n-lattice. Usually, any ambiguity in the description of a multilattice and its deformations is avoided by adopting essential descriptors, and we will mostly follow this practice, like for instance Ericksen (1980a), (1980b), (1982a), (1984). However, it must m be stressed that choosing n selects the conﬁguration space Dn+2 of the material, and thus restricts the conﬁgurations that the crystal can access in the model. For instance, assume the observed equilibrium conﬁguration of a crystal to be an essential n-lattice M for a given n; then, by choosing such an essential n we restrict also all the other (nonequilibrium) conﬁgurations for that crystalline substance to be n-lattices. In this way, in chapter 3 it was understood that 1-lattices were to remain 1-lattices in all deformations they might undergo. This assumption is often physically appropriate, but it should be emphasized that there are circumstances in which it is violated. In these cases it is necessary to abandon the restriction to essential descriptors, and use lattice vectors whose cell is not of minimum volume or, equivalently, to use a nonminimal number of shifts. This adds ‘degrees of freedom’ to the model, which are described by extra ‘internal variables’ whose presence helps accounting for the observed phenomena. See for instance §11.8.3. As an example, consider the description of the body-centered cubic 1lattice as a 2-lattice constituted by two primitive cubic 1-lattices (see (11.10)). The nonessential 2-lattice descriptors (˜ ea , ˜ p ) are needed if one has to model phase transitions in which the crystal passes from a 1-lattice structure to a 2-lattice structure, or vice versa. When the shift ˜ p is no longer subject to (11.10), it can be used as an extra independent variable that describes the movements of the center point of the cell independently of the deformation of the skeletal lattice generated by the ˜ ea . For instance, this kinematic possibility must be allowed in a model for the transition from a b.c.c. to a h.c.p. lattice conﬁguration, as observed in Li, Ti, Zn, or Hf (see Nishiyama (1978) pp. 68 and 344 for interesting kinematical details on such transitions). Other cases in which nonessential descriptions are relevant are implicitly discussed by Ericksen (1980a), (1982a), Parry (1981), Zanzotto (1992), (1996a). The problems connected with the adoption of nonessential descriptions of multilattices are just beginning to be examined in the literature, for instance by Ericksen (1998); here are some comments on two main points: (i) Once the selection of a given number n is made, and it is decided that the m relevant conﬁgurations of a crystal are given by descriptors εσ ∈ Dn+2 , the conﬁgurations that are nonessential n-lattices (as was the case for the b.c.c. lattice in (11.10) when described as a 2-lattice) are only ‘a few’. Indeed, as a consequence of Proposition 5.1 in Pitteri (1998),15 the nonessential 15

m Let the descriptors εσ ∈ Dn+2 have multilattice metric K ∈ Qm n+2 . Then the εσ are essential descriptors if and only if 1 ∈ Γn+2 is the only matrix µ ∈ Γn+2 , among those satisfying µt Kµ = K, such that its m-component is the identity in GL(3, Z).

11.2 THE GLOBAL SYMMETRY OF MULTILATTICES

313

m descriptors form lower dimensional submanifolds of Dn+2 or Qm n+2 . More m precisely, the matrices of the space Qn+2 obtained through (11.8) from nonessential descriptors εσ , σ = 1, . . . , n + 2, form submanifolds contained in linear subspaces of Qn+2 of strictly lower dimension than Qm n+2 . The nonessential submanifolds in D4m (n = 2, 2-lattices) are described, as an example, in (11.60). Relation (11.10), which, when satisﬁed for any vectors ˜ ea , makes the 2-lattice a 1-lattice, is but a special case of (11.60).16 m When the descriptors in Dn+2 belong to the nonessential submanifolds, the n-lattice M(P0 , εσ ) is actually an n -lattice with n < n. By letting m , one thus allows, in principle, for the possithe descriptors vary in Dn+2 bility that the crystal also assumes conﬁgurations that are n -lattices with some n < n, described as nonessential n-lattices. A class of possibilities is described by Ericksen (1998). With this proviso, the guideline of choosing essential descriptions is always a valid one, if it is applied to ‘the majority’ of the conﬁgurations a crystal can assume, rather than to some speciﬁc conﬁguration in which it may be observed.

(ii) The second point that needs to be stressed is that the use of nonessential m descriptors εσ ∈ Dn+2 is not devoid of problems. The main drawback is that Corollary 11.3 no longer holds. On the one hand, the ‘if’ part of the corollary remains true in the nonessential case if the skeletal lattice in the two descriptions is the same, but then there is no uniqueness17 of the matrix µ in (11.24). On the other hand, the ‘only if’ part of the corollary does not hold, because there are changes of nonessential descriptors that cannot be represented by a matrix µ ∈ Γn+2 with a nonessential n. This means that the group Γn+2 does not describe correctly the global symmetry of a nonessential n-lattice M(P0 , εσ ), which admits by deﬁnition an essential description as a n -lattice for a suitable n < n. On the other hand, in Γn+2 there are inﬁnitely many changes of nonessential descriptors which cannot be represented by elements of Γn +2 , roughly because the essential description of M(P0 , εσ ) has fewer internal degrees of freedom. It is not clear how to ‘ﬁt’ the two symmetries together, and this is one of the main problems that stands in the way of the creation of a uniﬁed kinematics for multilattices of various complexities.18 16 17

1-lattices are always essential, so nonessential submanifolds do not exist in B = D3m . This lack of uniqueness is related to the one mentioned in footnote 15. Consider for instance the primitive cubic 1-lattice R generated by three orthonormal vectors ea . ˜σ , σ = 1, . . . , 4 The same R can be considered as a 2-lattice generated by the vectors ε given by the analogue of (11.6) for ˜ e1 = 2e1 , ˜ e 2 = e2 , ˜ e 3 = e3

18

and

p = e1 ; ˜

(11.31)

this L(˜ ea ) is not maximal (the unit cell constructed on ˜ ea is twice the unit cell of the ˜1 , the identity cubic lattice vectors) and it does not have cubic symmetry. Since 2˜ p=e ˜σ is represented through equation transformation 1 of the nonessential descriptors ε (11.24) both by the identity matrix in Γ4 and by the matrix µ whose m-component is 1, while the α-component is −1, and the triple of integers la is (1, 0, 0). Since the symmetry of nonessential multilattices is an interesting open question, we give an explicit example of the problems that may arise with it, showing one of the

314

KINEMATICS OF MULTILATTICES

A ﬁnal note on the problems presented by the mechanics of multilattices is that the conﬁgurations or deformations experimentally observed in multilattices sometimes cannot be all described by any given choice of the number n. For instance, in various materials no single choice of skeletal m lattice, that is, of the conﬁguration space Dn+2 , can account for all the observed twinning deformations, as was remarked in §8.4.2. This nontrivial behavior of multilattices requires other models than elasticity for its interpretation, and Ericksen (1997) gives some novel ideas in this direction. For the reasons indicated above, in what follows we will always use essene m tial descriptions unless otherwise speciﬁed. We denote by Dn+2 Dn+2 and e m Qn+2 Qn+2 the subsets of multilattice descriptors and multilattice metrics, respectively, for essential n-lattices. The arguments in this subsection show that the nonessential n-lattice descriptors and metrics belong to nonm trivial lower-dimensional submanifolds of Dn+2 and Qm n+2 , respectively, and thus the inclusions above are indeed strict. 11.3 The aﬃne symmetry of multilattices We now go back to the symmetry issues raised by question (a ) in §11.2.1; for simplicity we restrict ourselves to the monatomic case, and only add some brief remarks on polyatomic multilattices. As the geometric (orthogonal) symmetry of simple lattices in §3.2, a classical subject of crystallography is the study of the aﬃne isometric symmetries of multilattices, which do not depend on the description used but only on the multilattices themselves. In this and the next section we will see that, for essentially described multilattices, the known results on aﬃne symmetry can be put in the perspective of the global symmetry studied in the previous section. This will give a natural way to phrase the questions related to symmetry in an elastic model of multilattice behavior, and to investigate the arithmetic symmetry of multilattices much like in the better known case of simple lattices. Recall that once an origin O is chosen in A3 , any isometry e ∈ E(3) is ‘missing’ symmetry operations. Consider the 1-lattice R of footnote 17. The orthogonal transformation deﬁned by 0 −1 0 Qea = mba eb for m = 1 0 0 (11.32) 0 0 1 is an isometry of R, and allows us to deﬁne a new set of 2-lattice descriptors for R: ˜σ , ¯σ = Q ε ε

¯ ea = Q˜ ea ,

p = Q˜ ¯ p.

(11.33)

Indeed, we do have ¯σ ) = M(P0 , ε ˜σ ) ; M(P0 , ε

(11.34)

¯σ = µτσ ε ˜τ , because however, there is no µ ∈ Γ4 for which (11.24) holds, that is, ε e2 , Q˜ e2 = −p = p − ˜ e1 , Q˜ e3 = ˜ e3 , Qp = ˜ e2 . Q˜ e1 = 2˜

(11.35)

The reason is that the translational group of the two descriptions is not the same in ea ) = R(˜ ea ). this example: R(¯ ea ) = QR(˜

11.3 THE AFFINE SYMMETRY OF MULTILATTICES

315

represented by a pair (t, Q), where Q ∈ O and t is the vector from O to e(O) (§2.4). Given an essentially described multilattice M(P0 , εσ ), for convenience we identify O with the base point P0 of M (see footnote 8). An isometry e mapping M(P0 , εσ ) to itself is called a symmetry operation of M. As in (11.11)–(11.12), any symmetry operation e = (t, Q) of ¯ 0, ε ¯σ ) for M: M gives new essential descriptors (P ¯ 0 := e(P0 ) = P0 + t, P

¯σ := Qεσ , and ε ¯ 0, ε ¯σ ) = e(M). M = M(P0 , εσ ) = M(P

(11.36) (11.37)

There are some characteristic features of the symmetry operations of multilattices that are not encountered in simple lattices. Indeed, if e is a symmetry operation for an aﬃne 1-lattice P0 + L, the essential translation of e, deﬁned in §2.4, necessarily belongs to L; equivalently, e can be thought of being produced by (0, Q) – which is an aﬃne isometry with a ﬁxed point – followed by a suitable lattice translation. This is not necessarily true for the symmetry operations of a general multilattice M, which, as a rule, do not need to ﬁx any point of A3 up to skeletal translations. That is, the essential translation of a symmetry operation e for M may not be a skeletal translation. If e = (t, Q) is such that its essential translation is not in the maximal skeletal lattice of M, and det Q = 1, e is called a screw rotation and its axis a screw axis for M; if det Q = −1, e is called a (screw) rotary reﬂection; in particular, if Q is a reﬂection (Q 2 = 1), e is called a glide reﬂection,19 and its invariant plane a glide plane for M. Some examples are given in §11.5.2; see also footnotes 22 and 27. Another distinctive feature of multilattices is the possible lack of central symmetry; that is, an n-lattice may not admit (t, −1) as a symmetry operation for any translation t. Indeed, depending on the given descriptors e εσ ∈ Dn+2 , there may or may not exist matrices µ ∈ Γn+2 whose action ¯σ := −εσ .20 on εσ given by (11.24) produces new equivalent descriptors ε However, for 1-lattices this always happens, with the aﬃne isometry (0, −1) and m = −1, if O = P0 – see also (3.20). We will see in §11.5.1 that also monatomic 2-lattices are always centrosymmetrical. 11.3.1 Space groups; crystal class and crystal system of a multilattice The space group S(M) of a monatomic multilattice M = M(P0 , εσ ) is the collection of all the aﬃne isometries mapping M to itself: S(M) = {e ∈ E(3) : e(M) = M}. 19

20

(11.38)

Since, as remarked below (2.46), the essential translation of e is the component of t along the kernel of Qt − 1, for any origin O a symmetry operation e = (t, Q) for M is a screw rotation if and only if Q is in O+ and the component of t along the axis of Q is not in the maximal skeletal lattice R of M; it is a glide reﬂection if Q is a reﬂection and the component of t on the mirror of Q is not in R. Notice that the matrix −1 is not in Γn+2 for any n > 2.

316

KINEMATICS OF MULTILATTICES

The following subgroups of S(M) are important: the normal commutative subgroup T (M) consisting of all the translations mapping M to itself: T (M) := S(M) ∩ T (3),

(11.39)

and the group P (M) of the operations Q appearing in (11.38)2 , that is, the group of all the orthogonal transformations each one of which preserves M when coupled with a suitable translation. Clearly, the groups S(M), T (M), P (M) above depend only on the lattice M and not on any choice of its descriptors. However, it is useful to introduce essential descriptors of M = M(P0 , εσ ) for discussing some of their properties. For instance, T (M) ⊂ Aﬀ (3) is immediately seen to be isomorphic to the maximal skeletal lattice R of M. Also, due to (11.45)2 below, one can check that P (M) is in general a (ﬁnite) nonholohedral crystallographic point group contained in the holohedry P (ea ) of the skeletal lattice R = R(ea ) of M: P (M) ≤ P (ea ) .

(11.40)

The group P (M) is called the point group of the multilattice M; as discussed in §3.2.2, the point group determines an orthogonal conjugacy class called the crystal class of M. Since P (M) may not be holohedral, the class of M can be any one of the thirty-two classes21 mentioned in Theorem 3.3. The crystal system of a multilattice M is customarily deﬁned as the least symmetric system to which the crystal class of M belongs. It need not coincide with the system of the maximal skeletal holohedry P (ea ) in (11.40), but this only happens in exceptional cases (excess skeletal symmetry, see for instance Landau et al. (1980) p. 410, Pitteri and Zanzotto (1998a), Fadda and Zanzotto (2001a). The geometric symmetry of simple lattices is based on the subdivision into crystal systems of the conﬁguration spaces B or Q> 3 ; it is possible to introduce an analogous classiﬁcation into crystal classes (and crystal e systems) for the conﬁguration spaces Dn+2 or Qen+2 of n-lattices. If M is a 1-lattice L = L(P0 , ea ), its class and system coincide, and its space group S(L) takes a particularly simple form which is worth mentioning and can for instance be inferred from (11.43) and (11.45) below. Indeed, for a 1-lattice the translation t appearing in (11.36) is necessarily a lattice translation (§11.3), hence there are neither screw rotations nor glide reﬂections in S(L): for any (t, Q) ∈ S(L), we necessarily have Q ∈ P (ea ) and t ∈ L(ea ). This means that the space group S(L) of any 1-lattice L(P0 , ea ) = P0 + L(ea ) is always given by the inﬁnite group S(L) = {(t, Q) : Q ∈ P (ea ), t ∈ L(ea )},

(11.41)

which is called the semidirect product (Mac Lane and Birkhoﬀ (1967)), 21

Recall that simple lattices can only realize the seven holohedral crystal classes (crystal systems) of Corollary 3.4. Multilattices, on the other hand, do realize, in theory and in nature, all the thirty-two crystal classes that are possible in three dimensions.

11.3 THE AFFINE SYMMETRY OF MULTILATTICES

317

under the multiplication of pairs in (2.45)1 , of the (maximal) translation group L(ea ) by the holohedry P (ea ). In general, space groups S(M) that are the semidirect product of the maximal skeletal lattice R of a multilattice M by its point group P (M) are called split or symmorphic; they are characteristic not only of simple lattices, but also of any multilattices admitting neither screw rotations nor rotary reﬂections.22 Most space groups however are not symmorphic, and their structure is determined constructively from the following basic property (see, for instance, Miller (1972), Schwarzenberger (1980), Farkas (1981), Senechal (1990)): Proposition 11.4 For any multilattice M, the space group S(M) is an inﬁnite and discrete subgroup of E(3) whose normal abelian translation subgroup T (M) is generated over Z by three independent vectors, and whose quotient group S(M)/T (M) is isomorphic to the (ﬁnite) crystallographic point group P (M) < O. Conversely, any subgroup of E(3) whose translation subgroup is as above, is the space group of some multilattice. Analogously to (3.19), if an aﬃne isometry e is applied to M, the space group S(M) changes to a conjugate in E(3): S(eM) = eS(M)e−1 .

(11.42)

Also, any dilation of a multilattice produce a new multilattice whose symmetry is not essentially diﬀerent from the initial one. This may support the choice of the classiﬁcation criterion adopted for space groups, which is conjugacy within the aﬃne group Aﬀ (3). By a known theorem of Bieberbach (1912), stating that any two isomorphic discrete subgroups of E(3) are aﬃnely conjugate, the classiﬁcation of space groups just mentioned coincides with the one based on group isomorphism (see Miller (1972), Janssen (1973), Farkas (1981), or Sternberg (1994)). The following classical result in crystallography, which is an analog of Theorem 3.3, was obtained at the end of the 1800s through the careful analysis of all the subgroups of E(3) with the properties of Proposition 11.4: Theorem 11.5 There are 219 conjugacy classes of space groups in Aﬀ(3); these are also isomorphism classes. The number increases to 230 if conjugacy is sought through elements of Aﬀ(3)+ . There are numerous descriptions of the space groups in the literature, for instance Burckhardt (1947), Miller (1972), Janssen (1973), Sternberg (1994), or the International Tables (1996). Here we only recall that among the 219 isomorphism classes, those of symmorphic groups are 73, as can be proved as a corollary of Theorem 3.6 (see Miller (1972)). Space groups that are conjugate in Aﬀ (3) but not in Aﬀ (3)+ are called 22

A screw rotation or rotary reﬂection is characterized by the fact that its T (M)-coset in S(M) does not contain any element with zero essential translation. This can be used to check the presence of such symmetries in S(M); see also footnote 27.

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KINEMATICS OF MULTILATTICES

enantiomorphic, and permit to distinguish right-handed from left-handed crystalline conﬁgurations.23 It is worth noticing that not all the space groups can be obtained from multilattices with too small n – for instance, we mentioned that all 2-lattices are centrosymmetrical (§11.5.1), while in general multilattices need not be so (§11.5.2). This is also shown explicitly by Fadda and Zanzotto (2000), (2001a), who describe the arithmetic symmetry classes of monatomic 2lattices in 2 and 3 dimensions, respectively. The latter reference gives a sound systematic procedure for enlarging the list of crystal structures contained in empirical collections like the Strukturberichte (1915-1940). In the polyatomic case the above deﬁnitions of space group, crystal class and crystal system of a multilattice remain the same, with the understanding that the aﬃne isometries must leave invariant the multilattice as well as each one of its atomic species.24 11.4 The arithmetic symmetry of multilattices As was done in §3.3 for simple lattices, in this section we brieﬂy discuss the notion of arithmetic symmetry of multilattices, which is not a classical subject in crystallography and so far has only been partially investigated. 11.4.1 Lattice groups of multilattices The following immediate consequence of (11.30) and (11.36) states that, as a special case of Corollary 11.3, the aﬃne symmetry operations of an n-lattice M all correspond to particular matrices µ in a ﬁnite subgroup of Γn+2 if essential descriptors are used.25 Corollary 11.6 Let M(P0 , εσ ) be a monatomic n-lattice in an essential description, and let K ∈ Qen+2 be the corresponding multilattice metric. An isometry e = (t, Q) is a symmetry operation for M, that is, e ∈ S(M), if 23

24

25

A space group S has enantiomorphs if and only if its normalizer in Aﬀ (3) is contained in Aﬀ (3)+ – see Yale (1968), Farkas (1981). In particular (but this need not be suﬃcient), S itself must lack elements e whose associated Q has negative determinant. Thus the space groups of centrosymmetrical crystals never admit enantiomorphs; moreover, it can be seen that the crystals admitting right- and left-handed conﬁgurations (also called enantiomorphs) are actually a proper subset of the crystals lacking central symmetry. For this reason, applying −1 to a conﬁguration of a crystal M(P0 , εσ ) should not be expected in general to give an enantiomorphic conﬁguration in crystals lacking central symmetry. However, when the space group of M(P0 , εσ ) does admit an enantiomorph, then the latter is indeed the space group of M(P0 , −εσ ). An example is given in §11.5.2. If for polyatomic multilattices also the permutation of atomic species are considered – see footnote 11 – the related theory of symmetry becomes identical with the crystallographic theory of color symmetry. As was remarked at the end of §11.2.3 for Corollary 11.3, the ‘if’ part of Corollary 11.6 holds also when the description is not essential (but the elements na 0 , f, µ representing a given symmetry operation may not be unique), while the ‘only if’ part fails.

11.4 THE ARITHMETIC SYMMETRY OF MULTILATTICES

319

and only if there exist a set of integers na0 and a matrix µ ∈ Γn+2 which are mutually independent and such that t := P0 e(P0 ) = pi(µ) + na0 ea

and

Qεσ = µτσ ετ ;

(11.43)

the last condition is equivalent to µ preserving the metric K: µt Kµ = K.

(11.44)

The integers na0 and the matrix µ ∈ Γn+2 uniquely determine the aﬃne symmetry e ∈ E(3) and vice versa. The index i(µ) in (11.43) is obtained as in Corollary 11.3. By (11.22) or (11.26), (11.43)3 can be written in the form Qea = mba eb ,

Qpi = αji pj + lai ea .

(11.45)

Formula (11.45) says that e = (t, Q) ∈ E(3) is a symmetry operation for the essential multilattice M(P0 , ea , pi ) if and only if the orthogonal tensor Q transforms the maximal skeletal lattice basis ea to an equivalent one according to (11.45)1 (that is, Q ∈ P (ea )), and satisﬁes the further conditions (11.45)2 . This proves the group-subgroup relation (11.40) between the point group P (M) and the maximal skeletal holohedry P (ea ) of M. In §11.5.2 low quartz is an example in which the two groups are actually distinct. In what follows, it will be convenient to use the notation P (εσ ) := P (M(P0 , εσ ))

(11.46)

for the point group of a multilattice with descriptors εσ . By analogy with the deﬁnitions (3.25)–(3.28) for 1-lattices, if M(P0 , εσ ) is an n-lattice with metric K we deﬁne the lattice group Λ(εσ ) of M, Λ(εσ ) < Γn+2 < GL(n + 2, Z),

(11.47)

to be the group of all the integral matrices µ ∈ Γn+2 acting isometrically on M, that is, such that (11.43)3 holds for a suitable Q ∈ O; as in the case of simple lattices, (11.44) implies that Λ(εσ ) can be equivalently deﬁned as the group of matrices µ preserving the multilattice metric K: Λ(εσ ) := {µ ∈ Γn+2 : µτσ ετ = Qεσ , Q ∈ O} = {µ ∈ Γn+2 : µt K µ = K}.

(11.48)

Therefore Λ(εσ ) is also denoted Λ(K). The analogs for multilattices of the transformation rules (3.26) for lattice groups are: Λ(ˆ µτσ ετ ) = µ ˆ−1 Λ(εσ )ˆ µ and Λ(Qεσ ) = Λ(εσ ),

(11.49)

for any µ ˆ ∈ Γn+2 and any Q ∈ O. By Corollary 11.6, when the description is essential each matrix µ ∈ Λ(εσ ) uniquely determines an element Q ∈ P (εσ ), and vice versa; indeed, the groups Λ(εσ ) and P (εσ ) are isomorphic, and Λ(εσ ) is necessarily ﬁnite. However, we will see in §11.4.3 that Λ(εσ ) carries more information than

320

KINEMATICS OF MULTILATTICES

P (εσ ): indeed, unlike the point group P (εσ ), the lattice group Λ(εσ ) allows one to reconstruct (the isomorphism class of) the space group S(M).26 In particular, through an analysis of the matrices µ ∈ Λ(εσ ) it is possible to see if S(M) contains any screw rotations or rotary inversions.27 This is not possible by analyzing the elements of the point group P (εσ ). 11.4.2 Fixed sets of lattice groups We introduce the ﬁxed set I(Λ) ⊂ Qn+2 of any subgroup Λ of Γn+2 : I(Λ) := {K ∈ Qn+2 : µt Kµ = K for all µ ∈ Λ} ,

(11.50)

which is the analogue of (3.57). Also corresponding O-invariant ﬁxed sets in the conﬁguration space Dn+2 can be introduced, as analogues of (3.58). Due to (11.49)1 and (11.29), the natural (arithmetic) equivalence relation for the lattice groups and for the metrics of n-lattices is conjugacy within Γn+2 , as is conjugacy in GL(3, Z) in the case of simple lattices. Then, similarly to §3.5, information on the arithmetic symmetry of deformable multilattices is obtained by studying the conjugacy properties of the lattice groups (arithmetic classes) in Γn+2 and the structure of their ﬁxed sets. For instance, it is possible to classify the arithmetic symmetry types of the e elements of Qen+2 or Dn+2 , obtaining an analog of the Bravais types in Q> 3 and B. As mentioned earlier in this chapter, these are interesting questions which are not classical in crystallography; they have not been systematically investigated and await further work. Here we only report some basic results which follow from the analysis of Pitteri (1998) and Pitteri and Zanzotto (1998a); they are analogues, respectively, of statements (3) and (2) in Proposition 3.5 and statement (1) in Proposition 3.8, to which we refer for comparison. We also address the reader to Fadda and Zanzotto (2000), (2001a) for explicit examples of the arithmetic classiﬁcation of monatomic 2-lattices in 2 and 3 dimensions, respectively. Proposition 11.7 (1) In the lattice group Λ(εσ ) < Γn+2 of essential εσ , any element µ has period 1, 2, 3, 4, or 6.28 (2) Λ < Γn+2 is a (subgroup of a) lattice group if and only if it is ﬁnite and it satisﬁes certain additional technical conditions. (3) A subgroup Λ < Γn+2 is a (subgroup of a) lattice group if and only if I(Λ) contains a multilattice metric K ∈ Qm n+2 , that is, a symmetric n+2 by n+2 matrix obtained through (11.8) from εσ satisfying (11.5). 26

27

28

See statement (1) in Proposition 11.8 for details. In the case of nonessential descriptors there is no one-to-one correspondence between the groups P (εσ ) and Λ(εσ ), and in general from the latter one can only reconstruct a subgroup of (the point group P (εσ ) and of) the space group of M(P0 , εσ ). For instance, a quick, suﬃcient but not necessary criterion for the absence of screw rotations and rotary inversions from the space group is that the α-component of the matrices µ belonging to the lattice group be permutation matrices. If the εσ are not essential, the result mentioned in footnote 15 implies that the period of the elements µ ∈ Λ(εσ ) is a multiple of the crystallographic periods 1, 2, 3, 4, 6.

11.4 THE ARITHMETIC SYMMETRY OF MULTILATTICES

321

Statement (1) derives from (11.22), (11.23), (11.43) and the crystallographic restrictions for Q ∈ P (εσ ). Statement (3) says that, unlike for simple lattices, it is not suﬃcient that I(Λ) be nonempty for Λ to be (included in) the lattice group of a multilattice. Here we do not report the technical conditions mentioned in statement (2), which are given explicitly in Pitteri (1998) or Pitteri and Zanzotto (1998a) and simply guarantee that I(Λ) contains a matrix K satisfying (11.5) and (11.8), as requested by (3); otherwise they do not seem very suggestive. Restrictive conditions are also needed for the multilattice counterparts of statements (2)–(4) in Proposition 3.8. An analogue of statement (1) in Proposition 3.5 does not hold because −1 ∈ / Γn+2 for any n > 2, nor is it in general possible to ﬁnd a matrix µ corresponding to −1 through equation (11.43). In this respect, however, the 2-lattices are special because −1 ∈ Γ4 , as detailed in §11.5.1. 11.4.3 Relation between the arithmetic and the space-group symmetries It is remarkable that, as for the simple lattices in §3.3, also for multilattices the geometric aﬃne symmetry is coarser than the arithmetic symmetry. This is shown by following proposition, which indicates certain properties e of the ﬁxed sets in Dn+2 or Qen+2 , thereby establishing that the classical aﬃne, or isomorphism, classes of space groups mentioned in Theorem 11.5 are less numerous than the arithmetic classes of lattice groups in Γn+2 . Proposition 11.8 (1) Two essential n-lattices whose lattice groups coincide have isomorphic space groups. More generally, their space groups are isomorphic if their lattice groups are conjugate in Γn+2 . (2) For n > 2 the converse of (1) does not hold: there are n-lattices whose space groups are isomorphic but whose essential descriptors give lattice groups that are not conjugate in Γn+2 . A proof of these statements is given below. We remark that for simple lattices statement (1) in Proposition 11.8 can be reversed. Thus two 1-lattices L and L have arithmetically equivalent lattice groups, that is, have the same Bravais lattice type (see (3.34)), if and only if their space groups are aﬃnely conjugate. This well known result (for instance in Janssen (1973) p. 120) can be used to deﬁne the Bravais lattice types as the conjugacy classes of the space groups of 1-lattices, as is sometimes done in the literature (Schwarzenberger (1972), Opechowski (1986)). A proof of statement (1) can be given as follows. Given two essential nlattices with arithmetically equivalent lattice groups in Γn+2 , by (11.49) we can always choose their descriptors so that they have the same lattice group, say Λ. So, P0 being an arbitrary point in A3 , suppose that two essential n-lattices M = M(P0 , εσ ), M = M(P0 , εσ ) have the same lattice group Λ, and let S = S(M ), S = S(M ) be the corresponding space groups. Notice that choosing the same base point for the two multilattices does not

322

KINEMATICS OF MULTILATTICES

change the conclusion because translating a multilattice changes its space group by conjugacy with the translation. For an element e = (t , Q ) ∈ S write equations (11.45)2 and (11.43)1 in the equivalent form (set p0 = 0): Q pi + t = pf (i) + nai ea ,

i = 0, . . . , n − 1,

(11.51)

and consider the map Φ : (t , Q ) → ({nai }, µ) from S to Z3n × Λ; here {nai } are the integers in (11.51), and µ is the unique element of Λ that corresponds to Q through equation (11.43)2 . Also, deﬁne the product ∗ of pairs ({nai }, µ) by the rule: ({(n1 )ai }, µ1 ) ∗ ({(n2 )ai }, µ2 ) = {(n1 )af2 (i) + (m1 )ab (n2 )bi }, µ1 µ2 ; (11.52) here, for r = 1, 2, fr is the permutation determined by µr and mr is the m-component of µr (see (11.22)). The transformation of Z3n deﬁned by (11.52) is an example of a Frobenius congruence – see Senechal (1980) or Jariˇc and Senechal (1984). The standard composition rules (2.45) for aﬃne maps together with (11.51), and the uniqueness assertion in Corollary 11.6, imply that the permutation associated with e1 e2 or µ1 µ2 is f1 ◦ f2 , and Φ (e1 e2 ) = Φ (e1 ) ∗ Φ (e2 ),

Φ (e1 ) = Φ (e2 ) ⇔ e1 = e2 ;

(11.53)

therefore Φ is a group isomorphism. Similarly we can construct a group isomorphism Φ : S → Z3n × Λ, and thus (Φ )−1 Φ : S → S is a group isomorphism too. An explicit construction of the aﬃne map of S to S , whose existence is guaranteed by the theorem of Bieberbach, can be found, for instance, in Pitteri (1998). Regarding statement (2), we give a 2-dimensional example, which can be easily made into a 3-dimensional one by adding a third vector e3 orthogonal to the basis e1 , e2 considered below. Take two mutually orthogonal vectors ea of diﬀerent length and, for n = 3 in the planar case (2-dimensional 3-lattices), consider the two sets of descriptors given by: (e1 , e2 , p1 , p2 ), (e1 , e2 , q1 , q2 ), (11.54) 1

p2 = e1 + βe2 ,

q1 = δe1 + γe2 ,

q2 = (1 − δ)e1 + γe2 ,

2

1

1

2

2

0<α< ,

1

p1 = e1 + αe2 ,

2

(11.55)

1

1

2

2

< β < 1, α $= 1 − β; 0 < γ < , 0 < δ < .

(11.56)

These two sets of descriptors give 2-dimensional essential 3-lattices whose primitive rectangular skeletal cell contains two extra points placed as shown in Fig. 11.1. Evidently, the space groups of the 2-dimensional multilattices M = M(e1 , e2 , p1 , p2 ) and M = M(e1 , e2 , q1 , q2 ) coincide and belong to the class denoted by P m in the standard notation of the International Tables (1996) (also Engel (1986)): S(M) = {(t, 1), (t , Reπ2 ) : t, t ∈ L(e1 , e2 )} = S(M ).

(11.57)

11.4 THE ARITHMETIC SYMMETRY OF MULTILATTICES

p2

e2

e2

323

q1 q2

p1 e1

e1

Figure 11.1 Two arithmetically inequivalent planar 3-lattices with the same space group

Furthermore one can check that the lattice groups are as follows: Λ(e1 , e2 , p1 , p2 ) = {1, µ ˆπe2 }, where µ ˆπe2 =



−1  0 0 0

0 1 0 0

−1 −1 0 0 1 0 0 1

Λ(e1 , e2 , q1 , q2 ) = {1, µ ¯πe2 },

  ∈ Γn+1 6



−1  0 0 0

0 1 0 0

−1 −1 0 0 0 1 1 0

(11.58)

 =µ ¯πe

2

(11.59)

(recall that here n = 3 and the matrices µ belong to the groups Γn+1 which are 2-dimensional analogs of the groups Γn+2 ). Both µ ˆπe2 and µ ¯πe2 represent π the operation Re2 belonging to the point group P (M) = P (M ) through equation (11.43)2 . The lattice groups in (11.58) are not conjugate in Γn+1 because their elements µ ˆπe2 and µ ¯πe2 are not so. This is due to the fact that the α-component (that is, the submatrix in the lower right corner) of µ ˆπe2 is the identity, and that the multiplication rules for the matrices in Γn+1 (see (11.21)–(11.23)) force the α-component of any conjugate of µ ˆπe2 in Γn+1 to π be the identity too, unlike the α-component of µ ¯ e2 . By statement (1) of Proposition 11.8, knowing the lattice group allows one to reconstruct a unique isomorphism class of space groups, as anticipated at the end of §11.4.1. This is because each matrix µ in a lattice group determines both an orthogonal operation Q and the index i(µ) in (11.43) – see the proof of statement (1). Thus, as the integral triple na0 is varied arbitrarily in (11.43)1,2 , we obtain the translations t of all the symmetries e = (t, Q) belonging to an entire R(M)-coset in the space group S(M(P0 , εσ )). This cannot be done starting from the point group alone. As already noticed earlier, Proposition 11.8 also shows that the lattice groups provide a ﬁner description of multilattice symmetry than space groups, because metrics or descriptors with the same lattice group always correspond to multilattices with isomorphic space groups, but not vice versa. This supports the interest in studying the arithmetic symmetry of multilattices, as certain phase transitions with a change of symmetry can only be detected through the analysis of the lattice groups and not by means of the space groups of the phases. These are analogues of the transitions that change the Bravais type but not the system of a simple lattice (see §3.5.3, Ericksen (1979), (1996b), Pitteri and Zanzotto (1996a)).

324

KINEMATICS OF MULTILATTICES

Remark 11.1 The fact that space-group symmetry may not be suﬃciently ﬁne for the classiﬁcation of crystalline structures had already been appreciated in the crystallographic literature, where additional criteria have been identiﬁed, for instance the one based on the so-called site-symmetry (see for instance Engel (1986), International Tables (1996)). The 2-dimensional 3-lattices in Fig 11.1, which are a simpliﬁed instance of the example given by Pitteri (1998), share the same space group but are discriminated by the site-symmetry. Pitteri and Zanzotto (1998a) give a more complex example of two 3-dimensional 3-lattices which are arithmetically inequivalent but share both space-group and site-symmetries. Therefore the classiﬁcation based on these criteria together is still coarser than the arithmetic one. Fadda and Zanzotto (2001a) detail the 29 arithmetic classes of 3dimensional 2-lattices, which coincide in this case with the ones resulting from the criterion of Koch and Fischer (1975) (also in International Tables (1996)) based on lattice complexes. Fadda and Zanzotto (2001a) give also arguments in favor of the arithmetic criterion for more complex multilattices. A general comparison of this with other proposed criteria remains to be investigated. Remark 11.2 Since the lattice group and the equivalence class of the space group S(M(P0 , εσ )) are independent of the base point P0 ; and since (footnote 4) we are mostly concerned with the energetic properties of multilattices, which are unaﬀected by translations, henceforth we eliminate the base point P0 from the notation, and indicate M(P0 , εσ ) just by M(εσ ). 11.5 Examples Here are some of the simplest examples of multilattices. The symmetry of diatomic 2-lattices, discussed in §11.5.1 has also been studied by Ericksen (1970) and Parry (1978); here we recover their earlier results from the general theory. Further examples and details can be found in Pitteri and Zanzotto (1998a), Fadda and Zanzotto (2000), (2001a,b). 11.5.1 Three-dimensional 2-lattices and hexagonal close-packed structures In general a 3-dimensional 2-lattice has essential descriptors (ea , p) ∈ D4 where, according to (11.5), p cannot be an integral combination of the lattice vectors. In D4m the nonessential submanifolds mentioned in §11.2.3 are not diﬃcult to characterize: they are given by the equations 1

p = ( ra + sa )ea ,

(11.60)

2

where the sa are integers and the numbers ra are either (1, 1, 1) or a permutation of (1, 1, 0) or of (1, 0, 0). By Corollary 11.3 and deﬁnitions (11.22)–(11.23), the global symmetry of monatomic 2-lattices is given by the group Γ4 < GL(4, Z) whose elements

11.5 EXAMPLES

325

µ have a submatrix α coinciding with an integer, still denoted by α, which is necessarily ±1. Explicitly:   l1 ! b 2 Γ4 :=  m a ll3  : (mba ) ∈ GL(3, Z), lb ∈ Z, α = ±1 . (11.61) 000

α

In the case of diatomic 2-lattices the symmetry is reduced because the permutations f mentioned in the proof of Corollary 11.3 must map each atomic species to itself (see also Proposition 11.2); for n = 2 this forces f to be the identity, so that the global symmetry in the diatomic case is given by the subgroup of Γ4 corresponding to α = 1. Recall that the global symmetry group Γ4 contains the diagonal matrix −1 (while −1 never belongs to Γn+2 for n > 2). This means that all the monatomic 2-lattices are centrosymmetrical because, by equation (11.43)2 , they are always left invariant by the aﬃne isometry represented by (p, −1) when the origin of the aﬃne space is chosen at a point of the lattice. By the discussion in footnote 23, this also implies that no monatomic 2-lattices can have enantiomorphic space groups. Remark 11.3 As for simple lattices (see Remark 3.1), not all conceivable types of multilattices are found in nature. For instance, of the 29 arithmetic classes of monatomic 3-dimensional 2-lattices detailed in the International Tables (1996) or by Fadda and Zanzotto (2001a), only 8 are actually realized: the structures of diamond, of the h.c.p. metals (α-Mg, α-Ti, α-Zn, etc.), of rhombohedral graphite, of the proposed bct5 Si, of β-Sn, of γ-Pu, of α-U and of β-Bi. The Strukturberichte (1915-1940) list only ﬁve of these, under the names A3–A5, A7 and A20. Here is an explicit example of lattice-group symmetry for a 2-lattice. In §3.7.2 we introduced the hexagonal close-packed (h.c.p.) structures, which are of great interest in metallurgy. They are obtained by piling close-packed nets of spheres as densely as possible, in such a way that the resulting arrangement be mirror symmetric with respect to any close-packed plane. The result is a 2-lattice M(ea , p) whose maximal skeletal lattice is generated by the hexagonal basis ea in (3.50), and whose shift p is a √3a c 2 1 1 , (11.62) p= , = e1 + e2 + e3 ; 2

6

2

3

3

2

here a = e1 and c = e3 are the lattice parameters of the hexagonal unit cell of Fig. 3.8(b)-(c). For later purposes, in these structures it is also convenient to introduce the orthonormal unit vectors i , j , k (which are not a lattice basis) such that: i = e1 /e1 ,

j = (e1 + 2e2 )/e1 + 2e2 ,

k = e3 /e3 .

The ideal h.c.p. structure, as deﬁned above, has ratio c 2√ 6∼ = = 1.633 , a 3

(11.63)

(11.64)

326

KINEMATICS OF MULTILATTICES

z

y x

Figure 11.2 The hexagonal close-packed (h.c.p.) structure

but in real crystals, for instance hexagonal metals, the value of this ratio can deviate considerably from the ideal one; usually the term h.c.p. applies also to these cases. In addition, in the literature it is customary to use for the h.c.p. lattices the same systems of indices as for the hexagonal simple lattices, that is, the Miller or Miller–Bravais indices deﬁned in §§3.7.3–3.7.4. Fig. 11.2 shows the h.c.p. 2-lattice M = M(ea , p) which, easily, cannot be described as a simple lattice because the point −p does not belong to M. The above description in terms of the εσ = (ea , p) is thus essential. Based on Corollary 11.6, the arithmetic symmetry of the h.c.p. structure is described by the lattice group Λ(εσ ) < Γ4 whose elements µ preserve the h.c.p. metric K (see (3.50) and (11.62)):   1 2 1 2 2 a

 − 1 a2  2 K= 0  1 2 a 2

−2a

0

a2

0

0

0

c2

1 2 c 2

0

1 2 c 2

2

1 2 a 2

a

  . 

(11.65)

+ 14 c2

It is not diﬃcult to see that Λ(εσ ) is generated by the inversion −1 ∈ Γ4 and by the matrices (recall (11.63)):     1 −1

0

µπi =  00 −10 −10 0

0

0

1 0 0 −1

1 −1 0 0 0 0

π  and µ 3 =  1 k 0

0 0 1 0

1 1 1 −1

,

(11.66)

which represent, according to (11.43) or (11.45), the orthogonal operations π

− 1, Riπ , and Rk3 ,

(11.67)

respectively; these tensors generate the full hexagonal holohedry P (ea ).29 This means that the point group of the h.c.p. 2-lattice (see below (11.38)) coincides with the entire skeletal holohedry. However, to obtain aﬃne symmetry operations for the h.c.p. lattice we must intend the orthogonal trans29

The rotations in P (ea ) form the group Hk listed in Table 3.1 – recall (11.63).

11.5 EXAMPLES

327

formations in P (ea ) = P (ea , p) as coupled with suitable translations, because the h.c.p. structure admits screw axes as well as glide reﬂections; thus the space group of the h.c.p. lattice is not symmorphic. For instance, it can be seen from (11.66) that both the sixfold ‘optic axis’ e3 and the twofold ‘basal axis’ e1 are screw because the corresponding translations p + R(ea ) all have a nonzero component along either axis. Also notice that, since this is a monatomic 2-lattice, in Λ(K) there are elements µ with α = −1 as well 2π/3 π/3 as α = 1, one of the latter being for instance µk = (µk )2 . 11.5.2 The structure of quartz as a 3-lattice Quartz (SiO2 ) is one of the most common minerals in the Earth’s crust, and its properties have been quite extensively investigated. It is a most interesting material from various points of view, theoretical as well as applicative, and we will use it to exemplify various aspects of the models we are adopting for multilattices. At low pressures quartz exhibits two stable phases, called ‘low’ (or trigonal, or α-) quartz and ‘high’ (or hexagonal, or β-) quartz; at room pressure, these phases are observed below and above about 574◦ C, respectively. Following James (1987), we describe the crystalline structure of both the quartz phases above by a monatomic 3-lattice, whose points are the positions of the Si atoms in the actual SiO2 lattice; for our purposes, in fact, the presence of the O atoms and the change of their positions relative to the 3-lattice of the Si atoms can be neglected. We refer to Yoo (1981) or Taylor (1984) for more details on the structure of SiO2 . Our main assumption is that in any conﬁguration of the SiO2 structure the positions of the Si atoms remain compatible with the deﬁnition of a 3-lattice; this implies that n = 3 in (11.6), so that the conﬁguration spaces are D5e and Qe5 . As was seen in §11.2.2, the global symmetry of any 3-lattice, and therefore the global symmetry of both the α- and β-quartz phases, is described by the group Γ5 < GL(5, Z) deﬁned by (11.21)–(11.23) for n = 3:  1 1    Γ5 :=  0 0 0 mba

A=

0

0 0

1

0

0

1

−1 ,

1

l1 l2 l21 l22 l31 l32

αji −1 0

 !  j  : m ∈ GL(3, Z), lbi ∈ Z, (α i ) ∈ A , (11.68) 

−1 ,

0

−1 1

,

1

0

−1 −1

,

0

1

−1 −1

0 , 1

1 0

" . (11.69)

In order to analyze the point-group and lattice-group symmetries of the quartz phases we must describe in detail their structure. In both α- and β-quartz the skeletal lattice type is (primitive) hexagonal (for instance in Dana (1962) vol. 3, p. 9). The hexagonal lattice vectors ea can be chosen as in (3.50), and the point group of the skeletal lattice is thus the hexagonal

328

KINEMATICS OF MULTILATTICES

holohedry P (ea ) generated by the elements in (11.67), with k parallel to e3 – see also footnote 29 and Table 3.1. To illustrate the actual dimensions, we give the values an and cn of a = e1 and c = e3 at normal conditions (room temperature and pressure) for α-quartz – see Levien et al. (1980): cn an = 4.9160 ˚ A , cn = 5.4054 ˚ A, = 1.0995 . (11.70) an However, we will rather focus on the conﬁgurations of α- and β-quartz at the transition temperature and room pressure. Let ea0 be the lattice vectors of β-quartz at the latter conditions, and set a0 = e10 = e20 and c0 = e30 ; the 3-lattice structure of β-quartz can be described by means of the shifts 1 2 1 1 p10 = e10 + e30 p20 = e20 + e30 , (11.71) 2

3

2

3

which, together with (3.50), give the multilattice metric K 0 :   1 2 1 2 1 2 2 a0

 − 12 a20   K0 =  0   1 a20 2

− 14 a20

− 2 a0

0

a 2 0

− 4 a0

a20

0

− 14 a20

0

c20

2 2 c 3 0

1 2 a 2 0 1 2 c 3 0

− 14 a20 1 2 a 2 0

2 2 c 3 0 1 2 c 3 0

1 2 a + 49 c20 4 0 − 18 a20 + 29 c20

− 18 a20 + 1 2 a + 4 0

2 2 c 9 0 1 2 c 9 0

   .  

(11.72)

We denote by ε0σ , σ = 1, . . . , 5 the descriptors (3.50) and (11.71), and show in Fig. 11.3 a projection of the 3-lattice M(ε0σ ) onto the basal plane e10 , e20 orthogonal to the optic axis e30 . Physically, the 3-lattice we are considering is obtained as follows from the actual structure of β-quartz (James (1987)): consider a regular hexagonal √ 3 planar honeycomb of edge d = 3 a, one cell of which is drawn by a plain line in the lower right part of Fig. 11.3. For each one of its vertices consider a left-handed circular helix of radius d2 and pitch c, whose axis goes through that vertex and is orthogonal to the plane of the honeycomb. With the help of that ﬁgure one can see that it is possible to arrange the left-handed helices in such a way that each one meets the neighboring three in equally spaced points along the helix itself, each point having the same projection onto the plane of the honeycomb as the third point following it. Each intersection point of the helices is the site of a Si atom. The circles in Fig. 11.3 are the projections of the helices in the plane of the honeycomb, which is the basal plane e10 , e20 of the 3-lattice. For any lattice point P in the basal plane, the positions relative to P of the two points following it in a counterclockwise rotation along a helix to which P belongs, are either (p20 − e20 , p10 ) or (p20 , p10 − e10 ), depending on the helix. Notice that the 3-lattice M(ε0σ ) of β-quartz is not centrosymmetrical, and this is reﬂected by the fact that its point group P (ε0σ ) lacks the central inversion −1. Indeed, there is no µ ∈ Γ5 that solves the equation −ε0σ = µτσ ε0τ , that is, equations (11.43) and (11.45) for Q = −1 and vectors ea0

11.5 EXAMPLES

329

e02 p02 p01 e01 basal plane

+

1c 3

+

2c 3

Figure 11.3 Projection of the Si atoms of left-handed hexagonal β-quartz onto the basal plane, and descriptors ε0σ = (e0a , p01 , p02 ) for the 3-lattice given by (3.50) and (11.71)

and pi0 given by (3.50) and (11.71). Geometrically, this corresponds to the fact that the aﬃne operation (0, −1) does leave invariant the hexagonal skeletal lattice L(ea0 ) but does not restore the motif: that is, (11.45)1 holds for m = −1 and Q = −1, but there is no choice of αji and lai to satisfy (11.45)2 . However, the two other generators of P (ea0 ) listed in (11.67) do satisfy (11.45) with suitable matrices µ ∈ Γ5 . Indeed, i being the unit π/3 vector of e10 , (11.43) holds if Q is either Rk or Riπ , and µ is, respectively,     π 3

µk

1 −1 0 0 0 0 0 0

 1 = 0

0 0 1 0 0

1 0 0 0 1 1 −1 −1 1 0

1 −1

0

 0 −1 0  π  , µi =  0 0 −1 0 0

0 0

0 0

1 0 0 −1 0 0 −1 −1 0 1

  . (11.73)

π/3

The matrices µk and µπi , which preserve the multilattice metric K 0 in (11.72), are the generators of the lattice group Λ(ε0σ ) of the β-quartz 3rπ/3 lattice M(ε0σ ), which has twelve elements:30 the six powers µk plus π rπ/3 the six elements µi µk for r = 1, . . . , 6. This lattice group corresponds, through (11.43) and (11.45), to the point group P (ε0σ ) < O+ generated π/3 by Rk and Riπ , which is the rotational subgroup Hk of the hexagonal holohedry P (ea0 ) (Table 5.9). The crystal class of P (ε0σ ) is called hexago30

James (1987) uses a diﬀerent but equivalent hexagonal basis: this accounts for the diﬀerent matrices µ in the lattice groups he obtains.

330

KINEMATICS OF MULTILATTICES

nal trapezohedral: this is the actual crystal class of β-quartz, so that the monatomic 3-lattice M(ε0σ ) gives a good approximation of the actual structure of this quartz phase. The low-symmetry α-quartz phase can also be described as a monatomic 3-lattice, with an hexagonal skeleton whose basis at the transition is given in terms of the transformation stretch UT ∈ Sym> :31 ea+ = UT ea0 ;

(11.74)

at the transition temperature θT and room pressure, UT is represented in the basis ea0 of β-quartz by the matrix (Berger et al. (1966))   UT = 

.9973

0

0

.9973

0

0

0

0

.9988

.

(11.75)

For p1 and p2 given by the analogue of (11.71), the shifts are p1+ = p1 + λ(e1+ + 2e2+ ) p2+ = p2 + λ(2e1+ + e2+ ), λ $= 0;

(11.76)

the descriptors ε+ ((11.74)–(11.76)) of the 3-lattice M(ε+ σ = σ ) of α-quartz, and its projection onto the basal plane e1+ , e2+ , are sketched in Fig. 11.4 for λ > 0. For a+ = e1+ = e2+ , c+ = e3+ , k = 32 λ − 14 and (ea+ , pi+ )

+ + + K44 =( 14 +3λ2 )a2+ + 49 c2+ , K45 =(− 18 + 32 λ+ 32 λ2 )a2+ + 29 c2+ , K55 =( 14 +3λ2 )a2+ + 19 c2+ ,

the corresponding multilattice metric is:  1 a2+

K+

 − 1 a2  2 +  0 =   1 2  2 a+

− 2 a2+

0

a2+

0

0 k a2+ 1 2 a 2 +

k a2+ π/3

k a2+

c2+

1 2 a 2 + k a2+ 2 2 c 3 +

2 2 c 3 + 1 2 c 3 +

+ K44 + K45

+ K45 + K55

1 2 a 2 + 1 2 c 3 +

(11.77)

    .   

(11.78)

π/3

Now Rk and µk no longer satisfy (11.43) and (11.45) for the given ε+ σ, while Riπ and µπi still do. However, the same equations are still satisﬁed by 2π/3 π/3 π/3 Rk = (Rk )2 and the following matrix (µk )2 :   0 −1

π 3

2

2π 3

(µk ) = µk

 1 −1 = 0 0 0 0

0 0

0 0 1 0 0

0 −1 1 0 1 0 0 1 −1 −1

 .

(11.79)

2π/3

The matrices µk and µπi thus generate the lattice group Λ(ε+ σ ) of α2rπ/3 quartz, which has six elements: the three powers µk plus the three 31

The stretch UT that gives the macroscopic change of shape and volume at the α-β transformation of SiO2 also describes the deformation of the unit cell of the skeletal lattice L(ea0 ), so in this case the Born rule (§6.2.1) holds. By (11.75) the skeletal lattice is (primitive) hexagonal in both phases, and the transformation stretch UT in (11.74)–(11.75) is symmetry-preserving for the hexagonal vectors ea0 of β-quartz.

11.5 EXAMPLES

331

e+ 2 p2+ p1+ e+ 1 basal plane

+

1c 3

+

2c 3

Figure 11.4 Projection as in Fig. 11.3 for the left-handed α-quartz structure and + + + descriptors ε+ σ = (ea , p1 , p2 ) for the 3-lattice given in (3.50), (11.76) with λ > 0 2rπ/3

elements µπi µk , for r = 1, 2, 3; recall that Λ(ε+ σ ) preserves the metric + K in (11.78). Through (11.43), Λ(ε+ σ ) corresponds to the point group 2π/3 + P (ε+ and Riπ , which gives the σ ) < O , generated by the elements Rk so-called trigonal trapezohedral class to which α-quartz belongs. The point group P (ε+ σ ) of α-quartz is a proper subgroup of the point group P (ε0σ ) of β-quartz, the analog holding for the lattice groups: 0 Λ(ε+ σ ) < Λ(εσ ). +

(11.80) Λ(ε+ σ ),

Thus the metric K in (11.78), which is left invariant by is not left invariant by all the elements of Λ(ε0σ ). This is related to the description of two essentially diﬀerent conﬁgurations in α-quartz, the so-called Dauphin´e twins. Following James (1987), we ﬁnd it instructive to illustrate how such twins of α-quartz arise from the deformation of the helices described earlier for β-quartz. If the radius of the helices is larger than h/2, then any point of intersection of helices with radius h/2 splits into two alternative intersections. The projection of these intersections onto the basal plane is shown in the lower right part of Fig. 11.4. To maintain the threefold rotational symmetry about the normal to the honeycomb, the Si atoms must still be evenly spaced on the helices on such intersections, each atom being on the same vertical (with respect to the basal plane) as the third atom following it on the helix. There are exactly two ways in which this can happen: on an arbitrarily selected helix the Si atoms must be placed on either the ﬁrst or the second of its possible intersections with the neighboring helices; this

332

KINEMATICS OF MULTILATTICES

forces the points on a neighboring helix to be all placed on the second or on the ﬁrst of their intersections, etc., up to completion of the whole structure. Fig. 11.4 shows one of the conﬁgurations obtained in this way, namely M(ε+ σ ) with shifts given by (11.76) for λ > 0. For ﬁxed λ, the other possi− − − + bility is given by the Dauphin´e twin M(ε− σ ), where the εσ = (ea , p1 , p2 ) + + have the same lattice vectors ea as εσ , and shifts p1− = p10 − λ(e1+ + 2e2+ ) ,

p2− = p20 − λ(2e1+ + e2+ ) ,

(11.81)

with the same λ as in (11.76). − One can prove that the twin multilattices M(ε+ σ ) and M(εσ ) are congruent – see (11.87) below – and that their skeletal lattices actually coincide; only their motif is not coincident. For this reason, the Dauphin´e twins are a typical example of motif twins in multilattices (also called merohedral twins), which share the skeleton and have motifs related by an operation belonging to the skeletal holohedry but not to the point group of the whole structure.32 Other well known motif twins in quartz, called Brazilian or optical twins, are given by the conﬁgurations obtained by the central inversion (0, −1), for −1 is an operation belonging to the skeletal holohedry 0 P (ea+ ) = P (ea0 ), but missing from both P (ε+ σ ) and P (εσ ). 0 The descriptors εσ given above are appropriate for left-handed β-quartz, that is, quartz whose helices wind up counterclockwise in the direction of k , the same being true for the descriptors ε+ σ and α-quartz. However, the construction of the left-handed helices described above for β-quartz can be done also with right-handed helices, thus obtaining a β-quartz conﬁguration with enantiomorphic space group.33 To picture the latter conﬁgurations, the triangles and the squares in Fig. 11.3 should be interchanged, and so should the factors 13 and 23 in (11.71). Since a set of descriptors for right-handed β-quartz is given by −ε0σ , the left- and right-handed conﬁgu32

33

+ 0 0 In the case of quartz P (ε+ σ ) < P (εσ ) < P (ea ) = P (ea ). In general, a multilattice M(εσ ) whose point group P (εσ ) is not holohedral is called merohedral, whence the name for these twins, which are often growth twins. When motif twinning can be mechanically produced or eliminated, it constitutes an example of the so-called shufﬂe twinning, which occurs by shuﬄe adjustments (see also §11.8.1) of the internal variables of the multilattice, leaving a suitable skeleton invariant. That the space groups of right-handed and left-handed β-quartz are enantiomorphic can be seen geometrically as follows, with the help of Fig. 11.3; we assume for simplicity the lattice parameters to be the same. Any element (t, Q ) ∈ S(M(ε0σ )) for π/3 which Q = Rk is a screw rotation of π/3 about the axis parallel to k through the center of one of the inﬁnitely many regular hexagons in the basal plane like the one drawn in Fig. 11.3, with essential translation t = (−1/3 + r)ck, r ∈ Z. Given any one of the axes above, by repeatedly applying this operation we obtain an orbit of points on a left-handed helix. The corresponding operation in right-handed β-quartz with the same lattice parameters produces an orbit on a right-handed helix of the same radius and pitch. According to (11.42), any isometry (or more generally aﬃne transformation) mapping the space group of right-handed into the one of left-handed β-quartz should map one of the left-handed orbits above into a right-handed one, and we know this to be impossible for aﬃne maps with positive determinant, because the latter cannot map any left-handed helix into a right-handed one.

11.6 WEAK-TRANSFORMATION NEIGHBORHOODS

333

rations with enantiomorphic space groups are exactly the Brazilian twins, that is, the motif twins obtained through the operation −1∈ P (ea0 )\P (ε0σ ). We recall, however, that −1 does not always produce enantiomorphic conﬁgurations for not centrosymmetrical crystals (footnote 23). The right-handed enantiomorphs of both the Dauphin´e twinned conﬁgurations of α-quartz can be obtained as for β-quartz above; also in this case they are motif twins generated by −1 and are called Brazilian twins. 11.6 Weak-transformation neighborhoods The following proposition is an analogue for multilattices of Proposition 4.1, and states that any n-lattice metric K has a neighborhood NK in the space Qen+2 such that the restriction to NK of the global symmetry given by the action of the group Γn+2 < GL(n + 2, Z) on Qen+2 reduces to the action of (that is, to the symmetry described by) the lattice group Λ(K) in (11.48). An equivalent result, which will not be given here, can be stated in terms of O+ -invariant (see footnote 8 in chapter 2) neighborhoods in the space e Dn+2 of multilattice descriptors. Proposition 11.9 Any K ∈ Qen+2 admits a neighborhood NK in Qn+2 such that µt NK µ = NK for all µ ∈ Λ(K), µt NK µ ∩ NK = ∅ for all µ ∈ Γn+2 \Λ(K). We address the reader to Pitteri (1985a) for a proof of this result, which can be used to cut down the domain and invariance of the constitutive equations for crystalline materials. Any neighborhood with the above properties will be called a weak-transformation neighborhood, wt-nbhd for short. Proposition 11.9 shows that also in the conﬁguration space Qen+2 (or of n-lattices wt-nbhds exist, and a result analogous to Corollary 4.2 on the local reduction of the symmetry of multilattices easily follows. An analysis of the wt-nbhds and ﬁxed sets along the lines of chapters 4 and 5 can thus be developed also for multilattices, and results have been recently obtained for the simplest cases (Fadda and Zanzotto (2000), (2001a,b). Here we make no attempt at a systematic treatment, and rather give an example of variants in the wt-nbhds for multilattices through a brief discussion of the conﬁgurations of quartz. As before, the descriptors ε0σ , σ = 1, . . . , 5 of left-handed β-quartz and the metric K 0 are given in (3.50), (11.71) and (11.72); also, the descrip+ tors ε+ are given in (11.74), σ for left-handed α-quartz and the metric K (11.76) and (11.78). We take the above conﬁguration of β-quartz as the ‘center’, and consider a wt-nbhd NK 0 of K 0 in the space Qe5 , as given by Proposition 11.9. The local symmetry in NK 0 is given by the lattice group Λ(K 0 ) = Λ(ε0σ ) < Γ5 , which is generated by the integral matrices π/3 µk and µπi in (11.73). In §11.5.2 we have also seen that the lattice group e Dn+2 )

334

KINEMATICS OF MULTILATTICES

0 Λ(K + ) = Λ(ε+ σ ) of α-quartz is strictly smaller than Λ(K ):

Λ(K + ) < Λ(K 0 ) < Γ5 ,

(11.82)

π/3

because, for instance, µk in (11.73) belongs to Λ(K 0 ) but not to Λ(K + ). This means that, for a small enough transition stretch UT in (11.74) and small enough λ in (11.76), the metric K + belongs to NK 0 . As in (4.38), consider the Λ(ε0σ )-orbit of K + under the action (11.29): {µt K + µ : µ ∈ Λ(ε0σ )};

(11.83)

one can verify that there are only two distinct variants in it: π π t K + and K − := µk3 K + µk3 −

(11.84)

+

(see (11.73)). In fact, K can be obtained from K not only by means of π/3 π/3 µk as in (11.84), but also through any element in the coset Λ(ε+ of σ )µk 0 0 + Λ(ε+ σ ) in Λ(εσ ) (which consists of all the elements of Λ(εσ ) not in Λ(εσ )), for instance through the twofold matrix µπk :   −1

µπk

0

0 0 1 0 0

 0 −1 = 0 0 0 0

0 0

−1 0 0 −1 0 0 1 0 0 1

 0 +  ∈ Λ(εσ )\Λ(εσ ) .

(11.85)

The metrics K + and K − correspond to the variant orbits O

+

O+ ε+ σ π τ + (µk ) σ ετ

+ := {Rε+ σ : R ∈ O },

:=

{R(µπk )τσ ε+ τ

(11.86)

: R ∈ O }, +

D5e

respectively, in the space of essential 3-lattice descriptors, and the neighborhood NK 0 corresponds to an O+ -invariant neighborhood Nε0σ of ε0σ in D5e . Since both K + and K − are in NK 0 , the variant orbits O+ ε+ σ and + 0 0 O+ (µπk )τσ ε+ are in N , together with the parent β-quartz orbit O εσ . It εσ τ is not diﬃcult to see that in this case Λ(K + ) = Λ(K − ), so that the two metrics K + and K − actually belong to the same ﬁxed set in NK 0 . Consider now the descriptors ε− σ introduced in §11.5.2 to describe the Dauphin´e twinned conﬁgurations of left-handed α-quartz; ε− σ have the same lattice vectors ea+ as ε+ , and shifts given by (11.81). A simple calculation σ − shows that the metric of ε− is K , because, for instance, σ π τ π + ε− σ = (µk ) σ Rk ετ ,

+ π τ + O+ ε− σ = O (µk ) σ ετ .

(11.87)

The variants K and K , and the corresponding orbits and O+ ε− σ, are thus called Dauphin´e twins. As mentioned in §11.5.2, in the mineralogical literature it is customary to indicate as Dauphin´e twin of the multilattice M(ε+ σ ), the multilattice π + M(Rkπ ε+ ) = R M(ε ), which by (11.87) and Corollary 11.3 coincides σ σ k + with M(ε− ). The relevant point here is that while ε and Rkπ ε+ σ σ σ belong + − to the same O -orbit in Nε0σ , the distinct descriptors εσ and Rkπ ε+ σ of +

−

hence

O+ ε+ σ

11.7 THE ENERGY OF A MULTILATTICE AND ITS INVARIANCE

335

+ M(Rkπ ε+ σ ) belong to distinct variant O -orbits in Nε0σ and have distinct e variant metrics in Q5 , due to (11.87). For this reason an elastic theory in which energy depends on the lattice metric is able to distinguish the Dauphin´e twinned conﬁgurations of α-quartz.

11.7 The energy of a multilattice and its invariance Let us consider the case in which the current conﬁguration of the crystal is given by a monatomic34 essential35 n-lattice M(εσ ), σ = 1, . . . , n + 2, the multilattice descriptors εσ = (ea , pi ) satisfying conditions (11.5). We use the results in §11.2 to derive the global invariance of multilattice constitutive equations. To do this, we ﬁrst assume the free energy density (per unit skeletal cell) of the multilattice, at any given temperature, to be determined by the current atomic positions, hence to be a smooth enough function36 φ¯ of the temperature θ and the current descriptors εσ = (ea , pi ): ¯ a , pi , θ) = φ(ε ¯ σ , θ). φ = φ(e (11.88) e The domain of φ¯ is in principle the whole space Dn+2 of essential multilattice descriptors εσ , together with some suitable interval I of temperatures. e is properly As was mentioned in footnote 35, the conﬁguration space Dn+2 m contained in the general conﬁguration space Dn+2 of all n-lattices. Galilean invariance37 applied to φ¯ is equivalent to the equality ¯ a , pi , θ) = φ(Qe ¯ φ(e (11.89) a , Qpi , θ), e for all (ea , pi ) ∈ Dn+2 , Q ∈ O+ and θ ∈ I. Since – see (2.10):

pi = pai ea ,

pai = C ab pbi ,

C ab = e a · e b ,

C ab Cbk = δka ,

(11.90)

with O-invariant coeﬃcients pai , (11.89) implies that φ depends on the current multilattice descriptors εσ only through their scalar products – that is, through the current multilattice metric K in (11.8) – and the sign s = sgn (e1 · e2 ∧ e3 ): ¯ a , pi , θ) := φ(K, ˆ φ = φ(e s, θ),

Kστ = εσ · ετ .

(11.91)

The function φˆ is deﬁned for all the matrices K in the set Qen+2 of essential 34 35

36 37

We assume the multilattice to be monatomic for simplicity; the changes for the polyatomic case should be obvious from the remarks in §§11.2.1 and 11.2.2. As discussed in §11.2.3, there are various problems related to the symmetry of nonessential descriptors, and we will assume that the allowed conﬁgurations of the crystal be only essential n-lattices. Since the nonessential descriptors are conﬁned to low-dimensional submanifolds of the conﬁguration spaces, and since for any multilattice it is always possible to ﬁnd essential descriptors, this is not too restrictive. An approach to the invariance of constitutive equations in a neighborhood of a nonessential conﬁguration, with interesting remarks about smoothness of the related elastic reduction, is proposed by Parry (1981), (1982b). For simplicity we use the same symbols as for the functions of simple lattices; there should be no danger of confusion, as the independent variables are diﬀerent. See footnote 8 in chapter 2.

336

KINEMATICS OF MULTILATTICES

n-lattice metrics. The free energy function in (11.91) can be equivalently represented through a pair of generally distinct functions φ = φˆs (K, θ),

s = + or s = − ,

(11.92)

deﬁned over the same domain Qen+2 × I. The two functions φˆ+ (K, θ) and φˆ− (K, θ) account for the diﬀerent mechanical responses of mirrorsymmetric crystals which can have diﬀerent energies when K is the same. Formula (11.92) can also be regarded as deﬁning over Qen+2 × I a doublevalued energy function for n-lattices. Since we assume φ to only depend on the positions of the multilattice points in the current conﬁguration M(εσ ), any set of essential descriptors generating M(εσ ) necessarily gives φ the same value. Corollary 11.3 says that the essential descriptors of M(εσ ) are determined up to linear transformations by elements of the matrix group Γn+2 . Combining this with (11.89) we obtain the following basic invariance38 for the energy function of essential multilattices: ¯ σ , θ) = φ(µ ¯ τ εσ , θ) = φ(µ ¯ τ Qετ , θ), φ(ε σ σ

(11.93)

e which must hold for all εσ ∈ Dn+2 , all µ ∈ Γn+2 , all Q ∈ O+ and θ ∈ I. Equivalently, denoting by mµ the m component of a matrix µ ∈ Γn+2 (see (11.22)), the invariance of the doublevalued function φˆs is given by

φˆs (K, θ) = φˆssm (µt Kµ, θ),

sm := sgn det mµ ,

(11.94)

for all K ∈ Qen+2 , all µ ∈ Γn+2 and all θ ∈ I. Formula (11.94) says that any µ ∈ Γn+2 such that det mµ > 0 belongs to the invariance group of each of the functions φˆ+ and φˆ− , while any µ such that det mµ < 0 transforms one function into the other. It is in this sense that the group Γn+2 gives the global invariance of the free energy and of the other thermodynamic potentials of any essential monatomic n-lattice. In §11.7.2 we give an example, based on quartz, of how the energy domain of a multilattice can be restricted to the wt-nbhds in Qen+2 mentioned in Proposition 11.9. As for simple lattices, this restriction reduces the energy invariance to the usual one given by the crystallographic space groups, as is assumed in the classical theories. 11.7.1 Minimizing out the internal variables of complex crystals Here we are mostly interested in describing twinning, microstructure formation, and other transition-related phenomena from a static point of view, that is, through analysis of the conﬁgurations that minimize the free energy 38

For polyatomic multilattices the invariance is reduced as mentioned before the proof of Corollary 11.3. Parry (1978) gives a method for constructing functions satisfying the analogue of (11.94) for the case of 2-dimensional diatomic 2-lattices, which was discussed in §11.5.1.

11.7 THE ENERGY OF A MULTILATTICE AND ITS INVARIANCE

337

of the crystal. For multilattices the variables involved in the minimization are the descriptors (ea , pi ). Now, such a free energy minimization may be conveniently done in steps,39 that is, by ﬁrst eliminating from the potential the internal variables pi ; this gives a (possibly multivalued) free energy function that depends on the sole lattice vectors ea – see for instance Ericksen (1980b), (1997), (2001b), Parry (1981), (1982b), (1984a), (1987). This is an interpretation of a procedure introduced into molecular theory by Born (1915) (see also Stakgold (1950), Born and Huang (1954), Ericksen (1980a)), who constructs the strain energy of a crystal from a molecular model by making the assumption that the internal variables are free to adjust instantly, so as to relax stress by reaching some (meta)stable equilibrium conﬁguration of the motif in the deformed crystal.40 This elimination procedure for the shifts pi has several interesting consequences. The most straightforward way to minimize out the shifts in the free energy, under the only assumption that it is bounded below, is to deﬁne the (internally) equilibrated free energy φE (ea , θ) of the multilattice by ¯ a , pi , θ); φE (ea , θ) = inf φ(e (11.95) pi

under mild assumptions for φ¯ the inﬁmum in (11.95) is actually attained, hence is a minimum. The function φE so deﬁned results O- and GL(3, Z)invariant, that is, it satisﬁes the analogue of (6.3): φE (ea , θ) = φE (mba Qeb , θ),

Q ∈ O,

m ∈ GL(3, Z).

(11.96)

In some circumstances the absolute minimization (11.95) can be too strict ¯ a , pi , θ) may be of an assumption, because also the relative minimizers of φ(e 39

40

This is a special case of the procedure of constructing ‘subpotentials’ in the search for (absolute) minimizers of the original potential, as presented, for instance, by Flory (1961) and Ericksen (1981a); see also Chipot and Kinderlehrer (1988) and Fonseca (1988) for a relation with the quasiconvex envelope of the energy function for elastic crystals. This adjustment is achieved through suitable shuﬄe movements of the atoms within the unit cell. Often, such shuﬄes are depicted as involving rotations of molecules, etc., as in the case of calcite by Klassen-Nekliudova (1964); however, the motif shufﬂes in a complex crystal are always given by suitable relative rigid translations of the constituent simple lattices. Such translations are described by the evolution to equilibrium of the internal variables pi for ﬁxed ea . We stress that the assumption of instantaneous shuﬄing of the motif to equilibrium is just a simpliﬁed model for the cases in which the internal variables have relaxation times that are very short compared to the characteristic time of evolution for the other variables in the model. Quartz is treated in this way (see James (1987), Ericksen (2001b)). However, the hypothesis of instantaneous equilibration may not be always appropriate; in fact, in some materials the internal variables can be slow to equilibrate, and this process may couple with those involving the macroscopic variables, such as the deformation of the skeletal lattice of the crystal. In these instances, rather than the equilibrium conditions (11.97) or (11.99) written below, other evolution laws for the internal parameters are better suited to describe the observed phenomena. An interesting example is mentioned in §11.8.3, following Bhattacharya et al. (1993). As noted by Germain (1986), there are also physically interesting cases in which the relaxation time of the internal variables can be very long. In these cases, as a better approximation, the internal variables are given ﬁxed values, with no evolution to equilibrium.

338

KINEMATICS OF MULTILATTICES

physical interest; they may become important, for instance, in the analysis of phase transitions involving the motif. Following Ericksen (1980b), we can then minimize the internal variables pi out of φ¯ by means of functions pi (ea , θ) such that the condition pi = ¯ ¯ a , pi , θ) ≥ φ(e ¯ a, ¯ φ(e pi (ea , θ), θ) (11.97) holds for any (ea , pi , θ) in a suitable open set. In this way one can deﬁne a new energy function φ¯E for the internally equilibrated crystal: ¯ a, ¯ φ¯E (ea , θ) = φ(e pi (ea , θ), θ). (11.98) ¯ Under mild smoothness assumptions on φ the minimizing function pi = ¯ pi (ea , θ) is a solution of the critical-point condition ∂ φ¯ (ea , pi , θ) = 0, (11.99) ∂pk which is sometimes used instead of (11.97) to eliminate the shifts pi from the energy function; in this case we obtain, through (11.98), an energy function for an internally equilibrated crystal in which also the metastable equilibria of the motif can be considered. For brevity we use the same notation for the solutions of (11.99) and of (11.97). In general, the functions pi = ¯ pi (ea , θ) satisfying the latter conditions are ¯i deﬁned for the same ea and θ may not unique. Such diﬀerent functions p produce diﬀerent energy functions φ¯E according to (11.98); in other words, the function φ = φ¯E (ea , θ) in (11.98) should be regarded, in general, as multivalued. This situation is considered for instance by Ericksen (1980a), Parry (1981), (1987), Sellers (1984), James (1986a), (1987).41 The multivaluedness disappears when only the absolute minimum is con¯ the funcsidered, as in (11.95). In this case, however, even for a smooth φ, tion φE is continuous but not necessarily smooth: when two (or more) relative minimizers coexist in a certain open set, and somewhere in that set exchange their role as absolute minimizers, there typically their ﬁrst derivatives are diﬀerent, and thus certain derivatives of the absolutely minimized energy suﬀer a jump. For instance, Parry (1981) shows how structural transitions in monatomic multilattices generally introduce jumps in the second derivatives of the potential φE . An example of the same eﬀect of the elimination of the internal parameters is given explicitly, in a diﬀerent context, by Truskinovsky and Zanzotto (1995), (1996). Therefore the prescription (11.95) for the internally equilibrated crystal produces a free energy function which has the same invariance (11.96) as the one of a simple lattice but not, in general, its smoothness. The above energies are very useful in elastic models of crystals when the details of the motif are not relevant for their behavior. This is the case of 41

An interesting task is to describe the general invariance of the multivalued function ¯ A related φ¯E which follows from the invariance (11.93) of the multilattice energy φ. problem is the treatment of symmetry-induced bifurcations for the nonunique criticalpoint functions ¯ pi (ea , θ), which can describe the equilibrium conﬁgurations of certain twins in crystals such as the Dauphin´e twins in quartz (§11.7.2).

11.7 THE ENERGY OF A MULTILATTICE AND ITS INVARIANCE

339

various complex crystalline materials, such as shape-memory alloys, which are eﬀectively modelled as simple lattices in much current literature, with an energy satisfying (11.96) (see chapters 7–10). In the sense speciﬁed by (6.3) and (11.96), GL(3, Z) describes the invariance of all crystals alike, and not only of simple lattices; this is remarked by Ericksen (1980b), who discusses some consequences of invariance and lack of smoothness of crystal energies (see also §7.1.1). The elimination of the internal variables outlined above, which generally produces multivalued elastic energy functions, has interesting analogies with the representation of stress-strain relations in implicit form; there, certain internal variables come out naturally in the representation near a point where such a relation cannot be solved for the stress in terms of the strain. We address the reader to Cardin (1991) for details, and to Cardin and Spera (1995) for an interesting application. 11.7.2 Local invariance of multilattice energies; the example of quartz The description of energy functions suitable for α- and β-quartz given here follows essentially James (1987), who considerably extended an earlier theory by Thomas and Wooster (1951); their theory is in turn equivalent to the linearization of the theory of elastic dielectrics of Toupin (1956). Unlike the ﬁrst, neither of the two latter references uses the crystalline structure explicitly. Here we stop before attacking two important issues: 1. The elimination of the ‘internal variables’ pi (James (1987)), which produces an elastic double-valued energy with invariance properties of the kind proposed earlier by Ericksen (1982b) and Sellers (1984). 2. The actual description of the α–β phase transformation, which is given by Ericksen (2001b) by means of an energetic analysis based on the geometric description of the two phases given in §11.5.2. A general, systematic scheme for weak phase changes in essential multilattices is being proposed by Pitteri (2002); for quartz this gives the α–β transition as one among a ﬁnite number of generic phase changes for β-quartz. As already mentioned, at low pressures quartz has two phases, α and β, whose stability depends on temperature and whose structures are described as 3-lattices in §11.5.2. We maintain the notation used there. Also, in agreement with the discussion of multilattice energies in §11.7, we assume the free energy density of the quartz 3-lattice in a generic conﬁguration to be a smooth enough function ¯ a , pi , θ) = φ(ε ¯ σ , θ) φ = φ(e

(11.100)

as in (11.88) for i = 1, 2, or σ = 1, . . . , 5, with domain D5e and invariance given by the groups O+ and Γ5 as speciﬁed in (11.93). Equivalently, the 3lattice energy is given by the doublevalued function φ = φˆs (K, θ) in (11.92) with the invariance in (11.94).

340

KINEMATICS OF MULTILATTICES

By analogy with the assumptions made in §6.4.2 for simple lattices, and in agreement with the observed stability properties of the α- and β-phases of quartz, we assume that, for θ > θT ≈ 574◦ C, φ¯ has an orbit of absolute minimizers O+ ε0σ (θ), and for θ < θT an orbit of absolute minimizers given 0 + by O+ ε+ σ (θ), where εσ (θ) and εσ (θ) are functions given by (3.50), (11.71) and by (3.50), (11.76), respectively. Then the invariance properties (11.93) of φ¯ imply that there exist for φ¯ two inﬁnite sets of minimizing O+ -orbits in D5e , respectively given, for θ > θT and θ < θT , by O+ µρσ ε0ρ (θ)

and O+ µρσ ε+ ρ (θ),

for (µρσ ) ∈ Γ5 .

(11.101)

In the same way φˆs (K, θ) is absolutely minimized by the elements µt K 0 (θ)µ of the Γ5 -orbit of K 0 (θ) for θ > θT , and by the elements µt K + (θ)µ of the Γ5 -orbit of K + (θ) for θ < θT , where K 0 (θ) and K + (θ) are the metrics of ε0σ (θ) and ε+ σ (θ), respectively. Each set of symmetry-related minimizers corresponds to a phase of quartz. The descriptors are given explicitly by + + ε+ σ (θ) = (ea (θ), pi (θ)),

ε0σ (θ) = (ea0 (θ), pi0 (θ)), 1

2

2

3

p10 (θ) = e10 (θ) + e30 (θ), p1+ (θ) = p2+ (θ) =

1 2 1 2

1

1

2

3

p20 (θ) = e20 (θ) + e30 (θ),

(11.102) (11.103)

2

e1+ (θ) + e3+ (θ) + λ(θ)(e1+ (θ) + 2e2+ (θ)), 3

1

e2+ + e3+ (θ) + λ(θ)(2e1+ (θ) + e2+ (θ))

(11.104)

3

(compare with the deﬁnitions (11.74), (11.71) and (11.76) of ea0 , ea+ , ε0σ and ε+ σ , and with the minimizers that are not symmetry-related in §6.4.2). As we have seen in §11.6, for small enough λ in (11.76) the O+ -orbit is contained in a Galilean-invariant wt-nbhd Nε0σ (θT ) ⊂ D5e of the higher-symmetry β-phase conﬁguration ε0σ (θT ) at the transition temperature (correspondingly, s is ﬁxed to the sign of e10 (θT ) · e20 (θT ) ∧ e30 (θT ), and the metric K + (θ) is contained in a neighborhood NK 0 (θT ) of the βquartz metric K 0 (θT )). We can thus restrict the domain of φ¯ to Nε0σ (θT ) , or, equivalently, select the sheet φˆs and restrict its domain to NK 0 (θT ) . In this way, in analogy with the discussion in §6.5.1, we obtain a localized nonlinear theory in which the allowed lattice deformations are ﬁnite but not too large, and symmetry is dictated by the ‘positive’ subgroup (det m = 1) of the lattice group Λ(ε0σ (θT )) generated by the matrices (11.73) – a ﬁnite group which in this case coincides with Λ(ε0σ (θT )). This greatly reduces the complexities of the theory while leaving much of its descriptive ability intact; indeed, many phenomena in the ﬁeld of phase transitions in multilattices (as for the weak transitions of simple lattices) occur with ‘ﬁnite but small’ changes in the multilattice descriptors. For instance, this is the case for motif transitions such as the α–β transition in quartz. As was already discussed in §11.6, the neighborhoods Nε0σ (θT ) or NK 0 (θT ) are enough for the description of Dauphin´e twinning in quartz. Indeed, two O+ ε+ σ (θ)

11.8 TWINNING IN MULTILATTICES

341

Dauphin´e twinned conﬁgurations of α-quartz are described by the couple of temperature-dependent O+ -orbits (wells) O+ ε+ σ (θ)

+ π ρ + and O+ ε− σ (θ) := O (µk ) σ ερ (θ),

(11.105)

which are both in Nε0σ (θT ) (see (11.85), (11.86)). Correspondingly, the metπ π t rics K + and K − = µk3 K + µk3 are both in NK 0 (θT ) ((11.84)). The energy φ¯ [φˆs ] in its reduced domain Nε0σ (θT ) [NK 0 (θT ) ] can then be used to model various aspects of the mechanics of the α–β transition of quartz, including the formation of the Dauphin´e twins and the elimination of the internal variables (James (1987), Ericksen (1982b), (2002c), Pitteri (2002)). 11.8 Twinning in multilattices The analysis of twinning presented in chapter 8 is well established for crystalline solids that can be described as simple lattices from the molecular point of view. Moreover, for simple lattices the Born rule connects the molecular and the continuum descriptions of the crystalline body, this being also true for twinning deformations. On the contrary, in certain twinning deformations of multilattices not all the points in the skeleton deform according to the twinning shear; this was already clear to the german mineralogists of the early 1900s, as mentioned in footnote 4 of chapter 8. This experimental fact calls for the Born rule to be weakened, as we brieﬂy discussed in §8.4.2. The displacements of the skeletal lattice points from the positions they would occupy if they did shear to their actual positions are usually called shuﬄe displacements. We shall refer to these as structural shuﬄes because they refer to the points of the skeleton. On the other hand, for many crystalline solids, for instance hexagonal metals, not only the deformation of the lattice is of interest, but also the displacements of individual atoms in the motif with respect to the skeletal lattice points, displacements that we call motif shuﬄes. In general, the movements of the individual atoms is such that the ﬁnal result is equivalent to a shearing of only a given fraction of the skeletal lattice points, that is, of only a part of the main structure, and to a shuﬄing of the rest, together with further shuﬄing in the motif. Schematically, motif shuﬄes are involved in twinning of structures that are not holohedral, that is, are such that some of the symmetry operations restoring the skeletal lattice (11.13) do not restore the whole multilattice (11.4). In these cases it is very common to observe twins in which the individuals are related by one of the symmetry operations of the skeleton but not of the motif. This is the case of Dauphin´e and Brazilian twins in quartz (§11.5.2), which are characterized by a diﬀerent, symmetry-related disposition of atoms about the same skeletal lattice. These twinning modes are called merohedral by Friedel (1926) and Cahn (1954). Due to the presence of both structural and motif shuﬄes, twinning in

342

KINEMATICS OF MULTILATTICES

multilattices is more diﬃcult to analyze than in simple lattices, one of the problems being a rationale of shuﬄe displacements. For instance, Kelly and Groves (1970) describe the complex situation of hexagonal close-packed metals, whose structure is yet relatively simple. A model for shuﬄes is provided by Jaswon and Dove (1957), and seems to apply to 2-lattices, providing also information on the amount of shear as a function of the composition plane. This mechanism is nevertheless too rough to account for the details of shuﬄe displacements in general. The model is considerably improved by Bilby and Crocker (1965) (also Bevis and Crocker (1968), (1969)), who provide formulae for the shuﬄe displacements in both the structural and motif cases, in our terminology. Since they only consider the conventional twinning operations, their results are included in the formulae of §11.8.1, which are obtained without using their assumption that the motif moves rigidly. Also, Bilby and Crocker (1965) are aware that kinematics alone is insuﬃcient to select the twins that actually occur, among all the kinematically possible ones, and that free energy minimization should be considered; but their kinematics does not indicate which variables should enter the free energy function. From the continuum point of view, it seems that the theory of elasticity may not suﬃce to describe complex crystals, and that a theory of structured continua should be used, following the lead of the Cosserats (see Capriz (1989), Maugin (1990)). In this case the shifts would be the molecular counterpart of the microstructural variables in the continuum theory. The analysis of such a continuum theory and, above all, of the counterpart of the Born rule for complex crystals is the subject of ongoing research (Ericksen (1997), (1999), (2000b), (2001c), (2002b)). Here we only sketch a special case: the characterization of Type-1 and Type-2 twins in §8.3.5 as those corresponding to the matrices m ∈ GL(3, Z) with m2 = 1 has an easy and immediate extension to multilattices; indeed, we analyze the twinning modes associated with elements µ ∈ Γn+2 such that µ2 = 1. We show how the theory states precisely what the shuﬄe displacements are, for any such µ. The related formulae include, in a form that involves multilattice descriptors and global symmetry elements, the ones of Bilby and Crocker (1965).42 The formulae in §11.8.1 are phrased in terms of multilattices, but can be also applied to (skeletal) simple lattices in which the twinning shear only transforms a sublattice. Analyzing these two cases separately would be less diﬃcult for the reader, but lenghty; instead, by way of example, in §11.8.2 we detail the twinning mode of βtin, which is a 2-lattice, using a lattice cell which is twice the unit cell, and introducing additional shifts to describe the positions of the atoms inside 42

We do not discuss here how the theory can be used to also analyze pseudotwinning, this being a mode of deformation which diﬀers somewhat from twinning and is analyzed by Cahn (1954) and Bolling and Richman (1965). A simpliﬁed version of pseudotwinning is modelled by Pitteri (1985b) by extending the symmetry of crystals to include color symmetry (footnote 11).

11.8 TWINNING IN MULTILATTICES

343

the enlarged cell.43 So, in this example, the twinning deformation consists of shear and of both structural and motif shuﬄes, the two types not being substantially diﬀerent and being treated at the same time and in the same way. The theory accounts not only for the directions of n and a and the amount of shear, but also for the shuﬄe displacements, in agreement with the data reported by Hall (1954), Klassen-Nekliudova (1964) and Kelly and Groves (1970). These results are presented in Table 8.1 and in Figure 11.5. As an additional example, we sketch analogous results for the (01¯12) twinning mode in hexagonal close-packed metals. With the help of Fig. 11.6 we associate one matrix µ ∈ Γ6 with that twinning mode, by means of which we correctly reproduce the atomic displacements when the lattice √ parameters ratio c/a is either bigger or smaller than 3; the displacements look quite diﬀerent in the two cases, as can be clearly seen in that ﬁgure. 11.8.1 A proposal for a class of twins For any given simple lattice conﬁguration, generated by a basis ea , one Type-1 and one Type-2 twin are associated to any tensor H ∈ G(ea0 ), H 2 = 1, with the exception of those in P (ea ) (§8.3.5). Moreover, these twins ﬁt the vast majority of the experimental data. Since for multilattices the group Γn+2 is a natural analogue of GL(3, Z) for simple lattices, we study the properties of pairwise homogeneous conﬁgurations associated with elements µ ∈ Γn+2 such that µ2 = 1. We proceed by constructing the multilattice analogue of a Type-1 twin, a similar construction being feasible for Type-2 twins. To do this we ﬁrst recall the explicit form (11.26) of the transformation (11.24) of multilattice descriptors associated with µ: ¯ ea = mba eb ,

¯ pi = αji pj + lai ea .

Consider a Type-1 twinned multilattice M locally and away from boundaries. According to experience, the crystal occupies two regions R1 and R2 as described in §8.1, and the conﬁguration of M in R2 can still be obtained from the one in R1 , up to a translation, by means of either the orthogonal Rnπ introduced in (2.32) or its negative, which is the mirror reﬂection across the composition plane. For multilattices, which may not be centrosymmetrical, these two operations need not be equivalent, and only one of them may be active in a twinning mode. We denote the twinning operation by R, this being either Rnπ or its negative, and recall that the multilattice in R2 is generated by the vectors ˜σ = Rεσ . ε (11.106) The assumption µ2 = 1 implies also m2 = 1. Therefore the basis ea transforms under the m component of µ as in (11.26), and we can still construct vectors a and n such that the twinning equation (8.24) holds when R or 43

´ Enlarged cells have been considered, for instance, by Koptsik and Evarestov (1980) and Ericksen (1982a).

344

KINEMATICS OF MULTILATTICES

its negative is the conventional operation Rnπ . Let us deﬁne Σi := RSpi − αji pj − lai ea ,

(11.107)

and notice that, by the condition µ2 = 1, mba lai + lbj αji = 0 and α2 = 1 .

(11.108)

Using (11.107) twice and (11.108), we have RS Σi = pi − αji RSpj − lai RSea = pi − αji (Σj + αkj pk + laj ea ) − lai mba eb =

−αji Σj

(11.109)

,

hence (11.107) is equivalent to Spi − RΣi = αji Rpj + lai Rea = S (pi + αji Σj ).

(11.110)

Therefore a set of shifts in R2 can be obtained by adding the vectors ∆i := −RΣi = αji S Σj

(11.111)

to the sheared shifts Spi . We conclude that the ∆i are the shuﬄe displacements, which all vanish if and only if the whole multilattice is restored by the twinning shear in R-symmetric positions. Unlike in Bilby and Crocker (1965), here the ∆i are expressed in terms of the multilattice descriptors εσ appearing in the constitutive equations, and of the matrix µ in their global symmetry group Γn+2 . Notice that, if µ satisﬁes (11.43)1 , hence corresponds to an element of the crystallographic space group of the conﬁguration represented by K, the twinning shear vanishes, and so do the Σi by (11.107). In this case the transformation leaves the multilattice invariant across the interface I, and the latter has no physical meaning. Therefore, in the twin deﬁnition we should disregard the matrices µ ∈ Γn+2 which satisfy (11.43)1 . This restriction still allows R to map the skeletal lattice generated by ea onto itself, while acting on the shifts in a nontrivial way. In this case the twinning operation produces in R2 a multilattice whose skeletal lattice is identical with the one in R1 but whose shifts diﬀer essentially from those in R1 . This happens, for instance, in Dauphin´e twinning in quartz (§11.5.2). For 2-lattices, that is, for n = 2, α is an integer, in fact ±1. Then (11.109) respectively reduces to RS Σ = ∓Σ. If α = −1, then Σ · n = 0; if α = 1, then Σ = k(2n − a) for some real number k. In this way we rationalize the X and Y mechanisms proposed by Jaswon and Dove (1957): the X mechanism corresponds to α = −1 and Σ parallel to the composition plane, whereas the Y mechanism corresponds to α = 1 and Σ having a component orthogonal to that plane. In fact the Y mechanism requires Σ to be parallel to n; we consider our condition to be compatible with it because in most cases, for n = 1, a is much smaller than 1, hence Σ is almost parallel to n.

11.8 TWINNING IN MULTILATTICES

345

11.8.2 Two examples Throughout this section the twinning operation, R, is −Rnπ ; this choice ﬁts the existing descriptions of both twinning modes analyzed here. Using Figure 11.5 as a reference, we see that β-tin has the structure of a body-centered tetragonal 2-lattice, generated by the vectors 1 1 e1 = (a, 0, 0), e2 = (0, a, 0), e3 = (a, a, c), p1 = 0, a, (x−1)c , (11.112) 2

2

for a suitable x, 0 < x < 1, in an orthonormal basis (i , j , k ). We introduce new vectors by the equalities v1 = e3 , v2 = (0, a, −c) = e1 + 2e2 − 2e3 , v3 = (0, 0, 2c) = −2(e1 + e2 ) + 4e3 ,

and (11.113)

which are a special instance of the general relation (3.6) selecting sublattice vectors. In this case the matrix (v ba ) in (3.6) has determinant 2. We introduce the two additional shifts 1 1 1 1 p2 = v2 + (x + 1)v3 and p3 = v3 p1 = (v2 + xv3 ) , (11.114) 2

2

2

2

thus using (va , p1 , p2 , p3 ) as (nonessential) 4-lattice descriptors for tin. As is detailed by Pitteri (1985b), the reported twinning mode of β-tin is associated with these descriptors and the matrix µ ∈ Γ6 such that 

1

(la1 ) = (la2 ) = (la3 ) = (0, −1, −1) ,    0

m =  −1

0

−1

−1

0

−1 

1

and α =  0

0

0

0

(11.115)

0

0

1.

1

0

(11.116)

First of all, by the general formulae in §8.3.5 the composition plane and the shear direction have indices (111) and [0¯ 11] with respect to the double cell, hence (see Fig. 11.5(a)) have indices (031) and [0¯13] with respect to the primitive tetragonal cell associated to the body-centered tetragonal basis ea in (11.112). The amount of shear s is (1 − 3κ2 )/2κ, κ = c/a; this is the theoretical shear proposed by Hall (1954), and for tin has the value 0.113, in some agreement with the value of 0.119 reported by the same author or Klassen-Nekliudova (1964). Secondly, from (8.24), (11.107), (11.115) and (11.116) we deduce 1

1

1

2

2

2

Σ1 = (1 − x)(v2 + v3 ), Σ2 = (1 − x)v2 , Σ3 = (1 − x)v3 ,

(11.117)

hence, by (11.110)1 , a set of shifts ¯ pi in the region R2 is 1

1

¯ p1 = Sp1 + (x − 1)R(v2 + v3 ), ¯ p2 = Sp2 + (x − 1)Rv2 , 2

1

and ¯ p3 = Sp3 + (x − 1)Rv3 .

2

(11.118)

2

This result is in good agreement with the mechanism for atomic displacements proposed by Hall (1954), as can be seen in Figure 11.5(b). There,

346

KINEMATICS OF MULTILATTICES

k R(v 2 + v 3 ) Rv2 Rv3 2 3 1

v3

p3

p2 p1 v2 j atoms that shear

(b)

(a)

Figure 11.5 (a) twinning in β-tin. (b) details the shuﬄe displacements: for each ¯i (dashed) are drawn; their diﬀerence agrees with shift pi the vectors Spi and p (11.118)

for i = 1, 2, 3, we have drawn ¯ pi and Sp i , and their diﬀerence agrees with (11.118); one can compare those diﬀerences with the directions of the second summands in (11.118), which are also drawn in the upper part. As a second example, we consider h.c.p. metals, which are 2-lattices generated by the lattice vectors ea in (3.50) and by the shift a √3a c 2 1 1 p1 = , , = e 1 + e2 + e3 2

6

2

3

3

2

(see §11.5.1). We introduce the new sublattice vectors √ v1 = e1 , v2 = (0, 3a, 0) = e1 + 2e2 and v3 = e3 ,

(11.119)

and notice that the matrix (v ba ) in (3.6) has determinant 2 also in this case. We introduce the additional shifts 2 1 1 1 1 p2 = v2 + v3 and p3 = (v1 + v2 ) p1 = (v1 + v2 + v3 ) , (11.120) 3

2

2

2

3

thus describing the structure as a nonessential 4-lattice. The most common twinning mode in h.c.p. metals corresponds to the descriptors above and the matrix µ ∈ Γ6 such that     1 0 0 0 0 1 (la1 ) = (0, −1, 0) a m = 0 0 −1  , (l 2 ) = (0, −1, 1) , and α =  0 1 0  . (la3 ) = (0, 0, −1) 0 −1 0 1 0 0 (11.121)

11.8 TWINNING IN MULTILATTICES

347

Rv 3

Rv 2 2

p-2 p-1

1

3

p-3

n

p2 p1

v3

v2

(a) c/a <

3

atoms in the plane of figure atoms above plane of figure

p3

Rv 3

√

Rv 2 2

p-2 1

p-1

3

p-3

n

(b)

c/a >

√

p2

3

atoms in the plane of figure atoms above plane of figure

p1 v3

v2 p3

Figure 11.6 (01¯ 12) twinning mode in hcp metals. Both in (a) and (b) atomic positions in the upper half plane before deformation are indicated by gray open and ﬁlled circles

348

KINEMATICS OF MULTILATTICES

Indeed, by the same procedure as for β-tin, the composition plane and the shear direction have indices (011) and [0¯ 11] with respect to the double cell, hence (see Fig. 11.6) Miller-Bravais hexagonal indices (01¯12) and [0¯111]. √ 2 The amount of shear is s = |3 − κ |/ 3κ, κ = c/a, and agrees with the data given by Hall (1954), Klassen-Nekliudova (1964), Barrett and Massalsky (1966), and Kelly and Groves (1970); an exception is the amount of shear for Ti in Hall (1954) and Klassen-Nekliudova (1964), which is 0.189 against 0.175 provided by the expression above, as well as by the other references. We address the reader to Pitteri (1985b) for more details. As we can see in Figure 11.6, a set of shifts ¯ pi in R2 , to be regarded as the left-hand sides of (11.110)1 , is given by ¯ p1 = R(p3 − v2 ), ¯ p2 = R(p2 − v2 − v3 ), ¯ p3 = R(p1 − v3 ).

(11.122)

Furthermore, by (11.107), 1

1

1

6

6

6

Σ1 = − v3 , Σ2 = − (v2 + v3 ), Σ3 = − v2 ,

hence

1

1

1

6

6

6

(11.123)

¯ p1 = Sp1 + Rv3 , ¯ p2 = Sp2 + R(v2 + v3 ), ¯ p3 = Sp3 + Rv2 . (11.124) This result is in good agreement with the mechanism for atomic displacements presented, for instance, by Kelly and Groves (1970), as can be seen in Fig. 11.6. There, for i = 1, 2, 3, we decompose ¯ pi − pi according to (11.124), that is, along the common direction of (S − 1)p i , which is that of a, and along the direction of the second summand in (11.124)i . There is good agreement between the theoretic prediction provided by (11.124) and the shuﬄe displacements drawn √ in Figure 11.6,√which look otherwise quite diﬀerent in the two cases c/a < 3 and c/a > 3. Other possible twins in a wt-nbhd are analyzed by Ericksen (2002c). 11.8.3 A model for stress relaxation Since even the presence of a single internal variable can successfully model phenomena that cannot be described by means of simple lattices, here we use a monatomic 2-lattice to sketch an interesting example of how multilattices can be useful in studying the dynamical behavior of an alloy, at least in a quasi-static approximation. The following rubber-like behavior has been observed in AuCd (at47.5%) as well as in other alloys, for instance InTl, CuZnAl, CuAlNi (see Lieberman et al. (1975), Bhattacharya et al. (1993)). This alloy transforms from cubic austenite to orthorhombic martensite, which is ﬁnely internally twinned. When an aged specimen containing ﬁne twins of two variants of martensite is pulled, large deformation occurs because the twin boundaries move, and one of the variants grows at the expense of the other until the specimen is completely de-twinned. If the loads are released immediately, then the twins appear again, and the strain is fully recovered. On the contrary, if

11.8 TWINNING IN MULTILATTICES

349

the loads are held ﬁxed for a suﬃciently long time and then released, no twins appear and no strain is recovered. There is a corresponding relaxation in cyclic loading: initially the material behaves pseudoelastically, with two distinct hysteresis loops in tension and compression, but these get closer and closer as the cycling continues, until they merge into a single loop. Lieberman et al. (1975) (also reported by Christian (1982)) suggest a mechanism based on twinning shuﬄes to explain the eﬀects above. The ordered alloy AuCd is a multilattice, hence its twinning modes involve both shear and shuﬄes, in general, as mentioned above. It is assumed that shear occurs as a fast response to the applied load, while shuﬄes obey a slower kinetics. The atoms do not shuﬄe when the twin boundaries move (and the specimen deforms) to accommodate the loads. If the loads are immediately released, the boundaries move back. On the contrary, if the loads are held, the atoms have time to shuﬄe by some thermally activated relaxation process, and the deformed conﬁguration becomes stable. This qualitative description has been turned into mathematics by Bhattacharya et al. (1993); we outline here their simplest model, cut down to the bones. We start from the kinematics of monatomic 2-lattices of §11.5.1, and the corresponding energetics along the lines of §11.7. In particular, we take the multilattice conﬁguration at the transformation temperature as a reference, and restrict the attention to a related wt-nbhd (§11.6). The energy invariance is thus governed by the lattice group of the (cubic) reference, and at a ﬁxed temperature below the transition temperature (not to appear in the notation henceforth) there is a ﬁnite number of martensitic energy wells (§11.7.2). We further restrict the attention to two such wells, and assume the skeletal lattice basis and the shift to vary each on a 1-dimensional manifold about the wells. In this case the free energy becomes a function φ = φ(;, p),

;, p ∈ R,

(11.125)

; and p denoting measures of strain and of shift on the aforementioned 1-dimensional manifolds, the latter normalized in such a way that the two wells correspond to (;+ , 1) and (;− , −1), respectively. The energy is assumed to satisfy the relations φ(;+ , 1) = φ(;− , −1) < φ(;, p)

(11.126)

for all (;, p) diﬀerent from either (;+ , 1) or (;− , −1) and, for calculations, Bhattacharya et al. (1993) actually use a function φ which is the minimum of two quadratic wells. For consistency we also assume the crystalline body to be a 1-dimensional bar with reference points x, with 0 ≤ x ≤ L, and deformation and shift ﬁelds y = y(x, t), p = p(x, t). Then the strain is ;=

∂u(x, t) , ∂x

u(x, t) = y(x, t) − x.

(11.127)

350

KINEMATICS OF MULTILATTICES

The simplest model of a slower change of the shift with respect to the skeleton is to assume the latter to equilibrate instantly at any given shift to minimize the elastic energy, while the shift evolves according to a gradientﬂow law. For a given time-dependent dead load σ(t) we have to solve L (1) min φ(;(x, t), p(x, t))dx − σ(t)u(L, t) for each t u(0,t)=0

(2) (3)

0

∂p ∂φ (x, t) = −κ ;(x, t), p(x, t) ∂t ∂p p(x, 0) = p0 (x), ;(x, 0) = ;0 (x),

(11.128) 0 ≤ x ≤ L.

Condition (1) is the instantaneous minimization of the elastic energy at given shift; the constraint u(0, t) = 0 eliminates the translational indeterminateness in the solution of the minimization problem. Condition (2) is the ordinary diﬀerential equation governing the evolution of the shift, and (3) are the initial conditions for the evolution. Calculations are feasible for the biquadratic choice of the function φ, and agree qualitatively with experience; for instance, pseudoelastic response and relaxation under cyclic loading are obtained, in spite of the crude approximations. A more reﬁned model in which strain satisﬁes the equilibrium equation away from twin boundaries, and the latter follow a kinetic equation, is also proposed by Bhattacharya et al. (1993); the corresponding dynamics agrees with experience in many respects, for instance in capturing the eﬀect of frequency and temperature, asymmetric loading and stabilization, as detailed there. We must be aware that, by the dynamical assumption made in the theory, the internal degree of freedom moves much more slowly than the skeletal structure. Therefore, from the physical point of view, what we call ‘shift’ here presumably does not refer to the basic crystalline structure itself, but rather to imperfections that are more or less periodically distributed, but with a much larger period than the elementary skeletal cell.

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Index (Aab ), 45 (t, A), 53 2/m, 71 4/mmm, 71 6/mmm, 71 a, aﬃne map, 53 B19 , 278 B2, 278 C, base-centered lattice, 71 C = (Cab ), metric, 45, 63 CG (H), 43 CG (h), 43 Dn , 52 E(3), 52 E(L), E(L(ea )), E(L(C)), 87 E(L)∗ , 87 e, isometry, 53 F , face-centered lattice, 71 G, 42 -orbit, see orbit G/H, 43 GL(3, Z), 44 I, body-centered lattice, 71 I(L), I(L(ea )), I(L(C)), 87 I(L)∗ , 87 J, 54 K, 304 K + , 330 K 0 , 328 K1 , 48 K2 , 49 M 18R, 277 mmm, 71 m¯ 3m, 71 NG (H), 43 O(3), 46 o(·), 214

P , primitive lattice, 71 P (M), P (εσ ), 316, 319 P (ea ), 66 R, 54 S(M), 316 SO(3), 46 T (3), 53 T (M), 316 W 1,∞ (R; R3 ), 288 Zn , 52 \, 41 ¯ 1, 71 ¯ 3m, 71 ∩, 41 ·, 41, 45 ∪, 41 :=, 41 =:, 41 #, 42 ⇔, 41 , . . . , , 41 ≤, 42 <, 42 →, 41 ⊕, 41 , 41 ≺, /, 97 ⇒, 41 →, 41 $, 42 , 41 , 41 uv + , 45 t , 45 −t , 45 ε+ σ , 330 ε0σ , 328

372

εσ , 304 σ, 56 B, 41, 62 D, 56 e Dn+2 , 314 m Dn+2 , 304 I, 56 L, L(ea ), 62 M, 303 M(P0 , εσ ), 304 M(P0 , . . . Pn−1 , ea ), 303 M(P0 , ea , p1 , . . . , pn−1 ), 304 M(Pi , ea ), 303 O, 46 O+ , 46 P, 153 Q> n , 45 Qen+2 , 314 Qm n+2 , 304 Qn , 45 Qn+2 , 304 α-γ, 96 ;, 56 κ, 59 η1 , 49 η2 , 49 φ, 56 ρ, ρAB , 153 θ, 56 a, 49 B, 55 C , 55 E , 55 ea , 62 F , 46, 54 n, 49 n ∗ , 55 P, 56 pi , 304 Rvω , 46 S , 48 T , 56 U , 47, 55 P, Q, 52 P0 , 304

INDEX

Γn+2 , 309 Λ(εσ ), Λ(K), 319 A3 , 52, 302 N, 41 R, 41 R3 , 41 Rn , 41 Z, 41 Aﬀ (3), 52, 302 Aut, 45 Sym, 45 Sym> , 45 A, 59 C, 190 L, 190 1-lattice(s), 303, 304 2-lattice(s), 318, 324 diatomic -, 324 monatomic -, 324 3-determined, 218 α-Hg, 86 α-Po, 86 α-quartz, 330 - descriptors, 330 - metric, 330 ab initio, 193 absolute temperature, 56 accommodation, see self-accommodation acoustic tensor, 205 action, 43, 44 - by conjugacy, 44 - of O on Sym, 150 - of GL(3, R) on Q> 3 , 46 reduced -, 153 actuation, 33 admissible function, 288 aﬃne - conjugacy classes of space groups, 317 - group, 52 - simple lattice(s), see 1-lattice(s)

INDEX

- space, 52, 302 - symmetry, 302 - transformation, 52, 302 Ag, 86, 98, 258 aggregate, see twin(s) algebra, 41, 45 algebraic equation, 37, 300 alloy(s) AuCd, 258, 348 CuAlNi, 31, 32, 258 CuZnAl, 276 InTl, 258 shape memory -, 32 steel(s), 98 TiNi, 279, 280 alum, 259 aluminum (Al), 86, 98 amount of shear, 49, 250 amplitude of jump, 55 anhydrite, 258 anisotropic, 182, 186, 188, 189, 192, 194, 213 area, see surface arithmetic - classes, see lattice, - types - group, 44, 309 - holohedry, see lattice, group(s) - symmetry, 62 - of n-lattices, 302, 318 atom, -ic, 22, 34–36, 38, 62, 98, 234, 301 - species, 301, 303, 309 Au, 86, 98 AuCd, 258, 348 austenite, -tic, 22, 31, 32, 180 automorphism, 43–45 availability, 59 axis - of rotation, 46 optic -, 84, 139, 328 β-Po, 86 β-quartz, 328

373

- descriptors, 328 - metric, 328 β-Sn (tin), 258, 325, 345 b.c.c., 76, 77 b.c.t., 82 Bain, 96, 234 Bain strain, stretch, see stretch ballistic free energy, 59 bands, 22, 31 - branching, 32 - tapering, 32 barothermometer, 261, 297 basal plane, 30, 84, 139, 327, 328 base point, 304 base-centered (C) lattice, 76 basis canonical - of Sym, 191 dual -, 46 Hilbert -, 213 integrity -, 213, 217 lattice -, 62 reduced -, 105 skeletal -, 304 BaTiO3 , 258 Baumhauer experiment, 30 bct5 Si, 325 Be, 258 beryl, 193 Bi, 258, 325 Bieberbach, 303, 322 bifurcation, 26, 150, 154, 177, 185, 186, 199–237, 338 - condition, 205 - point, 205 body, 54 - force, 59, 289 body-centered (I) lattice, 77 Born rule, 170, 245, 256, 330 failures of -, 171, 247, 258 boundary conditions, 22, 25, 59, 165, 173, 175, 178, 287–291 branch of equilibria, 185, 186, 199, 200, 205–208, 210 branching lemma, 236

374

Bravais lattices or lattice types, see lattice, - types Brazilian twins, 332, 341 Brillouin zones, 105 buckling, internal, 178 calcite, 30 calculus of variations, 23, 24, 58, 173, 178, 181, 289 canonical basis of Sym, 191 cardinality, 42 cassiterite, 30 Catastrophe Theory, 200 Cauchy, 166, 170 - relations, 193, 301 - stress tensor, 56 -Born hypothesis, see Born rule -Green tensors, 55 Cd, 258 cell, see unit cell centering, centered, 76–77 central inversion, see inversion centralizer, 43, 44 centrosymmetrical, 71, 97, 315, 318, 325 class(es), classiﬁcation arithmetic -, see lattice, - types conjugacy -, see conjugacy crystal -, see crystal classical question in crystallography, 63, 306 close-packed, 98, 301 cobalt (Co), 98 codimension, 207 cofactor(s), 292 coherent - deformation, 55 - interface, 241 - surface, 55 cohesion, 28 coincidence-site lattices, 37 color symmetry, 309, 318 commutation relation(s), 51, 91–94, 118

INDEX

commutative, see group compatibility, 32 Hadamard -, 55 composition, 23, 199, 229, 297 - plane, 30, 35, 241, 342–344 compound, see twin(s) compressibility, 187, 188 conditions matching -, 37, 40 structural -, 37 cone, 41 conﬁguration, 54 - space(s), 61, 62, 64, 304, 305 metastable -, 58 present -, 54 reference -, 54 stable -, 58, 175, 176, 239 congruence, 52 conjugacy, 42, 69 - class(es), 42, 62, 69, 72, 73 connection(s) (h, l)-, 268 conjugate -, 243 rank-1 -, 241 constituents of a multilattice, see n-lattice(s), constituent lattices constitutive function(s), 56, 165, 166, 168, 172–175, 178, 179, 181, 182, 184, 186, 190, 335–338 invariance of the -, 56, 168, 174, 181, 336 constitutive restrictions, 56, 60 contact twins, 29 continuum description, 53, 56, 166, 170, 172, 247, 258, 268, 341, 342 control, 23, 59, 60, 199, 203, 224, 260 conventional - operations, 35, 39, 252 - twins, 35, 39, 50, 252 convergence, 192, 289–291, 294, 298 copper (Cu), 86, 98, 258 coset, left -, right -, 42 covectors, 46 Cr, 258

INDEX

critical point, see point cross - product, 41 -ing twins, 296 -shaped, 30, 261 crystal, 28, 61, 303 - class(es), 69 - of a multilattice, 316 - invariance, 64, 166, 168 - system(s), 62, 69–71 of a multilattice, 316 multiphase -, 33, 58, 175–177, 203, 230, 232 single -, 28 crystallographic - indices, 99, 101 - plane, 35, 97 - point group, 66 - restriction, 68 - theory of martensite, 32, 297, 298 CuAlNi, 31, 32, 348 cube, 52 cubic, 29 - austenite, 31 - system and types, 84 cubic-to-, 212 monoclinic, 148 orthorhombic, 147 rhombohedral, 147 tetragonal, 122, 147, 224–226, 228, 299, 300 curve(s) of minimizers, 176, 177, 180, 183, 205, 210 CuZnAl, 276, 277, 348 cyclic group, 42, 52 d-determinacy, 218, 236 Dauphin´e twins, 36, 331, 332, 334 de-twin, 348 dead load, 165, 218, 288, 289, 350 decomposition i.i.-, 149, 150, 154–164 polar -, 47, 50

375

spectral -, 47 deformation, 54 - gradient, 54 density - of closest packing, 98 - per unit reference volume, 56, 59, 60 probability -, 166 description, see n-lattice(s), descriptors descriptive parameters, see n-lattice(s), descriptors descriptors, see n-lattice(s) determined, see d-determinacy diagonal, jointly -, 47 diagram, 89, 130, 148 bifurcation -, 221, 227 phase -, 33, 34, 224, 232, 276 diamond, 300, 325 diatomic 2-lattice(s), 324 diﬀraction, 30 diﬀusionless, see martensite, -ic dihedral, 52 direct - method, 58, 178, 289 - sum, 41 discontinuum, 303 homogeneous -, 303 regular -, 303 discrete, 44 - group, 44 - subset, 44 dislocations, 35 displacement, 54 continuous -, 36 displacive, see martensite, -ic dodecahedron, 52 domain(s), 45, 168 - in quartz, 36 constitutive -, 56, 168, 179, 181 fundamental -, 105 Wigner-Seitz -, 105 dual basis, vectors, 46

376

elastic, -ity, -ies, 190 - moduli, 190, 194–198 tensor of -, 190 elastomers, 24 elementary cell, see unit cell elimination - of shifts, 337 - of the passive coordinates, 236 enantiomorphic, 318, 325 energy - functional, 59, 60, 173, 288 - landscape, 185, 227, 232 - well(s), 26, 166, 176, 177, 182 free -, density, 56 Gibbs free -, density, 60 internal -, density, 56 sextic -, 214, 227, 228, 230 surface -, 40 entropy density, 56 environment, 23, 33, 56, 59, 60, 166, 187, 199, 260 equilibrium - branch, 185, 186, 199, 200, 205–208, 210 metastable -, 58 stable -, 58 equivalence - of pairs of wells, 269 arithmetical - of lattices, 74, 75 geometrical - of lattices, 70 equivariant, 236 essential - n-lattice descriptors, 314 - n-lattice metrics, 314 - description, 306 - descriptors, 306 - translation, 53 Euclidean group, 52, 302 excess free energy, 202, 234, 235 f.c.c., 76, 77, 98 face-centered (F ) lattice, 77 Fe, 86 feedback, 33

INDEX

ferroelectrics, 24 ﬁlms, 24 ﬁne scale, 24, 287, 289 ﬁnite subgroup(s) - of O, 51 - of Aut, 51 ﬁrst-order transformation or transition, see transition(s) ﬁxed set(s), 87, 88, 91, 94–95 local structure of -, see wt-nbhd(s) proper -, 87–89, 92 four-index notation, 101 frame indiﬀerence, 56 free energy, 56 ballistic -, 59 frequency, 36 Frobenius congruence, 322 function, 41 Lipschitz -, 288 functional, 59, 60, 173, 288 fundamental domain, 105 Galilean invariance, 56, 168, 174, 175, 184, 335, 340 Ge, 258 generators, 42 generic, -ity, 207 - rk-1 connection, 252, 269 geometric symmetry, 62, 302, 314 Gibbs free energy, see energy glide, 35 - plane, 315, 317 - reﬂection, 315 global symmetry, 62, 302, 306 gold (Au), 98 gradient, 54 -ﬂow, 170, 350 deformation -, 54 Gram matrix, gramian, 63 Grassmann notation, 52 group, 42 - action, 43 - homomorphism, 42, 43

INDEX

- isomorphism, 42, 45, 69, 88, 140, 316, 317, 321, 322 - representation, 45 -subgroup relation, 42, 112 aﬃne -, 52 arithmetic -, 44, 309 cyclic -, 42, 52 dihedral -, 52 discrete -, 44 Euclidean -, 52 lattice -, 73 - of n-lattices, 306 Laue -, 71 material symmetry -, 57, 174, 181 orthogonal -, 46 point -, 66, 316 quotient -, 43 reduced -, 153 special orthogonal -, 46 symmetry -, 44 symmorphic space -, 317 unimodular (sub)-, 44, 57, 58, 64, 174, 258, 259, 309 growth twins, see twin(s) gypsum, 30 (h, l)-connection, 268 (h, l)-twin, 268 h.c.p., 98 habit plane, 32, 298 Hadamard conditions, see compatibility heat-bath, 59, 60, 199 helix, 328, 331 Helmoltz free energy, see energy hessian, 175, 189, 218, 222, 223 hexagonal - close-packed, 325 - indices, 101 - metals, 30, 258, 259, 326, 343, 348 - system and type, 84 -to-rhombohedral, 149 Hf, 312

377

Hg, 39, 86, 258 high-symmetry, phase, see austenite, -tic Hilbert basis, see basis holohedry, -al, 66, 67, 69 - subgroup, 84, 117, 118 skeletal -, 316, 319, 326, 332 homogeneous - deformation, 54 - thermoelastic material, 56 homomorphism, see group honeycomb, 328 hydrostatic load, see pressure hysteresis, 24, 33, 185, 349 i.i., 150 icosahedron, 52 ideal crystal, 303 imagination, see twin(s) implicit function theorem, 205, 213, 214, 219, 222 In, 86 InTl, 32, 228, 234, 258, 299, 348 index, 42 indices, crystallographic hexagonal -, 101 Miller -, 99 Miller-Bravais -, 101 integrity basis, see basis interface, -cial, 32 - of low energy, 37 coherent -, 241 intergrowth, 28 internal energy, 56 internal variables, 304 invariance - under the action of a group, 44 Euclidean -, 56, 57 Galilean -, 56 invariant(s), 213, 217 - subgroup, 42 - subspace, 44 inversion, 21, 53, 58, 67, 71–73,

378

102, 119, 127, 138, 161, 168, 253, 320, 326, 328, 332 iron, 34, 35 - cross, 30 irreducible invariant subspace, 44, 149–164 isometry, 52 isomorphism, see group jacobian(s), 218, 219, 221, 222, 292 jump, 33, 36, 55, 200, 239, 241, 338 Kepler’s conjecture, 98 kernel, 45, 152, 206, 208, 209, 225, 245, 255, 315 Kurdjumov-Sachs, 234 L∞ weak∗ convergence, 290 Lagrange’s theorem, 42 lamellae, 34, 35 laminate, 32, 288, 293–295, 298 Lam´e coeﬃcients, 198 LaNbO4 , 39, 275 Landau potential, 209 landscape, energy -, 185, 227, 232 lattice, 62 - basis, 62 - correspondence, 96, 232, 273, 278 - group(s), 73 - of n-lattices, 306, 319 - metric, 63 - planes, 97 - point, 62, 166, 302 - rows, 97 - symmetry, 62, 74 - types, 62, 75, 76 - vectors - for a multilattice, 304 - for a simple lattice, 62 -invariant shear, 65 base-centered (C) -, 76 body-centered (I) -, 77 coincidence-site -, 37

INDEX

face-centered (F ) -, 77 primitive (P ) -, 76 reciprocal -, 63 skeletal -, 301, 304, 305 Laue groups, 71 Legendre-Hadamard inequality, 190 lengthscale, 170, 239, 287 Li, 86 Liapunov–Schmidt reduction, 208, 236 limit point, see turning point linear transformation, 45 Lipschitz functions, 288 load(s), 22, 24, 25, 32, 60, 165, 173, 175, 239, 260, 287–289, 348 local symmetry, 62, 302, 306 lost symmetry, 265, 266 low-symmetry, phase, see martensite, -tic lower semicontinuous, 289 m-component, 309 magnetostriction, 24 Mallard law, 285–286 map, 41 martensite, -tic, 22, 31, 32, 180 matching conditions, 37 material - point(s), 54 - surface, 55 - symmetry, 56, 57 - group, 57, 174, 181 - vector, 55 active -, 33 homogeneous thermoelastic -, 56 smart -, 33 matrix, 45 diagonal -, 120 Gram -, 63 integral -, 44 maximal, 73 - crystallographic group, 65–67 - lattice, 306 - lattice vectors, 306

INDEX

Maxwell temperature, 177, 185, 220, 228 measure H-, 290 Young -, 290 mercury, see Hg merohedral twins, 285, 332 metastable state, conﬁguration, 33, 58, 184 metric, 45, 63 Mg, 187, 258 micromagnetism, 24 microstructures, 22, 24, 32, 33, 165, 181, 239, 240, 263, 279, 287–300 Miller, Miller-Bravais, see indices Mineralogists’ Assumption, 247 minimizer, -ing, 58, 287, 288 - sequence, 58, 289, 290 absolute -, 58 relative -, 58 minors relations, 291–292 mirror - plane, - symmetry, 48, 247, 253 MISC, 223, 236 mode, twinning -, see twin(s) module, 62 modulus, -i - softening, 226 elastic -, 190, 194–198 monatomic, 301, 303 - 2-lattice(s), 324 monoclinic system and types, 79 motif, 301, 304 - shuﬄes, 341 - twins, 332 multilattice metric, see n-lattice(s), metric multilattice(s), see n-lattice(s) multiphase crystal, see crystal multivalued energy, 28, 169, 173, 337–339 n-lattice(s), 301, 303, 304 - descriptors, 304, 306

379

essential -, 306, 310 guidelines for choosing -, 312 indeterminateness in -, 306, 309 nonessential -, 311 - metric, 304 arithmetic symmetry of -, 318 constituent lattices of -, 303 deﬁnition of -, 303 essential -, 306 lattice group of -, 319 polyatomic -, 318 Na, 86 natural state, see state neighborhood, wt(weak-transformation -), 107, 108, 181, 183, 186, 190, 205, 263, 333, 340 nematic elastomers, 24 net(s), 97, 110 Neumann bands, 35 nickel (Ni), 98 nonconvex energy, 23, 59, 178–179 nonmonotone stress-strain relation, 23 norm, 41 normal - subgroup, 42 - vector, 46, 49, 55 normalizer, 43, 44 null Lagrangian, 292 objectivity, 56 observer, 54 octahedron, 52 optic axis, 84, 139, 328 optical twins, 332 orbit, 44, 47 order, 42, 48 - parameters, 209 zeroth -, 203, 235 orientation, 23, 27 - of a surface, 55

380

- relationships, 28, 29, 37, 38, 231, 232, 253, 276 -preserving, 56, 168, 245 origin in A3 , 53 orthogonal - group, 46 - sum, 41 - tensor, 46 orthohexagonal, 84, 101 orthorhombic, 31 - martensite, 31 - system and types, 80 oscillations, 289, 290 packing, see close-packed parameter(s), see order parent, phase, see austenite, -tic passive coordinates, elimination, 236 penetration twins, 29 perimeter, 295 period, 42, 48 period-two, 35, 39, 48 periodic structure, see crystal permutation matrix, 309 phase - diagram, 34 - transition or transformation, see transition(s) high-symmetry -, see austenite, -tic low-symmetry -, see martensite, -tic Piola–Kirchhoﬀ stress tensors, 56 pitchfork, 214–215 plane - of shear, 49 basal -, 30, 84, 139, 327, 328 composition -, 35, 241 crystallographic -, 35, 97 habit -, 32, 298 invariant -, 49 second undistorted -, 49 slip -, 34

INDEX

plastic ﬂow, 34 plate(s), 35 Platonic solids, 52 point - group, 66, 67, 316 - classiﬁcation, 69 - of a multilattice, 316 crystallographic -, 66, 68 - in aﬃne space, 52 bifurcation -, 205 critical -, 58, 183, 185 - condition, 59, 60, 204 event-, 57 lattice -, 61, 62, 166, 302 material -, 54 origin, 53 triple -, 33, 148, 203, 224, 232 polar decomposition, 47 polyatomic, 303, 318 polycrystal, 29 polynomial - energy, 201, 214, 217, 219, 221, 222, 226–231 - invariants, 213, 217, 219, 221, 222 potassium persulfate, 69 potential, 58 reduced -, 209 pressure, 59, 60, 165, 187, 188, 260 primitive (P ) lattice, 76 principal stretches, 298 product - phase, see martensite, -tic cross -, 41 scalar -, 41, 45 semidirect -, 317 tensor -, 45 wedge -, 41 proper - ﬁxed set, 87–89 - orthogonal, 52 - subgroup, 42 protoactinium, 86 pseudo-merohedry, 285 pyrite, 30

INDEX

quadratic form(s), 45 quartz, 31, 36, 187, 327 α-, 330 β-, 328 quasiconvex, -ity, 178, 289 quasicrystal, 303 quotient group, 43 rank, 45 -1 connection(s), 241, 263, 294 conjugate -, 243 reciprocal -, 243 -1 convex, -ity, 178, 289 rationality conditions for twins, 256 reciprocal lattice, 63 reconstructive, see transition(s) recovered strain, 348 reduced - bases, 105 - group, 153 - potential, 209 - problem, 208 reduction, see Liapunov–Schmidt reduction reference conﬁguration or state, 54 reﬂection, 48, 247 representation - of aﬃne transformations, 53 group -, see group reproducibility, 36 response, see constitutive function(s) restriction(s) constitutive -, see constitutive restrictions crystallographic -, 68 domain -, 108, 175, 179, 181–182, 186, 340 structural -, 292, 299, 300 reversibility, 32 rhombic basis, 80, 81 rhombohedral system and type, 83 Rochelle salt, 32 rock-salt, 193

381

rotary reﬂection, 315 rotation, 46 twofold -, or period-two -, or - of π, 48 row(s), 37, 97 running indices, 41 scalar product - of tensors, 45 euclidean -, 41 scale, see ﬁne scale screw - axis, 315, 317 - rotation, 315 second-order transformation or transition, see transition(s) self-accommodation, 22, 23, 299 semidirect product, 317 sequence, minimizing, see minimizer, - ing set, 41 ﬁxed -, see ﬁxed set(s) sextic, see energy shape - memory, 22, 33, 287 - strain vector, 298 shear - amplitude vector, 49 - elements, 49 amount of -, 49 invariant plane, 49 lattice-invariant -, 65 plane of -, 49 second undistorted plane, 49 simple -, 30, 32, 36, 48 shift vectors, shifts, 304 shuﬄe(s), 36 - twins, 332 motif -, 341 structural -, 240, 257, 341 Si, 86 silver (Ag), 98 simple lattice(s), see lattice simple shear, see shear

382

singularity theory, 200 SiO2 , 327 site-symmetry, 324 skeletal - holohedry, 316, 319, 326, 332 - lattice, 301, 304, 305 skeleton, 301, 304 slip, 34, 35 SMA, 22 smart material, 33 snapsprings, 220 softening, modulus -, 226 solid, 57 multiphase -, see crystal solid-state transition, see transition(s) space - group(s), 306, 315 - of a simple lattice, 316 aﬃne conjugacy classes -, 317 symmorphic -, 317 - of conﬁgurations, 62, 64, 304, 305 aﬃne -, 52 span, 41 special orthogonal group, 46 species, atomic, see atom, -ic speciﬁc heat, 59 split, 317 spontaneous symmetry breaking, see symmetry-induced bifurcations stabilizer, 44, 73 stable state, conﬁguration, 58, 175, 176, 239 stacking, 97 - faults, 98 - of spheres, 98 state, 54 natural -, 21, 22, 60, 287 stable -, see stable thermodynamic -, 33, 199 staurolite, 30 steel(s), 22 austenitic -, 98

INDEX

strain, 91 - tensor, 55, 190, 191 -temperature relation, 33 stratum, 211 stress, 31, 32 - relaxation, 301, 337, 348–350 -free - joint, 40, 260, 285, 286, 296 - state, 22, 27, 176, 183, 187, 239, 244, 252, 260, 292, 297 -strain relation, 23, 33, 179, 227, 339 Cauchy -, 56 interfacial -, 32 Piola -, 56 Piola second -, 56 stretch, 55 - tensor, 55 Bain -, 77, 96 symmetry-breaking -, 107, 108, 111–112 symmetry-preserving -, 87, 91–94 transformation -, 184, 330 structural - restrictions, 299, 300 - shuﬄes, 240, 257, 341 - transitions, see transition(s) - unit, 96, 232, 234 structure internal -, see motif laminar -, see laminate local - of wt-nbhds, see wt-nbhds periodic - or crystalline -, see crystal variant -, see variant(s), structures subcritical, 186 subgroup(s), 42 - of translations in E(3), 53 ﬁnite - of O, 51 - of Aut, 51 invariant -, or normal -, or self-conjugate -, 42 subpotential(s), 210, 337

INDEX

subspace invariant -, 44, 149–164 irreducible -, 44, 149–164 sum direct -, 41 orthogonal -, 41 summation convention, 41 supercritical, 186 surface - area, 295 - energy, 40, 239 - of discontinuity, 55, 241 material -, 55 Sylov’s theorem, 130 Sylvester’s inertia theorem, 48 symmetry, 43 - axis, 66 - element, 66 lost -, 266 - group, 44 - operation, 66, 315 - plane, 66 -breaking, see stretch -induced - bifurcations, 211–223, 338 - instabilities, 178–179 -preserving, see stretch arithmetic -, see arithmetic geometric -, see geometric symmetry global -, 64, 306 local -, 62, 306 material -, 56, 57 site-, 324 symmorphic, 317 systems, see crystal tacking, 290 tartaric acid, 69 temperature, absolute, 56 tensor product, 45 tensor(s), 45 - commuting with an orthogonal tensor, 51

383

- of period two, 48 - of the elasticities, 190 acoustic -, 205 Cauchy stress -, 56 Cauchy-Green -, 55 ﬁnite groups of -, 50 orthogonal -, 46 Piola stress -, 56 rank of -, 45 strain -, 55, 91 stretch -, 55 Terfenol (TbDyFe2 ), 258, 296 tetragonal system and types, 82 tetragonal-orthorhombicmonoclinic, 34, 148, 203 tetragonal-to-, 212 monoclinic, 225, 230 orthorhombic, 224 tetrahedron, 52 thermal - environment, - variable, 23, 56, 59, 166, 260 - expansion, 94, 171, 176, 184, 187–189 thermoelastic, -ity, 21, 23, 33, 38, 40, 58, 172, 174, 297 - material, homogeneous, 56 thin ﬁlms, 24 tin, 258, 325, 345 TiNi, 258, 279, 280 tinstone, 30 titanium (Ti), 98, 187, 258, 312, 325, 348 transformation(s), see transition(s) transition(s), 22, 24, 302, 312 α-γ -, 34, 77, 96 - stretch, 184 - for quartz, 330 b.c.c.-to-f.c.c. -, 34, 77 b.c.c.-to-h.c.p. -, 34, 312 ﬁrst-order -, 77, 200 weak -, 33, 186, 200, 224, 227, 230, 232 nonweak -, 110

384

reconstructive -, 201 second-order -, 33, 185, 200 structural -, 216, 338 triple point, 232 translation subgroup, 53 triclinic system and type, 79 trigonal, see rhombohedral triple point, 33, 148, 203, 224, 232 turning point, 213–214 twin(s), 28, 29, 239 (h, l)-, 268 - aggregate, 28, 39, 261 - bands, 22, 31, 32 - conditions, structural, 37 - equation, 31, 36, 244–248, 263 - frequency, 36 - in n-lattices, 341–348 - interface, 35, 253, 256 - law, 28, 31, 35 - of the imagination, 36 - operation, 28, 35, 240, 246, 247, 266, 275, 277, 280, 284 conventional -, 252 - reproducibility, 36 - shear, 30, 31, 240, 245–247 Brazilian -, 332, 341 classiﬁcation of -, 29 compound -, 31, 35, 255, 266 conventional -, 35, 50, 252 crossing -, 296 cyclic -, 296 Dauphin´e -, 31, 36, 331, 332, 334 deformation -, 29, 30 generic -, 252, 254 genuine -, 36 growth -, 29, 259–261, 297 contact -, 29 penetration -, 29 mechanical -, 33 merohedral -, 285, 332 motif -, 332 nonconventional generic -, 269, 275, 284 nongeneric -, 269, 276, 279 optical -, 332

INDEX

rationality conditions for -, 256 rotation -, 296 shuﬄe -, 31, 36, 332 tapering -, 40 transformation -, 29, 32, 263–268 Type 1 -, 35, 252, 265 Type 2 -, 35, 39, 252, 265 twinning, see twin(s) Type-1, Type-2, see twin(s) undistorted - reference conﬁguration, 57 - states of anisotropic solids, 189 second - plane, 49 unimodular (sub)group, see group unit cell, 63, 304 unstressed state, 21, 60 uranium (U), 39, 258, 325 van der Waals, 24, 179, 185 variant(s), - structures, 22, 23, 26, 33, 108, 110, 121–123, 128–146, 180, 182, 183, 185, 289, 294, 297 variation(s), -al, see calculus of variations vector(s), 41 dual -, 46 material -, 55 translation -, 52 vibrations, atomic, 62 Voigt notation, 192 volume - change, 299 - fraction, 32, 288 - integral, 59 - of unit cell, 63, 64, 84, 306 - preserving, 299 weak - ﬁrst-order, see transition(s) -∗ convergence, 290 -transformation, see wt-nbhd(s) wedge, 22, 32, 296, 300 - product, 41

INDEX

well(s), 166, 176, 177, 182, 199, 203, 233, 240, 245, 249, 263, 341 austenitic -, 183, 187 equivalent pairs of -, 269 martensitic -, 183, 187 N- problem, 292–295 Wigner-Seitz, see domain(s) work, 289 wt-nbhd(s), 107, 108, 181, 183, 186, 190, 205, 263, 333, 340 local structure of -, 113–120, 127–146, 333 X -interface, 299 -ray, 30, 63, 66, 71, 170, 247, 249, 259 Young, see measure zeroth-order, 203, 235 zinc (Zn), 98, 171, 258, 312, 325 zirconia (ZrO2 ), 34, 148, 203, 275 zirconium (Zr), 98, 148, 258 zones, see Brillouin

385