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108 Site Symmetry in Crystals Theory and Applications 2nd Edition By R. A. Evarestov and V P. Smirnov

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R. A. Evarestov V. P. Smirnov

Site Symmetry in Crystals Theory and Applications

Second Edition With 42 Figures

Springer

Professor Robert A. Evarestov Department of Chemistry, St. Petersburg University St. Peterhoff, Universitetiskii pr. 2, 198904 St. Petersburg, Russia

Professor Vyacheslav P. Smirnov

Department of Theoretical Physics, Institute of Fine Mechanics and Optics Sablinskaya 14, 197101 St. Petersburg, Russia

Series Editors:

Professor Dr., Dres. h. c. Manuel Cardona Professor Dr., Dres. h. c. Peter Fulde* Professor Dr., Dres. h. c. Klaus von Klitzing Professor Dr., Dres. h. c. Hans-Joachim Queisser Max-Planck-Institut fUr Festkorperforschung, Heisenbergstrasse I, 0-70569 Stuttgart, Germany * Max-Planck-Institut fUr Physik komplexer Systeme, Bayreuther Strasse 40, Haus 16,

0-01187 Dresden, Germany

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Dr.-Ing. Helmut K. V. Lotsch Springer-Verlag, Tiergartenstrasse 17,0-69121 Heidelberg, Germany

Library of Congress Cataloging-in-Publication Data.

Evarestov, R. A. (Robert Aleksandrovich). Site symmetry in crystals: theory and applications 1 R. A. Evarestov, V. P. Smirnov. - 2nd ed. p. cm. - (Springer series in solid·state sciences, ISSN 0171-1873; \08). Includes bibliographical references and index. TSBN-13: 978-3-540-61466-1 (softcover: alk. paper). I. Solid state physics. 2. Solid state chemistry. 3. Crystallography, Mathematical. 4. Symmetry (Physics). I. Smirnov, V. P. (Viacheslav Pavlovich) II. Title. III. Series. QC176.E8 1997 530.4' II-dc20 96-36380

ISSN 0171-1873 TSBN-13: 978-3-540-61466-1 DOl: 10.1007/978-3-642-60488-1

e-TSBN-13: 978-3-642-60488-1

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Preface to the Second Edition

The first edition of our monograph appeared in 1993 and has already sold out. As authors we are greatly satisfied that our book appeared to be interesting for solid-state scientists in different countries. For us it was a great pleasure to know that a second edition of our book was planned by Springer-Verlag. In preparing this new edition we took into account the following points.

During these years there was growing interest in the physical and chemical properties of recently discovered en modifications of carbon (n 2: 60) called fullerenes. Therefore we have added information about the symmetry groups of fullerenes C60 and C70 (Sect. 3.2.1), their irreducible (Sect. 3.7) and in­duced (Sect. 4.1) representations. The space group symmetry of fullerites (crystals consisting of fullerenes) with rotating molecules is treated in a new section (Sect. 10.3).

Recent publications on the theory of point defects in crystals use the super-cell model of an imperfect crystal, which is based on the periodical repeating of the defect in enlarged unit cells of the host crystal. Therefore we added (Sect. 6.1.4) information about the symmetry of the super-cell model of an imperfect crystal.

Section 8.5 contains some new results concerning the phonon properties of perovskite-like superconductors. The list of references and the subject index have been enlarged, and misprints and some errors have been corrected.

We are very indebted to Dr. H.K. V. Lotsch of Springer-Verlag for cooperation in the preparation of this edition.

St. Petersburg, July 1996 R. A. Evarestov V. P. Smirnov

Preface to the First Edition

The history of applications of space group theory to solid state physics goes back more than five decades. The periodicity of the lattice and the definition of a k-space were the corner-stones of this application. Prof. Volker Heine in Vol. 35 of Solid State Physics (1980) noted that, even in perfect crystals, where k-space methods are appropriate, the local properties (such as the charge densi­ty, bond order, etc.) are defined by the local environment of one atom. Natural­ly, "k-space methods" are not appropriate for crystals with point defects, sur­faces and interfaces, or for amorphous materials. In such cases the real-space approach favored by chemists to describe molecules has turned out to be very useful.

To span the gulf between the k-space and real space methods it is helpful to recall that atoms in crystalline solids possess a site symmetry defined by the symmetry of the local environment of the atom occupying the site. The site symmetry concept is familiar to crystallographers and commonly used by them in the description of crystalline structures. However, in the application of group theory to solid state physics problems, the site symmetry approach has been used only for the last ten to fifteen years. In our book Methods of Group Theory in the Quantum Chemistry of Solids published in Russian in 1987 by Leningrad University Press we gave the first results of this application to the theory of electronic structure of crystals.

In addition to the material of the Russian text, the present book takes into consideration the newly developed applications of the site symmetry approach such as the symmetry analysis in lattice dynamics, theory of second order phase transitions, and magnetically ordered and non-rigid crystals. Chapter 8 was written by Dr. Yu. Kitaev in collaboration with Dr. B. Bairamov and Dr. M. Limonov.

This text is intended mainly for researchers working in the physics and chemistry of solids. The authors assume that the reader has attended basic courses on group theory. This book should also be useful for those primarily interested in the applications of the site symmetry in crystals: these readers may start from Chap. 4, entitled "Site Symmetry and Induced Representations of Symmetry Groups", paying special attention to Sect. 4.5.1. In this section we explain how to use the tables of simple induced representations of space groups. Illustrative examples are given in Chaps. 5 - 9.

More than thirty years ago both of us heard lectures by Prof. Mariya I. Petrashen on group-theory applications in quantum mechanics. For many years she encouraged our work in this field. Thus we dedicate this book to our teacher and friend, Professor Petrashen.

VIII Preface to the First Edition

It is a great pleasure to have this opportunity to thank Professor M. Car­dona not only for the idea to write this book but also for good advice and hospitality during the visit of one of the authors (R. A. E.) to the Max­Planck-Institut fUr Festkorperforschung in Stuttgart.

We are especially indebted to Dr. H. K. V. Lotsch of Springer-Verlag for en­couragement and cooperation. We are very grateful to Dr. A. I. Panin and Dr. V. A. Veryazov for help in preparing the manuscript and the subject index and also for help in proofreading.

St. Petersburg, November 1992 R. A. Evarestov V. P. Smirnov

Contents

1. Introduction .............................................. .

2. Finite Groups and Their Representations ...................... 5 2.1 Elements of Group Theory ............................... 5

2.1.1 Groups. Generators and Generating Relations. Subgroups. Cosets. Invariant Subgroups. The Factor Group .................................. 5

2.1.2 Conjugate Elements and Classes. Factorization of Groups. . .. . . . . .. . . . .. . . . ... . . . .. . . . . .. . . .. . . . .. 7

2.1.3 Homomorphism and Isomorphism of Groups ......... 9 2.2 Elements of Group Representation Theory ................. 10

2.2.1 Representations of a Group. Equivalent, Reducible and Irreducible Representations. Orthogonality Relations. Representation Characters ........................... 10

2.2.2 Decomposition of Representations. Complex Conjugate Representations .................. 15

2.3 Generation of Representations ............................ 17 2.3.1 Direct Product of Representations .................... 17 2.3.2 Subduction of Representations ....................... 20 2.3.3 Induction of Representations ........................ 22 2.3.4 Little Group Method

of Irreducible Representation Generation .............. 26

3. Symmetry Groups and Their Representations .................. 31 3.1 The Euclidean Group and Its Subgroups ................... 31

3.1.1 Translation Group .................................. , 31 3.1.2 Rotation Group .................................... ' 32 3.1.3 Inversion Group ................................... 35 3.1.4 Full Orthogonal Group ............................. 35 3.1.5 Euclidean Group ................................... 36

3.2 Point Symmetry Groups ................................. 39 3.2.1 Symmetry Elements of Molecules

and Crystallographic Point Groups ................... 39 3.2.2 Site Symmetry Subgroups of Point Groups ............ 40

3.3 Space Groups .......................................... 43 3.3.1 Symmetry of a Model of an Infinite Crystal.

Symmorphic and Nonsymmorphic Space Groups ....... 43

X Contents

3.3.2 Symmetry of a Cyclic Model of a Crystal ............. 46 3.4 Site Symmetry in Space Groups ........................... 48

3.4.1 Crystallographic Orbits. Wyckoff Positions ............ 48 3.4.2 Oriented Site Symmetry Groups. Choice of Origin ..... 51 3.4.3 Crystal Structure Types. Crystals with Space Group Dlh. 54

3.5 Symmetry Operations in Quantum Mechanics .............. 55 3.5.1 Symmetry Group of a Quantum Mechanical System. . . . 55 3.5.2 Wigner's Theorem ................................. 56 3.5.3 Time-Reversal Symmetry ............................ 57

3.6 Irreducible Representations of Rotation and Full Orthogonal Groups ............................. 59

3.7 Representations of Point Groups .......................... 62 3.8 Representations of Space Groups .......................... 70

3.8.1 Irreducible Representations of the Translation Group. The Brillouin Zone ................................. 70

3.8.2 Stars of Wave Vectors. Little Group. Full Representations of Space Groups ................ 76

3.8.3 Small Representations of a Little Group. Projective Representations of Point Groups ............ 78

3.8.4 Double-Valued Representations of Space Groups ....... 79 3.8.5 Dependence of the Labeling of the Irreducible

Representations of a Space Group on the Setting ...... 81 3.8.6 Example: Irreducible Representations

of Space Group D lh. Compatibility Tables ........... 84

4. Site Symmetry and Induced Representations of Symmetry Groups 89 4.1 Induced Representations of Point Groups.

Correlation Tables ....................................... 89 4.2 Induced Representations of Space Groups .................. 91

4.2.1 Induction from Site Symmetry Subgroups of Space Groups ................................... 92

4.2.2 Induced Representations in the k-Basis. Band Representations ............................... 93

4.2.3 Simple and Composite Induced Representations ........ 97 4.3 Double-Valued Induced Representations .................... 99 4.4 Generation of the Simple Induced Representations

of the Space Group D lh ................................. 100 4.5 The Twenty-Four Most Common Space Groups: Crystal

Structures and Tables of Simple Induced Representations. . . .. 103 4.5.1 Tables of Simple Induced Representations and Their Use 103 4.5.2 Space Groups and Crystal Structures

with Cubic Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5.3 Space Groups and Crystal Structures

with Hexagonal and Trigonal Lattices . . . . . . . . . . . . . . . .. 111 4.5.4 Space Groups and Crystal Structures

with Tetragonal Lattices ............................. 114

Contents XI

4.5.5 Space Groups and Crystal Structures with Orthorhombic Lattices ......................... 117

4.5.6 Space Group Setting and Simple Induced Representations for Monoclinic Space Groups ........................ 121

5. Application of Induced Representations in the Electron Theory of Molecules and Crystals ................................... 125 5.1 Adiabatic and One-Electron Approximations ............... 125

5.1.1 Space Symmetry of the One-Electron Approximation Hamiltonian ...... 129

5.2 Induced Representations in the Electron Theory of Molecules 131 5.2.1 Canonical, Localized and Hybridized Molecular Orbitals 131 5.2.2 Localized Two-Center Bonds and Hybridized Orbitals

in AB4 and AB3 Molecules .......................... 136 5.2.3 Multicentered Bonds in the 1,6-C2B4H6 Molecule . . . . . . . 139 5.2.4 Canonical and Localized Orbitals

in the MnO';- Molecular Ion. . . . . . . . . . . . . . . . . . . . . . . . . 140 5.2.5 Localized Orbitals in the Tetrahedral Bi4 Molecule ..... 142

5.3 One-Electron Approximation for Crystals .................. 144 5.3.1 Crystalline Orbitals.

Degenerate and Nondegenerate Energy Bands. . . . . . . . .. 144 5.3.2 Equivalent Hamiltonians

for the Same Crystal Structures ...................... 146 5.3.3 k-p Perturbation Method in the Energy Band Theory.. 147 5.3.4 Zero-Slope Points of Energy Bands. . . . . . . . . . . . . . . . . .. 150 5.3.5 Energy Bands in the Neighborhood

of Degeneracy Points ............................... 152 5.3.6 Additional Degeneracy of Energy Bands Due

to the Reality of the Hamiltonian .................... 155 5.3.7 Density of States of an Energy Band ................. 155

5.4 Induced Representations and the Theory of Chemical Bonding in Crystals ...................................... :...... 158 5.4.1 Energy Band States and Localized Functions .......... 158 5.4.2 Localized Orbitals and Atomic States in Crystals ....... 159 5.4.3 Hybridized Orbitals in Crystals ...................... 160 5.4.4 Crystals with Space Group Oh ...................... 161 5.4.5 Crystals with Space Group O~ ...................... 162 5.4.6 Crystals with Space Group Dlh ...................... 163 5.4.7 One-Electron States in High-Tc Superconductors ....... 165

5.5 Energy Bands and Localized States ........................ 173 5.5.1 Localized Orbitals and Parameters of an Energy Band .. 173 5.5.2 Generation of Localized Functions in Crystals ......... 174 5.5.3 Interpolation Scheme Using Localized Functions ....... 175

5.6 Localized Orbitals in Molecular Models of Crystals ......... 179 5.6.1 Cluster Model of Perfect Crystals .................... 179 5.6.2 Cluster and Crystal Localized Orbitals ................ 180

XII Contents

5.6.3 Energy Bands of AgBr from Cluster Calculations of [Ag14Br 131 + ...•.....••....•..................... 181

5.6.4 Cyclic Model as a Molecular Model of Crystals. . . . . . . . 182 5.6.5 Localized Orbitals in the Cyclic Model ... . . . . . . . . . . . .. 183

6. Induced Representations in the Theory of Imperfect Crystals .... 185 6.1 Point Defects in Crystals ................................. 185

6.1.1 Single Defect Model ................................ 186 6.1.2 Cluster Model of Imperfect Crystals .................. 188 6.1.3 Cyclic Model of Imperfect Crystals ................... 189 6.1.4 Band Model of Imperfect Crystals ................... 189 6.1.5 Localized Orbitals in the Band Model of Point Defects. 191

6.2 Diperiodic Space Groups. Surface Electron States ........... 192 6.2.1 Diperiodic (Layer) Space Groups ..................... 192 6.2.2 Site Symmetry in Layer Groups ...................... 195 6.2.3 Irreducible Representations of Diperiodic Groups ...... 197 6.2.4 Induced Representations of Diperiodic Groups ......... 199 6.2.5 Use of Translational Symmetry in the Comparison

of Bulk and Surface Crystalline States ................ 201

7. Application of Induced Representations of Space Groups to Second Order Phase Transitions ........................... 205 7.1 Symmetry Rules in the Landau Theory

of Second Order Phase Transitions ........................ 205 7.2 Tensor Fields in Crystals and Induced Representations

of Space Groups. Tensor Fields for Space Group D lh ........ 207 7.3 Vibrational Field Representation and Phase Transitions

in High-Temperature Superconductors ...................... 210

8. Induced Representations of Space Groups in Phonon Spectroscopy of Crystals .......................... 213 8.1 Phonon Symmetry Analysis .............................. 213 8.2 Infrared and Raman Spectra Selection Rules ................ 214 8.3 Phonon Symmetry and Optical Spectra Selection Rules

in Semiconductor Superlattices ........................... 215 8.3.1 (GaAs)m(AlAs)n Superlattices ....................... 216 8.3.2 (Si)m(Ge)n Superlattices ............................. 221 8.3.3 Experimental Applications .......................... 221

8.4 Phonon Symmetry in High-Temperature Superconductors ..... 227 8.5 Phonon Symmetry in Diperiodic Systems .................. 233

9. Site Symmetry in Magnetic Crystals and Induced Corepresentations ............................... 237 9.1 Shubnikov Space Groups of Symmetry of Magnetic Crystals . 237 9.2 Site Symmetry in Magnetic Crystals ....................... 238 9.3 Corepresentations of Shubnikov Space Groups .............. 241

Contents XIII

9.4 Induced Corepresentations of Magnetic Space Groups ..... 244 9.5 Corepresentations of the Space Groups

of Antiferromagnetic La2Cu04 .......................... 247

10. Site Symmetry in Permutation - Inversion Symmetry Groups of Nonrigid Crystals ....................................... 251 10.1 Symmetry Groups of Nonrigid Crystals .................. 252

10.1.1 Labeling of Nuclei. Sampling of Coordinate Systems 252 10.1.2 Description of Permutation -

Inversion Symmetry Elements .................... 253 10.1.3 Coordinate Transformations Induced

by Permutation - Inversion Symmetry Elements ... 255 10.1.4 Site Symmetry Group of a Rotating Molecule

in a Nonrigid Crystal ............................ 256 10.1.5 Permutation - Inversion Group

of a Nonrigid Sodium Nitrate Crystal ............. 257 10.2 Irreducible Representations

of a Nonrigid Crystal Symmetry Group. . . . . . . . . . . . . . . . .. 260 10.2.1 Generation of Irreducible Representations .......... 260 10.2.2 Irreducible Representations

of a Site Symmetry Group ....................... 261 10.2.3 Classification of States .......................... 263

10.3 Generalized Symmetry of High-Temperature Phase of Fullerite C60 ••••••••••••••••••••••••••••••••••••••• 264 10.3.1 Permutation - Inversion Symmetry Group

of Fullerite C60 in the High-Temperature Phase .... 265 10.3.2 Irreducible Representations

of the Groups [nJ and Pc ........................ 265 10.3.3 Classification of States of Nonrigid Fullerite C60 .•• 266

References .................................................... 269

Subject Index ................................................. 277