i Applications of Group Theory to the Physics of Solids M. S. Dresselhaus 8.510J 6.734J SPRING 2002 ii Contents 1 Basic Mathematical Background 1 1.1 De¯nition of a Group . 1 1.2 Simple Example of a Group . 2 1.3 Basic De¯nitions . 4 1.4 Rearrangement Theorem . 6 1.5 Cosets . 6 1.6 Conjugation and Class . 8 1.6.1 Self-Conjugate Subgroups . 9 1.7 Factor Groups . 10 1.8 Selected Problems . 11 2 Representation Theory 15 2.1 Important De¯nitions . 15 2.2 Matrices . 17 2.3 Irreducible Representations . 18 2.4 The Unitarity of Representations . 20 2.5 Schur's Lemma (Part I) . 23 2.6 Schur's Lemma (Part 2) . 24 2.7 Wonderful Orthogonality Theorem . 27 2.8 Representations and Vector Spaces . 30 2.9 Suggested Problems . 31 3 Character of a Representation 33 3.1 De¯nition of Character . 33 3.2 Characters and Class . 34 3.3 Wonderful Orthogonality Theorem for Character . 36 3.4 Reducible Representations . 38 iii iv CONTENTS 3.5 The Number of Irreducible Representations . 40 3.6 Second Orthogonality Relation for Characters . 41 3.7 Regular Representation . 43 3.8 Setting up Character Tables . 47 3.9 Symmetry Notation . 51 3.10 Selected Problems . 73 4 Basis Functions 75 4.1 Symmetry Operations and Basis Functions . 75 4.2 Basis Functions for Irreducible Representations . 77 ^(¡n) 4.3 Projection Operators Pkl . 82 ^(¡n) 4.4 Derivation of Pk` . 83 4.5 Projection Operations on an Arbitrary Function . 84 4.6 Linear Combinations for 3 Equivalent Atoms . 86 4.7 Selected Problems . 92 5 Group Theory and Quantum Mechanics 95 5.1 Overview . 95 5.2 The Group of SchrÄodinger's Equation . 96 5.3 The Application of Group Theory . 98 5.4 Selected Problems . 100 6 Application to Crystal Field Splitting 103 6.1 Introduction . 103 6.2 Comments on the Form of Crystal Fields . 106 6.3 Characters for the Full Rotation Group . 109 6.4 Example of a Cubic Crystal Field Environment . 113 6.5 Comments on Basis Functions . 119 6.6 Characters for Other Symmetry Operators . 124 6.7 Selected Problems . 126 7 Application to Selection Rules 129 7.1 Summary of Important Results for Basis Functions . 131 7.2 Direct Product of Two Groups . 133 7.3 Direct Product of Two Irreducible Representations . 134 7.4 Characters for the Direct Product of Groups and Repre- sentations . 135 CONTENTS v 7.5 The Selection Rule Concept in Group Theoretical Terms 138 7.6 Selection Rules for Electric Dipole Transitions . 140 7.7 Selected Problems . 144 8 Electronic States of Molecules 147 8.1 Introduction . 147 8.2 General Concept of Equivalence . 151 8.3 Directed Valence Bonding . 153 8.4 Diatomic Molecules . 154 8.4.1 Homonuclear Diatomic Molecules in General . 154 8.4.2 The Hydrogen Molecule H2 . 156 8.4.3 The Helium Molecule He2 . 157 8.4.4 Heterogeneous Diatomic Molecules . 157 8.5 Electronic Orbitals for Multi-atomic Molecules . 162 8.5.1 The NH3 Molecule . 162 8.5.2 The CH4 Molecule . 163 8.5.3 The Hypothetical SH6 Molecule . 171 8.5.4 The SF6 Molecule . 175 8.5.5 The B12H12 Molecule . 178 8.6 Bond Strengths . 181 8.7 σ- and ¼-bonds . 184 8.8 Selected Problems . 192 9 Molecular Vibrations 195 9.1 Molecular Vibrations { Background . 195 9.2 Application of Group Theory to Molecular Vibrations . 197 9.3 Molecular Vibrations in H2O . 200 9.4 Overtones and Combination Modes . 203 9.5 Infrared Activity . 203 9.6 Vibrations for Linear Molecules . 206 9.6.1 The CO Molecule . 206 9.6.2 The O2 Molecule . 208 9.6.3 The CO2 Molecule . 209 9.6.4 The C2H2 Molecule . 210 9.7 Molecular Vibrations in Other Molecules . 212 9.7.1 Vibrations of the NH3 Molecule . 212 9.7.2 Vibrations of the CH4 Molecule . 214 vi CONTENTS 9.7.3 Vibrations of the B12H12 Molecule . 215 9.8 Raman E®ect . 218 9.8.1 The Raman E®ect for H2 . 221 9.8.2 The Raman E®ect for H2O . 221 9.8.3 The Raman E®ect for NH3 . 221 9.8.4 The Raman E®ect for CH4 . 222 9.8.5 The Raman E®ect for CO2 and C2H2 . 222 9.8.6 The Raman E®ect for Planar XH3 . 223 9.8.7 The Raman E®ect for B12H12 . 223 9.9 Rotational Energy Levels . 224 9.10 Vibrational-Rotational Interaction . 227 9.11 Wigner{Eckart Theorem and Selection Rules . 230 9.12 Selected Problems . 232 10 Permutation Groups 235 10.1 Introduction . 236 10.2 Classes of Permutation Groups . 239 10.3 Number of Irreducible Representations . 242 10.4 Basis Functions of Permutation Groups . 243 10.5 Pauli Principle in Atomic Spectra . 245 10.5.1 Two-Electron States . 246 10.5.2 Three-Electron States . 250 10.5.3 Four-Electron States . 255 10.5.4 Five-Electron States . 258 10.6 Discussion . 260 10.7 Selected Problems . 262 11 Transformation of Tensors 267 11.1 Introduction . 267 11.2 Independent Components of Tensors . 270 11.3 Tensors under Permutations . 271 11.4 Independent Components of Tensors . 276 11.5 Tensors Arising in Non-Linear Optics . 277 11.5.1 Cubic Symmetry { Oh . 277 11.5.2 Tetrahedral Symmetry { Td . 280 11.5.3 Hexagonal Symmetry . 281 11.5.4 Hexagonal Symmetry . 282 CONTENTS vii 11.6 Elastic Modulus Tensor . 283 11.6.1 Full Rotational Symmetry: 3D Isotropy . 284 11.6.2 Icosahedral Symmetry . 288 11.6.3 Cubic Symmetry . 289 11.6.4 Full Axial Symmetry . 291 11.6.5 Hexagonal Symmetry . 293 11.6.6 Other Symmetry Groups . 295 11.7 Selected Problems . 295 12 Space Groups 299 12.1 Simple Space Group Operations . 299 12.2 Space Groups and Point Groups . 306 12.3 Compound Space Group Operations . 308 12.4 Incompatibility of Five-Fold Symmetry . 311 12.5 Two Dimensional Space Groups . 315 12.5.1 Five Two-dimensional Bravais Lattices . 315 12.5.2 Notation . 315 12.5.3 Listing of the Space Groups . 316 12.5.4 2D Oblique Space Groups . 318 12.5.5 2D Rectangular Space Groups . 318 12.5.6 2D Square Space Group . 327 12.5.7 2D Hexagonal Space Groups . 335 12.6 Three Dimensional Space Groups . 335 12.6.1 Examples of Non-Symmorphic 3D Space Groups . 336 12.7 Selected Problems . 342 13 Group of the Wave Vector and Bloch's Theorem 345 13.1 Introduction . 345 13.2 Bloch's Theorem . 346 13.3 Group of the Wave Vector . 349 13.3.1 Reciprocal Lattice . 352 13.4 Simple Cubic Lattice . 353 13.5 High Symmetry Points and Axes . 359 13.6 Group Operations on Bloch Functions . 365 13.7 Compatibility Relations . 368 13.7.1 Irreducible Representations . 371 13.8 Selected Problems . 372 viii CONTENTS 14 Applications to Lattice Vibrations 375 14.1 Introduction . 375 14.2 Lattice Modes Relative to Molecular Vibrations . 379 14.3 Zone Center Phonon Modes . 381 14.3.1 In the NaCl Structure . 381 14.3.2 In the Perovskite Structure . 383 14.3.3 Phonons in the Diamond Lattice . 387 14.3.4 Phonons in the Zincblende Structure . 391 14.4 Lattice Modes Away From ~k = 0 . 392 ¼ 14.4.1 Phonons in NaCl at the X point k = a (100) . 393 14.4.2 Phonons in BaTi3 at the X point . 394 14.5 Phonons in Te and Quartz . 399 14.5.1 Phonons in Tellurium . 400 14.5.2 Phonons in ®-Quartz . 407 14.5.3 E®ect of Uniaxial Stress on Phonons . 414 14.6 Lattice Modes in High Tc Related Materials . 417 14.6.1 The K2NiF4 Structure . 417 14.6.2 Phonons in the YBa2Cu3O6 Structure . 419 14.6.3 In The YBa2Cu3O7 Structure . 421 14.7 Selected Problems . 424 15 Use of Standard Reference Texts 425 15.1 Introduction . 425 15.2 Determination of the Crystal Structure . 426.