M.S. Dresselhaus G. Dresselhaus A. Jorio. Group Theory. Application to the Physics of Condensed Matter. With 131 Figures and 219 Tables.

Transcription

1 M.S. Dresselhaus G. Dresselhaus A. Jorio Group Theory Application to the Physics of Condensed Matter With 131 Figures and 219 Tables 4) Springer

2 Contents Part I Basic Mathematics 1 Basic Mathematical Background: Introduction Definition of a Group Simple Example of a Group Basic Definitions Rearrangement Theorem Cosets Conjugation and Class Factor Groups Group Theory and Quantum Mechanics 11 2 Representation Theory and Basic Theorems Important Definitions Matrices Irreducible Representations The Unitarity of Representations Schur's Lemma (Part 1) Schur's Lemma (Part 2) Wonderful Orthogonality Theorem Representations and Vector Spaces 28 3 Character of a Representation Definition of Character Characters and Class Wonderful Orthogonality Theorem for Character Reducible Representations The Number of Irreducible Representations Second Orthogonality Relation for Characters Regular Representation Setting up Character Tables 40

3 X Contents 3.9 Schoenflies Symmetry Notation The Hermann Mauguin Symmetry Notation Symmetry Relations and Point Group Classifications 48 4 Basis Functions Symmetry Operations and Basis Functions Basis Functions for Irreducible Representations Projection Operators 1-3g'" ) Derivation of an Explicit Expression for fier' ) The Effect of Projection Operations on an Arbitrary Function Linear Combinations of Atomic Orbitals for Three Equivalent Atoms at the Corners of an Equilateral Triangle The Application of Group Theory to Quantum Mechanics 70 Part II Introductory Application to Quantum Systems 5 Splitting of Atomic Orbitals in a Crystal Potential Introduction Characters for the Full Rotation Group Cubic Crystal Field Environment for a Paramagnetic Transition Metal Ion Comments on Basis Functions Comments on the Form of Crystal Fields 92 6 Application to Selection Rules and Direct Products The Electromagnetic Interaction as a Perturbation Orthogonality of Basis Functions Direct Product of Two Groups Direct Product of Two Irreducible Representations Characters for the Direct Product Selection Rule Concept in Group Theoretical Terms Example of Selection Rules 106 Part III Molecular Systems 7 Electronic States of Molecules and Directed Valence Introduction General Concept of Equivalence Directed Valence Bonding Diatomic Molecules Homonuclear Diatomic Molecules Heterogeneous Diatomic Molecules 120

4 Contents XI 7.5 Electronic Orbitals for Multiatomic Molecules The NH 3 Molecule The CH4 Molecule The Hypothetical SH 6 Molecule The Octahedral SF6 Molecule u- and 7r-Bonds Jahn Teller Effect Molecular Vibrations, Infrared, and Raman Activity Molecular Vibrations: Background Application of Group Theory to Molecular Vibrations Finding the Vibrational Normal Modes Molecular Vibrations in H Overtones and Combination Modes Infrared Activity Raman Effect Vibrations for Specific Molecules The Linear Molecules Vibrations of the NH 3 Molecule Vibrations of the CH 4 Molecule Rotational Energy Levels The Rigid Rotator Wigner Eckart Theorem Vibrational Rotational Interaction 174 Part IV Application to Periodic Lattices 9 Space Groups in Real Space Mathematical Background for Space Groups Space Groups Symmetry Operations Compound Space Group Operations Translation Subgroup Symmorphic and Nonsymmorphic Space Groups Bravais Lattices and Space Groups Examples of Symmorphic Space Groups Cubic Space Groups and the Equivalence Transformation Examples of Nonsymmorphic Space Groups Two-Dimensional Space Groups D Oblique Space Groups D Rectangular Space Groups D Square Space Group D Hexagonal Space Groups Line Groups 204

5 XII Contents 9.5 The Determination of Crystal Structure and Space Group Determination of the Crystal Structure Determination of the Space Group Space Groups in Reciprocal Space and Representations Reciprocal Space Translation Subgroup Representations for the Translation Group Bloch's Theorem and the Basis of the Translational Group Symmetry of k Vectors and the Group of the Wave Vector Point Group Operation in r-space and k-space The Group of the Wave Vector Ok and the Star of k Effect of Translations and Point Group Operations an Bloch Functions Space Group Representations Symmorphic Group Representations Nonsymmorphic Group Representations and the Multiplier Algebra Characters for the Equivalence Representation Common Cubic Lattices: Symmorphic Space Groups The Point Points with k Compatibility Relations The Diamond Structure: Nonsymmorphic Space Group Factor Group and the 1' Point Points with k Finding Character Tables for all Groups of the Wave Vector 235 Part V Electron and Phonon Dispersion Relation 11 Applications to Lattice Vibrations Introduction Lattice Modes and Molecular Vibrations Zone Center Phonon Modes The NaC1 Structure The Perovskite Structure Phonons in the Nonsymmorphic Diamond Lattice Phonons in the Zinc Blende Structure Lattice Modes Away frorn k = Phonons in NaC1 at the X Point k = (ir/a)(100) Phonons in BaTiO 3 at the X Point Phonons at the K Point in Two-Dimensional Graphite 258

6 Contents Phonons in Te and a-quartz Nonsymmorphic Structures Phonons in Tellurium Phonons in the a-quartz Structure Effect of Axial Stress on Phonons XIII Electronic Energy Levels in a Cubic Crystals Introduction Plane Wave Solutions at k Symmetrized Plane Solution Waves along the L-Axis Plane Wave Solutions at the X Point Effect of Glide Planes and Screw Axes Energy Band Models Based on Symmetry Introduction k p Perturbation Theory Example of k p Perturbation Theory for a Nondegenerate Band Two Band Model: Degenerate First-Order Perturbation Theory Degenerate second-order k p Perturbation Theory Nondegenerate k p Perturbation Theory at a 3 Point Use of k p Perturbation Theory to Interpret Optical Experiments Application of Group Theory to Valley Orbit Interactions in Semiconductors Background Impurities in Multivalley Semiconductors The Valley Orbit Interaction Spin Orbit Interaction in Solids and Double Groups Introduction Crystal Double Groups Double Group Properties Crystal Field Splitting Including Spin Orbit Coupling Basis Functions for Double Group Representations Some Explicit Basis Functions Basis Functions for Other States Comments on Double Group Character Tables Plane Wave Basis Functions for Double Group Representations Group of the Wave Vector for Nonsymmorphic Double Groups 362

7 XIV Contents 15 Application of Double Groups to Energy Bands with Spin Introduction E(k) for the Empty Lattice Including Spin Orbit Interaction The k p Perturbation with Spin Orbit Interaction E(k) for a Nondegenerate Band Including Spin Orbit Interaction E(k) for Degenerate Bands Including Spin Orbit Interaction Effective g-factor Fourier Expansion of Energy Bands: Slater Koster Method Contributions at d = Contributions at d = Contributions at d = Summing Contributions through d = Other Degenerate Levels 397 Part VI Other Symmetries 16 Time Reversal Symmetry The Time Reversal Operator Properties of the Time Reversal Operator The Effect of T on E(k), Neglecting Spin The Effect of T on E(k), Including the Spin Orbit Interaction Magnetic Groups Introduction Types of Elements Types of Magnetic Point Groups Properties of the 58 Magnetic Point Groups 111 ic} Examples of Magnetic Structures Effect of Symmetry on the Spin Hamiltonian for the 32 Ordinary Point Groups Permutation Groups and Many-Electron States Introduction Classes and Irreducible Representations of Permutation Groups Basis Functions of Permutation Groups Pauli Principle in Atomic Spectra Two-Electron States Three-Electron States Four-Electron States Five-Electron States General Comments on Many-Electron States 451

8 Contents XV 18 Symmetry Properties of Tensors Introduction Independent Components of Tensors Under Permutation Group Symmetry Independent Components of Tensors: Point Symmetry Groups Independent Components of Tensors Under Full Rotational Symmetry Tensors in Nonlinear Optics Cubic Symmetry: Oh Tetrahedral Symmetry: Td Hexagonal Symmetry: D6h Elastic Modulus Tensor Full Rotational Symmetry: 3D Isotropy Icosahedral Symmetry Cubic Symmetry Other Symmetry Groups 474 A Point Group Character Tables 479 B Two-Dimensional Space Groups 489 C Tables for 3D Space Groups 499 C.1 Real Space 499 C.2 Reciprocal Space 503 D Tables for Double Groups 521 E Group Theory Aspects of Carbon Nanotubes 533 E.1 Nanotube Geometry and the (n, m) Indices 534 E.2 Lattice Vectors in Real Space 534 E.3 Lattice Vectors in Reciprocal Space 535 E.4 Compound Operations and Tube Helicity 536 E.5 Character Tables for Carbon Nanotubes 538 F Permutation Group Character Tables 543 References 549 Index 553