Space Group Representations: Theory, Tables and Applications

Description

This book is devoted to the construction of space group representations, their tabulation, and illustration of their use. Representation theory of space groups has a wide range of applications in modern physics and chemistry, including studies of electron and phonon spectra, structural and magnetic phase transitions, spectroscopy, neutron scattering, and superconductivity. The book presents a clear and practical method of deducing the matrices of all irreducible representations, including double-valued, and tabulates the matrices of irreducible projective representations for all 32 crystallographic point groups. One obtains the irreducible representations of all 230 space groups by multiplying the matrices presented in these compact and convenient to use tables by easily computed factors. A number of applications to the electronic band structure calculations are illustrated through real-life examples of different crystal structures. The book's content is accessible to both graduate and advanced undergraduate students with elementary knowledge of group theory and is useful to a wide range of experimentalists and theorists in materials and solid-state physics. Preface Contents Symbols 1 Scope and Overview References 2 Mathematical Preliminaries 2.1 Elements of Group Theory 2.1.1 Definition of Groups 2.1.2 Subgroups 2.1.3 Invariant Subgroup 2.1.4 Classes of Conjugate Elements 2.2 Matrices: Definitions and Notation 2.2.1 Special Types of Matrices 2.2.2 Equivalent Matrices 2.2.3 Reducible Matrices 2.2.4 Direct Matrix Product 2.3 Representation Theory 2.3.1 Matrix Groups 2.3.2 Group Representations 2.3.3 Regular Representation 2.3.4 Fundamental Theorems 2.3.5 Group Characters 2.3.6 Reality Properties 2.3.7 Direct Product of Representations References 3 Induced Representations 3.1 Regular Continuation from a Subgroup to a Group 3.2 Normalizer of a Representation of an Invariant Subgroup 3.3 Unitary Transformation for Elements of a Normalizer 3.4 Regular Continuation to the Normalizer 3.5 Projective Representations of the Factor Group of a Normalizer 3.6 Small Representations of a Normalizer 3.7 Special Case of an Abelian Invariant Subgroup 3.8 Main Theorem 3.9 Reality Properties of the Induced Representations 3.10 Algorithm for Constructing Irreducible Representations References 4 Projective Representations 4.1 The Factor System 4.2 Main Properties of Projective Representations 4.3 Regular Representation and Completeness Criterion 4.4 Group Generators 4.5 Character Factors 4.6 Covering Group and Its Representations 4.7 Irreducible Projective Representations 4.8 Outline of the Algorithm References 5 Representations of the Space Groups 5.1 Space Groups and Their Properties 5.2 Reciprocal Lattice and Brillouin Zone 5.3 Translation Symmetry and Its Consequences 5.4 Types of the k Vectors in the Brillouin Zone 5.5 Group of the Wavevector 5.6 Small Representations of the Group of the Wavevector 5.7 Irreducible Representations of Space Groups 5.8 Compatibility Relations References 6 Tables 6.1 Summary of Theory and Guide to Tables 6.1.1 Summary of Theory 6.1.2 Guide to Tables 6.2 Elements and Multiplication Tables of the Groups Oh and D6h 6.3 Groups C2, Ci and Cs 6.4 Group C3 6.5 Groups C4 and S4 6.6 Groups D2, C2h and C2v 6.7 Groups C6, S6 and C3h 6.8 Groups D3 and C3v 6.9 Groups D4, C4v and D2d 6.10 Group C4h 6.11 Group D2h 6.12 Groups D6, D3d, C6v and D3h 6.13 Group T 6.14 Group C6h 6.15 Group D4h 6.16 Groups O and Td 6.17 Group D6h 6.18 Group Th 6.19 Group Oh 6.20 Example of the Use of the Tables: The Space Group Oh7 References 7 Group Theory and Quantum Mechanics 7.1 Representation Spaces 7.1.1 Basis Functions 7.1.2 Expansion Theorem 7.1.3 Reduction by Idempotent Operators 7.1.4 Orthogonality of the Basis Functions 7.1.5 Criterion for Representation Functions 7.2 Classification of States 7.2.1 Hamiltonian of a System with Symmetry 7.2.2 Eigenspaces and Representations 7.3 Splitting of Eigenvalues 7.3.1 Schrödinger Perturbation Theory 7.3.2 Effect of Symmetry 7.4 Selection Rules References 8 Group Theory and Solid-State Physics 8.1 Electron in a Periodic Field 8.1.1 Mathematical Formulation of the Problem 8.1.2 The Ritz Method 8.2 Symmetrized Coordinate Functions 8.2.1 Construction of Symmetrized Functions 8.2.2 Matrix Elements of the Hamiltonian 8.3 Double-Valued Representations 8.4 Time-Reversal Symmetry 8.4.1 Time-Reversal Operator 8.4.2 Systems with Time-Reversal Symmetry 8.4.3 Criterion of Extra Degeneracy References 9 Energy Band Calculation 9.1 Basics of Band Calculations 9.2 The OPW Method 9.2.1 Main Formulas 9.2.2 Symmetrized OPW 9.2.3 Crystal Potential 9.2.4 Core Shifts 9.2.5 Generalized Eigenvalue Problem 9.3 Summary References 10 Energy Bands of Diamond 10.1 Crystal Structure 10.1.1 Symmetry of the Crystal 10.1.2 Reciprocal Lattice and Brillouin Zone 10.2 Energy Bands 10.2.1 Problems and Methods 10.2.2 Calculation Results 10.2.3 Comparison with Experiment 10.3 Summary References 11 Electronic States in Atoms and Solids 11.1 Correspondence Between Atomic and Crystal States 11.1.1 Background Material 11.1.2 Reducible Representation 11.1.3 Decomposition into Irreducible Components 11.2 An Example: The Pyrite Structure 11.2.1 Crystal Structure and Space Group Th6 11.2.2 Atomic and Crystal States 11.3 Summary References 12 Group-Theoretical Analysis of the Cr3Si Structure 12.1 Crystal Structure 12.1.1 Geometric Description 12.1.2 Space Group Oh3 12.1.3 Wigner–Seitz Cell 12.2 Small Representations of the Group of the Wavevector 12.2.1 Brillouin Zone 12.2.2 Structure of the Small Representations 12.2.3 Small Representations 12.3 Compatibility Relations 12.4 Time-Reversal Symmetry 12.4.1 Theory 12.4.2 Results 12.5 Correspondence Between Atomic and Crystal States 12.5.1 Method 12.5.2 Results 12.6 Concluding Remarks References 13 Symmetry in the Tight-Binding Method 13.1 General Method 13.1.1 Properties of Symmetry Operators 13.1.2 Problem Setup and Notation 13.1.3 Hermiticity of the Hamiltonian 13.1.4 Inversion Symmetry 13.1.5 Time-Reversal Symmetry 13.1.6 Relations Between the Matrix Components 13.1.7 Point Symmetry 13.1.8 Calculation of the Matrix Components 13.1.9 Determination of the Independent Parameters 13.1.10 Calculation Algorithm 13.2 Application to the Diamond Structure References 14 Interpolation of Energy Bands 14.1 Quasidiagonal Form of the Hamiltonian Matrix 14.2 Tight-Binding Method and kp-Method 14.3 An Example: The Diamond Structure 14.4 Electron Density of States References 15 Quasi-2D Dichalcogenides with the 2H-TaSe2 Structure 15.1 Introduction 15.2 Crystal Structure of 2H-TaSe2 15.3 Reciprocal Lattice and Brillouin Zone 15.4 Matrix Elements of the Hamiltonian 15.5 Hamiltonian Matrix at High Symmetry Points 15.6 Interim Conclusion References 16 Symmetrized Plane Waves 16.1 Projection Operator 16.2 Construction of Symmetrized Plane Waves References 17 Symmetrized Bloch Sums 17.1 Theory 17.1.1 Projection Matrix 17.1.2 Projection Matrix on Bloch Functions 17.1.3 Irreducible Representations of the Rotation Group 17.2 Algorithm 17.3 Summary References 18 Discussion and Outlook 18.1 Representation Theory of the Space Groups 18.2 Tables of the Space Group Representations 18.3 Applications of the Space Group Representations References Appendix A Frobenius-Schur Theorem and Its Applications A.1 Frobenius-Schur Theorem for Projective Representations A.2 Reality Criterion for Space Groups Appendix B Construction of Projective Representations of the Point Group T Appendix C More on Symmetry in Quantum Mechanics C.1 Discussion of Wigner's Theorem C.2 Projection Operator Corresponding to G and Gk Appendix D Factor System of Double-Valued Representations D.1 Composition of Two Rotations D.2 Euler-Rodrigues Parameters Appendix E Multiplication Table of the Point Group Oh Appendix F Projective Representations and Compatibility Relations for Oh7 Appendix G Factorization of the Characteristic Equation for Oh7 Index