Space Group Representations: Theory, Tables and Applications

✍ Scribed by Nikolai B. Melnikov, Boris I. Reser

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 343
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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