Introduction to Group Theory with Applications
Materials Science and Technology
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Introduction to Group Theory with Applications covers the basic principles, concepts, mathematical proofs, and applications of group theory. This book is divided into 13 chapters and begins with discussions of the elementary topics related to the subject, including symmetry operations and group concepts. The succeeding chapters deal with the properties of matrix representations of finite groups, the vibrations of molecular and crystals, vibrational wave function, selection rules, and molecular approximations. These topics are followed by reviews of the basic of quantum mechanics, crystal field theory, atomic physics, hybrid functions, and molecular orbital theory. The last chapters describe the symmetry of crystal lattices, the band theory of solids, and the full rotation group. This book will be of value to undergraduate mathematics and physics students.
Table of Contents
Preface
Acknowledgments
Chapter 1 Symmetry Operations
1-1 Introduction
1-2 Point Symmetry Operations
1-3 The Stereographic Projection
1-4 The 32 Crystallographic Point Groups
1-5 Related Considerations
1-6 Space Group Example
Notes
Problems
Chapter 2 Group Concepts
2-1 Introduction
2-2 Definition of a Group
2-3 Symmetry Operations Form a Group
2-4 Related Group Concepts
2-5 Isomorphism and Homomorphism
2-6 Special Kinds of Groups
2-7 More Involved Group Concepts (including a Factor Group of a Space Group)
Appendix to Chapter 2
Notes
Problems
Chapter 3 Matrix Representations of Finite Groups
3-1 Introduction
3-2 Representations
3-3 Irreducible Representations
3-4 Representations of a Factor Group
Appendix to Chapter 3
Notes
Problems
Chapter 4 Characters of Matrix Representations of Finite Groups
4-1 Properties of Characters of Irreducible Representations
4-2 Character Tables
4-3 Reduction of a Reducible Representation
4-4 Basis Functions
4-5 Examples—Neumann Principle
4-6 Atomic Positions
4-7 The Hamiltonian
Appendix to Chapter 4
Notes
Problems
Chapter 5 Vibrations of Molecules and Crystals
5-1 3N Degrees of Freedom
5-2 General Considerations
5-3 Number and Type of Normal Modes for Molecules
5-4 Internal Coordinates
5-5 Crystals
5-6 Eigenvectors and Symmetry Adapted Vectors
5-7 Projection Operators
5-8 Projection Operators Applied to Normal Coordinates
Notes
Problems
Chapter 6 Normal Modes (Direct Product and Selection Rules)
6-1 Direct Product of Irreducible Representations
6-2 Vibrational Wave Function
6-3 Selection Rules—Infrared and Raman
6-4 Molecular Approximations (Site Symmetry and Davydov Splitting)
Notes
Problems
Chapter 7 Quantum Mechanics
7-1 Atomic Wave Functions
7-2 Transformation of Functions
7-3 Eigenfunctions as Basis Functions
7-4 Proper Rotations and Angular Momentum
7-5 Perturbations
7-6 Matrix Elements (Selection Rules)
7-7 General Secular Equation Problem
Appendix to Chapter 7
Notes
Problems
Chapter 8 Crystal Field Theory (and Atomic Physics)
8-1 Rotations in Terms of Euler Angles
8-2 Representations of the Full Rotation Group
8-3 Reduction of Symmetry
8-4 Energy Level Diagrams (Correlation Diagrams)
8-5 Crystal Double Groups
8-6 Correlation Diagrams including Double Groups
8-7 Other Crystal Field Effects
Appendix to Chapter 8
Notes
Problems
Chapter 9 Hybrid Functions
9-1 Introduction
9-2 Simple Hybrid Functions and Bonding
9-3 Tetrahedral Hybridization
9-4 Other Hybrid Functions
9-5 π-Hybrid Functions
9-6 Comment on Hybrid Orbitals (Slater Determinant)
Notes
Problems
Chapter 10 Molecular Orbital Theory
10-1 Hydrogen Molecular Ion
10-2 Simple MO Theory
10-3 Transition Metal Complexes
10-4 LCAO-MO of π-Electrons in Conjugated Hydrocarbons
10-5 Woodward-Hoffman Rules
Notes
Problems
Chapter 11 Symmetry of Crystal Lattices
11-1 The Real Affine Group
11-2 Space Group
11-3 Translational Lattice
11-4 International Tables for X-Ray Crystallography, International Notation, etc.
11-5 Magnetic Groups (Color Groups)
Notes
Problems
Chapter 12 Band Theory of Solids
12-1 Translational Symmetry
12-2 Symmorphic Space Groups
12-3 Nonsymmorphic Space Groups
12-4 Spin-Orbit Effects on Bands
12-5 Time Reversal Symmetry
Notes
Problems
Chapter 13 The Full Rotation Group
13-1 The Homomorphism between the Special Unitary Group in Two Dimensions, SU(2), and the Three-Dimension Rotation Group
13-2 Irreducible Representations of SU(2)
13-3 Wigner Coefficients
13-4 Irreducible Tensor Operators
13-5 The Wigner-Eckart Theorem
13-6 Survey of 3j and Racah Coefficients
Notes
Problems
Appendix 1 Crystal Systems
Appendix 2 The 32 Point Groups
Appendix 3 Character Tables
Appendix 4 Space Groups
Appendix 5 Matrices, Vector Spaces, and Linear Operators
Appendix 6 Direct Product Tables
Appendix 7 Correlation Tables
Appendix 8 Spherical Harmonics
Appendix 9 Tanable-Sugano Diagrams
Appendix 10 Double Group Character Tables
Bibliography
Index
Product details
- No. of pages: 446
- Language: English
- Copyright: © Academic Press 1977
- Published: January 28, 1977
- Imprint: Academic Press
- eBook ISBN: 9781483191492
About the Author
Gerald Burns
Affiliations and Expertise
IBM Thomas J. Watson Research Center, Yorktown Heights, New York
About the Editors
Allen M. Alper
A. S. Nowick
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