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Physics | Primer for Point and Space Groups

Primer for Point and Space Groups

Liboff, Richard

2004, XIV, 220 p.

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This text stems from a course I have taught a number of times, attended by students of material science, electrical engineering, physics, chemistry, physical chemistry and applied mathematics. It is intended as an intro­ ductory discourse to give the reader a first encounter with group theory. The work concentrates on point and space groups as these groups have the principal application in technology. Here is an outline of the salient features of the chapters. In Chapter 1, basic notions and definitions are introduced including that of Abelian groups, cyclic groups, Sylow's theorems, Lagrange's subgroup theorem and the rearrangement theorem. In Chapter 2, the concepts of classes and direct products are discussed. Applications of point groups to the Platonic solids and non-regular dual polyhedra are described. In Chapter 3, matrix representation of operators are introduced leading to the notion of irreducible representations ('irreps'). The Great Orthogonal­ ity Theorem (GOT) is also introduced, followed by six important rules relating to dimensions of irreps. Schur's lemma and character tables are described. Applications to quantum mechanics are discussed in Chapter 4 including descriptions of the rotation groups in two and three dimensions, the symmetric group, Cayley's theorem and Young diagrams. The relation of degeneracy of a quantum state of a system to dimensions of irreps of the group of symmetries of the system are discussed, as well as the basis properties of related eigenfunctions.

Content Level » Lower undergraduate

Keywords » Abstract algebra - Group theory - Operator - Point group - Symmetry group - mechanics - quantum mechanics

Related subjects » Algebra - Electronics & Electrical Engineering - Materials - Physical Chemistry - Physics

Table of contents 

1 Groups and Subgroups.- 1.1 Definitions and Basics.- 1.2 Group Table.- 1.3 Rearrangement Theorem.- 1.4 Building Groups. Subgroups.- Summary of Topics for Chapter 1.- Problems.- 2 Classes and Platonic Solids.- 2.1 Conjugate Elements.- 2.2 Classes.- 2.3 Direct Product.- 2.4 Cnv and Dn Groups.- 2.5 Platonic Solids. T, O and I Groups.- Summary of Topics for Chapter 2.- Problems.- 3 Matrices, Irreps and the Great Orthogonality Theorem.- 3.1 Matrix Representations of Operators.- 3.2 Irreducible Representations.- 3.3 Great Orthogonality Theorem (GOT).- 3.4 Six Important Rules.- 3.5 Character Tables. Bases.- 3.6 Representations of Cyclic Groups.- Summary of Topics for Chapter 3.- Problems.- 4 Quantum Mechanics, the Full Rotation Group, and Young Diagrams.- 4.1 Application to Quantum Mechanics.- 4.2 Full Rotation Group O(3).- 4.3 SU(2).- 4.4 Irreps of O(3)+ and Coupled Angular Momentum States.- 4.5 Symmetric Group; Cayley’s Theorem.- 4.6 Young Diagrams.- 4.7 Degenerate Perturbation Theory.- Summary of Topics for Chapter 4.- Problems.- 5 Space Groups, Brillouin Zone and the Group of k.- 5.1 Cosets and Invariant Subgroups. The Factor Group.- 5.2 Primitive Vectors. Braviais Lattice. Reciprocal Lattice Space.- 5.3 Crystallographic Point Groups and Reciprocal Lattice Space.- 5.4 Bloch Waves and Space Groups.- 5.5 Application to Semiconductor Materials.- 5.6 Time Reversal, Space Inversion and Double Space Groups.- Summary of Topics for Chapter 5.- Problems.- 6 Atoms in Crystals and Correlation Diagrams.- 6.1 Central-Field Approximation.- 6.2 Atoms in Crystal Fields.- 6.3 Correlation Diagrams.- 6.4 Electric and Magnetic Material Properties.- 6.5 Tensors in Group Theory.- Summary of Topics for Chapter 6.- Problems.- 7 Elements of Abstract Algebra and the Galois Group.- 7.1 Integral Domains, Rings and Fields.- 7.2 Numbers.- 7.3 Irreducible Polynomials.- 7.4 The Galois Group.- Symbols for Chapter 7.- Summary of Topics for Chapter 7.- Problems.- Appendix A: Character Tables for the Point Groups.- Bibliography of Works in Group Theory and Allied Topics.

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