Primer for Point and Space Groups

by Liboff, Richard L.

• ISBN13: 9780387402482
• ISBN10: 0387402489
• Format: Hardcover
• Copyright: 2004-02-01
• Publisher: Springer Verlag

Summary

Written in the spirit of Liboff's acclaimed text on quantum mechanics, Primer for Point and Space Groups is an ideal introductory text for undergraduates in physics, engineering, materials science, chemistry. With its shrewd selection and arrangement of examples, the book traces at every turn the physical implications of abstract concepts. The book will therefore provide a solid background for those students who expect to use group theory in nuclear and particle physics and other specific applications. Among the many introductions to group theory currently available, few are as closely attuned to the needs of the applied scientist without compromise to the logic and lucidity of the presentation. Here are some of the applications covered in Liboff's Primer for Point and Space Groups: Applications to Quantum Mechanics:- Irreducible representations and degeneracy- Full rotation group; SU(2) group; angular momentum- Symmetry group and the wavefunction- Young diagrams and the wavefunction- Degenerate perturbation theory- Great Orthogonality Theorem Applications to Solid-State Physics:- Translation and Crystallographic Point Groups- Holohedral groups- Bloch waves and space groups- Bravais lattice- Energy -band eigenenergies- Seitz operator, (Translation and Rotation)- Glide-plane and screw operators- Diamond structure- Group of {bf k} and Star of $bf k $.- Space group of {bf k}- Time reversal and Space inversion effects on the- wavefunction and eigenenergies- Symmorphic group- Factor Group theorem Applications to Material Media:- Splitting of electron levels in crystals with symmetry- Correlation diagrams- Neumann's Principle- Polarizability- Piezoelectric effect- Classification of magnetic crystals- Black and white groups

Table of Contents

Preface	vii
1 Groups and Subgroups	1	(17)
1.1 Definitions and Basics	1	(2)
1.2 Group Table	3	(1)
1.3 Rearrangement Theorem	4	(3)
1.4 Building Groups. Subgroups	7	(7)
Summary of Topics for Chapter 1	14	(1)
Problems	15	(3)
2 Classes and Platonic Solids	18	(18)
2.1 Conjugate Elements	18	(1)
2.2 Classes	19	(1)
2.3 Direct Product	20	(1)
2.4 Cnv and Dn Groups	20	(5)
2.5 Platonic Solids. T, O and I Groups	25	(7)
Summary of Topics for Chapter 2	32	(1)
Problems	33	(3)
3 Matrices, Irreps and the Great Orthogonality Theorem	36	(18)
3.1 Matrix Representations of Operators	36	(5)
3.2 Irreducible Representations	41	(1)
3.3 Great Orthogonality Theorem (GOT)	42	(2)
3.4 Six Important Rules	44	(2)
3.5 Character Tables. Bases	46	(2)
3.6 Representations of Cyclic Groups	48	(3)
Summary of Topics for Chapter 3	51	(1)
Problems	52	(2)
4 Quantum Mechanics, the Full Rotation Group, and Young Diagrams	54	(35)
4.1 Application to Quantum Mechanics	54	(4)
4.2 Full Rotation Group O(3)	58	(2)
4.3 SU(2)	60	(5)
4.4 Irreps of O(3)+ and Coupled Angular Momentum States	65	(3)
4.5 Symmetric Group; Cayley's Theorem	68	(5)
4.6 Young Diagrams	73	(7)
4.7 Degenerate Perturbation Theory	80	(3)
Summary of Topics for Chapter 4	83	(1)
Problems	84	(5)
5 Space Groups, Brillouin Zone and the Group of k	89	(47)
5.1 Cosets and Invariant Subgroups. The Factor Group	89	(3)
5.2 Primitive Vectors. Braviais Lattice. Reciprocal Lattice Space	92	(4)
5.3 Crystallographic Point Groups and Reciprocal Lattice Space	96	(7)
5.4 Bloch Waves and Space Groups	103	(17)
5.5 Application to Semiconductor Materials	120	(4)
5.6 Time Reversal, Space Inversion and Double Space Groups	124	(6)
Summary of Topics for Chapter 5	130	(1)
Problems	131	(5)
6 Atoms in Crystals and Correlation Diagrams	136	(24)
6.1 Central-Field Approximation	136	(2)
6.2 Atoms in Crystal Fields	138	(4)
6.3 Correlation Diagrams	142	(2)
6.4 Electric and Magnetic Material Properties	144	(11)
6.5 Tensors in Group Theory	155	(3)
Summary of Topics for Chapter 6	158	(1)
Problems	159	(1)
7 Elements of Abstract Algebra and the Galois Group	160	(34)
7.1 Integral Domains, Rings and Fields	160	(5)
7.2 Numbers	165	(6)
7.3 Irreducible Polynomials	171	(7)
7.4 The Galois Group	178	(9)
Symbols for Chapter 7	187	(1)
Summary of Topics for Chapter 7	188	(1)
Problems	189	(5)
Appendix A: Character Tables for the Point Groups	194	(18)
Appendix B: Irreps for the Oh and Doh Groups, their Dimensions and Notations	212	(1)
Bibliography of Works in Group Theory and Allied Topics	213	(4)
Index	217