Primer for Point and Space Groups

ISBN-13: 9780387402482 / Angielski / Twarda / 2003 / 220 str.

Richard L. Liboff; R. Liboff

Opis

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."

Recenzje

From the reviews:

"I teach a course which involves symmetry and point groups so I was especially pleased to have been asked to do this review. ... The material is covered succinctly but in surprising depth for the size of the volume. ... The text is clear throughout and the nomenclature is consistent. ... It is more likely to provide a solid background for researchers and postgraduate students entering a new field or undergraduates dealing with advanced topics which are especially group theory intensive." (Contemporary Physics, Vol. 46 (4), July-August, 2005)

"This small volume may prove most useful to the lecturer in symmetry and/or group theory who will find many brief, interesting, and unusual ideas and approaches to many of different topics. ... The individual chapters are followed by interesting questions that are designed to extend the applications discussed in the chapter. ... Again these questions will surely prove useful to the lecturer for future lectures and exams!" (Gary J. Long, Fernande Grandjean, Physicalia, Vol. 57 (3), 2005)

"Liboff's book provides a general introduction to the subject with applications in quantum mechanics and solid state physics. ... A distinguishing feature of the book is the generous use of figures and illustrations ... . The book could be used in a short course on group theory or to supplement courses on quantum mechanics or solid state physics. It also provides a relatively painless introduction to the subject for those who need to learn it later in life." (Serdar Kuyucak, The Physicist, Vol. 41 (5), 2004)

"Written in the spirit of Liboff's text on quantum mechanics, the book is an introductory text for undergraduates in physics, engineering, materials science, and chemistry. With its selection and arrangement of examples, the book traces at every turn the physical implications of abstract concepts. It provides a solid background for those students who expect to use group theory in nuclear and particle physics and other specific applications." (Zentralblatt für Didaktik der Mathematik, Mai, 2004)

"This text, a part of the series Undergraduate Texts in Contemporary Physics, is intended as an introductory book on group theory for undergraduates in materials science, electrical engineering, physics, chemistry, physical chemistry, and applied mathematics. ... Overall, this text will probably be useful as a reference to those in physics, materials science, and applied mathematics ... . The text would be ... useful for those involved in the advanced study of solid state materials, as this tends to be the focus of the work." (Christopher Stromberg, Theoretical Chemistry Accounts, Vol. 115 (1), 2006)

Spis treści

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.