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- Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
While group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals.
Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics.
The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research.
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Seitenzahl: 489
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Preface
1 Introduction
1.1 Symmetries in Solid-State Physics and Photonics
1.2 A Basic Example: Symmetries of a Square
Part One: Basics of Group Theory
2 Symmetry Operations and Transformations of Fields
2.1 Rotations and Translations
2.2 Transformation of Fields
3 Basics Abstract Group Theory
3.1 Basic Definitions
3.2 Structure of Groups
3.3 Quotient Groups
3.4 Product Groups
4 Discrete Symmetry Groups in Solid-State Physics and Photonics
4.1 Point Groups
4.2 Space Groups
4.3 Color Groups and Magnetic Groups
4.4 Noncrystallographic Groups, Buckyballs, and Nanotubes
5 Representation Theory
5.1 Definition of Matrix Representations
5.2 Reducible and Irreducible Representations
5.3 Characters and Character Tables
5.4 Projection Operators and Basis Functions of Representations
5.5 Direct Product Representations
5.6 WIGNER–ECKART Theorem
5.7 Induced Representations
6 Symmetry and Representation Theory in k-Space
6.1 The Cyclic BORNVON KÁRMÁN Boundary Condition and the BLOCH Wave
6.2 The Reciprocal Lattice
6.3 The BRILLOUIN Zone and the Group of the Wave Vector k
6.4 Irreducible Representations of Symmorphic Space Groups
6.5 Irreducible Representations of Nonsymmorphic Space Groups
Part Two: Applications in Electronic Structure Theory
7 Solution of the SCHRÖDINGER Equation
7.1 The SCHRÖDINGER Equation
7.2 The Group of the SCHRÖDINGER Equation
7.3 Degeneracy of Energy States
7.4 Time-Independent Perturbation Theory
7.5 Transition Probabilities and Selection Rules
8 Generalization to Include the Spin
8.1 The PAULI Equation
8.2 Homomorphism between
SU
(2) and
SO
(3)
8.3 Transformation of the Spin–Orbit Coupling Operator
8.4 The Group of the PAULI Equation and Double Groups
8.5 Irreducible Representations of Double Groups
8.6 Splitting of Degeneracies by Spin–Orbit Coupling
8.7 Time-Reversal Symmetry
9 Electronic Structure Calculations
9.1 Solution of the Schrödinger Equation for a Crystal
9.2 Symmetry Properties of Energy Bands
9.3 Symmetry-Adapted Functions
9.4 Construction of Tight-Binding Hamiltonians
9.5 Hamiltonians Based on Plane Waves
9.6 Electronic Energy Bands and Irreducible Representations
9.7 Examples and Applications
Part Three: Applications in Photonics
10 Solution of Maxwell’s Equations
10.1 MAXWELL’S Equations and the Master Equation for Photonic Crystals
10.2 Group of the Master Equation
10.3 Master Equation as an Eigenvalue Problem
10.4 Models of the Permittivity
11 Two-Dimensional Photonic Crystals
11.1 Photonic Band Structure and Symmetrized Plane Waves
11.2 Group Theoretical Classification of Photonic Band Structures
11.3 Supercells and Symmetry of Defect Modes
11.4 Uncoupled Bands
12 Three-Dimensional Photonic Crystals
12.1 Empty Lattice Bands and Compatibility Relations
12.2 An example: Dielectric Spheres in Air
12.3 Symmetry-Adapted Vector Spherical Waves
Part Four: Other Applications
13 Group Theory of Vibrational Problems
13.1 Vibrations of Molecules
13.2 Lattice Vibrations
14 Landau Theory of Phase Transitions of the Second Kind
14.1 Introduction to Landau’s Theory of Phase Transitions
14.2 Basics of the Group Theoretical Formulation
14.3 Examples with GTPack Commands
Appendix A: Spherical Harmonics
A.1 Complex Spherical Harmonics
A.2 Tesseral Harmonics
Appendix B: Remarks on Databases
B.1 Electronic Structure Databases
B.2 Molecular Databases
B.3 Database of Structures
Appendix C: Use of MPB together with GTPack
C.1 Calculation of Band Structure and Density of States
C.2 Calculation of Eigenmodes
C.3 Comparison of Calculations with MPB and
Mathematica
Appendix D: Technical Remarks on GTPack
D.1 Structure of GTPack
D.2 Installation of GTPack
References
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
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e1
Wolfram Hergert and R. Matthias Geilhufe
Group Theory in Solid State Physics and Photonics
Problem Solving with Mathematica
Authors
Prof. Wolfram HergertMartin Luther University Halle-WittenbergVon-Seckendorff-Platz 106120 HalleGermany
Dr. R. Matthias GeilhufeNorditaRoslagstullsbacken 2310691 StockholmSweden
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.:
applied for
British Library Cataloguing-in-Publication Data:
A catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.
©2018 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN 978-3-527-41133-7ePDF ISBN 978-3-527-41300-3ePub ISBN 978-3-527-41301-0Mobi ISBN 978-3-527-41302-7oBook ISBN 978-3-527-69579-9
CoverDesign Formgeber, Mannheim, GermanyTypesetting le-tex publishing services GmbH, Leipzig, Germany
Preface
Symmetry principles are present in almost all branches of physics. In solid-state physics, for example, we have to take into account the symmetry of crystals, clusters, or more recently detected structures like fullerenes, carbon nanotubes, or quasicrystals. The development of high-energy physics and the standard model of elementary particles would have been unimaginable without using symmetry arguments. Group theory is the mathematical approach used to describe symmetry. Therefore, it has become an important tool for physicists in the past century.
In some cases, understanding the basic concepts of group theory can become a bit tiring. One reason is that exercises connected to the definitions and special structures of groups as well as applications are either trivial or become quickly tedious, even if the concrete calculations are mostly elementary. This occurs, especially, when a textbook does not offer additional help and special tools to assist the reader in becoming familiar with the content. Therefore, we chose a different approach for the present book. Our intention was not to write another comprehensive text about group theory in solid-state physics, but a more applied one based on the Mathematica package GTPack. Therefore, the book is more a handbook on a computational approach to group theory, explaining all basic concepts and the solution of symmetry-related problems in solid-state physics by means of GTPack commands. With the length of the manuscript in mind, we have, at some points, omitted longer and rather technical proofs. However, the interested reader is referred to more rigorous textbooks in those cases and we provide specific references. The examples and tasks in this book are supposed to encourage the reader to work actively with GTPack.
GTPack itself provides more than 200 additional modules to the standard Mathematica language. The content ranges from basic group theory and representation theory to more applied methods like crystal field theory and tight-binding and plane-wave approaches to symmetry-based studies in the fields of solid-state physics and photonics. GTPack is freely available online via GTPack.org. The package is designed to be easily accessible by providing a complete Mathematica style documentation, an optional input validation, and an error strategy. Therefore, we believe that also advanced users of group theory concepts will benefit from the book and the Mathematica package. We provide a compact reference material and a programming environment that will help to solve actual research problems in an efficient way.
In general, computer algebra systems (CAS) allow for a symbolic manipulation of algebraic expressions. Modern systems combine this basic property with numerical algorithms and visualization tools. Furthermore, they provide a programming language for the implementation of individual algorithms. In principle, one has to distinguish between general purpose systems like, e.g., Mathematica and Maple, and systems developed for special purposes. Although the second class of systems usually has a limited range of applications, it aims for much better computational performance. The GAP system (Groups, Algorithms, and Programming) is one of these specialized systems and has a focus on group theory. Extensions like the system Cryst, which was built on top of GAP, are specialized in terms of computations with crystallographic groups.
Nevertheless, for this book we decided to use Mathematica, as Mathematica is well established and often included in the teaching of various Physics departments worldwide. At the Department of Physics of the Martin Luther University Halle-Wittenberg, for example, specialized Mathematica seminars are provided to accompany the theoretical physics lectures. In these courses, GTPack has been used actively for several years.
During the development of GTPack, two paradigms were followed. First, in the usual Mathematica style, the names of commands should be intuitive, i.e., from the name itself it should become clear what the command is supposed to be applied for. This also implies that the nomenclature corresponds to the language physicists usually use in solid-state physics. Second, the commands should be intuitive in their application. Unintentional misuse should not result in longer error messages and endless loop calculations but in an abort with a precise description of the error itself. To distinguish GTPack commands from the standard Mathematica language, all commands have a prefix GT and all options a prefix GO. Analogously to Mathematica itself, commands ending with Q result in logical values, i.e., either TRUE or FALSE. For example, the new command GTGroupQ[list] checks if a list of elements forms a group.
The combination of group theory in physics and Mathematica is not new in its own sense. For example, the books of EL-BATANOUNY and WOOTEN [1] and MCCLAIN [2] also follow this concept. These books provide many code examples of group theoretical algorithms and additional material as a CD or on the Internet. However, in contrast to these books, we do not concentrate on the presentation of algorithms within the text, but provide well-established algorithms within the GTPack modules. This maintains the focus on the application and solution of real physics problems. References for the implemented algorithms are provided whenever appropriate.
In addition to applications in solid-state physics we also discuss photonics, a field that has undergone rapid development over the last 20 years. Here, instead of discussing the symmetry properties of the Schrödinger, Pauli, or Dirac equations, Maxwell’s equations are in the focus of consideration. Analogously to the periodic crystal lattice in solids, periodically structured dielectrics are discussed. GTPack can be applied in a similar manner to both fields.
The book itself is structured as follows. After a short introduction, the basic aspects of group theory are discussed in Part One. Part Two covers the application of group theory to electronic structure theory, whereas Part Three is devoted to its application to photonics. Finally, in Part Four two additional applications are discussed to demonstrate that GTPack will be helpful also for problems other than electronic structure and photonics.
GTPack has a long history in terms of its development. In this context, we would like to thank Diemo Ködderitzsch, Markus Däne, Christian Matyssek, and Stefan Thomas for their individual contributions to the package. We would especially like to acknowledge the careful work of Sebastian Schenk, who contributed significantly to the implementation of the documentation system. Furthermore, we would like to thank Kalevi Kokko, Turku University Finland, who provided a silent work place for us on several occasions. At his department, we had the opportunity to concentrate on both the book and the package and many parts were completed in this context. This was a big help. We acknowledge general interest and support from Martin Hoffmann and Arthur Ernst. Also we would like to thank Wiley-VCH, especially Waltraud Wüst, Martin Preuss and Stefanie Volk.
Lastly, we would like to thank our families for their patience and support during this long-term project.
Stockholm and Halle (Saale), October 2017
R. Matthias Geilhufe, Wolfram Hergert
