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Group Theory in Solid State Physics and Photonics: Problem Solving with Mathematica [Pehme köide]

, (University Halle-Wittenberg, Germany)
  • Formaat: Paperback / softback, 377 pages, kõrgus x laius x paksus: 244x170x18 mm, kaal: 748 g
  • Ilmumisaeg: 20-Jun-2018
  • Kirjastus: Blackwell Verlag GmbH
  • ISBN-10: 352741133X
  • ISBN-13: 9783527411337
Teised raamatud teemal:
Group Theory in Solid State Physics and Photonics: Problem Solving with MathematicaSuurem pilt
  • Formaat: Paperback / softback, 377 pages, kõrgus x laius x paksus: 244x170x18 mm, kaal: 748 g
  • Ilmumisaeg: 20-Jun-2018
  • Kirjastus: Blackwell Verlag GmbH
  • ISBN-10: 352741133X
  • ISBN-13: 9783527411337
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783527411337.html
Märksõnad:
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.
Preface vii
1 Introduction
1(8)
1.1 Symmetries in Solid-State Physics and Photonics
4(2)
1.2 A Basic Example: Symmetries of a Square
6(3)
Part One Basics of Group Theory
9(740)
2 Symmetry Operations and Transformations of Fields
11(22)
2.1 Rotations and Translations
11(2)
2.1.1 Rotation Matrices
13(3)
2.1.2 Euler Angles
16(2)
2.1.3 Euler--Rodrigues Parameters and Quaternions
18(5)
2.1.4 Translations and General Transformations
23(2)
2.2 Transformation of Fields
25(8)
2.2.1 Transformation of Scalar Fields and Angular Momentum
26(1)
2.2.2 Transformation of Vector Fields and Total Angular Momentum
27(1)
2.2.3 Spinors
28(5)
3 Basics Abstract Group Theory
33(19)
3.1 Basic Definitions
33(6)
3.1.1 Isomorphism and Homomorphism
38(1)
3.2 Structure of Groups
39(7)
3.2.1 Classes
40(2)
3.2.2 Cosets and Normal Divisors
42(4)
3.3 Quotient Groups
46(2)
3.4 Product Groups
48(4)
4 Discrete Symmetry Groups in Solid-State Physics and Photonics
52(31)
4.1 Point Groups
52(7)
4.1.1 Notation of Symmetry Elements
52(4)
4.1.2 Classification of Point Groups
56(3)
4.2 Space Groups
59(10)
4.2.1 Lattices, Translation Group
59(3)
4.2.2 Symmorphic and Nonsymmorphic Space Groups
62(3)
4.2.3 Site Symmetry, Wyckoff Positions, and Wigner-Seitz Cell
65(4)
4.3 Color Groups and Magnetic Groups
69(6)
4.3.1 Magnetic Point Groups
69(3)
4.3.2 Magnetic Lattices
72(1)
4.3.3 Magnetic Space Groups
73(2)
4.4 Noncrystallographic Groups, Buckyballs, and Nanotubes
75(8)
4.4.1 Structure and Group Theory of Nanotubes
75(4)
4.4.2 Buckminsterfullerene C60
79(4)
5 Representation Theory
83(50)
5.1 Definition of Matrix Representations
84(4)
5.2 Reducible and Irreducible Representations
88(6)
5.2.1 The Orthogonality Theorem for Irreducible Representations
90(4)
5.3 Characters and Character Tables
94(611)
5.3.1 The Orthogonality Theorem for Characters
96(2)
5.3.2 Character Tables
98(1)
5.3.3 Notations of Irreducible Representations
98(4)
5.3.4 Decomposition of Reducible Representations
102(3)
5.4 Projection Operators and Basis Functions of Representations
105(7)
5.5 Direct Product Representations
112(8)
5.6 Wigner--Eckart Theorem
120(3)
5.7 Induced Representations
123(10)
6 Symmetry and Representation Theory in k-Space
133(616)
6.1 The Cyclic Born--von Karman Boundary Condition and the Bloch Wave
133(3)
6.2 The Reciprocal Lattice
136(1)
6.3 The Brillouin Zone and the Group of the Wave Vector k
137(5)
6.4 Irreducible Representations of Symmorphic Space Groups
142(1)
6.5 Irreducible Representations of Nonsymmorphic Space Groups
143(6)
Part Two Applications in Electronic Structure Theory
149(2)
7 Solution of the Schrodinger Equation
151(1)
7.1 The Schrodinger Equation
151(2)
7.2 The Group of the Schrodinger Equation
153(1)
7.3 Degeneracy of Energy States
154(3)
7.4 Time-Independent Perturbation Theory
157(2)
7.4.1 General Formalism
159(1)
7.4.2 Crystal Field Expansion
160(4)
7.4.3 Crystal Field Operators
164(5)
7.5 Transition Probabilities and Selection Rules
169(8)
8 Generalization to Include the Spin
177(1)
8.1 The Pauli Equation
177(1)
8.2 Homomorphism between SU(2) and SO(3)
178(2)
8.3 Transformation of the Spin--Orbit Coupling Operator
180(3)
8.4 The Group of the Pauli Equation and Double Groups
183(3)
8.5 Irreducible Representations of Double Groups
186(3)
8.6 Splitting of Degeneracies by Spin--Orbit Coupling
189(4)
8.7 Time-Reversal Symmetry
193(4)
8.7.1 The Reality of Representations
193(1)
8.7.2 Spin-Independent Theory
194(2)
8.7.3 Spin-Dependent Theory
196(1)
9 Electronic Structure Calculations
197(1)
9.1 Solution of the Schrodinger Equation for a Crystal
197(1)
9.2 Symmetry Properties of Energy Bands
198(2)
9.2.1 Degeneracy and Symmetry of Energy Bands
200(1)
9.2.2 Compatibility Relations and Crossing of Bands
201(2)
9.3 Symmetry-Adapted Functions
203(1)
9.3.1 Symmetry-Adapted Plane Waves
203(2)
9.3.2 Localized Orbitals
205(5)
9.4 Construction of Tight-Binding Hamiltonians
210(2)
9.4.1 Hamiltonians in Two-Center Form
212(4)
9.4.2 Hamiltonians in Three-Center Form
216(8)
9.4.3 Inclusion of Spin--Orbit Interaction
224(1)
9.4.4 Tight-Binding Hamiltonians from ab initio Calculations
225(2)
9.5 Hamiltonians Based on Plane Waves
227(3)
9.6 Electronic Energy Bands and Irreducible Representations
230(6)
9.7 Examples and Applications
236(15)
9.7.1 Calculation of Fermi Surfaces
236(2)
9.7.2 Electronic Structure of Carbon Nanotubes
238(2)
9.7.3 Tight-binding Real-Space Calculations
240(5)
9.7.4 Spin--Orbit Coupling in Semiconductors
245(2)
9.7.5 Tight-Binding Models for Oxides
247(4)
Part Three Applications in Photonics
251(48)
10 Solution of Maxwell's Equations
253(16)
10.1 Maxwell's Equations and the Master Equation for Photonic Crystals
254(3)
10.1.1 The Master Equation
254(2)
10.1.2 One- and Two-Dimensional Problems
256(1)
10.2 Group of the Master Equation
257(2)
10.3 Master Equation as an Eigenvalue Problem
259(1)
10.4 Models of the Permittivity
260(9)
10.4.1 Reduced Structure Factors
264(2)
10.4.2 Convergence of the Plane Wave Expansion
266(3)
11 Two-Dimensional Photonic Crystals
269(18)
11.1 Photonic Band Structure and Symmetrized Plane Waves
270(6)
11.1.1 Empty Lattice Band Structure and Symmetrized Plane Waves
270(3)
11.1.2 Photonic Band Structures: A First Example
273(3)
11.2 Group Theoretical Classification of Photonic Band Structures
276(3)
11.3 Supercells and Symmetry of Defect Modes
279(4)
11.4 Uncoupled Bands
283(4)
12 Three-Dimensional Photonic Crystals
287(12)
12.1 Empty Lattice Bands and Compatibility Relations
287(4)
12.2 An example: Dielectric Spheres in Air
291(2)
12.3 Symmetry-Adapted Vector Spherical Waves
293(6)
Part Four Other Applications
299(32)
13 Group Theory of Vibrational Problems
301(18)
13.1 Vibrations of Molecules
301(9)
13.1.1 Permutation, Displacement, and Vector Representation
302(3)
13.1.2 Vibrational Modes of Molecules
305(2)
13.1.3 Infrared and Raman Activity
307(3)
13.2 Lattice Vibrations
310(9)
13.2.1 Direct Calculation of the Dynamical Matrix
312(2)
13.2.2 Dynamical Matrix from Tight-Binding Models
314(1)
13.2.3 Analysis of Zone Center Modes
315(4)
14 Landau Theory of Phase Transitions of the Second Kind
319(12)
14.1 Introduction to Landau's Theory of Phase Transitions
320(4)
14.2 Basics of the Group Theoretical Formulation
324(2)
14.3 Examples with GTPack Commands
326(5)
14.3.1 Invariant Polynomials
326(1)
14.3.2 Landau and Lifshitz Criterion
327(4)
Appendix A Spherical Harmonics
331(6)
A.1 Complex Spherical Harmonics
332(2)
A.1.1 Definition of Complex Spherical Harmonics
332(1)
A.1.2 Cartesian Spherical Harmonics
332(1)
A.1.3 Transformation Behavior of Complex Spherical Harmonics
333(1)
A.2 Tesseral Harmonics
334(3)
A.2.1 Definition of Tesseral Harmonics
334(1)
A.2.2 Cartesian Tesseral Harmonics
335(1)
A.2.3 Transformation Behavior of Tesseral Harmonics
336(1)
Appendix B Remarks on Databases
337(4)
B.1 Electronic Structure Databases
337(2)
B.1.1 Tight-Binding Calculations
337(1)
B.1.2 Pseudopotential Calculations
338(1)
B.1.3 Radial Integrals for Crystal Field Parameters
339(1)
B.2 Molecular Databases
339(1)
B.3 Database of Structures
339(2)
Appendix C Use of MPB together with GTPack
341(4)
C.1 Calculation of Band Structure and Density of States
341(1)
C.2 Calculation of Eigenmodes
342(1)
C.3 Comparison of Calculations with MPB and Mathematica
343(2)
Appendix D Technical Remarks on GTPack
345(4)
D.1 Structure of GTPack
345(1)
D.2 Installation of GTPack
346(3)
References 349(10)
Index 359
Wolfram Hergert, extraordinary professor in Computational Physics, is member of the Theoretical Physics group at University Halle-Wittenberg, Germany. Main subjects of his work are solid state theory, electronic and magnetic structure of nanostructures and photonics. Prof. Hergert has experience in teaching group theory and in applying Mathematica to physical problems. He has published in renowned journals, like Nature and Physical Review Letters, and edited a books on Computational Materials Science and Mie Theory. He is also coauthor of a book on Quantum Theory.

Matthias Geilhufe studied physics at the Martin Luther University Halle-Wittenberg (Germany) with specialization in theoretical and computational physics. From 2012-2015 he was employed as a PhD student at the Max Planck Institute of Microstructure Physics in Halle. In 2015 he obtained his PhD at the Martin Luther University Halle-Wittenberg. Currently, he is working at the Nordita Institute in Stockholm, Sweden. His work is based on the investigation of electronic and magnetic properties of complex materials. For his research, methods based on group theory or density functional theory are applied.