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Group Theory in Solid State Physics and Photonics Problem Solving with Mathematica

Author: Wolfram Hergert; R Matthias Geilhufe
Publisher: Newark John Wiley & Sons, Incorporated 2018
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Cover -- Main title -- Copyright page -- Contents -- 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 -- 2.1.1 Rotation Matrices -- 2.1.2 Euler Angles -- 2.1.3 Euler-Rodrigues Parameters and Quaternions -- 2.1.4
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Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Wolfram Hergert; R Matthias Geilhufe
ISBN: 9783527413003 3527413006 9783527411337 352741133X
OCLC Number: 1040274651
Description: 1 Online-Ressource (382 Seiten)
Contents: 1 Preface 2 Introduction I Basics of group theory 3 Symmetry operations and transformations of fields 4 Basic abstract group theory 5 Discrete symmetry groups for solid state physics and photonics 6 Representation theory 7 Symmetry in k-space II Applications in electronic structure theory 8 Solution of the Schroedinger equation 9 Generalization to include the spin 10 Electronic energy bands III Applications in photonics 11 Solution of Maxwell's equations 12 Twodimensional photonic crystals 13 Threedimensional photonic crystals 14 Other Applications A Mathematica Package Reference B Connection of the group theory package to MPB and MEEP

Abstract:

While group theory is well established, this textbook raises two completely new aspects: gaining a better understanding by focusing on problem solving, making extensive use of Mathematica; offering a  Read more...

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    schema:description "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 Born-von 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 -- 7.4.1 General Formalism -- 7.4.2 Crystal Field Expansion -- 7.4.3 Crystal Field Operators -- 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 -- 8.7.1 The Reality of Representations -- 8.7.2 Spin-Independent Theory -- 8.7.3 Spin-Dependent Theory -- 9 Electronic Structure Calculations -- 9.1 Solution of the Schrödinger Equation for a Crystal -- 9.2 Symmetry Properties of Energy Bands -- 9.2.1 Degeneracy and Symmetry of Energy Bands -- 9.2.2 Compatibility Relations and Crossing of Bands -- 9.3 Symmetry-Adapted Functions -- 9.3.1 Symmetry-Adapted Plane Waves -- 9.3.2 Localized Orbitals -- 9.4 Construction of Tight-Binding Hamiltonians -- 9.4.1 Hamiltonians in Two-Center Form -- 9.4.2 Hamiltonians in Three-Center Form" ;
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    schema:description "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 -- 14.3.1 Invariant Polynomials -- 14.3.2 Landau and LifshitzCriterion -- Appendix A Spherical Harmonics -- A.1 Complex Spherical Harmonics -- A.1.1 Definition of Complex Spherical Harmonics -- A.1.2 Cartesian Spherical Harmonics -- A.1.3 Transformation Behavior of Complex Spherical Harmonics -- A.2 Tesseral Harmonics -- A.2.1 Definition of Tesseral Harmonics -- A.2.2 Cartesian Tesseral Harmonics -- A.2.3 Transformation Behavior of Tesseral Harmonics -- Appendix B Remarks on Databases -- B.1 Electronic Structure Databases -- B.1.1 Tight-Binding Calculations -- B.1.2 Pseudopotential Calculations -- B.1.3 Radial Integrals for Crystal Field Parameters -- 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 -- EULA" ;
    schema:description "9.4.3 Inclusion of Spin-Orbit Interaction -- 9.4.4 Tight-Binding Hamiltonians from ab initio Calculations -- 9.5 Hamiltonians Based on Plane Waves -- 9.6 Electronic Energy Bands and Irreducible Representations -- 9.7 Examples and Applications -- 9.7.1 Calculation of Fermi Surfaces -- 9.7.2 Electronic Structure of Carbon Nanotubes -- 9.7.3 Tight-binding Real-Space Calculations -- 9.7.4 Spin-Orbit Coupling in Semiconductors -- 9.7.5 Tight-Binding Models for Oxides -- Part Three Applications in Photonics -- 10 Solution of Maxwell's Equations -- 10.1 Maxwell's Equations and the Master Equation for Photonic Crystals -- 10.1.1 The Master Equation -- 10.1.2 One- and Two-Dimensional Problems -- 10.2 Group of the Master Equation -- 10.3 Master Equation as an Eigenvalue Problem -- 10.4 Models of the Permittivity -- 10.4.1 Reduced Structure Factors -- 10.4.2 Convergence of the Plane Wave Expansion -- 11 Two-Dimensional Photonic Crystals -- 11.1 Photonic Band Structure and Symmetrized Plane Waves -- 11.1.1 Empty Lattice Band Structure and Symmetrized Plane Waves -- 11.1.2 Photonic Band Structures: A First Example -- 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 -- 13.1.1 Permutation, Displacement, and Vector Representation -- 13.1.2 Vibrational Modes of Molecules -- 13.1.3 Infrared and Raman Activity -- 13.2 Lattice Vibrations -- 13.2.1 Direct Calculation of the Dynamical Matrix -- 13.2.2 Dynamical Matrix from Tight-Binding Models -- 13.2.3 Analysis of Zone Center Modes" ;
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