Space Group Representations : Theory, Tables and Applications / / Nikolai B. Melnikov and Boris I. Reser

• Autore: Melnikov Nikolai B.
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (343 pages)
• Disciplina: 548.7
• Soggetto topico: Space groups
• Persona (resp. second.): ReserBoris I.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- Symbols -- 1 Scope and Overview -- References -- 2 Mathematical Preliminaries -- 2.1 Elements of Group Theory -- 2.1.1 Definition of Groups -- 2.1.2 Subgroups -- 2.1.3 Invariant Subgroup -- 2.1.4 Classes of Conjugate Elements -- 2.2 Matrices: Definitions and Notation -- 2.2.1 Special Types of Matrices -- 2.2.2 Equivalent Matrices -- 2.2.3 Reducible Matrices -- 2.2.4 Direct Matrix Product -- 2.3 Representation Theory -- 2.3.1 Matrix Groups -- 2.3.2 Group Representations -- 2.3.3 Regular Representation -- 2.3.4 Fundamental Theorems -- 2.3.5 Group Characters -- 2.3.6 Reality Properties -- 2.3.7 Direct Product of Representations -- References -- 3 Induced Representations -- 3.1 Regular Continuation from a Subgroup to a Group -- 3.2 Normalizer of a Representation of an Invariant Subgroup -- 3.3 Unitary Transformation for Elements of a Normalizer -- 3.4 Regular Continuation to the Normalizer -- 3.5 Projective Representations of the Factor Group of a Normalizer -- 3.6 Small Representations of a Normalizer -- 3.7 Special Case of an Abelian Invariant Subgroup -- 3.8 Main Theorem -- 3.9 Reality Properties of the Induced Representations -- 3.10 Algorithm for Constructing Irreducible Representations -- References -- 4 Projective Representations -- 4.1 The Factor System -- 4.2 Main Properties of Projective Representations -- 4.3 Regular Representation and Completeness Criterion -- 4.4 Group Generators -- 4.5 Character Factors -- 4.6 Covering Group and Its Representations -- 4.7 Irreducible Projective Representations -- 4.8 Outline of the Algorithm -- References -- 5 Representations of the Space Groups -- 5.1 Space Groups and Their Properties -- 5.2 Reciprocal Lattice and Brillouin Zone -- 5.3 Translation Symmetry and Its Consequences -- 5.4 Types of the k Vectors in the Brillouin Zone -- 5.5 Group of the Wavevector. ; 5.6 Small Representations of the Group of the Wavevector -- 5.7 Irreducible Representations of Space Groups -- 5.8 Compatibility Relations -- References -- 6 Tables -- 6.1 Summary of Theory and Guide to Tables -- 6.1.1 Summary of Theory -- 6.1.2 Guide to Tables -- 6.2 Elements and Multiplication Tables of the Groups Oh and D6h -- 6.3 Groups C2, Ci and Cs -- 6.4 Group C3 -- 6.5 Groups C4 and S4 -- 6.6 Groups D2, C2h and C2v -- 6.7 Groups C6, S6 and C3h -- 6.8 Groups D3 and C3v -- 6.9 Groups D4, C4v and D2d -- 6.10 Group C4h -- 6.11 Group D2h -- 6.12 Groups D6, D3d, C6v and D3h -- 6.13 Group T -- 6.14 Group C6h -- 6.15 Group D4h -- 6.16 Groups O and Td -- 6.17 Group D6h -- 6.18 Group Th -- 6.19 Group Oh -- 6.20 Example of the Use of the Tables: The Space Group Oh7 -- References -- 7 Group Theory and Quantum Mechanics -- 7.1 Representation Spaces -- 7.1.1 Basis Functions -- 7.1.2 Expansion Theorem -- 7.1.3 Reduction by Idempotent Operators -- 7.1.4 Orthogonality of the Basis Functions -- 7.1.5 Criterion for Representation Functions -- 7.2 Classification of States -- 7.2.1 Hamiltonian of a System with Symmetry -- 7.2.2 Eigenspaces and Representations -- 7.3 Splitting of Eigenvalues -- 7.3.1 Schrödinger Perturbation Theory -- 7.3.2 Effect of Symmetry -- 7.4 Selection Rules -- References -- 8 Group Theory and Solid-State Physics -- 8.1 Electron in a Periodic Field -- 8.1.1 Mathematical Formulation of the Problem -- 8.1.2 The Ritz Method -- 8.2 Symmetrized Coordinate Functions -- 8.2.1 Construction of Symmetrized Functions -- 8.2.2 Matrix Elements of the Hamiltonian -- 8.3 Double-Valued Representations -- 8.4 Time-Reversal Symmetry -- 8.4.1 Time-Reversal Operator -- 8.4.2 Systems with Time-Reversal Symmetry -- 8.4.3 Criterion of Extra Degeneracy -- References -- 9 Energy Band Calculation -- 9.1 Basics of Band Calculations -- 9.2 The OPW Method. ; 9.2.1 Main Formulas -- 9.2.2 Symmetrized OPW -- 9.2.3 Crystal Potential -- 9.2.4 Core Shifts -- 9.2.5 Generalized Eigenvalue Problem -- 9.3 Summary -- References -- 10 Energy Bands of Diamond -- 10.1 Crystal Structure -- 10.1.1 Symmetry of the Crystal -- 10.1.2 Reciprocal Lattice and Brillouin Zone -- 10.2 Energy Bands -- 10.2.1 Problems and Methods -- 10.2.2 Calculation Results -- 10.2.3 Comparison with Experiment -- 10.3 Summary -- References -- 11 Electronic States in Atoms and Solids -- 11.1 Correspondence Between Atomic and Crystal States -- 11.1.1 Background Material -- 11.1.2 Reducible Representation -- 11.1.3 Decomposition into Irreducible Components -- 11.2 An Example: The Pyrite Structure -- 11.2.1 Crystal Structure and Space Group Th6 -- 11.2.2 Atomic and Crystal States -- 11.3 Summary -- References -- 12 Group-Theoretical Analysis of the Cr3Si Structure -- 12.1 Crystal Structure -- 12.1.1 Geometric Description -- 12.1.2 Space Group Oh3 -- 12.1.3 Wigner-Seitz Cell -- 12.2 Small Representations of the Group of the Wavevector -- 12.2.1 Brillouin Zone -- 12.2.2 Structure of the Small Representations -- 12.2.3 Small Representations -- 12.3 Compatibility Relations -- 12.4 Time-Reversal Symmetry -- 12.4.1 Theory -- 12.4.2 Results -- 12.5 Correspondence Between Atomic and Crystal States -- 12.5.1 Method -- 12.5.2 Results -- 12.6 Concluding Remarks -- References -- 13 Symmetry in the Tight-Binding Method -- 13.1 General Method -- 13.1.1 Properties of Symmetry Operators -- 13.1.2 Problem Setup and Notation -- 13.1.3 Hermiticity of the Hamiltonian -- 13.1.4 Inversion Symmetry -- 13.1.5 Time-Reversal Symmetry -- 13.1.6 Relations Between the Matrix Components -- 13.1.7 Point Symmetry -- 13.1.8 Calculation of the Matrix Components -- 13.1.9 Determination of the Independent Parameters -- 13.1.10 Calculation Algorithm. ; 13.2 Application to the Diamond Structure -- References -- 14 Interpolation of Energy Bands -- 14.1 Quasidiagonal Form of the Hamiltonian Matrix -- 14.2 Tight-Binding Method and kp-Method -- 14.3 An Example: The Diamond Structure -- 14.4 Electron Density of States -- References -- 15 Quasi-2D Dichalcogenides with the 2H-TaSe2 Structure -- 15.1 Introduction -- 15.2 Crystal Structure of 2H-TaSe2 -- 15.3 Reciprocal Lattice and Brillouin Zone -- 15.4 Matrix Elements of the Hamiltonian -- 15.5 Hamiltonian Matrix at High Symmetry Points -- 15.6 Interim Conclusion -- References -- 16 Symmetrized Plane Waves -- 16.1 Projection Operator -- 16.2 Construction of Symmetrized Plane Waves -- References -- 17 Symmetrized Bloch Sums -- 17.1 Theory -- 17.1.1 Projection Matrix -- 17.1.2 Projection Matrix on Bloch Functions -- 17.1.3 Irreducible Representations of the Rotation Group -- 17.2 Algorithm -- 17.3 Summary -- References -- 18 Discussion and Outlook -- 18.1 Representation Theory of the Space Groups -- 18.2 Tables of the Space Group Representations -- 18.3 Applications of the Space Group Representations -- References -- Appendix A Frobenius-Schur Theorem and Its Applications -- A.1 Frobenius-Schur Theorem for Projective Representations -- A.2 Reality Criterion for Space Groups -- Appendix B Construction of Projective Representations of the Point Group T -- Appendix C More on Symmetry in Quantum Mechanics -- C.1 Discussion of Wigner's Theorem -- C.2 Projection Operator Corresponding to G and Gk -- Appendix D Factor System of Double-Valued Representations -- D.1 Composition of Two Rotations -- D.2 Euler-Rodrigues Parameters -- Appendix E Multiplication Table of the Point Group Oh -- Appendix F Projective Representations and Compatibility Relations for Oh7 -- Appendix G Factorization of the Characteristic Equation for Oh7 -- Index.
• Titolo autorizzato: Space Group Representations
• ISBN: 3-031-13991-7
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996503462003316
• Lo trovi qui: Univ. di Salerno