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Aperiodic order . Volume 1 A mathematical invitation / / Michael Baake, Uwe Grimm [[electronic resource]]
| Aperiodic order . Volume 1 A mathematical invitation / / Michael Baake, Uwe Grimm [[electronic resource]] |
| Autore | Baake Michael |
| Pubbl/distr/stampa | Cambridge : , : Cambridge University Press, , 2013 |
| Descrizione fisica | 1 online resource (xvi, 531 pages) : digital, PDF file(s) |
| Disciplina | 548.7 |
| Collana | Encyclopedia of mathematics and its applications |
| Soggetto topico |
Aperiodic tilings
Quasicrystals - Mathematics |
| ISBN |
1-316-18318-1
1-316-18367-X 1-316-18377-7 1-316-18448-X 1-316-18473-0 1-316-18403-X 1-139-02525-2 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Cover; Half-title; Series information; Title page; Copyright information; Table of contents; Foreword; Preface; Chapter 1 Introduction; Chapter 2 Preliminaries; 2.1. Point sets; 2.2. Voronoi and Delone cells; 2.3. Groups; 2.4. Perron-Frobenius theory; 2.5. Number-theoretic tools; Chapter 3 Lattices and Crystals; 3.1. Periodicity and lattices; 3.2. The crystallographic restriction; 3.3. Root lattices; 3.4. Minkowski embedding; Chapter 4 Symbolic Substitutions and Inflations; 4.1. Substitution rules; 4.2. Hulls and their properties; 4.3. Symmetries, invariant measures and ergodicity
4.4. Metallic means sequences4.5. Period doubling and paper folding; 4.6. Thue-Morse substitution; 4.7. Rudin-Shapiro and Kolakoski sequences; 4.8. Complexity and further directions; 4.9. Block substitutions; Chapter 5 Patterns and Tilings; 5.1. Patterns and local indistinguishability; 5.2. Local derivability; 5.3. Repetitivity and finite local complexity; 5.4. Geometric hull; 5.5. Proximality; 5.6. Symmetry and inflation; 5.7. Local rules; Chapter 6 Inflation Tilings; 6.1. Ammann-Beenker tilings; 6.2. Penrose tilings and their relatives; 6.3. Square triangle and shield tilings 6.4. Planar tilings with integer inflation multiplier6.5. Examples of non-Pisot tilings; 6.6. Pinwheel tilings; 6.7. Tilings in higher dimensions; 6.8. Colourful examples; Chapter 7 Projection Method and Model Sets; 7.1. Silver mean chain via projection; 7.2. Cut and project schemes and model sets; 7.3. Cyclotomic model sets; 7.4. Icosahedral model sets and beyond; 7.5. Alternative constructions; Chapter 8 Fourier Analysis and Measures; 8.1. Fourier series; 8.2. Almost periodic functions; 8.3. Fourier transform of functions; 8.4. Fourier transform of distributions 8.5. Measures and their decomposition8.6. Fourier transform of measures; 8.7. Fourier-Stieltjes coefficients of measures on S1; 8.8. Volume averaged convolutions; Chapter 9 Diffraction; 9.1. Mathematical diffraction theory; 9.2. Poisson's summation formula and perfect crystals; 9.3. Autocorrelation and diffraction of the silver mean chain; 9.4. Autocorrelation and diffraction of regular model sets; 9.5. Pure point diffraction of weighted Dirac combs; 9.6. Homometric point sets; Chapter 10 Beyond Model Sets; 10.1. Diffraction of the Thue-Morse chain 10.2. Diffraction of the Rudin-Shapiro chain10.3. Diffraction of lattice subsets; 10.4. Visible lattice points; 10.5. Extension to Meyer sets; Chapter 11 Random Structures; 11.1. Probabilistic preliminaries; 11.2. Bernoulli systems; 11.3. Renewal processes on the line; 11.4. Point processes from random matrix theory; 11.5. Lattice systems with interaction; 11.6. Random tilings; Appendix A The Icosahedral Group; Appendix B The Dynamical Spectrum; References; List of Definitions; List of Examples; List of Remarks; Index |
| Record Nr. | UNINA-9910466882903321 |
Baake Michael
|
||
| Cambridge : , : Cambridge University Press, , 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Aperiodic order . Volume 1 A mathematical invitation / / Michael Baake, Uwe Grimm [[electronic resource]]
| Aperiodic order . Volume 1 A mathematical invitation / / Michael Baake, Uwe Grimm [[electronic resource]] |
| Autore | Baake Michael |
| Pubbl/distr/stampa | Cambridge : , : Cambridge University Press, , 2013 |
| Descrizione fisica | 1 online resource (xvi, 531 pages) : digital, PDF file(s) |
| Disciplina | 548.7 |
| Collana | Encyclopedia of mathematics and its applications |
| Soggetto topico |
Aperiodic tilings
Quasicrystals - Mathematics |
| ISBN |
1-316-18318-1
1-316-18367-X 1-316-18377-7 1-316-18448-X 1-316-18473-0 1-316-18403-X 1-139-02525-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Cover; Half-title; Series information; Title page; Copyright information; Table of contents; Foreword; Preface; Chapter 1 Introduction; Chapter 2 Preliminaries; 2.1. Point sets; 2.2. Voronoi and Delone cells; 2.3. Groups; 2.4. Perron-Frobenius theory; 2.5. Number-theoretic tools; Chapter 3 Lattices and Crystals; 3.1. Periodicity and lattices; 3.2. The crystallographic restriction; 3.3. Root lattices; 3.4. Minkowski embedding; Chapter 4 Symbolic Substitutions and Inflations; 4.1. Substitution rules; 4.2. Hulls and their properties; 4.3. Symmetries, invariant measures and ergodicity
4.4. Metallic means sequences4.5. Period doubling and paper folding; 4.6. Thue-Morse substitution; 4.7. Rudin-Shapiro and Kolakoski sequences; 4.8. Complexity and further directions; 4.9. Block substitutions; Chapter 5 Patterns and Tilings; 5.1. Patterns and local indistinguishability; 5.2. Local derivability; 5.3. Repetitivity and finite local complexity; 5.4. Geometric hull; 5.5. Proximality; 5.6. Symmetry and inflation; 5.7. Local rules; Chapter 6 Inflation Tilings; 6.1. Ammann-Beenker tilings; 6.2. Penrose tilings and their relatives; 6.3. Square triangle and shield tilings 6.4. Planar tilings with integer inflation multiplier6.5. Examples of non-Pisot tilings; 6.6. Pinwheel tilings; 6.7. Tilings in higher dimensions; 6.8. Colourful examples; Chapter 7 Projection Method and Model Sets; 7.1. Silver mean chain via projection; 7.2. Cut and project schemes and model sets; 7.3. Cyclotomic model sets; 7.4. Icosahedral model sets and beyond; 7.5. Alternative constructions; Chapter 8 Fourier Analysis and Measures; 8.1. Fourier series; 8.2. Almost periodic functions; 8.3. Fourier transform of functions; 8.4. Fourier transform of distributions 8.5. Measures and their decomposition8.6. Fourier transform of measures; 8.7. Fourier-Stieltjes coefficients of measures on S1; 8.8. Volume averaged convolutions; Chapter 9 Diffraction; 9.1. Mathematical diffraction theory; 9.2. Poisson's summation formula and perfect crystals; 9.3. Autocorrelation and diffraction of the silver mean chain; 9.4. Autocorrelation and diffraction of regular model sets; 9.5. Pure point diffraction of weighted Dirac combs; 9.6. Homometric point sets; Chapter 10 Beyond Model Sets; 10.1. Diffraction of the Thue-Morse chain 10.2. Diffraction of the Rudin-Shapiro chain10.3. Diffraction of lattice subsets; 10.4. Visible lattice points; 10.5. Extension to Meyer sets; Chapter 11 Random Structures; 11.1. Probabilistic preliminaries; 11.2. Bernoulli systems; 11.3. Renewal processes on the line; 11.4. Point processes from random matrix theory; 11.5. Lattice systems with interaction; 11.6. Random tilings; Appendix A The Icosahedral Group; Appendix B The Dynamical Spectrum; References; List of Definitions; List of Examples; List of Remarks; Index |
| Record Nr. | UNINA-9910795917503321 |
|
Baake Michael
|
||
| Cambridge : , : Cambridge University Press, , 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Aperiodic order . Volume 1 A mathematical invitation / / Michael Baake, Uwe Grimm [[electronic resource]]
| Aperiodic order . Volume 1 A mathematical invitation / / Michael Baake, Uwe Grimm [[electronic resource]] |
| Autore | Baake Michael |
| Edizione | [1st ed.] |
| Pubbl/distr/stampa | Cambridge : , : Cambridge University Press, , 2013 |
| Descrizione fisica | 1 online resource (xvi, 531 pages) : digital, PDF file(s) |
| Disciplina | 548.7 |
| Collana | Encyclopedia of mathematics and its applications |
| Soggetto topico |
Aperiodic tilings
Quasicrystals - Mathematics |
| ISBN |
1-316-18318-1
1-316-18367-X 1-316-18377-7 1-316-18448-X 1-316-18473-0 1-316-18403-X 1-139-02525-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Cover; Half-title; Series information; Title page; Copyright information; Table of contents; Foreword; Preface; Chapter 1 Introduction; Chapter 2 Preliminaries; 2.1. Point sets; 2.2. Voronoi and Delone cells; 2.3. Groups; 2.4. Perron-Frobenius theory; 2.5. Number-theoretic tools; Chapter 3 Lattices and Crystals; 3.1. Periodicity and lattices; 3.2. The crystallographic restriction; 3.3. Root lattices; 3.4. Minkowski embedding; Chapter 4 Symbolic Substitutions and Inflations; 4.1. Substitution rules; 4.2. Hulls and their properties; 4.3. Symmetries, invariant measures and ergodicity
4.4. Metallic means sequences4.5. Period doubling and paper folding; 4.6. Thue-Morse substitution; 4.7. Rudin-Shapiro and Kolakoski sequences; 4.8. Complexity and further directions; 4.9. Block substitutions; Chapter 5 Patterns and Tilings; 5.1. Patterns and local indistinguishability; 5.2. Local derivability; 5.3. Repetitivity and finite local complexity; 5.4. Geometric hull; 5.5. Proximality; 5.6. Symmetry and inflation; 5.7. Local rules; Chapter 6 Inflation Tilings; 6.1. Ammann-Beenker tilings; 6.2. Penrose tilings and their relatives; 6.3. Square triangle and shield tilings 6.4. Planar tilings with integer inflation multiplier6.5. Examples of non-Pisot tilings; 6.6. Pinwheel tilings; 6.7. Tilings in higher dimensions; 6.8. Colourful examples; Chapter 7 Projection Method and Model Sets; 7.1. Silver mean chain via projection; 7.2. Cut and project schemes and model sets; 7.3. Cyclotomic model sets; 7.4. Icosahedral model sets and beyond; 7.5. Alternative constructions; Chapter 8 Fourier Analysis and Measures; 8.1. Fourier series; 8.2. Almost periodic functions; 8.3. Fourier transform of functions; 8.4. Fourier transform of distributions 8.5. Measures and their decomposition8.6. Fourier transform of measures; 8.7. Fourier-Stieltjes coefficients of measures on S1; 8.8. Volume averaged convolutions; Chapter 9 Diffraction; 9.1. Mathematical diffraction theory; 9.2. Poisson's summation formula and perfect crystals; 9.3. Autocorrelation and diffraction of the silver mean chain; 9.4. Autocorrelation and diffraction of regular model sets; 9.5. Pure point diffraction of weighted Dirac combs; 9.6. Homometric point sets; Chapter 10 Beyond Model Sets; 10.1. Diffraction of the Thue-Morse chain 10.2. Diffraction of the Rudin-Shapiro chain10.3. Diffraction of lattice subsets; 10.4. Visible lattice points; 10.5. Extension to Meyer sets; Chapter 11 Random Structures; 11.1. Probabilistic preliminaries; 11.2. Bernoulli systems; 11.3. Renewal processes on the line; 11.4. Point processes from random matrix theory; 11.5. Lattice systems with interaction; 11.6. Random tilings; Appendix A The Icosahedral Group; Appendix B The Dynamical Spectrum; References; List of Definitions; List of Examples; List of Remarks; Index |
| Record Nr. | UNINA-9910828054403321 |
|
Baake Michael
|
||
| Cambridge : , : Cambridge University Press, , 2013 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Mechanics and mathematics of crystals : selected papers of J. L. Ericksen / editors Millard F. Beatty, Michael A. Hayes
| Mechanics and mathematics of crystals : selected papers of J. L. Ericksen / editors Millard F. Beatty, Michael A. Hayes |
| Autore | Ericksen, Jerald L. |
| Pubbl/distr/stampa | Hackensack, NJ. : World Scientific, c2005 |
| Descrizione fisica | xxxi, 621 p. ; 26 cm |
| Disciplina | 548.7 |
| Altri autori (Persone) |
Beatty, Millard F.
Hayes, M. A. |
| Soggetto (Persona) | Ericksen, Jerald L. |
| Soggetto topico | Crystallography, Mathematical |
| ISBN | 9812562834 |
| Classificazione |
AMS 01A75
AMS 82-03 AMS 82D-25 LC QD911.E75 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991001510079707536 |
|
Ericksen, Jerald L.
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||
| Hackensack, NJ. : World Scientific, c2005 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
N-dimensional crystallography / R. L. E. schwarzenberger
| N-dimensional crystallography / R. L. E. schwarzenberger |
| Autore | SCHWARZENBERGER, R. L. E. |
| Pubbl/distr/stampa | Boston [etc.] : Pitman advanced publishing, copyr. 1980 |
| Descrizione fisica | 139 p. : ill. ; 24 cm |
| Disciplina | 548.7 |
| Collana | Research notes in mathematics |
| Soggetto topico | Cristallografia |
| ISBN | 0-273-08468-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISA-990003246110203316 |
|
SCHWARZENBERGER, R. L. E.
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||
| Boston [etc.] : Pitman advanced publishing, copyr. 1980 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
N-dimensional crystallography / R. L. E. Schwarzenberger
| N-dimensional crystallography / R. L. E. Schwarzenberger |
| Autore | Schwarzenberger, R. L. E. |
| Pubbl/distr/stampa | San Francisco : Pitman Advanced Publ. Program, c1980 |
| Descrizione fisica | 139 p. : ill. ; 25 cm. |
| Disciplina |
512.2
548.7 |
| Collana | Pitman research notes in mathematics series, ISSN 02693674 ; 41 |
| Soggetto topico | Crystallography |
| ISBN | 0822484684 |
| Classificazione | AMS 20H15 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNISALENTO-991001163619707536 |
|
Schwarzenberger, R. L. E.
|
||
| San Francisco : Pitman Advanced Publ. Program, c1980 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. del Salento | ||
| ||
Numerical problems in crystallography / / M. A. Walsh
| Numerical problems in crystallography / / M. A. Walsh |
| Autore | Walsh M. A. |
| Edizione | [1st ed. 2021.] |
| Pubbl/distr/stampa | Singapore : , : Springer, , [2021] |
| Descrizione fisica | 1 online resource (XV, 387 p. 176 illus., 10 illus. in color.) |
| Disciplina | 548.7 |
| Soggetto topico |
Crystallography, Mathematical
Crystallography |
| ISBN | 981-15-9754-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-9910484189803321 |
|
Walsh M. A.
|
||
| Singapore : , : Springer, , [2021] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Space Group Representations : Theory, Tables and Applications / / Nikolai B. Melnikov and Boris I. Reser
| Space Group Representations : Theory, Tables and Applications / / Nikolai B. Melnikov and Boris I. Reser |
| Autore | Melnikov Nikolai B. |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (343 pages) |
| Disciplina | 548.7 |
| Soggetto topico | Space groups |
| ISBN | 3-031-13991-7 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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. |
| Record Nr. | UNINA-9910637723103321 |
|
Melnikov Nikolai B.
|
||
| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Space Group Representations : Theory, Tables and Applications / / Nikolai B. Melnikov and Boris I. Reser
| Space Group Representations : Theory, Tables and Applications / / Nikolai B. Melnikov and Boris I. Reser |
| Autore | Melnikov Nikolai B. |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (343 pages) |
| Disciplina | 548.7 |
| Soggetto topico | Space groups |
| ISBN | 3-031-13991-7 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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. |
| Record Nr. | UNISA-996503462003316 |
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Melnikov Nikolai B.
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| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
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Symmetry (group theory) and mathematical treatment in chemistry / / Takashiro Akitsu, editor
| Symmetry (group theory) and mathematical treatment in chemistry / / Takashiro Akitsu, editor |
| Autore | Takashiro Akitsu |
| Edizione | [1st ed.] |
| Pubbl/distr/stampa | IntechOpen, 2018 |
| Descrizione fisica | 1 online resource (186 pages) : illustrations |
| Disciplina | 548.7 |
| Soggetto topico |
Crystallography, Mathematical
Crystals - Mathematical models Group theory |
| Soggetto non controllato |
Physical Sciences
Engineering and Technology Chemistry Inorganic Chemistry |
| ISBN |
1-83881-324-1
1-78923-315-1 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Altri titoli varianti | Symmetry |
| Record Nr. | UNINA-9910317806503321 |
|
Takashiro Akitsu
|
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| IntechOpen, 2018 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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