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Symmetry and Condensed Matter Physics: A Computational Approach by M. El-Batanouny, F. Wooten

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Symmetry and Condensed Matter Physics: A Computational Approach written by M. El-Batanouny , M. El-Batanouny, Boston University and F. Wooten , F. Wooten, University of California, Davis. Unlike existing texts, this book blends for the first time three topics in physics - symmetry, condensed matter physics and computational methods - into one pedagogical textbook. It includes new concepts in mathematical crystallography, experimental methods capitalizing on symmetry aspects, non-conventional applications such as Fourier crystallography, color groups, quasicrystals and incommensurate systems, as well as concepts and techniques behind the Landau theory of phase transitions. Ideal for graduate students in condensed matter physics, materials science, and chemistry.

Symmetry and Condensed Matter Physics: A Computational Approach written by M. El-Batanouny and F. Wooten cover the following topics.

  • 1. Symmetry and physics
    1.1 Introduction
    1.2 Hamiltonians, eigenfunctions, and eigenvalues
    1.3 Symmetry operators and operator algebra
    1.4 Point-symmetry operations
    1.5 Applications to quantum mechanics

  • 2. Symmetry and group theory
    2.1 Groups and their realizations
    2.2 The symmetric group
    2.3 Computational aspects
    2.4 Classes
    2.5 Homomorphism, isomorphism, and automorphism
    2.6 Direct- or outer-product groups

  • 3. Group representations: concepts
    3.1 Representations and realizations
    3.2 Generation of representations on a set of basis functions

  • 4. Group representations: formalism and methodology
    4.1 Matrix representations
    4.2 Character of a matrix representation
    4.3 Burnside’s method
    Computational projects

  • 5. Dixon’s method for computing group characters
    5.1 The eigenvalue equation modulo p
    5.2 Dixon’s method for irreducible characters
    5.3 Computer codes for Dixon’s method
    Appendix 1 Finding eigenvalues and eigenvectors
    Appendix 2
    Computation project

  • 6. Group action and symmetry projection operators
    6.1 Group action
    6.2 Symmetry projection operators
    6.3 The regular projection matrices: the simple characteristic

  • 7. Construction of the irreducible representations
    7.1 Eigenvectors of the regular Rep
    7.2 The symmetry structure of the regular Rep eigenvectors
    7.3 Symmetry projection on regular Rep eigenvectors
    7.4 Computer construction of Irreps with da >1
    7.5 Summary of the method

  • 8. Product groups and product representations
    8.1 Introduction
    8.2 Subgroups and cosets
    8.3 Direct outer-product groups
    8.4 Semidirect product groups
    8.5 Direct inner-product groups and their representations
    8.6 Product representations and the Clebsch–Gordan series
    8.7 Computer codes
    8.8 Summary

  • 9. Induced representations
    9.1 Introduction
    9.2 Subduced Reps and compatibility relations
    9.3 Induction of group Reps from the Irreps of its subgroups
    9.4 Irreps induced from invariant subgroups
    9.5 Examples of Irrep induction using the method of little-groups
    Appendix Frobenius reciprocity theorem and other useful theorems
    Exercises 261

  • 10. Crystallographic symmetry And Space - Groups
    10.1 Euclidean space
    10.2 Crystallography
    10.3 The perfect crystal
    10.4 Space-group operations: the Seitz operators
    10.5 Symmorphic and nonsymmorphic space-groups
    10.6 Site-symmetries and the Wyckoff notation
    10.7 Fourier space crystallography

  • 11. Space -Groups: Irreps
    11.1 Irreps of the translation group
    11.2 Induction of Irreps of space-groups

  • 12. Time-reversal symmetry: color groups and the Onsager relations
    12.1 Introduction
    12.2 The time-reversal operator in quantum mechanics
    12.3 Spin-1/2 and double-groups
    12.4 Magnetic and color groups
    12.5 The time-reversed representation: theory of corepresentations
    12.6 Theory of crystal fields
    12.7 Onsager reciprocity theorem (Onsager relations) and transport properties

  • 13. Tensors and tensor fields
    13.1 Tensors and their space-time symmetries
    13.2 Construction of symmetry-adapted tensors
    13.3 Description and classification of matter tensors
    13.4 Tensor field representations

  • 14. Electronic properties of solids
    14.1 Introduction
    14.2 The one-electron approximations and self-consistent-field theories
    14.3 Methods and techniques for band structure calculations
    14.4 Electronic structure of magnetically ordered systems
    Appendix 1 Derivation of the Hartree–Fock equations
    Appendix 2 Holstein–Primakoff (HP) operators

  • 15. Dynamical properties of molecules, solids, and surfaces
    15.1 Introduction
    15.2 Dynamical properties of molecules
    15.3 Dynamical properties of solids
    15.4 Dynamical properties of surfaces
    Appendix 1 Coulomb interactions and the method of Ewald summations
    Appendix 2 Electronic effects on phonons in insulators and semiconductors

  • 16. Experimental measurements and selection rules
    16.1 Introduction
    16.2 Selection rules
    16.3 Differential scattering cross-sections in the Born approximation
    16.4 Light scattering spectroscopies
    16.5 Photoemission and dipole selection rules
    16.6 Neutron and atom scattering spectroscopies

  • Phase transitions
    17.1 Phase transitions and their classification
    17.2 Landau theory of phase transitions: principles
    17.3 Construction and minimization techniques for ?F

  • 18. Incommensurate systems And quasi - crystals
    18.1 Introduction
    18.2 The concept of higher-dimensional spaces: superspaces and superlattices
    18.3 Quasi-crystal symmetry: the notion of indistinguishability and the classification of space-groups
    18.4 Two-dimensional lattices, cyclotomic integers, and axial stacking

  • Bibliography

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