Representations of crystallographic groups

Janssen, T.

doi:10.1107/97809553602060000629

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 34-71
https://doi.org/10.1107/97809553602060000629

Chapter 1.2. Representations of crystallographic groups

1.2. Representations of crystallographic groups (pp. 34-71) | html | pdf | chapter contents |
T. Janssen
- 1.2.1. Introduction (pp. 34-35) | html | pdf |
- 1.2.2. Point groups (pp. 35-46) | html | pdf |
  - 1.2.2.1. Finite point groups in one, two and three dimensions (pp. 35-36) | html | pdf |
  - 1.2.2.2. Representations of finite groups (pp. 36-37) | html | pdf |
  - 1.2.2.3. General tensors (pp. 37-38) | html | pdf |
  - 1.2.2.4. Orthogonality relations (pp. 38-39) | html | pdf |
  - 1.2.2.5. Characters (pp. 39-40) | html | pdf |
  - 1.2.2.6. The representations for point groups in one, two and three dimensions (pp. 40-42) | html | pdf |
  - 1.2.2.7. Tensor representations (pp. 42-43) | html | pdf |
  - 1.2.2.8. Projective representations (pp. 43-45) | html | pdf |
  - 1.2.2.9. Double groups and their representations (pp. 45-46) | html | pdf |
- 1.2.3. Space groups (pp. 46-51) | html | pdf |
  - 1.2.3.1. Structure of space groups (pp. 46-47) | html | pdf |
  - 1.2.3.2. Irreducible representations of lattice translation groups (p. 47) | html | pdf |
  - 1.2.3.3. Irreducible representations of space groups (pp. 47-49) | html | pdf |
  - 1.2.3.4. Characterization of space-group representations (pp. 49-50) | html | pdf |
  - 1.2.3.5. Double space groups and their representations (pp. 50-51) | html | pdf |
- 1.2.4. Tensors (pp. 51-53) | html | pdf |
  - 1.2.4.1. Transformation properties of tensors (pp. 51-52) | html | pdf |
  - 1.2.4.2. Invariants (p. 52) | html | pdf |
  - 1.2.4.3. Clebsch–Gordan coefficients (pp. 52-53) | html | pdf |
- 1.2.5. Magnetic symmetry (pp. 53-56) | html | pdf |
  - 1.2.5.1. Magnetic point groups (p. 53) | html | pdf |
  - 1.2.5.2. Magnetic space groups (pp. 53-54) | html | pdf |
  - 1.2.5.3. Transformation of tensors (p. 54) | html | pdf |
  - 1.2.5.4. Time-reversal operators (pp. 54-55) | html | pdf |
  - 1.2.5.5. Co-representations (pp. 55-56) | html | pdf |
- 1.2.6. Tables (pp. 56-62) | html | pdf |
- 1.2.7. Introduction to the accompanying software Tenχar (pp. 62-70) | html | pdf |
  M. Ephraïm, T. Janssen, A. Janner and A. Thiers
  - 1.2.7.1. Overview (pp. 62-64) | html | pdf |
  - 1.2.7.2. Tensors (pp. 64-66) | html | pdf |
  - 1.2.7.3. Characters (pp. 66-67) | html | pdf |
  - 1.2.7.4. Algorithms (pp. 67-70) | html | pdf |
    - 1.2.7.4.1. Construction of a basis (p. 67) | html | pdf |
    - 1.2.7.4.2. Action of the generators of the point group G on the basis (p. 67) | html | pdf |
    - 1.2.7.4.3. Diagonalization of the action matrix and determination of the invariant tensor (pp. 67-68) | html | pdf |
    - 1.2.7.4.4. Determination of the vector representation (p. 68) | html | pdf |
    - 1.2.7.4.5. Determination of tensor products and their decomposition (p. 68) | html | pdf |
    - 1.2.7.4.6. Invariant tensors (pp. 68-70) | html | pdf |
- 1.2.8. Glossary (pp. 70-71) | html | pdf |
- References | html | pdf |
- Tables
  - Table 1.2.2.1. Character table for $[S_3\sim D_3]$ (p. 40) | html | pdf |
  - Table 1.2.2.2. Character table of $[D_{4}]$ (p. 44) | html | pdf |
  - Table 1.2.3.1. Choices of $[{\bf k}]$ in the fundamental domain of and the elements of $[K_{\bf k}]$ (p. 49) | html | pdf |
  - Table 1.2.3.2. Strata of irreducible representations of and (p. 49) | html | pdf |
  - Table 1.2.3.3. Characteristic values of $[\lambda_{i}]$ for the projective irreps of $[K_{\bf k}]$ for the point group (p. 50) | html | pdf |
  - Table 1.2.5.1. Character of the representations corresponding to the electric and magnetic fields in point groups , and (p. 54) | html | pdf |
  - Table 1.2.6.1. Finite point groups in three dimensions (p. 57) | html | pdf |
  - Table 1.2.6.2. Crystallographic point groups in three dimensions (p. 57) | html | pdf |
  - Table 1.2.6.3. Irreducible representations for cyclic groups $[C_{n}]$ (p. 57) | html | pdf |
  - Table 1.2.6.4. Irreducible representations for dihedral groups $[D_{n}]$ (p. 57) | html | pdf |
  - Table 1.2.6.5. Irreducible representations and character tables for the 32 crystallographic point groups in three dimensions (pp. 58-60) | html | pdf |
  - Table 1.2.6.6. Direct products with $[\{E,\bar{1}\}]$ and $[\{E,1'\}]$ (p. 60) | html | pdf |
  - Table 1.2.6.7. Extra representations of double groups (p. 61) | html | pdf |
  - Table 1.2.6.8. Projective spin representations of the 32 crystallographic point groups (p. 61) | html | pdf |
  - Table 1.2.6.9. Number of free parameters of some tensors (p. 62) | html | pdf |
  - Table 1.2.6.10. Irreducible projective representations of the 32 crystallographic point groups (pp. 63-64) | html | pdf |
  - Table 1.2.6.11. Special points in the Brillouin zones in three dimensions (pp. 65-66) | html | pdf |
  - Table 1.2.6.12. Magnetic point groups (p. 66) | html | pdf |
  - Table 1.2.7.1. Data connected with the character table for point group (p. 67) | html | pdf |
  - Table 1.2.7.2. Calculation with characters (p. 69) | html | pdf |

Chapter 1.2. Representations of crystallographic groups

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