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

Chapter 1.2. Representations of crystallographic groups

Contents

  • 1.2. Representations of crystallographic groups  (pp. 34-71) | html | pdf | chapter contents |
    • 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 |
      • 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 [Pnma] and the elements of [K_{\bf k}]   (p. 49) | html | pdf |
      • Table 1.2.3.2. Strata of irreducible representations of [Pmm2] and [Pmmm]   (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 [mmm]   (p. 50) | html | pdf |
      • Table 1.2.5.1. Character of the representations corresponding to the electric and magnetic fields in point groups [222], [2'2'2] and [2'mm']   (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 [4mm]   (p. 67) | html | pdf |
      • Table 1.2.7.2. Calculation with characters  (p. 69) | html | pdf |