International Tables for Crystallography (2011). Vol. A1, ch. 1.6, pp. 44-56
doi: 10.1107/97809553602060000795

Chapter 1.6. Relating crystal structures by group–subgroup relations

Contents

  • 1.6. Relating crystal structures by group–subgroup relations  (pp. 44-56) | html | pdf | chapter contents |
    • 1.6.1. Introduction  (p. 44) | html | pdf |
    • 1.6.2. The symmetry principle in crystal chemistry  (p. 44) | html | pdf |
    • 1.6.3. Bärnighausen trees  (pp. 44-46) | html | pdf |
    • 1.6.4. The different kinds of symmetry relations among related crystal structures  (pp. 46-52) | html | pdf |
      • 1.6.4.1. Translationengleiche maximal subgroups  (pp. 46-47) | html | pdf |
      • 1.6.4.2. Klassengleiche maximal subgroups  (pp. 47-48) | html | pdf |
      • 1.6.4.3. Isomorphic maximal subgroups  (p. 48) | html | pdf |
      • 1.6.4.4. The space groups of two structures having a common supergroup  (p. 49) | html | pdf |
      • 1.6.4.5. Can a structure be related to two aristotypes?  (pp. 49-50) | html | pdf |
      • 1.6.4.6. Treating voids like atoms  (pp. 50-51) | html | pdf |
      • 1.6.4.7. Large families of structures. Prediction of crystal-structure types  (pp. 51-52) | html | pdf |
    • 1.6.5. Handling cell transformations  (pp. 52-54) | html | pdf |
    • 1.6.6. Comments concerning phase transitions and twin domains  (p. 54) | html | pdf |
    • 1.6.7. Exercising care in the use of group–subgroup relations  (pp. 54-55) | html | pdf |
    • References | html | pdf |
    • Figures
      • Fig. 1.6.3.1. Scheme of the formulation of the smallest step of symmetry reduction connecting two related crystal structures  (p. 45) | html | pdf |
      • Fig. 1.6.4.10. Bärnighausen tree of hettotypes of the ReO 3 type  (p. 52) | html | pdf |
      • Fig. 1.6.4.1. Bärnighausen tree for the family of structures of pyrite  (p. 47) | html | pdf |
      • Fig. 1.6.4.2. The structures of AlB 2 , ZrBeSi and CaIn 2   (p. 48) | html | pdf |
      • Fig. 1.6.4.3. Both hettotypes of the AlB 2 type have the same space-group type and a doubled c axis, but the space groups are different due to different origin positions relative to the origin of the aristotype  (p. 48) | html | pdf |
      • Fig. 1.6.4.4. The group–subgroup relation rutile–trirutile  (p. 49) | html | pdf |
      • Fig. 1.6.4.5. The unit cells of β-K 2 CO 3 and β-Na 2 CO 3   (p. 49) | html | pdf |
      • Fig. 1.6.4.6. Group–subgroup relations among some modifications of the alkali metal carbonates  (p. 50) | html | pdf |
      • Fig. 1.6.4.7. The structure of the high-pressure modification Si-XI is related to the structures of both Si-II and Si-V  (p. 50) | html | pdf |
      • Fig. 1.6.4.8. The connected coordination octahedra in ReO 3 and FeF 3 (VF 3 type)  (p. 50) | html | pdf |
      • Fig. 1.6.4.9. Derivation of the FeF 3 structure either from the ReO 3 type or from the hexagonal closest packing of spheres  (p. 51) | html | pdf |
      • Fig. 1.6.5.1. Relative orientations of a cubic to a rhombohedral cell (hexagonal setting) (left) and of a rhombohedral cell (hexagonal setting) to a monoclinic cell (the monoclinic cell has an acute angle β)  (p. 53) | html | pdf |
    • Tables
      • Table 1.6.4.1. Crystal data for FeF 3 at different pressures  (p. 51) | html | pdf |