Transformations of coordinate systems

Wondratschek, H.; Aroyo, M.I.; Souvignier, B.; Chapuis, G.

doi:10.1107/97809553602060000923

International Tables for Crystallography (2015). Vol. A. ch. 1.5, pp. 75-105
https://doi.org/10.1107/97809553602060000923

Chapter 1.5. Transformations of coordinate systems

1.5. Transformations of coordinate systems (pp. 75-105) | html | pdf | chapter contents |
H. Wondratschek, M. I. Aroyo, B. Souvignier and G. Chapuis
- 1.5.1. Origin shift and change of the basis (pp. 75-83) | html | pdf |
  H. Wondratschek and M. I. Aroyo
  - 1.5.1.1. Origin shift (pp. 75-76) | html | pdf |
  - 1.5.1.2. Change of the basis (pp. 76-83) | html | pdf |
  - 1.5.1.3. General change of coordinate system (p. 83) | html | pdf |
- 1.5.2. Transformations of crystallographic quantities under coordinate transformations (pp. 83-87) | html | pdf |
  H. Wondratschek and M. I. Aroyo
  - 1.5.2.1. Covariant and contravariant quantities (p. 83) | html | pdf |
  - 1.5.2.2. Metric tensors of direct and reciprocal lattices (p. 84) | html | pdf |
  - 1.5.2.3. Transformation of matrix–column pairs of symmetry operations (p. 84) | html | pdf |
  - 1.5.2.4. Augmented-matrix formalism (pp. 84-86) | html | pdf |
  - 1.5.2.5. Example: paraelectric-to-ferroelectric phase transition of GeTe (pp. 86-87) | html | pdf |
- 1.5.3. Transformations between different space-group descriptions (pp. 87-90) | html | pdf |
  G. Chapuis, H. Wondratschek and M. I. Aroyo
  - 1.5.3.1. Space groups with more than one description in this volume (pp. 87-88) | html | pdf |
  - 1.5.3.2. Examples (pp. 88-90) | html | pdf |
    - 1.5.3.2.1. Transformations between different settings of P2₁/c (pp. 88-90) | html | pdf |
    - 1.5.3.2.2. Transformation between the two origin-choice settings of I4₁/amd (p. 90) | html | pdf |
- 1.5.4. Synoptic tables of plane and space groups (pp. 91-106) | html | pdf |
  B. Souvignier, G. Chapuis and H. Wondratschek
  - 1.5.4.1. Additional symmetry operations and symmetry elements (pp. 91-95) | html | pdf |
    - 1.5.4.1.1. Determining the type of a symmetry operation (pp. 91-92) | html | pdf |
    - 1.5.4.1.2. Cosets without additional types of symmetry elements (p. 92) | html | pdf |
    - 1.5.4.1.3. Examples with additional types of symmetry elements (pp. 92-95) | html | pdf |
  - 1.5.4.2. Synoptic table of the plane groups (p. 95) | html | pdf |
  - 1.5.4.3. Synoptic table of the space groups (pp. 95-106) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 1.5.1.1. The coordinates of the points X (or Y) with respect to the old origin O are x (y), and with respect to the new origin they are $[{\bi x}']$ $[({\bi y}')]$ (p. 75) | html | pdf |
  - Fig. 1.5.1.2. Monoclinic centred lattice, projected along the unique axis (p. 80) | html | pdf |
  - Fig. 1.5.1.3. Body-centred cell I with a_I, b_I, c_I and a corresponding primitive cell P with a_P, b_P, c_P (p. 80) | html | pdf |
  - Fig. 1.5.1.4. Face-centred cell F with a_F, b_F, c_F and a corresponding primitive cell P with a_P, b_P, c_P (p. 80) | html | pdf |
  - Fig. 1.5.1.5. Tetragonal lattices, projected along $[[00\bar{1}]]$ (p. 81) | html | pdf |
  - Fig. 1.5.1.6. Unit cells in the rhombohedral lattice (p. 81) | html | pdf |
  - Fig. 1.5.1.7. Hexagonal lattice projected along $[[00\bar{1}]]$ (p. 82) | html | pdf |
  - Fig. 1.5.1.8. Hexagonal lattice projected along $[[00\bar{1}]]$ (p. 82) | html | pdf |
  - Fig. 1.5.1.9. Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (p. 82) | html | pdf |
  - Fig. 1.5.1.10. Rhombohedral lattice with primitive rhombohedral cell a_rh, b_rh, c_rh and the three centred monoclinic cells (p. 82) | html | pdf |
  - Fig. 1.5.2.1. Illustration of the transformation of symmetry operations $[({\bi W}, {\bi w})]$ , also called a `mapping of mappings' (p. 84) | html | pdf |
  - Fig. 1.5.3.1. Three possible cell choices for the monoclinic space group $[P2_{1}/c]$ (14) with unique axis b (p. 87) | html | pdf |
  - Fig. 1.5.3.2. Two possible origin choices for the orthorhombic space group Pban (50) (p. 88) | html | pdf |
  - Fig. 1.5.3.3. General-position diagram of the space group R3m (160) showing the relation between the hexagonal and rhombohedral axes in the obverse setting: $[{\bf a}_{\rm rh}]$ = $[\textstyle{{1}\over{3}}(2{\bf a}_{\rm hex}+{\bf b}_{\rm hex}+{\bf c}_{\rm hex})]$ , $[{\bf b}_{\rm rh}]$ = $[\textstyle{{1}\over{3}}(-{\bf a}_{\rm hex}+{\bf b}_{\rm hex}+{\bf c}_{\rm hex})]$ , $[{\bf c}_{\rm rh}]$ = $[\textstyle{{1}\over{3}}(-{\bf a}_{\rm hex}]$ $[-2{\bf b}_{\rm hex}+{\bf c}_{\rm hex})]$ (p. 88) | html | pdf |
  - Fig. 1.5.4.1. Symmetry-element diagram for space group P23 (195) (p. 95) | html | pdf |
- Tables
  - Table 1.5.1.1. Selected 3 × 3 transformation matrices $[{\bi P}]$ and $[{\bi Q} = {\bi P}^{ -1}]$ (pp. 77-80) | html | pdf |
  - Table 1.5.3.1. Transformation of reflection-condition data for P12₁/c1 to P112₁/a (p. 89) | html | pdf |
  - Table 1.5.4.1. Additional symmetry operations and their locations if the translation vector t is inclined to the symmetry axis or symmetry plane (p. 93) | html | pdf |
  - Table 1.5.4.2. Additional symmetry operations due to a centring vector t and their locations (p. 94) | html | pdf |
  - Table 1.5.4.3. List of plane-group symbols (p. 96) | html | pdf |
  - Table 1.5.4.4. List of space-group symbols for various settings and cells (pp. 97-105) | html | pdf |

Chapter 1.5. Transformations of coordinate systems

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