International
Tables for Crystallography Volume A Space-group symmetry Edited by Th. Hahn © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A. ch. 11.1, p. 810
Section 11.1.1. Coordinate triplets and symmetry operations
a
Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany |
The coordinate triplets of a general position, as given in the space-group tables, can be taken as a shorthand notation for the symmetry operations of the space group. Each coordinate triplet corresponds to the symmetry operation that maps a point with coordinates x, y, z onto a point with coordinates . The mapping of x, y, z onto is given by the equations: If, as usual, the symmetry operation is represented by a matrix pair, consisting of a matrix W and a column matrix w, the equations read with W is called the rotation part and the translation part; w is the sum of the intrinsic translation part (glide part or screw part) and the location part (due to the location of the symmetry element) of the symmetry operation.
Example
The coordinate triplet stands for the symmetry operation with rotation part and with translation part Matrix multiplication yields
Using the above relation, the assignment of coordinate triplets to symmetry operations given as pairs (W, w) is straightforward.