International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 780-781
Section 3.3.3.1. Derivation of the space group from the short symbol^{a}Universität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and ^{b}Institut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany |
Because the short international symbol contains a set of generators, it is possible to deduce the space group from it. With the same distinction between generators and indicators as for point groups, the modified point-group symbol directly gives the rotation parts W of the generating operations (W, w).
The modified symbols of the generators determine the glide/screw parts of w. To find the location parts of w, it is necessary to inspect the product relations of the group. The deduction of the set of complete generating operations can be summarized in the following rules:
The origin that is selected by these rules is called the `origin of the symbol' (Burzlaff & Zimmermann, 1980). It is evident that the reference to the origin of the symbol allows a very short and unique notation of all desirable origins by appending the components of the origin of the symbol to the short space-group symbol, thus yielding the so-called expanded Hermann–Mauguin symbol. The shift of origin can be performed easily, for only the translation parts have to be changed. The components of the transformed translation part can be obtained by [cf. Section 1.5.2.3 and equation (1.5.2.13) ] Applications can be found in Burzlaff & Zimmermann (2002).
Examples: Deduction of the generating operations from the short symbol
Some examples for the use of these rules are now described in detail. It is convenient to describe the symmetry operation (W, w) by the corresponding coordinate triplets, i.e. using the so-called shorthand notation, cf. Section 1.2.2 . The coordinate triplets can be interpreted as combinations of two constituents: the first one consists of the coordinates of a point in general position after the application of W on x, y, z, while the second corresponds to the translation part w of the symmetry operation. The coordinate triplets of the symmetry operations are tabulated as the general position in the space-group tables (in some cases a shift of origin is necessary). If preference is given to full matrix notation, Table 1.2.2.1 may be used. The following examples contain, besides the description of the symmetry operations, references to the numbering of the general positions in the space-group tables of this volume; cf. Sections 2.1.3.9 and 2.1.3.11 . Centring translations are written after the numbers, if necessary.
References
Burzlaff, H. & Zimmermann, H. (1980). On the choice of origins in the description of space groups. Z. Kristallogr. 153, 151–179.Burzlaff, H. & Zimmermann, H. (2002). On the treatment of settings of space groups and crystal structures by specialized short Hermann–Mauguin space-group symbols. Z. Kristallogr. 217, 135–138.