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. 789-790
Section 3.3.3.5. Systematic absences^{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 |
Hermann (1928a) emphasized that the short symbols permit the derivation of systematic absences of X-ray reflections caused by the glide/screw parts of the symmetry operations. If describes the X-ray reflection and is the matrix representation of a symmetry operation, the matrix can be expanded as follows:The absence of a reflection is governed by the relation (i) = and the scalar product (ii) . A reflection h is absent if condition (i) holds and the scalar product (ii) is not an integer. The calculation must be made for all generators and indicators of the short symbol. Systematic absences, introduced by the further symmetry operations generated, are obtained by the combination of the extinction rules derived for the generators and indicators.
Example: Space group
The generators of the space group are the integral translations and the centring translation , the rotation 2 in direction [100]: and the rotation 2 in direction : . The combination of the two generators gives the operation corresponding to the indicator, namely , which represents a fourfold screw rotation in the direction [001].
The integral translations imply no restriction because the scalar product is always an integer. For the centring, condition (i) with holds for all reflections (integral condition), but the scalar product (ii) is an integer only for . Thus, reflections hkl with are absent. The screw rotation 4 has the screw part . Only 00l reflections obey condition (i) (serial extinction). An integral value for the scalar product (ii) requires . The twofold axes in the directions [100] and do not imply further absences because .
Detailed discussion of the theoretical background of conditions for possible general reflections and their derivation is given in Chapter 1.6 .
References
Hermann, C. (1928a). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.