International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.3, pp. 789-790

Section 3.3.3.5. Systematic absences

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@krist.uni-erlangen.de

3.3.3.5. Systematic absences

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Hermann (1928a[link]) 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 [{\bi h} = (hkl)] describes the X-ray reflection and [({\bi W},{\bi w})] is the matrix representation of a symmetry operation, the matrix can be expanded as follows:[({\bi W},{\bi w}) = ({\bi W}, {\bi w}_{g} + {\bf w}_{l}) = ({\bi W},\openup3pt\left(\matrix{\hfill w_{g,1}\cr \hfill w_{g,2}\cr \hfill w_{g,3}\cr}\right) + {\bi w}_{l}).]The absence of a reflection is governed by the relation (i) [{\bi h} \cdot {\bi W}] = [{\bi h}] and the scalar product (ii) [{\bi h} \cdot {\bi w}_{g} = hw_{g,1} + kw_{g,2} + lw_{g,3}]. 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 [D_{4}^{10} = I4_{1}22\ (98)]

The generators of the space group are the integral translations and the centring translation [x+\textstyle{1\over 2},y+\textstyle{1\over 2} ,z+\textstyle{1\over 2}], the rotation 2 in direction [100]: [x,\overline{y},\overline{z}] and the rotation 2 in direction [\hbox{[}1\overline{1}0\hbox{]}]: [\overline{y},\overline{x},\overline{z} -\textstyle{1\over 4}]. The combination of the two generators gives the operation corresponding to the indicator, namely [\overline y, x, z+\textstyle{1\over 4}], 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 [{\bi W} = {\bi I}] holds for all reflections (integral condition), but the scalar product (ii) is an integer only for [h + k + l = 2n]. Thus, reflections hkl with [h + k + l \ne 2n] are absent. The screw rotation 4 has the screw part [{\bi w}_{g} = \pmatrix{0\cr 0\cr \textstyle{1\over 4}}]. Only 00l reflections obey condition (i) (serial extinction). An integral value for the scalar product (ii) requires [l = 4n]. The twofold axes in the directions [100] and [\hbox{[}1\overline{1}0\hbox{]}] do not imply further absences because [{\bi w}_{g} = {\bi o}].

Detailed discussion of the theoretical background of conditions for possible general reflections and their derivation is given in Chapter 1.6[link] .

References

Hermann, C. (1928a). Zur systematischen Strukturtheorie I. Eine neue Raumgruppensymbolik. Z. Kristallogr. 68, 257–287.








































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