The application of eigensymmetries of face forms to anomalous scattering and twinning by merohedry in X-ray diffraction

The face form (crystal form) {hkl} which corresponds to an X-ray reflection hkl is considered. The eigensymmetry (inherent symmetry) of such a face form can be used to derive general results on the intensities of the corresponding X-ray reflections. Two cases are treated. (i) Non-centrosymmetric crystals exhibiting anomalous scattering: determination of reflections hkl for which Friedel's rule is strictly valid, i.e. I(hkl) = I((h) over bar(k) over bar(l) over bar) (Friedel pair, centric reflection), or violated, i.e. I(hkl) not equal I((h) over bar(k) over bar(l) over bar) (Bijvoet pair, acentric reflection). It is shown that those reflections hkl strictly obey Friedel's rule, for which the corresponding face form {hkl} is centrosymmetric. If the face form {hkl} is non-centrosymmetric, Friedel's rule is violated due to anomalous scattering. (ii) Crystals twinned by merohedry: determination of reflections hkl, the intensities of which are affected (or not affected) by the twinning. It is shown that the intensity is affected if the twin element is not a symmetry element of the eigensymmetry of the corresponding face form {hkl}. The intensity is not affected if the twin element belongs to the eigensymmetry of {hkl} ('affected' means that the intensities of the twin-related reflections are different for different twin domain states owing to differences either in geometric structure factors or in anomalous scattering or in both). A simple procedure is presented for the determination of these types of reflections from Tables 10.1.2.2 and 10.1.2.3 of International Tables for Crystallography, Vol. A [Hahn & Klapper (2002). International Tables for Crystallography, Vol. A, Part 10, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer]. The application to crystal-structure determination of crystals twinned by merohedry (reciprocal space) and to X-ray diffraction topographic mapping of twin domains (direct space) is discussed. Relevant data and twinning relations for the 63 possible twin laws by merohedry in the 26 merohedral point groups are presented in Appendices A to D.