The application of eigensymmetries of face forms to X-ray diffraction intensities of crystals twinned by `reticular merohedry'

This paper is an extension of a previous treatment of `twins by merohedry' with full lattice coincidence [Σ = 1, Klapper & Hahn (2010). Acta Cryst. A66, 327–346] to `twins by reticular merohedry' with partial lattice coincidence (Σ > 1). Again, the sets of symmetrically equivalent reflections {hkl} are considered as sets of equivalent faces (face forms) {hkl}, and the behaviour of the oriented eigensymmetries of these forms under the action of a twin operation is used to determine the X-ray reflection sets, the intensities of which are affected or not affected by the twinning. The following cases are treated: rhombohedral obverse/reverse Σ3 twins, cubic Σ3 (spinel) twins, tetragonal Σ5 twins (twin elements m′(120), 2′[10]) and hexagonal Σ7 twins (m′(120), 2′[20]). For each case the twin laws for all relevant point groups are defined, and the twin diffraction cases A (intensity of twin-related reflection sets not affected), B1 (intensity affected), B2 (intensity affected only by anomalous scattering) and S (single, i.e. non-coincident reflection sets) are derived for all twin laws. A special treatment is provided for the cubic Σ3 twins, where the cubic face forms first have to be split into up to four rhombohedral subforms with a threefold axis along one of the four cube 〈111〉 directions, here [111]. These subforms exhibit different twin diffraction cases analogous to those derived for the rhombohedral obverse/reverse Σ3 twins. A complete list of the split forms and their diffraction cases for all cubic point groups and all Σ3 twin elements is given. The application to crystal structure determination of crystals twinned by reticular merohedry and to X-ray topographic mapping of twin domains is discussed.