Diffraction enhancement of symmetry and modular structures.

Diffraction enhancement of symmetry (DES) is the phenomenon according to which the symmetry of the diffraction pattern of a crystal can be higher than the point symmetry of the structure that has produced it. The most well known example is that of Friedel's law, which is however violated in the presence of resonant scattering. This phenomenon is addressed in monoarchetypal modular structures and it is shown that a sufficient condition for DES is that both the module and the family of stacking vectors are invariant under an isometry that is not a symmetry operation for the structure.

Layer groups: Brillouin-zone and crystallographic databases on the Bilbao Crystallographic Server.

The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.

Groupoid description of modular structures.

Modular structures are crystal structures built by subperiodic (zero-, mono- or diperiodic) substructures, called modules. The whole set of partial operations relating substructures in a modular structure build up a groupoid; modular structures composed of identical substructures are described by connected groupoids, or groupoids in the sense of Brandt. A general approach is presented to describe modular structures by Brandt's groupoids and how to obtain the corresponding space groups, in which only the partial operations that have an extension to the whole crystal space appear.

The staurolite enigma solved.

Staurolite has been long considered an enigma because of its remarkable pseudosymmetry and the frequent twinning. Staurolite gives two twins whose occurrence frequency seems to contradict the condition of lattice restoration requested by the reticular theory of twinning, in that the more frequent one (Saint Andrews cross twin) has a twin index of 12, whereas the less frequent one (Greek cross twin) has a twin index of 6. The hybrid theory of twinning shows that the former is actually a hybrid twin with two concurrent sublattices and an effective twin index of 6.

Twinning of aragonite - the crystallographic orbit and sectional layer group approach.

The occurrence frequency of the {110} twin in aragonite is explained by the existence of an important substructure (60% of the atoms) which crosses the composition surface with only minor perturbation (about 0.2 Å) and constitutes a common atomic network facilitating the formation of the twin. The existence of such a common substructure is shown by the C2/c pseudo-eigensymmetry of the crystallographic orbits, which contains restoration operations whose linear part coincides with the twin operation.

Analysis of the structural continuity in twinned crystals in terms of pseudo-eigensymmetry of crystallographic orbits.

The reticular theory of twinning gives the necessary conditions on the lattice level for the formation of twins. The latter are based on the continuation, more or less approximate, of a substructure through the composition surface. The analysis of this structural continuity can be performed in terms of the eigensymmetry of the crystallographic orbits corresponding to occupied Wyckoff positions in the structure.

Effects of merohedric twinning on the diffraction pattern.

In merohedric twinning, the lattices of the individuals are perfectly overlapped and the presence of twinning is not easily detected from the diffraction pattern, especially in the case of inversion twinning (class I). In general, the investigator has to consider three possible structural models: a crystal with space-group type H and point group P, either untwinned (H model) or twinned through an operation t in vector space (t-H model), and an untwinned crystal with space group G whose point group P' is obtained as an extension of P through the twin operation t (G model). In 71 cases, consideration of the reflection conditions may directly rule out the G model; in seven other cases the reflection conditions suggest a space group which does not correspond to the extension of H by the twin operation and the structure solution or at least the refinement will fail.

Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6.

This paper gives classification results for crystallographic groups in dimensions up to 6 which refine earlier enumeration results. Based on the classification data, the asymptotic growth of the number of space-group types is discussed. The classification scheme for crystallographic groups is revisited and a new classification level in between that of geometric and arithmetic crystal classes is introduced and denoted as harmonic crystal classes.