Bernd Souvignier

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. Enantiomorphic pairs are determined on all classification levels from space-group types to crystal families and the enantiomorphic pairs of fixed-point-free space groups are given. A general algorithm to compute enantiomorphic pairs is described.

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

A general approach to the analysis of structural continuity in twins is presented and applied to the known twins in melilite.

The four-dimensional magnetic point and space groups

Abstract This paper describes the classification of magnetic point and space groups which are also referred to as antisymmetry groups or black-and-white groups. These groups play an important role in the description of discrete point sets in which the points are not only characterized by their spatial coordinates but also by an additional property taking one of two possible values (e.g. spin up or down). Each operation of a magnetic group may or may not switch the value of this additional property. In this paper, the methods for classifying magnetic groups in arbitrary dimensions are described in an algorithmic fashion. Results of the full classification in four-dimensional space are given and the application of the magnetic groups in this dimension to quasicrystals is indicated.

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 6.0. However, this is still not sufficient to explain the observed higher occurrence frequency of the Saint Andrews twin. The (pseudo)-eigensymmetry of the crystallographic orbits of staurolite has been analysed and it was found that the whole substructure built on anions is restored (with small deviations) by both twin laws, which explains why twinning is frequent in staurolite. On the other hand, 45% of the cation sites are quasi-restored in the Saint Andrews cross twin, against only 19% for the Greek cross twin: this difference finally explains the different occurrence frequencies of the two twins.

About the concept and definition of "noncrystallographic symmetry"

Abstract The definition of “noncrystallographic symmetry” given in Volume B of the International Tables for Crystallography actually corresponds to the concept of “local symmetry”. A new definition of “noncrystallographic symmetry” is proposed, which fully complies with that of “crystallographic symmetry” in Volume A of the International Tables for Crystallography.

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. Furthermore, the local analysis of the composition surface in the aragonite structure shows that the structure is built from slices which are fixed by the twin operation, confirming and reinforcing the crystallographic orbit analysis of the structural continuity across the composition surface.

Crystallographic orbits analysis of staurolite twins

Staurolite is an enigmatic mineral characterized by a high degree of pseudo-symmetry, which frequently occurs twinned. It gives two twins with different occurrence frequency, the 90o or Greek cross (lower frequency) and the 60o or Saint Andrews cross (higher frequency). To date no explanation has been found for their different occurrence frequency. We have analyzed the structure of staurolite in terms of the pseudo-eigensymmetry of the crystallographic orbits building this structure [1]. The union of the crystallographic orbits based on oxygen atoms has a pseudo-cubic eigensymmetry which contains the twin operations of both twins: the operations restore, with good approximation, the whole set of oxygen atoms, which justifies the high frequency of twinning of this mineral but does not discriminate between the two twins. On the other hand, a subset of the tetrahedral cations has a pseudoeigensymmetry which contains the twin operation of the Saint Andrews cross, but not that of the Greek cross. Also, a subset of the octahedral cations has a pseudo-eigensymmetry which contains the twin operation of the Greek cross and a larger subset has an eigensymmetry which contains the twin operation of the Saint Andrews. The substructure approximately restored by the twin operation is thus more extensive for the Saint Andrews cross, which justifies its higher occurrence frequency.

Effects of merohedric twinning on the diffraction pattern. Erratum and corrigenda

A number of corrections are made to the article by Nespolo et al. [Acta Cryst. (2014), A70, 106–125].

Structural analysis of twinning: the example of melilite

The reticular theory of twinning, in its last extension known as theory of hybrid twins [1], provides the necessary conditions for the formation of a twinned crystal. These are based on the degree of restoration of the lattice under the action of the twin operation. The lattice nodes which are restored define a common sublattice across the interface between the individuals constituting the twin and a pre-requisite for the formation of the twin is given by the condition that the proportion of restored lattice nodes is high (at least 1/6 according to an empirical rule). But since the atomic structure does not possess the full symmetry of its lattice, the lattice restoration cannot discriminate twins of different compounds with the same type of lattice. This requires a structural approach [2] providing the sufficient conditions for the formation of a twin, which can be obtained by the following procedure. 1) Determine the intersection of the lattices of the individuals: this is the largest sublattice common to the twinned individualsand is called the twin lattice; 2) find the largest subgroup H of the space group G of the individual which is compatible with the twin lattice: this is obtained as the intersection of the space groups of the individuals (of the same type) in their respective orientations; 3) split the occupied Wyckoff positions of G into those of H and determine the crystallographic orbits under the action of H; 4) identify those crystallographic orbits under H which are restored by the twin operation, within a reasonable tolerance (pseudorestored): this occurs when the eigensymmetry of the crystallographic orbit contains, exactly or as a pseudo-symmetry, the twin operation; 5) for a crystallographic orbit not restored by the twin operation, check whether it is mapped to a different orbit such that the union of these orbits is invariant under the twin operation: in this case, the eigensymmetry group of this union of orbits contains the twin operation, and it is this union of orbits which is restored. The proposed procedure is illustrated by applying it to the (100), (010) and (120) twins of melilite for which the eigensymmetry and restoration of crystallographic orbits is analysed .

Point groups in crystallography

Two dual spaces are extensively used in crystallography: the point space E, hosting the crystal pattern; and the vector space V, where face normals and reciprocal-lattice vectors are defined. The term “point group” is used in crystallography to indicate four different types of groups in these two spaces. 1) Morphological point groups in V; they can be obtained by determining subgroups of maximal holohedries (holohedries not in group-subgroup relation): this gives 21 and 136 point groups in V2 and V3, respectively, which are classified into 10 and 32 pointgroup types (on the basis of which geometric crystal classes are defined) falling into 9 and 18 abstract iso-