Volume 221, Issue 1 - Mathematical and Theoretical Crystallography

After an illustrative example for the ‘symmetry of symmetry’ the group-theoretical concept of normalizers is introduced by a series of definitions. Subsequently, this concept is applied to space groups for which Euclidean and affine normalizers are discussed. Their implications on point configurations, Wyckoff positions and coordinate descriptions of crystal structures are explained. The derivation of all equivalent descriptions of a crystal structure with the aid of the tables of normalizers given in the International Tables for Crystallography, Vol. A is elucidated by several examples.

The Bilbao Crystallographic Server is a web site with crystallographic databases and programs available on-line at www.cryst.ehu.es. It has been operating for about six years and new applications are being added regularly. The programs available on the server do not need a local installation and can be used free of charge. The only requirement is an Internet connection and a web browser. The server is built on a core of databases, and contains different shells. The innermost one is formed by simple retrieval tools which serve as an interface to the databases and permit to obtain the stored symmetry information for space groups and layer groups. The k -vector database includes the Brillouin zones and the wave-vector types for all space groups. As a part of the server one can find also the database of incommensurate structures. The second shell contains applications which are essential for prob lems involving group-subgroup relations between space groups ( e.g. subgroups and supergroups of space groups, splittings of Wyckoff positions), while the third shell contains more sophisticated programs for the computation of space-group representations and their correlations for group-subgroup related space groups. There are also programs for calculations focused on specific problems of solid-state physics. The aim of the article is to report on the current state of the server and to provide a brief description of the accessible databases and crystallographic computing programs. The use of the programs is demonstrated by illustrative examples.

The geometric theory of twinning was developed almost a century ago. Despite its age, it still represents the fundamental approach to the analysis and interpretation of twinned crystals, in both the direct and the reciprocal space. In recent years, this theory has been extended not only in its formalism (group-subgroup analysis, chromatic symmetry) but also in its classification of special cases that were not recognized before. The geometrical theory of twinning is thus reviewed here with emphasis on lattice aspects and recent developments. The classification of various types of twins starts with Friedel’s well-known scheme, which distinguishes four cases according to whether the twin index n is equal to or larger than 1 and whether the obliquity ω is equal to or larger than 0: the computation of these quantities is discussed in detail. It is shown that the concept of obliquity is not sufficient to characterize the pseudo-symmetry of a lattice, and the consequent twinning, in the case of manifold twins. The application of the theory of coincidence-site lattices to twinning is presented. Finally, the effect of twinning on the diffraction pattern is illustrated with a number of examples.

Transformation properties of tensors under the action of the full magnetic group SO (3) ×  E o are specified by the intrinsic symmetry which defines how the tensor transforms under the action of the proper rotation group SO (3) and by their parity which defines how the tensor transforms under the action of the group of inversions E o = { e , i , e ′, i ′}, where i denotes the space inversion, e ′ the magnetic inversion and i ′ = i . e ′ their combination. There exist therefore four types of tensors to each intrinsic symmetry whose transformation properties under the group SO (3) are identical while their transformation under the group E o determines their parities with respect to the three inversions. As a result of this separation, transformation properties of tensors of the same intrinsic symmetry under the action of magnetic groups of the same oriented Laue class are strongly correlated by relations for which we propose the name Opechowski's magic relations in homage to late Prof. Opechowski. The origin of these relations is explained and tables which describe them are presented. It is found, in particular, that the number of different forms of tensor decompositions under the action of groups of oriented Laue class of magnetic point groups equals the number of one-dimensional real irreducible representations of the proper rotation group which generates the Laue class. Accordingly, there exists also the same number of allowed forms of tensors which do not vanish.

The present review deals with some selected topics pertaining to the theory of OD structures of layers founded by K. Dornberger-Schiff. An attention is paid mainly to the notion of VC (Vicinity Condition) structures and to the position of OD (Order-Disorder) structures within the VC structures. Explained is also the notion of MDO (Maximum Degree of Order) structures and indicated their derivation. The importance of a suitable axial setting for periodic OD structures is illustrated by an OD interpretation of the structure of the mineral kermesite. Based on X-ray studies of the mineral cronstedtite, the choice of suitable crystals for X-ray studies, reasons for the occurrence of ‘ghost’ peaks in Fourier maps and the refinement strategy for OD structures, is also discussed.

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 graph theory prediction of bond valence distributions in a crystal network is reformulated in a form more familiar to chemists, that of symmetry-adapted functions. The graph theory terms relevant to this exposition are explained, including graph matrices, graph vector spaces and graph automorphism groups. It was found that the vertex and edge sets of a chemical graph form representations of the graph automorphism group. These representations may be expressed as a sum of irreducible representations using the group character table in the usual way. Alternatively the classes of the individual orbits spanned may be used to derive these irreducible representations. The orthogonal cut and cycle subspaces of the edge space can be defined so as to also transform as a sum of irreducible representations. Hence appropriate symmetry-adapted orthogonal basis vectors may be chosen for the vertex and edge spaces. Removing the usual Equal Valence Rule constraint creates a solution space, a range of possible solutions for a particular prob lem. This solution space may be found by combining a set of contributions from the cut space vectors fixed by the atom valences, with a convex polytope in the cycle space which is defined by the non-negativity of the individual bond valences. The complete procedure is illustrated for the β-Ga 2 O 3 structure.

Topological transformations in nets resulting from the insertion or deletion of edges or vertices are analyzed through the analogous operations performed on their quotient graphs. The role of strong rings and cages of the net is emphasized. It is shown that closed trails of the oriented quotient graph define the topology of 3-periodic nets derived from regular, vertex and edge transitive, 4-periodic minimal nets.

In this paper we look for densest ball packings of Euclidean space E 3 to given symmetry groups. We restrict our investigation to the 36 space groups of the cubic crystal system, and we search for only those packings where the group acts simply transitively on the balls. In order to find the centre of a ball and its radius for the case of an optimal packing we will apply an algorithm and the corresponding computer program, which was developed by the second author [21, 12]. In the list of our results we will give the coordinates of the ball centre and the radius, moreover, we will compute the density of the optimal packing and display the corresponding D-V cell for each space group above (see also [14, 15]).