Hans Wondratschek

Bilbao Crystallographic Server: I. Databases and crystallographic computing programs

Abstract 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.

Bilbao Crystallographic Server : Useful Databases and Tools for Phase-Transition Studies

Bilbao Crystallographic Server is a web site with crystallographic programs and databases available on-line. The programs give access to general information related to space groups (generators, general positions, Wyckoff positions, irreducible representations), group-subgroup or group-supergroup pairs of space groups, and/or results on specific crystal structures. The utility of the programs is illustrated by treating phase-transition problems related to structural pseudosymmetry.

Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups.

The Bilbao Crystallographic Server is a web site with crystallographic programs and databases freely available on-line (http://www.cryst.ehu.es). The server gives access to general information related to crystallographic symmetry groups (generators, general and special positions, maximal subgroups, Brillouin zones etc.). Apart from the simple tools for retrieving the stored data, there are programs for the analysis of group-subgroup relations between space groups (subgroups and supergroups, Wyckoff-position splitting schemes etc.). There are also software packages studying specific problems of solid-state physics, structural chemistry and crystallography. This article reports on the programs treating representations of point and space groups. There are tools for the construction of irreducible representations, for the study of the correlations between representations of group-subgroup pairs of space groups and for the decompositions of Kronecker products of representations.

PSEUDO: a program for a pseudo- symmetry search

The program PSEUDO provides tools for the systematic search of pseudosymmetry, based on group±subgroup relations between space groups (Igartua et al., 1996). For a crystal structure L, speci®ed by its space group U, the cell parameters and the coordinates of the atoms in the asymmetric unit, the program searches for pseudosymmetry among all minimal supergroups Gk > U of the group U. The interpretation of a structural pseudosymmetry as a small distortion of a higher symmetric (prototype) structure H allows the following. (i) The prediction of phase transitions at higher temperature. If the distortion is small enough, it can be expected that the crystal acquires the more symmetric con®guration at a higher temperature after a phase transition (Igartua et al., 1996, 1999). (ii) The search for new ferroelastic and ferroelectric materials. Polar structures having atomic displacements smaller than 1 AE with respect to a hypothetical non-polar con®guration are considered as possible ferroelectrics (Abrahams & Keve, 1971; Kroumova et al., 2000). (iii) The detection of false symmetry assignments (overlooked symmetry) in crystal structure determination.

Brillouin-zone database on the Bilbao Crystallographic Server.

The Brillouin-zone database of the Bilbao Crystallographic Server (http://www.cryst.ehu.es) offers k-vector tables and figures which form the background of a classification of the irreducible representations of all 230 space groups. The symmetry properties of the wavevectors are described by the so-called reciprocal-space groups and this classification scheme is compared with the classification of Cracknell et al. [Kronecker Product Tables, Vol. 1, General Introduction and Tables of Irreducible Representations of Space Groups (1979). New York: IFI/Plenum]. The compilation provides a solution to the problems of uniqueness and completeness of space-group representations by specifying the independent parameter ranges of general and special k vectors. Guides to the k-vector tables and figures explain the content and arrangement of the data. Recent improvements and modifications of the Brillouin-zone database, including new tables and figures for the trigonal, hexagonal and monoclinic space groups, are discussed in detail and illustrated by several examples.

Splitting of Wyckoff positions (orbits)

SummaryA crystal structure is composed of one or more sets of atoms where the atoms of each set are symmetrically equivalent under a space group (‘regular system of atoms’ or ‘orbit of atoms’). For a characterization of the crystal structure it is sufficient to specify one atom of each set (each orbit) by the coordinates of its centre.Relations between crystal structures in crystal chemistry or in continuous phase transitions manifest themselves in group (G)-subgroup (H) relations: H < G. Atoms which are equivalent under G may become non-equivalent under the subgroup H, i.e., ‘the orbit may split’. The splitting is the same for all orbits of a Wyckoff position but may be different for orbits of different Wyckoff positions.In this paper, the possible ways of splitting of orbits and the conditions for this splitting are analyzed. Relevant factors for the splitting are: the index of the group-subgroup relation G-H, the order of the point group of G, the order of the site-symmetry groups of the orbit in G and, to a lesser degree, in the subgroup H. Also the kind of group-subgroup relation plays a role. The splitting is most restricted if H is a normal subgroup of G. The splitting conditions are expressed in an equation which is illustrated by examples.ZusammenfassungEine Kristallstruktur besteht aus Sätzen symmetrisch gleichwertiger Atome (Orbits von Atomen), die ineinandergestellt sind. Ihre Charakterisierung geschieht durch die Angabe der Koordinaten je eines Atoms (bzw. dessen Schwerpunkts) jedes Satzes sowie der Raumgruppe der Struktur. Wird die Symmetrie zum Vergleich mit verwandten Kristall-strukturen oder bei einer kontinuierlichen Phasenumwandlung reduziert (Gruppe-Untergruppe-Beziehung G-H), so können Atome, die unter G gleichwertig sind, in H ungleichwertig werden (Aufspalten des Orbits). Dieses Aufspalten geschieht für Orbits der gleichen Punktlage (Wyckoff position) in gleicher Weise, kann aber für verschiedene Punktlagen verschieden sein.Es werden die Bedingungen untersucht, die über Möglichkeiten eines solchen Aufspaltens entscheiden. Bestimmende Grössen sind: Die Ordnung der Punktgruppe der Ausgangsraumgruppe G; die Lagesymmetrie der Atome (Punkte) in G und, weniger wichtig, diejenige in der Untergruppe H, sowie die Art der Gruppe—Untergruppe—Beziehung. Am geringsten sind die Möglichkeiten der Aufspaltung, wenn H Normal-teiler von.G ist.Beispiele veranschaulichen die abgeleiteten Gesetzmässigkeiten.

SUPERGROUPS - a computer program for the determination of the supergroups of the space groups

The problem of the determination of the supergroups of a given space group is of rather general interest. It is useful in phase-transition problems, in the search for overlooked symmetries in a crystal structure determination, or in the detection of pseudosymmetries as a tool for predicting higher-temperature phase transitions (Igartua et al., 1996). In all these applications, it is not suf®cient to know the space-group types of the supergroups of a given group; rather, it is necessary to have available all the different supergroups Gj > H which are isomorphic to G, and are of the same index [i]. In the literature, there are very few papers treating the supergroups of space groups in any detail (Koch, 1984; Wondratschek & Aroyo, 2001). In the International Tables for Crystallography, Vol. A, Space Group Symmetry (1983), one ®nds only listings of minimal supergroups of space groups, which, in addition, are not explicit: they only provide for each space group H the list of those space-group types in which H occurs as a maximal subgroup, i.e to which minimal supergroups of H belong. It is rather dif®cult to determine all supergroups Gj > H if only the types of the minimal supergroups are known. The program SUPERGROUPS solves this problem for a given ®nite index [i].

Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-Hoc Committee on the Nomenclature of Symmetry

Standard symbols representing crystal families, two- and three-dimensional Bravais-lattice types and arithmetic classes are recommended for use by the IUCr. The six crystal families are designated by lower-case letters. The family letter in the symbol of each of the 14 lattice types is followed by an upper-case letter to distinguish different lattice types within the family. Arithmetic classes are indicated by modified symbols of the corresponding symmorphic space groups.

Possible superlattices of extraordinary orbits in 3-dimensional space9

Extraordinary orbits are distinguished from other crystallographic orbits by additional translations. A systematic derivation of all extraordinary orbits requires the knowledge of all possible superlattices which may be assigned to extraordinary orbits. These superlattices are listed in this paper. The corresponding additional laws of integral extinctions are discussed.

The extraordinary orbits of the 17 plane groups

In the preceding paper an orbit has been defined as the set of all points equivalent to a starting point under the symmetry operations of a (generating) space group R. An orbit was called extraordinary if it admits symmetry translations which are not translations of the generating space group R. A procedure was given for deriving systematically and listing all extraordinary orbits of a space group.