D. Orobengoa

AMPLIMODES: symmetry-mode analysis on the Bilbao Crystallographic Server

AMPLIMODES is a computer program available on the Bilbao Crystallographic Server that can perform a symmetry-mode analysis of any distorted structure of displacive type. The analysis consists in decomposing the symmetry-breaking distortion present in the distorted structure into contributions from different symmetry-adapted modes. Given the high- and the low-symmetry structures, AMPLIMODES determines the atomic displacements that relate them, defines a basis of symmetry-adapted modes, and calculates the amplitudes and polarization vectors of the distortion modes of different symmetry frozen in the structure. The program uses a mode parameterization that is as close as possible to the crystallographic conventions, expressing all quantities for the asymmetric unit of the low-symmetry structure. Distorted structures are often related to their higher-symmetry counterparts by temperature- and/or pressure-driven phase transitions, ferroic phase transitions being a particular example. The automatic symmetry-mode analysis performed by AMPLIMODES can be very useful for establishing the driving mechanisms of such structural phase transitions or the fundamental instabilities at the origin of the distorted phases.

Mode crystallography of distorted structures

The description of displacive distorted structures in terms of symmetry-adapted modes is reviewed. A specific parameterization of the symmetry-mode decomposition of these pseudosymmetric structures defined on the setting of the experimental space group is proposed. This approach closely follows crystallographic conventions and permits a straightforward transformation between symmetry-mode and conventional descriptions of the structures. Multiple examples are presented showing the insight provided by the symmetry-mode approach. The methodology is shown at work, illustrating its various possibilities for improving the characterization of distorted structures, for example: detection of hidden structural correlations, identification of fundamental and marginal degrees of freedom, reduction of the effective number of atomic positional parameters, quantitative comparison of structures with the same or different space group, detection of false refinement minima, systematic characterization of thermal behavior, rationalization of phase diagrams and various symmetries in families of compounds etc. The close relation of the symmetry-mode description with the superspace formalism applied to commensurate superstructures is also discussed. Finally, the application of this methodology in the field of ab initio or first-principles calculations is outlined. At present, there are several freely available user-friendly computer tools for performing automatic symmetry-mode analyses. The use of these programs does not require a deep knowledge of group theory and can be applied either a posteriori to analyze a given distorted structure or a priori to parameterize the structure to be determined. It is hoped that this article will encourage the use of these tools. All the examples presented here have been worked out using the program AMPLIMODES [Orobengoa et al. (2009). J. Appl. Cryst. 42, 820-833].

A new computer tool at the Bilbao Crystallographic Server to detect and characterize pseudosymmetry

Abstract New ferroelectrics can be predicted by considering the existence of pseudosymmetry with respect to a higher symmetry structure using the so-called atomic displacement method and investigating the minimal supergroups of the given structure’s space group. This analysis can be performed with the new version of the computer program PSEUDO, located at the Bilbao Crystallographic Server. After defining the procedures for the detection and quantification of pseudosymmetry, we present the new program, illustrating its use with worked cases of polar structures from the literature which are either known or reported as possible ferroelectrics.

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.

Comparison of structures applying the tools available at the Bilbao Crystallographic Server

A quantitative comparison of similar crystal structures is often convenient to cross-check different experimental and/or theoretical structural models of the same phase coming from different sources. It is also important for the identification of different phases with the same symmetry, and it is fundamental for the still open problem of the classification of structures into structure types. In most cases, even if the setting of its space group is fixed, there is more than one equivalent description for a given structure. The existence of various equivalent structure descriptions makes the comparison of different structural models a non-trivial task in general. To deal with it, the program COMPSTRU has been developed, available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The program measures the similarity between two structures having the same space-group symmetry (or space groups that form an enantiomorphic pair) with the same or different compositions, and under the condition that the sequence of the occupied Wyckoff positions is the same in both structures (isopointal structures). The efficiency and utility of the program are demonstrated by a number of illustrative examples. It is also shown how the program can be used to outline different structure types within a set of isopointal structures.

Double crystallographic groups and their representations on the Bilbao Crystallographic Server

A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (this http URL). The double crystallographic groups are required in the study of physical systems whose Hamiltonian includes spin-dependent terms. In the symmetry analysis of such systems, instead of the irreducible representations of the space groups, it is necessary to consider the single- and double-valued irreducible representations of the double space groups. The new section includes databases of symmetry operations (DGENPOS) and of irreducible representations of the double (point and space) groups (REPRESENTATIONS DPG and REPRESENTATIONS DSG). The tool DCOMPATIBILITY RELATIONS provides compatibility relations between the irreducible representations of double space groups at different k-vectors of the Brillouin zone when there is a group-subgroup relation between the corresponding little groups. The program DSITESYM implements the so-called site-symmetry approach, which establishes symmetry relations between localized and extended crystal states, using representations of the double groups. As an application of this approach, the program BANDREP calculates the band representations and the elementary band representations induced from any Wyckoff position of any of the 230 double space groups, giving information about the properties of these bands. Recently, the results of BANDREP have been extensively applied in the description and the search of topological insulators.

Modes vs. modulations: Symmetry-mode analysis of commensurate modulated structures compared with the superspace method

We know from Landau theory that the natural language to deal with distorted structures is the one of symmetry-adapted modes. A symmetry-mode decomposition is both useful for investigating the mechanisms responsible of these phases and for pure crystallographic purposes. Some symmetry-mode components are usually dominant, introducing a parameter hierarchy or reducing in practice the crystallographic degrees of freedom. Several computer tools are now freely available allowing structure refinements directly using symmetry-mode collective coordinates. Alternatively, the superspace formalism is a very efficient method for similar purposes. By means of several examples we compare the two approaches.

Symmetry mode analysis of the phase transitions in Rb2ZnBr4

Abstract Crystal structures of low temperature phases III, IV and V of Rubidium Zinc Bromide have been determined by single crystal X-ray diffraction at 90 K, 85 K, and 30 K, respectively. The analysis of the structures in terms of symmetry-adapted modes of the high temperature prototype phase permits the identification of the modes responsible for each phase and to assess its thermal evolution. The primary mode responsible for the incommensurate and lock-in phases remains dominant throughout the whole transition sequence, maintaining its structure and slightly increasing its amplitude as temperature decreases. The two monoclinic phases IV and V are presumably due to the instability of a second mode associated with a single active irreducible representation. However, the presence with similar weights of several symmetry-adapted distortions of different symmetry indicates a strong hybridization in phase III of the softer normal modes. A symmetry-mode description of the structures of phase IV and V fully consistent with a simple Landau model, can then only be obtained if phase III is taken as effective parent phase. Besides the insight on the microscopic origin of the different phases, the symmetry modes were used as crystallographic coordinates in the refinement of the structure of phase V. The result shows the advantages of using these non-conventional collective structural parameters instead of individual atomic coordinates. The refinable parameters can be reduced to the amplitudes of those symmetry modes with significant weight in the distortions.

Symmetry studies of ferroic structures with the Bilbao Crystallographic Server

The Bilbao Crystallographic Server [1] is a free web site with crystallographic databases and programs available at http://www.cryst.ehu.es. The server is built on a core of databases that contains data of the International Tables for Crystallography, Vol. A (Space-group Symmetry), Vol. A1 (Symmetry Relations between Space Groups) and Vol. E (Subperiodic Groups). More specialized crystallographic software is also available and distributed in shells according to different topics: group-subgroup relations of space groups, solid-state physics and crystal chemistry applications or representation theory. The aim of the contribution is to report on the databases and tools available on the server facilitating the symmetrymode analysis of distorted structures of displacive type, the ferroic structures being a particular case. Starting from the experimental structures of the highand low symmetry phases, it is possible to determine the global structural distortion that relates the two phases. The symmetry-modes compatible with the studied symmetry break are calculated, their orthonormalization permits the decomposition of the global distortion into symmetry-mode contributions, and the determination of the corresponding polarization vectors. This type of analysis allows the determination of the correlated atomic displacements that correspond to the structural instabilities at the origin of the ferroic distortion, i.e. the so-called primary modes, and to distinguish them from secondary ones, weaker distortions of limited relevance for the transition mechanism. The server also offers online tools for the evaluation of the pseudosymmetry of a given structure with respect to a supergroup of its space group [2]. The detection of structural pseudosymmetry as the consequence of a small distortion of a higher symmetry, described by a supergroup of the crystal space group, is a powerful method for the prediction of new ferroelectric and ferroelastic materials. For example, polar structures with pseudosymmetry related to a hypothetical non-polar configuration can be considered as good candidates for ferroelectrics. There are also computer tools for systematic studies of the possible transition paths of phase transitions with no groupsubgroup relation between their phases [3]. The method is based on the assumption that the transformation involves, at least locally, an intermediate “hypothetical” configuration corresponding to a common space subgroup of the two end phases. The structure-dependent characterization of the proposed mechanisms and an evaluation of their plausibility is achieved by the analysis of lattice strains

Bilbao Crystallographic Server: computer determination of non-characteristic orbits

25th European Crystallographic Meeting, ECM 25, İstanbul, 2009 Acta Cryst. (2009). A65, s 309 Page s 309 FA4-MS10-P09 Bilbao Crystallographic Server: Computer Determination of Non-characteristic Orbits. Mois I. Aroyoa, Danel Orobengoaa, Massimo Nespolob. aDepartment of Solid State Physics, University of the Basque Country, Bilbao, Spain. bUniversity of Nancy, Institut Jean Barriol, Nancy, France. E-mail: mois.aroyo@ehu.es A crystallographic orbit is called characteristic if its intrinsic symmetry (given by the so-called eigensymmetry group E) is that of the original or generating space group G. The orbit is called non-characteristic if it displays higher symmetry. The interest in the determination of noncharacteristic orbits and their eigensymmetry groups is based on their importance in crystal structure determination, crystal physics and structural chemistry. Both theoretical discussions and tabulations of non-crystallographic orbits and their eigensymmetry groups can be found in the literature (e.g. see Engel et al. [1], and the literature therein). The aim of this contribution is to report on the development of an algorithmic procedure for the determination of the non-characteristic orbits of the space groups and their eigensymmetry groups. The procedure is based on the following basic arguments: (a) Consider a non-characteristic orbit of a point X of the space group G with an eigensymmetry group E. In the eigensymmetry group, the point X belongs to the orbit that does not split during the symmetry reduction from E to G, only its site symmetry group is reduced. The crystallographic orbits in E that satisfy this non-splitting condition are possible solutions for the non-characteristic orbits of G. (b) The number n of points per primitive unit cell of a noncharacteristic orbit limits the index of the translation lattices of E and G. Thus, the eigensymmetry group of an orbit of G could be only among the supergroups of G with a reduction of the primitive unit cell that is smaller than n. (c) Consider a space group Z intermediate between G and the eigensymmetry group E of a non-characteristic orbit of point X. The orbit of X does not split neither for the symmetry reduction from E to G, nor for the reduction from Z to G, i.e. non-characteristic orbits may appear as solutions for several supergroups of G. Obviously, these supergroups are group-subgroup related, and the eigensymmetry group is the supergroup of highest index with respect to G. The algorithmic procedure is implemented in the computing program NONCHAR that is available on the Bilbao Crystallographic Server (www.cryst.ehu.es) [2] The method for the determination of the non-characteristic orbits and their eigensymmetry groups is based on the crystallographic databases and computer tools available on the Bilbao Crystallographic Server. The efficiency of the program is increased taking into account the close relationships between the concepts of lattice complexes and limiting complexes, and those of non-characteristic orbits [3]. [1]. Engel P., Matsumoto T., Steinmann G., Wondratschek H. The Non-characteristic Orbits of the Space Groups. Z. Kristallogr., Suppl. Issue No.1, Oldenbourg, Muenchen, 1984. [2] Aroyo M.I., Perez-Mato J.M., Capillas C., Kroumova E., Ivantchev S., Madariaga G., Kirov A., Wondratschek H., Z. Kristallog. 2006, 221, 15-27. [3]. Koch E., Fischer W. Acta Cryst. A 1985, 41, 421426.