Understanding the symmetries described by subperiodic groups – frieze, rod and layer groups – has been instrumental in predicting various properties (band structures, optical absorption, Raman ...spectra, diffraction patterns, topological properties etc.) of `low‐dimensional' crystals. This knowledge is crucial in the tailored design of materials for specific applications across electronics, photonics and materials engineering. However, there are materials that have the property of being periodic only in one direction and whose symmetry cannot be described by the subperiodic rod groups. Describing the symmetry of these materials necessitates the application of line group theory. This paper gives an overview of subperiodic groups while briefly introducing line groups in order to acquaint the crystallographic community with these symmetries and direct them to pertinent literature. Since line groups are generally not subperiodic, they have thus far remained outside the realm of symmetries traditionally considered in crystallography, although there are numerous `one‐dimensional' crystals (i.e. monoperiodic structures) possessing line group symmetry.
Since they do not belong to the class of subperiodic groups, line groups thus far have been beyond the scope of symmetries traditionally studied in crystallography, although they describe the symmetry of numerous `one‐dimensional' crystals (i.e. monoperiodic structures). Therefore, together with an overview of frieze, rod and layer groups, a brief introduction to line groups is presented.
MAGNEXT is a new computer program available from the Bilbao Crystallographic Server (http://www.cryst.ehu.es) that provides symmetry‐forced systematic absences or extinction rules of magnetic ...nonpolarized neutron diffraction. For any chosen Shubnikov magnetic space group, the program lists all systematic absences, and it can also be used to obtain the list of the magnetic space groups compatible with a particular set of observed systematic absences. Absences corresponding to specific ordering modes can be derived by introducing effective symmetry operations associated with them. Although systematic extinctions in neutron diffraction do not possess the strong symmetry‐resolving power of those in nonmagnetic crystallography, they can be important for the determination of some magnetic structures. In addition, MAGNEXT provides the symmetry‐adapted form of the magnetic structure factor for different types of diffraction vectors, which can then be used to predict additional extinctions caused by some prevailing orientation of the atomic magnetic moments. This program, together with a database containing comprehensive general information on the symmetry operations and the Wyckoff positions of the 1651 magnetic space groups, is the starting point of a new section in the Bilbao Crystallographic Server devoted to magnetic symmetry and its applications.
A new section of databases and programs devoted to double crystallographic groups (point and space groups) has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). 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 DCOMPREL 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 of and the search for topological insulators.
A new section of computer tools devoted to the double crystallographic groups has been implemented in the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The section includes databases of symmetry operations and irreducible representations of the double point and space groups and programs that compute the compatibility relations, generate relevant information related to the site‐symmetry approach, and calculate band representations and elementary band representations induced from any Wyckoff position of any double space group.
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.
The program COMPSTRU, available on the Bilbao Crystallographic Server (http://www.cryst.ehu.es), for the comparison of structures is presented. The utility of the program is discussed and illustrated with several examples.
The International Tables Symmetry Database de la Flor, Gemma; Kroumova, Eli; Hanson, Robert M. ...
Journal of applied crystallography,
December 2023, 2023-12-01, 20231201, Letnik:
56, Številka:
6
Journal Article
Recenzirano
Odprti dostop
The International Tables Symmetry Database (https://symmdb.iucr.org/), which is part of International Tables for Crystallography, is a collection of individual databases of crystallographic ...space‐group and point‐group information with associated programs. The programs let the user access and in some cases interactively visualize the data, and some also allow new data to be calculated `on the fly'. Together these databases and programs expand upon and complement the symmetry information provided in International Tables for Crystallography Volume A, Space‐Group Symmetry, and Volume A1, Symmetry Relations between Space Groups. The Symmetry Database allows users to learn about and explore the space and point groups, and facilitates the study of group–subgroup relations between space groups, with applications in determining crystal‐structure relationships, in studying phase transitions and in domain‐structure analysis. The use of the International Tables Symmetry Database in all these areas is demonstrated using several examples.
The International Tables Symmetry Database (https://symmdb.iucr.org/), part of International Tables for Crystallography, is a collection of individual databases of crystallographic space‐group and point‐group information. Programs are provided to access and interactively visualize the data, and also allow new data to be calculated `on the fly', facilitating the in‐depth study of group–subgroup relations, domain structures and twinning, families of related crystal structures and phase transitions, and the prediction of new crystal structures.
The section of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) dedicated to subperiodic groups contains crystallographic and Brillouin‐zone databases for the layer groups. The ...crystallographic databases include the generators/general positions (GENPOS), Wyckoff positions (WYCKPOS) and maximal subgroups (MAXSUB). The Brillouin‐zone database (LKVEC) offers k‐vector tables and Brillouin‐zone figures of all 80 layer groups which form the background of the classification of their irreducible representations. The symmetry properties of the wavevectors are described applying the so‐called reciprocal‐space‐group approach and this classification scheme is compared with that of Litvin & Wike (1991), Character Tables and Compatibility Relations of the Eighty Layer Groups and Seventeen Plane Groups. New York: Plenum Press. The specification of independent parameter ranges of k vectors in the representation domains of the Brillouin zones provides a solution to the problems of uniqueness and completeness of layer‐group representations. The Brillouin‐zone figures and k‐vector tables are described in detail and illustrated by several examples.
The set of databases of the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) including crystallographic data on generators, general positions, Wyckoff positions, maximal subgroups and Brillouin‐zone figures and k‐vector tables for all 80 layer groups are discussed in detail and illustrated.
The numerous structures that have been reported for the different phases of the lead zirconate titanate system, PbZr1−xTixO3 (PZT), are analysed by means of a systematic symmetry‐mode analysis. The ...distortion corresponding to the order parameter has been separated out and expressed in all phases in a comparable form. The fact that the physical origin of the PZT phases is an unstable threefold degenerate polar mode, plus in some cases an unstable octahedral tilting mode, produces structural correlations between the different phases. These correlations had remained unnoticed until now but are directly observable in a mode parameterization. They can be used both to characterize the evolution of the order parameters through the phase diagram and as a stringent test of the reported structural models. It is further shown that the activity of a single polar mode yields a specific feature in the mode decomposition of the monoclinic phases. This single‐mode signature can be observed in the majority of the monoclinic structures proposed, making the others questionable. In fact, this internal constraint is satisfied by PZT to such a high degree that it drastically reduces the number of effective structural degrees of freedom. It is conjectured that this type of structural constraint beyond space‐group symmetry can be a rather general property of low‐symmetry distorted structures. As shown here, its existence can be detected and assessed by a symmetry‐mode analysis, if considered in relation to the single underlying multidimensional order parameter.
Symmetry database of International Tables online Kroumova, Eli; de la Flor, Gemma; Ashcroft, Nicola J. ...
Acta crystallographica. Section A, Foundations and advances,
08/2021, Letnik:
77, Številka:
a2
Journal Article
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.
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