Extension of three-dimensional space-group generators to the (3+1) case

1999 ◽  
Vol 32 (3) ◽  
pp. 452-455
Author(s):  
Kazimierz Stróż
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Group ◽  
Space Groups ◽  
Symmetry Operations ◽  
Three Dimensional Space

A method of building up the generators of 775 (3+1)-dimensional superspace groups is proposed. The generators are based on the conventional space-group generators selected by Wondratschek and applied in theInternational Tables for Crystallography(1995, Vol. A). By the method, the generation of (3+1) space groups is found to be easier, the description of symmetry operations is closer to that used for the conventional space groups, and ambiguities in the (3+1) group notation are avoided.

Intergrowth polytypoids as modulated structures: a superspace description of the Sr n (Nb,Ti) n O3n + 2 compound series

2001 ◽  
Vol 57 (4) ◽  
pp. 471-484 ◽  
Author(s):  
L. Elcoro ◽  
J. M. Perez-Mato ◽  
R. L. Withers
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Separation Distance ◽  
Space Group ◽  
Space Groups ◽  
Cell Parameters ◽  
Average Structure ◽  
Layer Stacking ◽  
Compound Series ◽  
Three Dimensional Space

A new, unified superspace approach to the structural characterization of the perovskite-related Sr n (Nb,Ti) n O3n + 2 compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)6Ti6O20 compound demonstrates the suitability of the proposed formalism.

scholarly journals From space to superspace and back: Superspace Group Finder

2008 ◽  
Vol 41 (6) ◽  
pp. 1182-1186 ◽  
Author(s):  
Ivan Orlov ◽  
Lukas Palatinus ◽  
Gervais Chapuis
Keyword(s):  
Crystal Structure ◽  
Dimensional Space ◽  
Flexible Structures ◽  
Three Dimensional ◽  
Real Space ◽  
Space Group ◽  
Space Groups ◽  
Group A ◽  
Modular Composition ◽  
Three Dimensional Space

The symmetry of a commensurately modulated crystal structure can be described in two different ways: in terms of a conventional three-dimensional space group or using the superspace concept in (3 +d) dimensions. The three-dimensional space group is obtained as a real-space section of the (3 +d) superspace group. A complete network was constructed linking (3 + 1) superspace groups and the corresponding three-dimensional space groups derived from rational sections. A database has been established and is available at http://superspace.epfl.ch/finder/. It is particularly useful for finding common superspace groups for various series of modular (`composition-flexible') structures and phase transitions. The use of the database is illustrated with examples from various fields of crystal chemistry.

SPGGEN: a computer program for retrieval of space-group information in several settings and generator-containing space-group symbols

2016 ◽  
Vol 49 (4) ◽  
pp. 1370-1376 ◽  
Author(s):  
Uri Shmueli
Keyword(s):  
Computer Program ◽  
Dimensional Space ◽  
Three Dimensional ◽  
General Setting ◽  
Detailed Comparison ◽  
Space Group ◽  
Space Groups ◽  
Group Number ◽  
Group Information ◽  
Three Dimensional Space

A brief outline of the algorithm for the derivation of a space group is followed by a detailed description of the explicit space-group symbols here employed. These space-group symbols are unique insofar as they contain explicitly the generators of the space group dealt with. Next, the implementation of the above in a computer program,SPGGEN, is briefly discussed and the options presented by the program are outlined. Briefly, these options are (i) conventional derivation of the space group from an explicit symbol, including a user-defined one; (ii) such derivation from the conventional space-group number only; (iii) introduction of a general setting into the derivation; (iv) introduction of a Cartesian setting into the derivation; and (v) treatment of some non-conventional settings of orthorhombic space groups. This is followed by a detailed comparison withInternational Tables for Crystallography, Vol. A, and by examples of the output ofSPGGEN. A complete tabulation of the explicit three-dimensional space-group symbols is readily accessed.

Zur Theorie der kristallographischen Orbits

1983 ◽  
Vol 163 (3-4) ◽  
Author(s):  
Peter Engel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Group ◽  
Space Groups ◽  
Theoretical Procedure ◽  
Three Dimensional Space

AbstractA group theoretical procedure to determine the non-characteristic orbits is described. For the 230 three-dimensional space group types all non-characteristic orbits were determined. Two examples for the space groups

A computer program to derive (3+1)-dimensional symmetry operations from two-line symbols

1997 ◽  
Vol 30 (1) ◽  
pp. 73-78 ◽  
Author(s):  
Z.-Q. Fu ◽  
H.-F. Fan
Keyword(s):  
Computer Program ◽  
Crystal Structures ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Direct Methods ◽  
Basic Space ◽  
Space Group ◽  
Symmetry Operations ◽  
Three Dimensional Space

A computer program has been written for the derivation of (3 + 1)-dimensional symmetry operations from the two-line symbols. The derivation is based on the concept of generators {[Γ(Rv E ), ∊v , s v , τv , q)|v = 1, NG}, in which {[Γ(Rv E ), s v )|v = 1, NG} denotes the set of generators of the basic space group represented by the upper line. The program, called SPGR4D, is written in Fortran77 and based on the program by Burzlaff & Hountas (1982). [J. Appl. Cryst. (1982), 15, 464–467] for the derivation of symmetry operations in three-dimensional space. SPGR4D has been incorporated into a new version of the direct-methods program DIMS for solving incommensurate modulated crystal structures.

Influence of Chemical and Spatial Constraints on the Structures of Inorganic Compounds

1997 ◽  
Vol 53 (3) ◽  
pp. 381-393 ◽  
Author(s):  
I. D. Brown
Keyword(s):  
Dimensional Space ◽  
Inorganic Compounds ◽  
Three Dimensional ◽  
Bond Valence ◽  
Graph Connectivity ◽  
Site Symmetry ◽  
High Symmetry ◽  
Space Group ◽  
Bond Valence Model ◽  
Three Dimensional Space

The restrictions imposed by both chemistry and three-dimensional space on the structures of inorganic crystals can be analysed using the bond-valence model and space-group theory. The bond-valence model is used to construct a bond graph (connectivity table) from which the multiplicities and possible site symmetries of each atom can be assigned. These are matched to Wyckoff positions of the three-dimensional space groups, selecting the matching space group with the highest possible symmetry. High-symmetry structures such as NaCl, perovskite and garnet are readily derived from the chemical formula and, with a little more effort, the same can be done for structures of intermediate symmetry such as wurtzite, corundum and rutile. For other compounds a relationship between the site symmetry and the multiplicity of an atom can severely restrict the possible structures.

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