These are tables which accompany Stokes, H. T., Campbell, B. J. & Hatch, D. M. (2007). "Order Parameters for Phase Transitions to Structures with One-Dimensional Incommensurate Modulations," Acta Cryst. A 63, 365-373 (2007). Download PDF reprint
Table 1. Generators of the (3+1)-dimensional superspace groups
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Table 2. Incommensurate IRs of the (3+1)-dimensional superspace
extensions of the 230 crystallographic space groups
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Table 3. Incommensurate isotropy subgroups of the (3+1)-dimensional
superspace extensions of the 230 crystallographic space groups
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Incommensurately modulated distortions in crystalline solids are induced by irreducible representations (IRs) associated with k-vectors with irrational components. For one-dimensional modulations, the resulting symmetries are described by the 775 (3+1)-dimensional superspace groups. For a discussion of these superspace groups, see Janssen et al. (2004) and references therein. We give a list of these superspace groups in Table 1. For each superspace group, we give the generators in terms of their action on the dimensionless coordinates x1,x2,x3,x4 of an arbitrary point in superspace. The generators are consistent with the settings in Jana2000 (Petricek et al., 2000) and in Superspace Group Finder (Orlov & Chapuis, 2005).
There are 5508 IRs associated with irrational k-vectors among the 230 crystallographic space groups. We give a list of these IRs in Table 2. The symbols for the IRs are from Cracknell et al. (1979) which are an extension of the tables of Miller and Love (1967). Some of the symbols contain upper-case Greek letters: SM (sigma), DT (delta), and LD (lambda). GP denotes "general point". For complex IRs (type 2 and 3), we form physically irreducible representations from the direct sum of the IR and its complex conjugate. These are denoted by putting two IR symbols together. For example, LD3LD4 denotes the direct sum of LD3 and LD4 (when LD3 and LD4 are complex conjugates of each other), and LD1LD1 denotes the direct sum of LD1 and itself (when LD1 is equivalent to its own complex conjugate).
We extend the 230 crystallographic space groups to (3+1)-dimensional superspace by including translations along the t axis. For each IR of a three-dimensional crystallographic space group, we identify a corresponding IR of the extended space group. In this table, we give the matrices onto which the IRs map operators of the extended space group.
In column 1, we give the crystallographic space group which has been extended to (3+1)-dimensional superspace.
In column 2, we give the IR symbol.
In column 3, we give the coordinates of the k-vector. Irrational components are denoted by lower-case Greek letters alpha, beta, gamma (abbreviated in the table by a,b,g, respectively). As explained in Stokes, Campbell & Hatch (2007), the IR matrices have been truncated to include only the rows and columns which are relevant to a one-dimensional modulation in the direction of k.
In column 5, we give the IR matrix for the operator x1,x2,x3,x4+delta which is simply a translation delta along the a4 axis.
In column 4, we identify the image of the group H. See Stokes, Campbell & Hatch (2007) for a discussion of the group H. The image is simply the collection of distinct matrices onto which the operators in H are mapped by the IR. We have been able to bring every image to a form identical to images found in Stokes and Hatch (1988) for IRs associated with rational k-vectors. We use here the symbols for those images from Stokes and Hatch (1988). In many cases, the image of H is reducible and is a direct sum of two or more images. For example, A1aA2a denotes the direct sum of images A1a and A2a.
Beginning with column 6, we list the generating operators of H along with their IR matrices.
We find 7799 nonequivalent isotropy subgroups belonging to the 5508 IRs associated with irrational k-vectors among the 230 crystallographic space groups extended to (3+1)-dimensional superspace. We give a list of these isotropy subgroups in Table 3. For each isotropy subgroup, we give the IR and k-vector, as in the table of IRs.
In column 1, we give the crystallographic space group which has been extended to (3+1)-dimensional superspace as the parent group.
In column 2, we give the IR symbol.
In column 3, we give the coordinates of the k-vector.
In column 4, we give the direction of the order parameter. The symbols come from Stokes and Hatch (1988) and refer to the image of group H given in the IR table. For reducible images, the symbol contains a direction for each image. For example, the direction P1P1=(a,0,a,0) for image B8aB8a denotes a direction P1=(a,0) for the first image B8a and a direction P1=(a,0) for the second image B8a. The direction P1Z=(a,0) for image A1aA2a denotes a direction P1=(a) for A1a and a direction Z=(0) for A2a.
In column 5, we give the superspace-group symmetry of the isotropy subgroup.
In column 6, we give the basis vectors of the lattice in the setting of the superspace group in terms of the basis vectors of the lattice in the setting of the parent space group.
In column 7, we give the origin of the superspace group relative to the origin of the parent space group, in terms of the basis vectors of the lattice in the setting of the parent space group.
We use settings for the 230 crystallographic parent space groups from International Tables for Crystallography (1992). When more than one setting is given, we use (1) unique axis b, cell choice 1 for monoclinic space groups, (2) hexagonal axes for R-centered trigonal space groups, and (3) origin choice 2. Note that the settings for the superspace groups always use origin choice 1. This sometimes leads to isotropy subgroup origins that are entirely due to this difference in settings. For example, there are two origin choices for space group 126 P4/nnc. In our table of isotropy subgroups, we see that the isotropy subgroup for IR LD1 is 126.1 P4/nnc(00g) with origin at (3/4,3/4,3/4,0). The parent space group uses origin choice 2 and the superspace group uses origin choice 1. These two origins differ by (3/4,3/4,3/4,0).
International Tables for Crystallography (1992), Vol. A, edited by T. Hahn. Dordrecht: Kluwer Academic.
Janssen, T., Janner, A., Loouenga-Vos, A. & de Wolff, P. M. (2004). International Tables for Crystallography, Vol. C, edited by E. Prince, pp. 907-945. Dordrecht: Kluwer Academics.
Miller, S. C. & Love, W. F. (1967). Tables of Irreducible Representations of Space Group and Co-Representations of Magnetic Space Groups. Boulder: Pruett.
Orlov, I. P. & Chapuis, G. (2005). Superspace Group Finder. http://superspace.epfl.ch/groups/
Petricek, V., Dusek, M. & Palatinus, L. (2000). Jana2000. The Crystallographic Computing System. http://www-xray.fzu.cz/jana/Jana2000/jana.html.
Stokes, H. T. & Hatch, D. M. (1988) Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientific.
Stokes, H. T., Campbell, B. J. & Hatch, D. M. (2007). Acta Cryst. A, in press.