International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. E, ch. 1.2, pp. 15-16
Section 1.2.9. Symmetry operations^{a}Freelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and ^{b}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA |
The coordinate triplets of the General position of a subperiodic group may be interpreted as a shorthand description of the symmetry operations in matrix notation as in the case of space groups [see Sections 2.2.3 , 8.1.5 and 11.1.1 of IT A (2005)]. The geometric description of the symmetry operations is found in the subperiodic group tables under the heading Symmetry operations. These data form a link between the subperiodic group diagrams (Section 1.2.6) and the general position (Section 1.2.11). Below the geometric description we give the Seitz notation (Burns & Glazer, 1990) of each symmetry operation using the subindex notation of Zak et al. (1969).
The numbering of the entries in the blocks Symmetry operations and General position (first block below Positions) is the same. Each listed coordinate triplet of the general position is preceded by a number between parentheses (p). The same number (p) precedes the corresponding symmetry operation. For all subperiodic groups with primitive lattices, the two lists contain the same number of entries.
For the nine layer groups with centred lattices, to the one block of General positions correspond two blocks of Symmetry operations. The numbering scheme is applied to both blocks. The two blocks correspond to the two centring translations below the subheading Coordinates, i.e. . For the Positions, the reader is expected to add these two centring translations to each printed coordinate triplet in order to obtain the complete general position. For the Symmetry operations, the corresponding data are listed explicitly with the two blocks having the subheadings `For (0, 0, 0)+ set' and `For (1/2, 1/2, 0)+ set', respectively.
The designation of symmetry operations for the subperiodic groups is the same as for the space groups. An entry in the block Symmetry operations is characterized as follows:
Details of this symbolism are given in Section 11.1.2 of IT A (2005).
Examples
References
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]Burns, G. & Glazer, A. M. (1990). Space Groups for Solid State Scientists, 2nd ed. New York: Academic Press.
Zak, J., Casher, A., Glück, M. & Gur, Y. (1969). The Irreducible Representations of Space Groups. New York: W. A. Benjamin.