International Tables for Crystallography

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Crystallographic viewpoints in the classification of space-group representations
M. I. Aroyo and H. Wondratschek. International Tables for Crystallography (2006). Vol. B, ch. 1.5, pp. 162-188  [ doi:10.1107/97809553602060000553 ]

Abstract

The k vectors are vectors in reciprocal space and play an important role in the description of space-group representations. Chapter 1.5 deals with the classification of these k vectors with special regard to crystallographic points of view. In 1941, Wintgen found that the k vectors of any space group can be classified in a natural way analogous to the classification of the Wyckoff positions of the symmorphic space groups. This is possible by introducing the so-called reciprocal-space group, which is isomorphic to a symmorphic space group. The symmetry types of k vectors correspond to the Wyckoff positions of this symmorphic space group and the tables of the Wyckoff positions in of International Tables for Crystallography present the classification directly. In this chapter, the basic concepts of representations of space groups are defined and the reciprocal-space group is introduced. The sometimes complicated Brillouin zone and its representation domain may be replaced by the often much simpler conventional unit cell of the reciprocal lattice and its asymmetric unit. The different k vectors of the same symmetry type are characterized by parameters which correspond to the coordinates of the representative points of the Wyckoff positions. The ranges of these parameters are chosen in such a way that each k vector is listed exactly once in the k-vector table of the space group. The Wintgen classification is applied in several examples (space groups [Im\overline{3}m], [Ia\overline{3}d], [Im\overline{3}], [Ia\overline{3}], [I4/mmm], [I4/mcm], [I4/amd], [I4/acd], [Fmm2] and [Fdd2]) and compared with the usual classification.


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About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The multi-volume series comprises articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials.