InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 1.4, p. 114
## Section 1.4.1. Introduction U. Shmueli
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Crystallographic symmetry, as reflected in functions on reciprocal space, can be considered from two complementary points of view.

We start the next section with a brief discussion of the point-group symmetries of associated direct and reciprocal lattices. The weighted reciprocal lattice is then briefly introduced and the relation between the values of the weight function at symmetry-related points of the weighted reciprocal lattice is discussed in terms of the Fourier expansion of a periodic function in crystal space. The remaining part of Section 1.4.2 is devoted to the formulation of the Fourier series and its coefficients (values of the weight function) in terms of space-group-specific symmetry factors. Section 1.4.3 then explains the basis for an automated generation of simplified geometrical structure-factor formulae, which are presented for all the two- and three-dimensional space groups in Appendix A1.4.3. This is a revised version of the structure-factor tables given in Sections 4.5–4.7 of Volume I (*IT* I, 1952). Appendix A1.4.4 contains a reciprocal-space representation of the 230 crystallographic space groups and some explanatory material related to these space-group tables is given in Section 1.4.4; the latter are interpreted in terms of the two viewpoints discussed above. The tabular material given in this chapter is compatible with the direct-space symmetry tables given in Volume A (*IT* A, 1983) with regard to the space-group settings and choices of the origin.

Most of the tabular material, the new symmetry-factor tables in Appendix A1.4.3 and the space-group tables in Appendix A1.4.4 have been generated by computer with the aid of a combination of numeric and symbolic programming techniques. The algorithm underlying this procedure is briefly summarized in Appendix A1.4.1. Appendix A1.4.2 deals with computer-adapted space-group symbols, including the set of symbols that were used in the preparation of the present tables.

Computer generation of symmetry information is not new. However, we can quote the Bilbao Crystallographic Server (*e.g.* Aroyo *et al.*, 2006) as a rich source of symmetry information which is readily accessible from the Internet.

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