International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.4, p. 114   | 1 | 2 |

## Section 1.4.1. Introduction

U. Shmuelia

### 1.4.1. Introduction

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Crystallographic symmetry, as reflected in functions on reciprocal space, can be considered from two complementary points of view.

 (1) One can assume the existence of a certain permissible symmetry of the density function of crystalline (scattering) matter, a function which due to its three-dimensional periodicity can be expanded in a triple Fourier series (e.g. Bragg, 1966 ), and inquire about the effects of this symmetry on the Fourier coefficients – the structure factors. Since there exists a one-to-one correspondence between the triplets of summation indices in the Fourier expansion and vectors in the reciprocal lattice (Ewald, 1921 ), the above approach leads to consequences of the symmetry of the density function which are relevant to the representation of its Fourier image in reciprocal space. The symmetry properties of these Fourier coefficients, which are closely related to the crystallographic experiment, can then be readily established. This traditional approach, the essentials of which are the basis of Sections 4.5–4.7 of Volume I (IT I, 1952 ), and which was further developed in the works of Buerger (1949, 1960 ), Waser (1955), Bertaut (1964) and Wells (1965), is one of the cornerstones of crystallographic practice and will be followed in the present chapter, as far as the basic principles are concerned. (2) The alternative approach, proposed by Bienenstock & Ewald (1962), also presumes a periodic density function in crystal space and its Fourier expansion associated with the reciprocal. However, the argument starts from the Fourier coefficients, taken as a discrete set of complex functions, and linear transformations are sought which leave the magnitudes of these functions unchanged; the variables on which these transformations operate are h, k, l and ϕ – the Fourier summation indices (i.e., components of a reciprocal-lattice vector) and the phase of the Fourier coefficient, respectively. These transformations, or the groups they constitute, are then interpreted in terms of the symmetry of the density function in direct space. This direct analysis of symmetry in reciprocal space will also be discussed.

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.

### References

International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Dordrecht: Reidel.
International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
Aroyo, M. I., Kirov, A., Capillas, C., Perez-Mato, J. M. & Wondratschek, H. (2006). Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups. Acta Cryst. A62, 115–128.
Bertaut, E. F. (1964). On the symmetry of phases in the reciprocal lattice: a simple method. Acta Cryst. 17, 778–779.
Bienenstock, A. & Ewald, P. P. (1962). Symmetry of Fourier space. Acta Cryst. 15, 1253–1261.
Bragg, L. (1966). The Crystalline State, Vol. I, A General Survey. London: Bell.
Buerger, M. J. (1949). Crystallographic symmetry in reciprocal space. Proc. Natl Acad. Sci. USA, 35, 198–201.
Buerger, M. J. (1960). Crystal Structure Analysis. New York: John Wiley.
Ewald, P. P. (1921). Das `reziproke Gitter' in der Strukturtheorie. Z. Kristallogr. Teil A, 56, 129–156.
Waser, J. (1955). Symmetry relations between structure factors. Acta Cryst. 8, 595.
Wells, M. (1965). Computational aspects of space-group symmetry. Acta Cryst. 19, 173–179.