Chapter 1.4 Symmetry in reciprocal space

Reciprocal space

Second Online Edition (2010)

Part 1. General relationships and techniques

  1. U. Shmueli1,
  2. S. R. Hall2,
  3. R. W. Grosse-Kunstleve3

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000761

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Shmueli, U., Hall, S. R. and Grosse-Kunstleve, R. W. 2010. Symmetry in reciprocal space. International Tables for Crystallography. B:1:1.4:114–174.

Author Information

  1. 1

    School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel

  2. 2

    Crystallography Centre, University of Western Australia, Nedlands 6907, WA, Australia

  3. 3

    Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Mailstop 4-230, Berkeley, CA 94720, USA

Publication History

  1. Published Online: 1 JUN 2010


The original practical purpose of this chapter was the automated derivation of simplified geometrical structure-factor formulae for all the 17 plane groups and 230 space groups. These expressions were derived by hand in older editions of International Tables. The present chapter also contains an introduction to symmetry in reciprocal space and a summary of computer-readable space-group symbols. Section 1.4.1 describes two viewpoints of crystallographic symmetry as reflected in functions on reciprocal space, elaborated on further in Section 1.4.4. In Section 1.4.2, a brief discussion of the point-group symmetry of the reciprocal lattice is presented. This is followed by a derivation of the effect of the space-group symmetry of the crystal on the magnitude and phase of the structure factor and, subsequently, by a derivation of a space-group-specific expression of electron density. Section 1.4.3 then explains the basis for the 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 followed by a section on space-group symmetry in reciprocal space (Section 1.4.4), which outlines the viewpoints on symmetry in reciprocal space that were mentioned briefly in Section 1.4.1. The outcome of Section 1.4.4 is Appendix A1.4.4, which lists for each space group the indices of symmetry-related reflections accompanied by the corresponding space-group-dependent phase shifts. The computer generation of these results is done by a series of programs, the input to the first being a computer-readable space-group symbol and the output of the last being, for most space groups, an expression appearing in Appendix A1.4.3 or in one of the tables of reciprocal-space ‘equivalent positions’ appearing in Appendix A1.4.4. This process is outlined in Appendix A1.4.1. Appendix A1.4.2 presents the computer-readable space-group symbols which were actually employed in the generation of the tables in Appendix A1.4.3 and Appendix A1.4.4, and another set of space-group symbols applicable to several modern crystallographic software packages.


  • symmetry;
  • reciprocal space;
  • Fourier summations;
  • structure factors;
  • structure-factor formulae;
  • Bravais lattices;
  • Hall symbols;
  • Fourier space;
  • symmetry factors;
  • affine transformations;
  • direct-space transformations