Chapter 2.1 Statistical properties of the weighted reciprocal lattice

Reciprocal space

Second Online Edition (2010)

Part 2. Reciprocal space in crystal-structure determination

  1. U. Shmueli1,
  2. A. J. C. Wilson2

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000763

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Shmueli, U. and Wilson, A. J. C. 2010. Statistical properties of the weighted reciprocal lattice. International Tables for Crystallography. B:2:2.1:195–214.

Author Information

  1. 1

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

  2. 2

    St John’s College, Cambridge, England

Publication History

  1. Published Online: 1 JUN 2010


This chapter has two purposes: (i) it gives an introduction to the principles of probability, which plays an important role in most methods of structure determination, and (ii) it describes the application of such techniques to the first phase of structure determination – the resolution of space-group ambiguities, where present. The introduction in Section 2.1.1 justifies briefly the application of statistical methods to the structure-factor function. In Sections 2.1.2 and 2.1.3, a discussion is presented of the average intensity of general reflections and of zones and rows, with particular attention to the relation of these averages to symmetry elements present in the crystal. This is followed in Section 2.1.4 by mathematical preliminaries to the basics of the calculus of probabilities and the central-limit theorem is introduced. In Section 2.1.5, probability density functions (p.d.f.’s) are derived which allow one to determine whether or not the crystal is centrosymmetric. These p.d.f.’s are widely applicable to structures with a large number of not too dissimilar atoms in the asymmetric unit. Such functions for structures containing dispersive scatterers and noncrystallographic centres of symmetry are also presented, and distributions of sums, averages and ratios of intensities are discussed in Section 2.1.6. All these p.d.f.’s are based on the central-limit theorem and are termed ideal p.d.f.’s. However, these ideal p.d.f.’s are no longer applicable when an outstandingly heavy atom is present in the asymmetric unit. Two approaches are discussed in this chapter which may resolve this difficulty: (i) the correction-factor approach (Section 2.1.7), well known from classical probability as Gram–Charlier or Edgeworth p.d.f.’s, and (ii) the Fourier method (Section 2.1.8), only recently introduced to crystallography. The Gram–Charlier p.d.f.’s depend on even moments of the magnitude of the structure factor. General expressions for such moments of all orders are given for space groups of low symmetry, and the first four even moments are tabulated for all 230 space groups. The Fourier p.d.f.’s depend on their characteristic functions, or their Fourier transforms. Expressions for the atomic contributions to the characteristic function are tabulated for space groups up to and including the cubic space group equation image. Higher cubic space groups can be satisfactorily treated with the correction-factor p.d.f.’s. A comparison of these two non-ideal methods indicates that in the instances in which departures from the central-limit theorem predictions are large, the Fourier method is definitely superior to the correction-factor method.


  • weighted reciprocal lattice;
  • crystallographic statistics;
  • probability density functions;
  • systematic absences;
  • average intensities;
  • characteristic functions;
  • cumulant-generating functions;
  • central-limit theorem;
  • centring;
  • probability density distributions;
  • cumulative distribution functions;
  • structure factors;
  • correction-factor approach;
  • Fourier method;
  • distribution functions;
  • Fourier–Bessel series