Chapter 1.2 The structure factor

Reciprocal space

Second Online Edition (2010)

Part 1. General relationships and techniques

  1. P. Coppens

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000759

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Coppens, P. 2010. The structure factor. International Tables for Crystallography. B:1:1.2:10–23.

Author Information

  1. Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260‐3000, USA

Publication History

  1. Published Online: 1 JUN 2010



This chapter summarizes the mathematical development of the structure‐factor formalism. It starts with the definition of the structure factor appropriate for X‐ray and neutron scattering, and includes the derivation of the appropriate expressions. Beyond the isolated‐atom case, the atom‐centred spherical harmonic (multipole) model is treated in detail. Expressions for thermal motion in the harmonic approximation and for treatments including anharmonicity are given and their relative merits are discussed.


  • structure factors;
  • X‐ray scattering;
  • magnetic scattering;
  • spherical harmonic expansion;
  • orbital products;
  • temperature factors;
  • rigid‐body motion;
  • cumulant expansion;
  • Gram–Charlier series;
  • one‐particle potential model