Chapter 4.1 Thermal diffuse scattering of X-rays and neutrons

Reciprocal space

Second Online Edition (2010)

Part 4. Diffuse scattering and related topics

  1. B. T. M. Willis

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000773

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Willis, B. T. M. 2010. Thermal diffuse scattering of X-rays and neutrons. International Tables for Crystallography. B:4:4.1:484–491.

Author Information

  1. Department of Chemistry, Chemistry Research Laboratory, University of Oxford, Mansfield Road, Oxford OX1 3TA, England

Publication History

  1. Published Online: 1 JUN 2010


Thermal motion of the atoms in a crystal leads to a reduction in the intensities of the Bragg reflections and to a diffuse distribution of non-Bragg scattering (known as thermal diffuse scattering) in the rest of reciprocal space. Thermal diffuse scattering gives information about the lattice dynamics of the crystal, i.e. about the amplitudes and frequencies of the small oscillatory displacements of the atoms from their equilibrium positions which arise from thermal motion. This chapter gives a brief account of the classical Born–von Kármán treatment of lattice dynamics underlying the interpretation of experimental data on thermal diffuse scattering. This is followed by the theory of the scattering of X-rays and neutrons by thermal vibrations in a single crystal. The thermal diffuse scattering of X-rays or neutrons is used principally for the measurement of frequency versus wavevector (phonon dispersion relations). Although the first measurements were obtained with X-rays, these experiments are carried out more readily with neutrons where the particle energies are comparable with phonon energies. A summary is given of the derivation of interatomic force constants and of elastic constants from the dispersion relations.


  • thermal diffuse scattering;
  • neutron scattering;
  • X-ray scattering;
  • Einstein model;
  • Debye model;
  • lattice-dynamical model;
  • elastic constants;
  • phonon dispersion relations