1.9 Atomic displacement parameters

Physical properties of crystals

Second Online Edition (2013)

Part 1. Tensorial aspects of physical properties

  1. W. F. Kuhs

Published Online: 19 DEC 2013

DOI: 10.1107/97809553602060000908

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Kuhs, W. F. 2013. Atomic displacement parameters. International Tables for Crystallography. D:1:1.9:231–245.

Author Information

  1. GZG Abt. Kristallographie, Goldschmidtstrasse 1, 37077 Göttingen, Germany

Publication History

  1. Published Online: 19 DEC 2013


The theory of lattice dynamics shows that the atomic thermal Debye–Waller factor is related to the atomic displacements. In the harmonic approximation, these are fully described by a fully symmetric second-order tensor. Anharmonicity and disorder, however, cause deviations from a Gaussian distribution of the atomic displacements around the atomic position. A generalized description of atomic displacements therefore also involves first-, third-, fourth- and even higher-order displacement terms.The description of the properties of these tensors is the purpose of this chapter. The number of independent tensor coefficients depends on the site symmetry of the atom and are given in tables. The symmetry restrictions according to the site symmetry are tabulated for second- to sixth-rank thermal motion tensors. A selection of representation surfaces of higher-rank tensors showing the distribution of anharmonic deformation densities is given at the end of the chapter.


  • Gram–Charlier series;
  • atomic displacement;
  • cumulants;
  • invariants;
  • quasimoments;
  • representation surface;
  • site symmetry;
  • site-symmetry restrictions;
  • tensor contraction;
  • tensor expansion