Chapter 9.8 Incommensurate and commensurate modulated structures

Mathematical, physical and chemical tables

First Online Edition (2006)

Part 9. Basic structural features

  1. T. Janssen1,
  2. A. Janner1,
  3. A. Looijenga-Vos2,
  4. P. M. de Wolff3

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000624

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Janssen, T., Janner, A., Looijenga-Vos, A. and de Wolff, P. M. 2006. Incommensurate and commensurate modulated structures. International Tables for Crystallography. C:9:9.8:907–955.

Author Information

  1. 1

    Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands

  2. 2

    Roland Holstlaan 908, NL-2624 JK Delft, The Netherlands

  3. 3

    Meermanstraat 126, 2614 AM, Delft, The Netherlands

Publication History

  1. Published Online: 1 JAN 2006


This chapter discusses incommensurate and commensurate modulated structures. After an introduction to the concepts used to describe modulated structures, including the basic ideas of higher-dimensional crystallography, four-dimensional space groups and displacive and occupational modulation, superspace-group determination is outlined. Tables of Bravais lattices, geometric and arithmetic crystal classes, and superspace groups (with reflection conditions) are presented and their use is illustrated with different examples.


  • arithmetic crystal classes;
  • basic structural features;
  • Bravais classes;
  • centring reflection conditions;
  • composite crystal structures;
  • crystallographic systems;
  • higher-dimensional crystallography;
  • holohedry;
  • incommensurate modulated structures;
  • lattices;
  • Laue classes;
  • modulated crystal structures;
  • modulations;
  • occupation modulations;
  • point groups;
  • structure factors;
  • superspace groups