Chapter 9.3 Further properties of lattices

Space-group symmetry

First Online Edition (2006)

Part 9. Crystal lattices

  1. B. Gruber

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000519

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Gruber, B. 2006. Further properties of lattices. International Tables for Crystallography. A:9:9.3:756–760.

Author Information

  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, CZ-11800 Prague 1, Czech Republic

Publication History

  1. Published Online: 1 JAN 2006


In this chapter, several less well known aspects of lattices, concerning mainly their classification, are treated. In Section 9.3.1, the fact that the cell fulfilling a + b + c = min (Buerger cell) is generally not unique in the lattice is discussed. To achieve uniqueness, various additional conditions must be added. Four of them are shown. They minimize or maximize either the surface or the ‘deviation’ |90° − α| + |90° − β| + |90° − γ| of the cell given above. One of these unique cells coincides with the generally used Niggli cell, which thus has a significant geometrical property. In Section 9.3.2, the 44 lattice characters (see Section 9.2.5) are discussed. These lattice characters represent a finer partition of lattices that the 14 Bravais types. Although the original definition of lattice characters is perhaps rather vague, they can be rigorously introduced using the topological concepts of connectedness and convexity. In Section 9.3.3, another important division of lattices, into the 24 Delaunay sorts (symmetrische Sorten) is discussed. However, the Delaunay sorts are not compatible with the lattice characters. The search for a common subdivision of both leads finally to a very detailed division of lattices into 127 genera. Their definition is based on the decomposition of five-dimensional polyhedra into their interior, hyperfaces, edges and vertices. Genera form a remarkably strong bond between lattices and can be considered as building blocks for various classifications of lattices. In Section 9.3.4, it is found that the conditions characterizing the conventional cells of the 14 Bravais types (see Section 9.1.7) are only necessary. To make them sufficient as well, they have to be extended to a more comprehensive system. Particularly interesting relations appear between Bravais types mI and hR. In Section 9.3.5, topological methods (connectedness) are used for the set of points whose coordinates are parameters of the conventional cell to obtain a subdivision of the Bravais types into 22 conventional characters. They form a superdivision of the lattice characters. The concept of convexity, however, is not applicable in this case. In Section 9.3.6, a simple formula for the number of sublattices of arbitrary index of an n-dimensional (n ≥ 1) lattice is given.


  • reduced bases;
  • lattices;
  • Niggli cell;
  • Buerger cell;
  • reduced cells;
  • Niggli images;
  • lattice characters;
  • conventional characters;
  • symmetrische Sorten;
  • conventional cells;
  • sublattices