International Tables for Crystallography

Access to online content requires a licence


Crystal lattices
H. Burzlaff, H. Grimmer, B. Gruber, P. M. de Wolff and H. Zimmermann. International Tables for Crystallography (2016). Vol. A, ch. 3.1, pp. 698-718  [ doi:10.1107/97809553602060000929 ]

Abstract

The chapter starts with a brief revision of the basic definitions of lattices and their bases followed by a discussion of the symmetry rules for the selection of the conventional crystallographic bases and unit cells. In Section 3.1.2 crystal lattices are described and classified algebraically by their standard bases, topologically by their Voronoi domains and in terms of symmetry by Delaunay types and their Bravais types of lattices. The Delaunay reduction procedure is described and illustrated, and the resulting classification of lattices into the 24 Delaunay sorts (`Symmetrische Sorte') is presented. Reduced bases (reduced cells) are described in Section 3.1.3. The treatment starts with the definition of the reduced basis and the reduced form in terms of the metric tensor and lists the conditions (both main and special) for reduced bases of type-I cells and type-II cells. This is followed by a detailed and systematic geometric explanation of these conditions. The resulting 44 lattice characters are defined and tabulated, and the relations between the lattice characters and the conventional cell parameters of the 14 Bravais lattices are listed and discussed. In Section 3.1.4, several less known aspects of lattices, concerning mainly their classification, are treated. It is shown that the 44 lattice characters can be rigorously introduced using the topological concepts of connectedness and convexity. A very detailed division of lattices into 127 genera appears as a result of the search for a common subdivision of the classifications in Delaunay sorts and lattice characters. In Section 3.1.4.4, it is found that the conditions characterizing the conventional cells of the 14 Bravais types of lattices are only necessary and to make them sufficient they have to be extended to a more comprehensive system. It is shown which Bravais types appear as limiting cases of other, more general types. The subdivision of the Bravais types of lattices into 22 conventional characters and the problem of sublattices of arbitrary index of an n-dimensional (n ≥ 1) lattice are discussed in the last two sections of the chapter.


Access, prices and ordering

International Tables for Crystallography is available online as a full set of volumes through Wiley.

set

If you have already registered and are using a computer listed in your registration details, please email support@iucr.org for assistance.

About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The multi-volume series comprises articles and tables of data relevant to crystallographic research and to applications of crystallographic methods in all sciences concerned with the structure and properties of materials.