International Tables for Crystallography

Lattice complexes
W. Fischer and E. Koch. International Tables for Crystallography (2016). Vol. A, ch. 3.4, pp. 792-825  [ doi:10.1107/97809553602060000932 ]


In Section 3.4.1, the concept of lattice complexes and limiting complexes is introduced and compared with the concept of orbit types and non-characteristic orbits. To this end it is necessary to differentiate strictly between the two terms `point configuration' and `crystallographic orbit', both of which have often been used with two slightly different meanings: (1) for sets of points that are equivalent with respect to a given space group, i.e. in the mathematical sense of `orbit'; (2) for such sets of points, but detached from their generating space groups. A `lattice complex' is defined as a set of point configurations that may be generated within one type of Wyckoff set. Furthermore, the following items are introduced and illustrated by examples: Wyckoff position, Wyckoff set, type of Wyckoff set, limiting complex, comprehensive complex, Weissenberg complex, degrees of freedom of a lattice complex, and reference symbols of the lattice complexes. In Section 3.4.2, the concept of characteristic and non-characteristic orbits is introduced and compared with the concept of lattice complexes and limiting complexes. In Section 3.4.3, the descriptive symbols of lattice complexes are introduced, their properties are described and their interpretation is demonstrated by numerous examples. Tables and give the explicit assignment of the Wyckoff positions of all plane groups and space groups, respectively, to Wyckoff sets and to lattice complexes. For each Wyckoff position, the reference symbol of the corresponding lattice complex is tabulated. In addition, a descriptive symbol is given that describes the arrangement of points in the corresponding point configurations. It refers directly to the coordinate description of the Wyckoff position. Section 3.4.4 gives a short introduction to some applications of lattice complexes: (i) The knowledge of the assignment of the Wyckoff positions to lattice complexes considerably facilitates the study of geometrical properties of point configurations. (ii) Relations between crystal structures with different symmetries are often discernible because the corresponding Wyckoff positions either belong to the same lattice complex or because a limiting-complex relationship exists. (iii) Wyckoff positions belonging to the same lattice complex show analogous reflection conditions. (iv) If a phase transition of a crystal is connected with a group–subgroup transition, comparison of the lattice complexes corresponding to the Wyckoff positions of the original space group on the one hand and of its various subgroups on the other hand very often shows which of these subgroups are suitable for the low-symmetry modification. (v) Many incorrect space-group assignments to crystal structures could be avoided by simply looking at the lattice complexes (and their descriptive symbols) that correspond to the Wyckoff positions occupied by the different kinds of atoms.

Access, prices and ordering

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


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

About International Tables for Crystallography

International Tables for Crystallography is the definitive resource and reference work for crystallography. The series consists of nine volumes and 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.