Lattice complex

Gittercomplex (Ge).

Definition

A lattice complex is the set of all point configurations that may be generated within one type of Wyckoff set.

The name lattice complex comes from the fact that an assemblage of points that are equivalent with respect to a group of symmetry operations including lattice tranlations can be visualized as a set of equivalent lattices.

If a first lattice complex formes a true subset of a second one, i.e. if each point configuration of the first lattice complex also belongs to the second one, then the first one is called a limiting complex of the second one and the second complex is called a comprehensive complex of the first one.

Lattice complexes are called invariant if they can occupy a parameterless position in a space group. The points in an invariant lattice complex can be split ino different assemblages of equivalent points, the sum of which constitute a variant lattice complex (also termed a lattice complex with degrees of freedom). Variant lattice complexes are classified into univariant, bivariant and trivariant according to the number of parameters that can be varied independently.

There exist 67 lattice complexes for which the multiplicity foes not decrease for any special values of the coordinates. These lattice complexes can simulate invariant lattice complexes as limiting complexes. They were first recognized by Weissenberg, who called them Hauptgitter. Today they are known as Weissenberg complexes.

History

Paul Niggli introduced in 1919 the term lattice complex to indicate a set of crystallographically equivalent atoms in a crystal structure, like the Na atoms in NaCl or the C atoms in diamond.

See also

Chapter 14 of International Tables of Crystallography, Section A