# 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. All Wyckoff positions, Wyckoff sets and types of Wyckoff sets that generate the same set of point configurations are assigned to the same lattice complex.

Concretely, two Wyckoff positions are assigned to the same lattice complex if there is a suitable transformation that maps the point configurations of the two Wyckoff positions onto each other and if their space groups belong to the same crystal family. The 72 (in E2) or 1731 (in E3) Wyckoff positions are classified in 51 (E2) or 1128 (E3) types of Wyckoff sets. They are assigned to 30 (E2) or 402 (E3) lattice complexes.

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 translations can be visualized as a set of equivalent lattices.

## Classification

If a lattice complex can be generated in different space-group types, one of them stands out because its corresponding Wyckoff positions show the highest site symmetry. This is called the characteristic space-group type of the lattice complex. The characteristic space-group type and the corresponding oriented site symmetry express the common symmetry properties of all point configurations of a lattice complex. In the symbol of a lattice complex, however, instead of the site symmetry, the Wyckoff letter of one of the Wyckoff positions with that site symmetry is given. This Wyckoff position is called the characteristic Wyckoff position of the lattice complex.

If a first lattice complex forms 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 into 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 does 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.

## Symbols of invariant lattice complexes

An invariant lattice complex in its characteristic Wyckoff position is designated by a capital letter, with an appropriate superscript when necessary. Lattice complexes from different crystal families but with the same coordinate description for their characteristic Wyckoff positions receive the same descriptive symbol.

 Symbol Coordinates of equivalent points in the cell Multiplicity A 000, 0$\frac{1}{2}$$\frac{1}{2}$ 2 B 000, $\frac{1}{2}$0$\frac{1}{2}$ 2 C 000, $\frac{1}{2}$$\frac{1}{2}$0 2 D (Diamond structure) 000, $\frac{1}{4}$$\frac{1}{4}$$\frac{1}{4}$, 0$\frac{1}{2}$$\frac{1}{2}$, $\frac{1}{2}$0,$\frac{1}{2}$, $\frac{1}{2}$$\frac{1}{2}$0,$\frac{3}{4}$$\frac{3}{4}$$\frac{1}{4}$, $\frac{3}{4}$$\frac{1}{4}$$\frac{3}{4}$, $\frac{1}{4}$$\frac{3}{4}$$\frac{3}{4}$ 8 E (Hexagonal close packing; Italian: Esagonale) $\frac{1}{3}$$\frac{2}{3}$$\frac{1}{4}$, $\frac{2}{3}$$\frac{1}{3}$$\frac{3}{4}$ 2 F 000, 0$\frac{1}{2}$$\frac{1}{2}$, $\frac{1}{2}$0$\frac{1}{2}$, $\frac{1}{2}$$\frac{1}{2}$0 4 G (Graphene) $\frac{1}{3}$$\frac{2}{3}$0, $\frac{2}{3}$$\frac{1}{3}$0 (hexagonal cell) 2 I 000, $\frac{1}{2}$$\frac{1}{2}$$\frac{1}{2}$ 2 J ('Jack') 0$\frac{1}{2}$$\frac{1}{2}$, $\frac{1}{2}$0$\frac{1}{2}$, $\frac{1}{2}$$\frac{1}{2}$0 3 J* = J+$\frac{1}{2}$$\frac{1}{2}$$\frac{1}{2}$J 0$\frac{1}{2}$$\frac{1}{2}$, $\frac{1}{2}$0$\frac{1}{2}$, $\frac{1}{2}$$\frac{1}{2}$0,$\frac{1}{2}$00, 0$\frac{1}{2}$0, 00$\frac{1}{2}$ 6 M J in rhombohedral cell 3 N (Kagome net) $\frac{1}{2}$00, 0$\frac{1}{2}$0, $\frac{1}{2}$$\frac{1}{2}$0 (hexagonal cell) 2 P 000 1 +Q $\frac{1}{2}$00, 0$\frac{1}{2}$$\frac{2}{3}$, $\frac{1}{2}$$\frac{1}{2}$$\frac{1}{3}$ (hexagonal cell) 3 R $\frac{1}{3}$$\frac{2}{3}$$\frac{2}{3}$, $\frac{2}{3}$$\frac{1}{3}$$\frac{1}{3}$ (hexagonal cell) 3 'R $\frac{1}{3}$$\frac{2}{3}$$\frac{1}{3}$, $\frac{2}{3}$$\frac{1}{3}$$\frac{2}{3}$ (hexagonal cell) 3 S 0$\frac{1}{4}$$\frac{3}{8}$, $\frac{5}{8}$$\frac{1}{2}$$\frac{1}{4}$, $\frac{3}{4}$$\frac{1}{8}$0, $\frac{1}{2}$$\frac{3}{4}0$$\frac{7}{8}$ + permutations 12 'S S·-1 12 S* S + 'S 24 T $\frac{1}{8}$$\frac{1}{8}$$\frac{1}{8}$, $\frac{3}{8}$$\frac{3}{8}$$\frac{1}{8}$, $\frac{1}{8}$$\frac{5}{8}$$\frac{5}{8}$, $\frac{7}{8}$$\frac{7}{8}$$\frac{1}{8}$, $\frac{7}{8}$$\frac{5}{8}$$\frac{3}{8}$ + permutations 16 +V $\frac{1}{4}$$\frac{1}{8}$0, $\frac{3}{4}$$\frac{3}{8}$0, $\frac{3}{4}$$\frac{5}{8}$$\frac{1}{2}$, $\frac{7}{8}$$\frac{1}{2}$$\frac{1}{4}$ + permutations 12 -V +V·-1 12 V* +V + -V 24 W $\frac{1}{2}$$\frac{1}{4}$0, 0$\frac{1}{2}$$\frac{3}{4}$ + permutations 6 W' $\frac{1}{2}$$\frac{1}{2}$$\frac{1}{2}$W 6 W* W + W' 12 +Y $\frac{1}{8}$$\frac{1}{8}$$\frac{1}{8}$, $\frac{7}{8}$$\frac{5}{8}$$\frac{3}{8}$ + permutations 4 +Y' $\frac{1}{2}$$\frac{1}{2}$$\frac{1}{2}$+Y 4 -Y +Y·-1 4 -Y' $\frac{1}{2}$$\frac{1}{2}$$\frac{1}{2}$-Y 4 +Y* +Y++Y' 8 -Y* -Y+-Y' 8 Y** +Y*+-Y* 16

## Symbols of variant lattice complexes

The symbol of a lattice complex with degrees of freedom may contain up to four parts:

• a shift vector;
• the distribution symmetry;
• a central part, that normally is the symbol of an invariant lattice complex;
• the site-set symbol, which represents the set of points obtained by splitting a single point in an invariant lattice complex; these points are equivalent under the site-symmetry group of the point that they replace.

The distribution symmetry, which is not a group, is the set of symmetry operations sufficient to specify the orientations of all the subsets of points obtained from a single point when generating a variant lattice complex from an invariant or limiting lattice complex.

## 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.