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.

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 Wykcoff letter of one of the Wyckoff positions with that site symmetry is given. This Wyckoff position is called the characteristic Wykcoff position of the lattice complex.

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.

Symbols of invariant lattice complexes

An invariant lattice complex in its characteristic Wyckoff position is designated by a capitol letter, with a some superscript when necessary. Lattice complexes from different crystal familiies but iwht 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[math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]	2
B	000, [math]\frac{1}{2}[/math]0[math]\frac{1}{2}[/math]	2
C	000, [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]0	2
D (Diamond structure)	000. [math]\frac{1}{4}[/math][math]\frac{1}{4}[/math][math]\frac{1}{4}[/math], 0[math]\frac{1}{2}[/math][math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math]0,[math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]0, [math]\frac{3}{4}[/math][math]\frac{3}{4}[/math][math]\frac{1}{4}[/math], [math]\frac{3}{4}[/math][math]\frac{1}{4}[/math][math]\frac{3}{4}[/math], [math]\frac{1}{4}[/math][math]\frac{3}{4}[/math][math]\frac{3}{4}[/math]	8
E (Hexagonal close packing; Italian: Esagonale)	[math]\frac{1}{3}[/math][math]\frac{2}{3}[/math][math]\frac{1}{4}[/math], [math]\frac{2}{3}[/math][math]\frac{1}{3}[/math][math]\frac{3}{4}[/math]	2
F	000, 0[math]\frac{1}{2}[/math][math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math]0[math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]0	4
G (Graphene)	[math]\frac{1}{3}[/math][math]\frac{2}{3}[/math]0, [math]\frac{2}{3}[/math][math]\frac{1}{3}[/math]0	2
I	000, [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]	2
J ("Jack")	0[math]\frac{1}{2}[/math][math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math]0[math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]0	3
J* = J+[math]\frac{1}{2}[/math][math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]J	0[math]\frac{1}{2}[/math][math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math]0[math]\frac{1}{2}[/math], [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]0 [math]\frac{1}{2}[/math]00, 0[math]\frac{1}{2}[/math]0, 00[math]\frac{1}{2}[/math]	6
M	J in rhombohedral cell	3
N (Kagome Net)	[math]\frac{1}{2}[/math]00, 0[math]\frac{1}{2}[/math]0, [math]\frac{1}{2}[/math][math]\frac{1}{2}[/math]0	2
P	000	1

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