# The geometric coordination concept

### Background

The geometric coordination concept is the simplest of the three models. The basic idea is, that each point in space belongs to the nearest atoms. The mathematic basics are already described by Voronoi [1908]. Niggli [1927] used the concept in crystallography by defining the 'Wirkungsbereich'. Frank and Kasper [1958] used the concept in intermetalic systems and Ruban [1985] defined an inverse construction of the same polyhedra.

### How to construct a Wirkungsbereich or Frank-Kasper-Polyhydra

To construct a Wirkungsbereich, it is necessary, to draw lines from a center atom to all atoms around. Then in the middle of each of this lines, we construct a perpendicular plane to the vector along the line.

The smallest possible convex polyhedra is the Wirkungsbereich or Frank-Kasper-Polyhedra.

A two-dimensional picture is shown below.

A two-dimensional picture of the Wirkungsbereich

All five atoms in the polyhedra, which contribute a face are coordinated and are included in the geometrical coordination. Atom A only shares a single point, but still is treated as lying in the coordination. Atom B is lying shortly outside the coordination.

For practical purposes, the geometrical coordination may be reduced to a chemical coordination by only counting electronegative and electropositive neighbours and ignoring faces between two electronegative or two electropositive neighbours.

Last Updated: 26 October 1996