# 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