A practical crystallographic algorithm for determining the distribution of spatial polyhedra among coordination spheres

Abstract

We propose a method for determining the population of atoms (sites) on the coordination spheres in fcc crystals. It is shown that only seven spatial polyhedra, containing 6, 8, 12, 24, 24, 24, and 48 vertices may be chosen as a basis for the characteristics of coordination sphere filling. A number of spheres may be represented as a superposition of spatial polyhedra and the coordination number is the sum of the vertices of all the basis polyhedra. The method may be implemented as an automatic process to search for neighbors in any coordination sphere. The basis polyhedra proposed — regular and semiregular Platonic and Archimedic figures — are duals of simple forms which correspond to the symmetry classes considered.