An extension of the Voronoi analysis of crystal structures

The Voronoi method of constructing polyhedra about all atoms in a given crystal structure is extended to provide a general, quantitative and concise means of discriminating crystal structures from one another. Although applicable to all crystal structural types, the methodology is developed here by reference to binary non-molecular inorganic systems, A_mB_n, corresponding mainly to ionic crystals and intermetallic compounds. Consideration is given to the geometrical forms of the Voronoi polyhedra, with AB₂ compounds singled out for illustrative purposes. Two methods are proposed for summarizing structural geometry: (i) the use of representative points for elements of Voronoi polyhedra (i.e. corners, edges and faces) and relating these to space-group symmetry; (ii) calculation of the ratio of the volumes of Voronoi polyhedra, V^V_A/V^V_B. Topological structural properties (i.e. numbers of A...A, A...B and B...B pairwise interactions), by comparison, may be quantified by the indices I_AA, I_AB, I_BA and I_BB, whose meanings are defined and elucidated. By adopting statistically disordered structures as reference points, indices I′_AA, I′_AB,I′_BA and I′_BB are defined, which lead to the generation of two-dimensional topological structure diagrams with I′_AA and I′_BB as axes. Alternatively, geometrical-topological diagrams may be generated, with V^V_A/V^V_B and I′_AA — I′_BB as axes. The straightforward modifications required for a chemical–geometrical rather than a topological–geometrical analysis are also described. It is anticipated that the technique will find widespread application in solid-state chemistry and materials science.