Analysis of voids in crystal structures: the methods of `dual' crystal chemistry

The theoretical basics of the analysis of voids in crystal structures by means of Voronoi-Dirichlet polyhedra (VDP) and of the graph theory are stated. Topological relations are considered between VDPs and atomic domains in a crystal field. These relations allow the separation of two non-intersecting topological subspaces in a crystal structure, whose connectednesses are defined by two finite `reduced' graphs. The first, `direct', subspace includes the atoms (VDP centres) and the network of interatomic bonds (VDP faces), the second, `dual', one comprises the void centres (VDP vertices) and the system of channels (VDP edges) between them. Computer methods of geometrical-topological analysis of the `dual' subspace are developed and implemented within the program package TOPOS. They are designed for automatically restoring the system of channels, visualizing and sizing voids and void conglomerates, dimensional analysis of continuous void systems, and comparative topological analysis of `dual' subspaces for various substances. The methods of analysis of `dual' and `direct' subspaces are noted to differ from each other only in some details that allows the term `dual' crystal chemistry to be introduced. The efficiency of the methods is shown with the analysis of compounds of different chemical nature: simple substances, ionic structures, superionic conductors, zeolites, clathrates, organic supramolecular complexes.