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A graph theoretical approach to the analysis, comparison, and enumeration of crystal structures
(2008)
As an alternative approach to lattices and space groups, this work explores graph theory as a means to model crystal structures. The approach uses quotient graphs and nets - the graph theoretical equivalent of cells and lattices - to represent crystal structures. After a short review of related work, new classes of cycles in nets are introduced and their ability to distinguish between non-isomorphic nets and their computational complexity are evaluated. Then, two methods to estimate a structure’s density from the corresponding net are proposed. The first uses coordination sequences to estimate the number of nodes in a sphere, whereas the second method determines the maximal volume of a unit cell. Based on the quotient graph only, methods are proposed to determine whether nets consist of islands, chains, planes, or penetrating, disconnected sub-nets. An algorithm for the enumeration of crystal structures is revised and extended to a search for structures possessing certain properties. Particular attention is given to the exclusion of redundant nets and those, which, by the nature of their connectivity, cannot correspond to a crystal structure. Nets with four four-coordinated nodes, corresponding to sp3 hybridised carbon polymorphs with four atoms per unit cell, are completely enumerated in order to demonstrate the approach. In order to render quotient graphs and nets independent from crystal structures, they are reintroduced in a purely graph-theoretical way. Based on this, the issue of iso- and automorphism of nets is reexamined. It is shown that the topology of a net (that is the bonds in a crystal) constrains severely the symmetry of the embedding (that is the crystal), and in the case of connected nets the space group except for the setting. Several examples are studied and conclusions on phases are drawn (pseudo-cubic FeS2 versus pyrite; α- versus β- quartz; marcasite- versus rutile-like phases). As the automorphisms of certain quotient graphs stipulate a translational symmetry higher than an arbitrary embedding of the corresponding net would show, they are examined in more detail and a method to reduce the size of such quotient graphs is proposed. Besides two instructional examples with 2-dimensional graphs, the halite, calcite, magnesite, barytocalcite, and a strontium feldspar structures are discussed. For some of the structures it is shown that the quotient graph which is equivalent to a centred cell is reduced to a quotient graph equivalent to the primitive cell. For the partially disordered strontium feldspar, it is shown that even if it could be annealed to an ordered structure, the unit cell would likely remain unchanged. For the calcite and barytocalcite structures it is shown that the equivalent nets are not isomorphic.