Algebraic Description of Coordination Sequences and Exact TopologicalDensities for Zeolites

Abstract

Coordination sequences (CS) have been calculated for all approved zeolite topologies, all dense SiO 2 polymorphs and 16 selected non-tetrahedral structures and the algebraic structure of these CS's has been analyzed. Two algebraic descriptions of coordination sequences are presented. One description uses periodic sets of quadratic equations and is already established in the literature. The second description employs generating functions, which are well known in combinatorics but are used here for the first time in connection with coordination sequences. The algebraic analysis based on generating functions turns out to be more powerful than the other approach. Based on the algebraic analyses, exact topological densities are derived and tabulated for all the structures investigated. In addition, 'n-dimensional sodalite' is observed to have an especially simple n-dimensional graph.

From 2D hyperbolic forests to 3D Euclidean entangled thickets

“…This confirms that there is only one component in T 3 . The coordination sequence [31][32][33] for either of the two components is identical to that of the Y * graph, supporting the contention that each component is a deformation of that network. The full graph thus consists of a pair of entangled (and distorted) +Y * or a pair of −Y * nets.…”

Sphere Packings and Space Filling by Congruent Simple Polyhedra

“…A similar situation holds for the 12-coordinated closest packings for which the f.c.c, lattice packing has the lowest topological density (O'Keeffe, 1995;Conway & Sloane, 1997). Remarkably, the topological density of foams that are duals of the Frank-Kasper phases are significantly higher (Grosse-Kunstleve et al, 1996) even though they have lower coordination numbers. Thus, for the type I clathrate structure (dual of the A15 structure), with average coordination number 13.5, Poo = 187/36 = 5.19... and for the type II hydrate structure (dual of the C15 structure), with average coordination number 13.33, p~ = 2371/432 = 5.48 ....…”

Topological Signatures of Medium Range Order in Amorphous Semiconductor Models

“…Effective algorithms for determining the coordination sequences [11] and circuit symbols [4,12] have already been presented in the literature. Here we outline a simpler algorithm for determining the circuit symbols.…”