Modeling Self-Organization Processes in Crystal Forming Systems. Symmetry and Topology Codes of Cluster Self-Assembly of Crystal Structure of Na₄₄Tl₇ (Na₆Tl)

Abstract—The geometrical and topological analysis of the crystal structure of intermetallide Na₄₄Tl₇ (Na₆Tl, a = 24.154 Å, V = 14091.8 ų, space group F-43m) is carried out. The algorithms of the combinatorial-topological analysis, which ensure the recovery of the symmetrical and topological code (program) of the cluster self-assembly of the crystal structure of an intermetallide, are developed. The topological type of the basic 3D network for two types of cluster precursors corresponds to a simple cubic 3D network P_c with CN = 6 and basic 2D network of type 4⁴. There are eight cluster precursors in the unit cell: four K86 and four K50. The cluster precursor K86 made from 86 atoms is formed from eight icosahedra i-TlNa₁₂ linked by the apices. The center of the cluster precursor K86 is at the position 4a (0, 0, 0) with the point symmetry g = –43m. The 50-atomic cluster precursor K50 consists of six i-TlNa₁₂ icosahedra. The center of the cluster precursor K50 is in the partial position 4b (0, 0, 0) with the point symmetry g = –43m. The symmetrical and topological codes of the self-assembly of 3D structures from nanocluster precursors K86 and K50 are reconstructed in the following form: primary chain → microlayer → microframework.