Geometric and topological analysis of icosahedral structures of samson Mg2Zn11 (cP39) Phases, K6Na15Tl18H (cP40), and Tm3In7Co9 (cP46): Nanocluster precursors, self-assembly mechanism, and superstructure ordering

Geometric and topological analysis and 3D reconstruction of self-assembly of icosahedral structures of Samson Mg₂Zn₁₁ clusters (space group Pm`3]Pm\bar 3, cP39, 10 compounds) and the K<sub>6</sub>Na<sub>15</sub>Tl<sub>18</sub>H and Tm<sub>3</sub>In<sub>7</sub>Co<sub>9.29</sub> structures were performed by computer methods (the TOPOS program package). The complete decomposition of the 3D graph of the crystal structures into cluster substructures showed the existence of the crystal-forming nanocluster precursor A comprising 45 atoms (A-45). The S-6 cluster spacers were identified in Mg<sub>2</sub>Zn<sub>11</sub>, and the S-7 cluster spacers were found in K<sub>6</sub>Na<sub>15</sub>Tl<sub>18</sub>H. In Tm<sub>3</sub>In<sub>7</sub>Co<sub>9.29</sub>, the S-6 and S-7 cluster spacers with the centers statistically occupying the same position were determined. The A-45, S-6 (octahedron), and S-7 (centered octahedron) clusters have symmetry `3]m\bar 3m. The A-45 nanocluster contains an inner Zn(Zn)₁₂ template icosahedron and an external quasi-spherical shell composed of 32 atoms (deltahedron D32). A-45 is equivalent to the Bergman cluster used as the approximant of the local structure of quasicrystals. For deltahedron D32, the existence of a hierarchical structure was identified as a result of self-assembly involving two types of cyclic clusters: K-7 with an atom in the center of the sixth ring and three-atom cyclic clusters K-3. The atoms of the K-3 and K-7 clusters occupy all possible positions over the 12 vertices and 20 faces of an icosahedron and thereby form an edge net of bonds made of triangles. For the K₆(Na₁₄MTl₁₈) structures (M = Mg, An, Cd, Hg), the cluster nature of superstructure ordering of three chemically different atoms (14Na, M, and 18Tl) over 33 positions of the Zn atoms in the unit cell of the basis Mg₂Zn₁₁(Mg₆Zn₃₃) structure was considered.