Cluster Self-Organization of Intermetallic Systems: K3, K4, and K6 Clusters-Precursors for the Self-Assembly of Y₈Rh₁₂Sn₂₀-oS40, Lu₁₆Zn₂₀Ge₂₄-oS60, and Ba₈Ir₁₆In₅₂-oS76 Crystal Structures

Abstract— Using computer methods (the ToposPro software package), combinatorial-topological analysis and modeling of the self-assembly of Y₈Rh₁₂Sn₂₀-oS40 (a = 4.387 Å, b = 26.212 Å, c = 7.155Å, V = 822.77 ų), Lu₁₆Zn₂₀Ge₂₄-oS60 (a = 4.179 Å, b = 18.368 Å, c = 15.050 Å, V = 1155.24 ų), and Ba₈Ir₁₆In₅₂-oS76 (a = 4.485 Å, b = 29.052 Å, c = 13.687 Å, V = 1783.63 ų) crystal structures with the Cmc2₁(36) space group are carried out. For Y₈Rh₁₂Sn₂₀-oS40, 18 variants for identifying cluster structures with the number of clusters N = 1 (variant 1), 2 (variant 11), and 3 (6 variants) are established. The variant of the self-assembly of a crystal structure with the participation of clusters-precursors forming the packing in the form of the K6 = 0@6(YSn₃Rh₂) and tetrahedra K4 = 0@4(YSn₂Rh) double tetrahedra is considered. For Lu₁₆Zn₂₀Ge₂₄-oS60, 66 variants are established for identifying cluster structures with the number of clusters N = 1 (1 variant), 2 (25 variants), 3 (20 variants), and 4 (20 variants). The variant of the self-assembly of a crystal structure with the participation of K3(D1) = 0@3(Lu Ge₂), K3(D2) = 0@3(Lu Zn Ge), K3(D3) = 0@3(Lu Zn Ge), K3(D4) = 0@3Lu Zn Ge), and K3(D5) = 0@3(GeZn₂) 3-atom clusters-precursors forming the packing is considered. For Ba₈Ir₁₆In₅₂-oS76, 129 variants are established of the allocation of cluster structures with the number of clusters N = 2 (36 variants) and N = 3 (103 variants). The variant of the self-assembly of a crystal structure with the participation of the clusters-precursors forming the packing is considered: K6 = 0@6(BaIn5) pentagonal pyramids, K4a = 0@4(BaRhIn2) tetrahedra and K4b = 0@4(Rh₂In2) tetrahedra, K3 = 0@4(RhIn₂) rings, and In spacer atoms. The symmetry and topological code of the self-assembly processes of the Y₈Rh₁₂Sn₂₀-oS40, Lu₁₆Zn₂₀Ge₂₄-oS60, and Ba₈Ir₁₆In₅₂-oS76 crystal structure from K3, K4, and K6 clusters-precursors in the following form is reconstructed: primary chain → layer → framework.