Cluster Self-Organization of Intermetallic Systems: Clusters-Precursors K4, K6, and K7 for the Self-Assembly of Crystal Structures Y₂₀Cu₂₀Mg₆₄-oC104, Y₂₀Cu₂₀Mg₅₂-oC92, and Y₃(NiAl₃)Ge₂-hP9

Abstract—Using computer methods (ToposPro software package), a combinatorial topological analysis and modeling of the self-assembly of the following crystal structures are carried out: Y₂₀Cu₂₀Mg₆₄-oC104 (a = 4.136 Å, b = 19.239 Å, c = 29.086 Å, V = 2314.45 ų, Cmcm), Y₂₀Cu₂₀Mg₅₂-oC92 (a = 4.097 Å, b = 19.279 Å, c = 25.790 Å, V = 2037.30 ų, Cmcm), and Y₃(NiAl₃)Ge₂-hP9 (a = b = 6.948 Å, c = 4.156 Å, V = 173.78 5 ų, P-62m). For the Y₂₀Cu₂₀Mg₆₄-oC104 crystal structure, 52 variants of the cluster representation of the 3D atomic network with the 3, 4, and 5 structural units are established. Four crystallographically independent structural units in the form of a tetrahedron are determined: tetrahedron K4 = 0@CuMg3, tetrahedron K4 = 0@YMg3, tetrahedron K4 = 0@YCuMg₂, and a supratetrahedron K6 = 0@YCu2Mg3. A variant of self-assembly with the participation of hexamers from six linked structural units is considered (K4B+ K4C)(K4A+ K6)(K4B+ K4C). For the Y₂₀Cu₂₀Mg₆₄-oC92 crystal structure, 27 variants of cluster representation of the 3D atomic mesh with 3, 4, and 5 structural units are established. Three crystallographically independent structural units in the form of a tetrahedron are determined: tetrahedron K4 = 0@YCuMg₂, cluster K6 = 0@6(Y₂Mg₄) in the form of double tetrahedrons YMg₃, and a nine-atom supratetrahedron K9 = Mg@Y₂Cu₂Mg₄ consisting of two YMg₂Cu and two YMg₃ tetrahedrons. A variant of the self-assembly involving trimers of three structural units K4+ K6+ K9 is considered. For the Y₃(NiAl₃)Ge₂-hP9 crystal structure, eight variants of decomposition of the 3D atomic mesh into cluster structures with the participation of two structural units are established. A variant of the self-assembly with the participation of packing generatrices of seven-atom clusters-precursors K7 = 0@Y₃(NiAl₃) with the participation of Ge atoms-spacers is considered. The symmetry and topological code of the self-assembly processes of 3D-structures is reconstructed from clusters-precursor in the following form: primary chain → layer → framework.