Cluster Self-Organization of Intermetallic Systems: Cluster Precursors K4, K5, and K9 for the Self-Assembly of Zr₇₂P₃₆-oS108, Zr₁₈Ni₂₂-tI40, and Zr₄Ni₄-oS8 Crystal Structures

Abstract—Using computer methods (ToposPro software package), a combinatorial topological analysis and modeling of the self-assembly of Zr₇₂P₃₆-oS108 (a = 29.509 Å, b = 19.063 Å, c = 3.607 Å, V = 2029.49 ų, Cmmm), Zr₁₈Ni₂₂-tI40 (a = b = 9.880 Å, c = 6.610 Å, V = 645.23 ų, I4/m, and Zr₄Ni₄-oS8 (a = 3.271 Å, b = 9.931 Å, c = 4.107 Å, V = 133.43 ų, Cmcm) crystal structures are carried out. For the crystal structure of Zr₇₂P₃₆-oS108, 40 variants of the cluster representation of the 3D atomic net with the number of structural units 5, 6, and 7 are established. Structural units in the form of a pyramid K5 = 0@PZr₄, tetrahedron K4 = 0@Zr₄, and supratetrahedron K9 = Zr(Zr₄P₄) of four connected tetrahedra. For the crystal structure of Zr₁₈Ni₂₂-tI40 also defined supratetrahedra K9 = Ni(Zr₄Ni₄) are defined. For the crystal structure of Zr₄Ni₄-oS8, the tetrahedral cluster precursor K4 = Zr₂Ni₂ is defined. The symmetry and topological code of the processes of self-assembly of 3D structures from cluster precursors is reconstructed in the following form: primary chain → layer → framework.