Cluster Self-Organization of Crystal-Forming Systems: New Three-Layer (K155 = Al@Al6Pd8@Pd12Al30@Pd8Сo18Al72) and Double-Layer (K55 = Co@Al12@Co12Al30) Precursor Clusters for the Self-Assembly of the Pd112Co204Al684-cP1000 Crystal Structure

Geometric and topological analysis of the Pd112Co204Al684-cP1000 crystal structure with the sp. gr. Pa-3, a = 24.433 Å, and V = 14587.24 Å3 is performed using the ToposPro software package. Metal precursor clusters of crystalline structures are determined using an algorithm for decomposing structural graphs into cluster structures and by constructing a basic grid of the structure in the form of a graph whose nodes correspond to the position of the centers of precursor clusters $$S_{3}^{0}.$$ A total of 26 906 variants of the cluster representation of a 3D atomic mesh with the number of structural units ranging from 3 to 12 are established. The self-assembly of the crystal structure from new three-layer K155(4a) = Al@Al6Pd8)@Pd12Al30@Pd8Co18Al72 and bilayer precursor clusters K55(4b) = Co@Al12@Co12Al30 with symmetry g = –3 is considered. In the unit cell, positions 4a are occupied by Al atoms, which are the central atoms of the 15-atom polyhedron K15(4a) = Al@Al8Pd6, and positions 4b are occupied by Co atoms, which are the central atoms of the 13-atom icosahedron K13(4b) = Co@Al12. The symmetric and topological code of the processes of self-assembly of 3D structures from precursor clusters K155 and K55 is reconstructed as follows: primary chain → microlayer → microframework. Al atoms are established as spacers occupying voids in the 3D framework of the K155 and K55 nanoclusters.