Symmetry and topology code of cluster self-assembly of icosahedral framework structures of boron and borides

The topological types of suprapolyhedral clusters composed of i-B₁₂ icosahedra have been modeled. The models of icosahedral supraclusters have been used in analysis of the crystal structures of boron and borides (the TOPOS program package). To identify nanocluster precursors in crystal structures, there have been used special algorithms for partitioning structural graphs into disjoint substructures and constructing a basis 3D network of the structure as a graph with the nodes corresponding to the positions of the centroids of the cluster precursors. The cluster self-assembly have been modeled for 25 types of icosahedral framework structures of boron—B₁₂-hR ₁₂, (B₁₂)2(B₂)₂-oP ₂₈, (B₁₂)₄B₂-P50, B₁₉₆-tP ₁₉₆, and B₃₃₃-hR ₃₃₃; binary borides—(B₁₂)O₂-hR ₁₄, (B₁₂)P₂-hR ₁₄, and (B₁₂)(CBC)-hR ₁₅; templated metal borides—Na₂(B₁₂)₂B₆-oI ₆₄, Mg₂(B₁₂)B₂-oI ₆₈, Tb(B₁₂)(B₄)-mI ₆₀, Al₄(B₁₂)₄B₈-oC ₈₈, (B₁₂)₄(Si4)₄-oI ₆₄, (B₁₂)₄B₂Be₄-tP ₅₈, Ti₂(B₁₂)₄B₂-tP ₅₂, Sc₁₂B₁₈₀-tP ₁₉₂, Cu₄Sc₁₂B₁₈₀-tP ₁₉₂, Si_1.5Sc₉B₁₇₈-tP ₂₁₆, Mg₂₈B₃₆₀-oP ₃₈₈, Al₂₈B₃₅₂-oP ₃₈₄, Si₂₈B₃₅₂-oP ₃₀₆, Y₂₄(B₁₅₆)₈(B₃₉)₈-cF1944, Sc₁₀B₃₁₅-hR339, and Li₂₄B₃₁₅-hR336. The symmetry and topology code of the crystal structure self-assembly from nanocluster precursors in the form of primary chain → microlayer → microframework has been completely restored. Frequency analysis of various topological and symmetry pathways for the formation and evolution of cluster precursors makes it possible to elucidate crystal-formation trends in inorganic systems at the microscopic level.