Compositions and structures of nanoparticles of tetrahedral symmetry

The formulas for calculation of the composition of nanoparticles with symmetry group Td are reported. The numbers of atoms in the shells of tetrahedral symmetry are determined by three structurally invariant numbers and the “quantum number” of the group order n. Four classes of all possible nanostructures with symmetry group Td are revealed: Cϑ + 12z, where z = 0, 1, 2, ..., and Cϑ is C4, C6, C8, C10, or C14. The sum rule for the coordination numbers of the shell sites located on the axes of symmetry is obtained. The six-shell periodicity of the number of occupied sites located on the symmetry axes and in symmetry planes is revealed.