Compositions of the shells of nanoparticles with pentagonal symmetry

A method of calculation of the number of atoms in the structure of nanoforms with pentagonal symmetry (fullerenes, nanoparticles, clusters) depending on the arrangement of atoms on the symmetry elements of the I (h) group has been developed. The formulas for calculation of the number of particles in all possible shells, including multilayer ones, are reported. The numbers of atoms in the shells of pentagonal symmetry are determined by four structurally invariant numbers and the "quantum number" of the order n of the group. The classification of all possible atomic shells S theta + 60z (z = 0, 1, ...) is presented, and the constructions of the basic shells S-theta (theta = 12, 20, 30, 50, 60) are given. For each basic shell, the sum rule is met: the sum of the coordination numbers of the elements of subshells is equal to 60. In clusters with magic numbers, basic shells are periodically repeated. In addition to the known shells of nanostructures, the formulas of new structures that are expected to be stable-B20O30, B60O90 (B2O3), and B90O13010+ (borate)-are reported for the first time. The same is valid for similar compounds of Group III elements.