Compositions of nanoparticles of cubic symmetry

The formulas for calculation of the number of particles in structures of cubic symmetry O (h) are reported. The numbers of atoms in the shells of cubic symmetry are determined by four structurally invariant numbers and the "quantum number" of the order n of the group. The classification of the shells of cubic symmetry is presented, and eight classes of shells are revealed. A key role in this classification is played by basic shells; in the case of close-packed spheres, these basic shells are repeated every six layers. The sum of all coordination numbers of all atoms of subshells is 24. The possibility of the existence of new fullerenes and nanoparticles of elemental boron and its oxides is considered.