Compositions and structures of nanoparticles of pentagonal axial symmetry D 5d : Nanotubes

The method of determination of the structure and the number of atoms in the shells of nanoparticles as a function of the arrangement of atoms at the symmetry elements of a symmetry group has been developed. The formulas for the calculation of the number of particles with symmetry group D 5d are reported. The number of particles in these shells is determined by three structurally invariant numbers and the 'quantum number' of the group order n. The classification of all possible nanostructures with symmetry group D 5d is given: C c+10z , z = 0, 1, 2, ..., where the basic shells are C c = C 2, C 10, C 12. The sum rule has been obtained for the coordination numbers of shell sites located at symmetry axes. Pentagonal axial nanoparticles are shown to be the initial shells for obtaining (5,5) and (10,10) armchair nanotubes or (5,0) and (10,0) zigzag nanotubes. The general formula of these nanotubes closed with icosahedral and dodecahedral caps is N 20+10p , N 60+10p (p = 1, 2, ...). The graphical constructions of all classes of nanoparticles and nanotubes of the pentagonal axial type are reported.