Compositions and Structures of Nanoparticles
of Trigonal Symmetry D_3d: Nanotubes

E. F. Kustov^a, V. I. Nefedov^b, A. V. Kalinin^a, and M. S. Karelina^b

^a Moscow Power Engineering Institute (Technical University), Krasnokazarmennaya ul. 14, Moscow, 111250 Russia

^b Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences,
Leninskii pr. 31, Moscow, 119991 Russia

Received December 6, 2007

Abstract—A method has been developed for the determination of the structure and number of atoms in the
shells of nanoparticles as a function of the arrangement of atoms at the symmetry elements of a symmetry
group. The formulas for calculation of the number of particles of symmetry D_3d have been reported. It has been
shown that the number of atoms in trigonal shells is determined by three structurally invariant numbers and the
quantum number of the group order n. All possible nanostructures of symmetry D_3d have been classified: C_+6z,
z = 0, 1, 2, , where the basic shells are C= C₆, C₈, and C₁₄. A sum rule has been obtained for the coordination
numbers of the shell sites located on symmetry axes. Trigonal nanoparticles are parent ones for obtaining (3,0),
(6,0), and (9,0) nanotubes of trigonal type. The general formulas of these nanotubes with icosahedral, dodeca-
hedral, and cubic caps are N_{8 + 12p}, N_{20 + 24p}, and N_{60 + 36p} (p = 1, 2, ), respectively. The graphical constructions
of all classes of trigonal nanoparticles and nanotubes are reported.

DOI: 10.1134/S0036023608070188

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