Classification of two-shell nanotubes with commensurate structures of shells

A classification of two-shell carbon nanotubes with commensurate structures of shells is proposed. The classification is based on the concept of equivalence classes as a set of shells with chiral indices of the (kf, kg) type, where f and g are the chiral indices of the equivalence class and k is the index of the shell diameter. All two-shell nanotubes with commensurate shells that are characterized by the chiral indices (k₁f₁, k₁g₁) and (k₂f₂, k₂g₂), where k₁ and k₂ are integral numbers, make up a family of nanotubes of different radii but with equal geometric parameters (such as the intershell distance, the unit cell length of the nanotube, and the difference between the chiral angles of the shells). The geometric parameters of nanotubes are calculated for a number of families, the distribution of different types of two-shell nanotubes with commensurate shells over the outer-shell radii is determined, and the threshold forces required to induce relative motion of nanotube shells are evaluated. The possible use of two-shell nanotubes with commensurate shells in nanostructures is discussed.