Discrete and continuum models for calculating the phonon spectra of carbon nanotubes

The vibration spectrum of perfect carbon nanotubes is studied using a two-parametric potential which includes pairwise and three-particle interatomic interactions. This potential was proposed by Keating and allows one to take into account the elasticity of pairwise interatomic bonds and the elasticity associated with a change in the angle between directional interatomic bonds in covalent crystals. Using the Keating potential, the vibration spectrum of a graphite monolayer is calculated and fitted to the vibration spectrum of crystalline graphite, thereby determining the parameters of the potential. With these parameters, the phonon spectra of perfect monolayer graphite nanotubes are calculated. A continuum model, in which a monolayer nanotube is represented as an elastic cylindrical shell of a finite thickness, is also discussed. Within this model, the vibration spectrum of a nanotube is calculated numerically in the long-wavelength limit as a function of the radius and thickness of the nanotube.