Phonons in narrow carbon nanotubes

Accurate calculations of the phonon dispersion relations, phonon density of states, and phonon eigenvectors of the narrow single-wall carbon nanotubes in optimized geometry are carried out. The method applied is based on the force constants for graphene which reflect the long-range character of the dynamical matrix. Further, the relaxation and symmetry imposed modifications of the force constants are performed and the calculations are carried out by means of the fully symmetry implemented POLSym code. Shortcomings of the widely used frozen phonon model are overcome. The results obtained are compared to the Raman scattering measurements on the zeolite-grown nanotubes.