Continuum model for long-wavelength optical phonons in cylinders: Application to carbon nanotubes

A continuum model for long-wavelength nonpolar optical phonons in systems with cylindrical symmetry is presented. The eigenfrequencies and modes for hollow cylinders with free surfaces are obtained, choosing parameters that model single-wall carbon nanotubes. A strong mixing of longitudinal and transverse modes is found for nonzero wave vectors; in fact, the mixing is present for certain modes even at the center of the Brillouin zone. The results show the need to treat on an equal footing all optical modes in these systems of reduced dimensionality.