Lattice vibrations in single-wall carbon nanotubes

We present a study of lattice vibrations in single-wall carbon nanotubes. The study is based on a force-constant model, where the force constants are obtained from a general invariant potential that includes up to fourth-nearest-neighbor interactions. Applying the model, we obtain phonon dispersions and density of states for various types of nanotubes. General low-frequency modes such as ring modes and longitudinal modes are also identified and investigated. They are almost independent of the chiral angle but significantly dependent on the radius. The radius dependence is used for scaling, leading to a universal density of states at low frequencies.