Chirality dependence of the radial breathing phonon mode density in   single wall carbon nanotubes
  A mass and spring model is used to calculate the phonon mode dispersion for single wall carbon nanotubes (SWNTs) of arbitrary chirality. The calculated dispersions are used to determine the chirality dependence of the radial breathing phonon mode (RBM) density. Van Hove singularities, usually discussed in the context of the single particle electronic excitation spectrum, are found in the RBM density of states with distinct qualitative differences for zig zag, armchair and chiral SWNTs. The influence the phonon mode density has on the two phonon resonant Raman scattering cross-section is discussed. 
A. N. Vamivakas, Y. Yin, A. G. Walsh, M. S. Unlu, B. B. Goldberg, and   A. K. Swan