Phonon Dispersion in Chiral Single Wall Carbon Nanotubes

The phonon dispersion of chiral single wall carbon nanotubes has been obtained from $6\times 6$ dyanimic matrix. The present manuscript is the extension of G. D. Mahan and Gun Sang Jeon's work on armchair and zigzag nanotubes. [see P. R. B., {\bf 70}, 075405 (2004)]. We use spring and mass model with the proper phonon potential suggested by \cite{M2}. We can calculate the dispersion of single wall carbon nanotubes near $\Gamma$ point with arbitrary chirality. The results are compatible with Mahan et. al's results for armchair and zigzag SWNTs.