From second-order Raman scattering we find that the maximum in the phonon density of states of single-walled nanotubes softens by 30 ${\mathrm{cm}}^{-1}$ over the value of graphite. A comparison of the phonon-dispersion curves of nanotubes in the literature shows that the ab initio approach best describes this softening. The excitation-energy dependence of the D and $D^{\star }$ modes in single-walled nanotubes is 38 and 90 ${\mathrm{cm}}^{-1}/\mathrm{eV},$ respectively, similar to the multiwalled species.

• Received 18 May 1999

DOI:https://doi.org/10.1103/PhysRevB.61.4542