Vibrational spectra of double-wall carbon nanotubes

The vibrational spectra of double-wall carbon nanotubes (DWNTs) are studied theoretically within a continuum model in this paper. The phonon dispersion relations are derived numerically and analytically, and two radial breathing modes (RBMS) are calculated analytically, which agree with experimental data [Phys. Rev. B 66, 075416 (2002)]. By comparison of the RBM frequencies with the ones of isolated single-wall carbon nanotubes, we find that there is a systematic upward shift for DWNTs RBM frequencies due to the interlayer van der Waals interactions. In case of counterphase modes, the upshift magnitude of RBM frequencies increases with an increase of the outer-layer radius R. However, for inphase RBMs, the upshift magnitude of RBM frequencies may increase or decrease with an increase of R. We then discuss four acoustic phonons and find that there is no transverse acoustic phonon in DWNTs. Finally, the general phonon dispersion relations are calculated.