Phonons in multiwalled carbon nanotubes

The phonons in multi-walled carbon nanotubes (MWNTs) are studied within a force-constant model and a continuum model. Special attention is given the low-frequency modes that are most affected by nesting of the separate layers into a MWNT. First, a MWNT has four acoustic branches, which determine its elastic properties and low-temperature specific heat. Secondly, due to the interlayer interactions, the so-called breathing modes of the separate layers are modified into breathing-like modes of the MWNTs. The results of the calculation of the Raman intensity agree well with an available experimental Raman spectra of a double-walled nanotube sample.