Specific features of low-frequency vibrational dynamics and low-temperature heat capacity of double-walled carbon nanotubes

A continuous model has been constructed for low-frequency dynamics of a double-walled carbon nanotube. The formation of the low-frequency part of the phonon spectrum of a double-walled nanotube from phonon spectra of its constituent single-walled nanotubes has been considered in the framework of the proposed approach. The influence of the environment on the phonon spectrum of a single double-walled carbon nanotube has been analyzed. A combined method has been proposed for estimating the coefficients of the van der Waals interaction between the walls of the nanotube from the spectroscopic data and the known values of the elastic moduli of graphite. The low-temperature specific heat has been calculated for doublewalled carbon nanotubes, which in the field of applicability of the model ($T<$ 35 K) is substantially less than the sum of specific heats of two individual single-walled nanotubes forming it.