Commensurate double-walled carbon nanotubes: Symmetry and phonons

For periodic double-walled carbon nanotubes stable configurations and full symmetry groups are found. Using this, the phonon dispersions and displacement eigenvectors are calculated and assigned by the complete set of conserved quantum numbers. The calculated infrared and Raman active modes match the measured values nicely. The approximate analytical frequency-diameter dependence for the breathing like, high energy, and rigid layer modes is derived within the perturbative model (justified by the weak interlayer interaction). The high energy modes are scarcely influenced by the interlayer interaction. The low temperature specific heat is below that of the ingredient (single-walled) layers, while the sound velocities fall close.