Using continuous models in the analysis of vibrational spectra for carbon nanotubes

Rod models of longitudinal, torsional, and bending vibrations are used to find the natural vibration spectrum of a carbon nanotube. The spectrum of natural radial vibrations is found using the membrane theory of cylindrical shells. The coefficients of these models are chosen by comparing the results obtained on the basis of the micromodel with the Keating interaction potential in the framework of the long-wave approximation and the continuous model. It is shown that the spectra of longitudinal, radial, and torsional vibrations of the carbon nanotube are of the same order of magnitude (the minimum frequency is about $10^{11}$ Hz), whereas the natural frequency spectrum for the bending vibrations is of two orders of magnitude less (the minimum frequency is about $10^{9}$ Hz). These spectra belong to the super-high frequency range.