General expressions for the dispersion relation of acoustic radial (breathing) modes in cylinders and cylindrical shells of general anisotropic crystals

We present here a study of the radial modes for infinitely long cylinders and cylindrical shells of general anisotropic crystals. The elastic coefficients matrix includes twenty-one independent constants. We obtain expressions in closed form for the dispersion relation, valid for any anisotropic material. In the case of the lowest breathing mode of a thin cylindrical shell we obtain a simple analytical formula. This can be used to obtain a first estimate of the breathing mode frequency in nanotubes for any material.