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

TL;DR

This paper derives general closed-form expressions for the dispersion relations of acoustic radial modes in anisotropic cylinders and shells, providing analytical formulas useful for estimating nanotube breathing mode frequencies.

Contribution

It introduces universal formulas for radial acoustic modes in anisotropic cylinders, applicable to any material with twenty-one elastic constants, including a simple estimate for nanotube frequencies.

Findings

Closed-form dispersion relations for anisotropic cylinders.

Analytical formula for the lowest breathing mode of thin shells.

Applicable to nanotubes of arbitrary anisotropic materials.

Abstract

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.