Quantized vibrational modes of nanospheres and nanotubes in the elastic continuum model

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Abstract
The properties of nanoscale spheres and tubes are of recent interest due to the discovery of the fullerene molecule and the carbon nanotube. These carbon structures can be modeled as nanoscale spherical or cylindrical shells. In this article, these nanostructures are treated in the thin shell approximation with the elastic properties taken to be those of the graphene sheet. A quantization prescription is applied to the classical elastic modes to facilitate the first calculations of the quantum-mechanical normalizations of selected modes. These modes are shown to be amenable to the study of electron-phonon interactions. Indeed, electron-phonon interaction Hamiltonians are derived. Moreover, it is shown for such a tube of finite length that the electron-phonon interaction strength depends on the axial position. As a special case it is shown that the dispersion relation for the clamped tube depends on the length of the tube. In this article we consider both the vibrational frequencies and the mode quantization for both spherical shell and the nanotube using realistic material parameters.

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