Flexure modes in carbon nanotubes

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Abstract
Phonons are calculated for single wall carbon nanotubes. Eigenvalues and eigenvectors are presented for armchair and zig-zag tubes. The model contains just three adjustable spring constants: two for first and second nearest neighbor directed bonds, and a third for radial bond-bending interactions. There are four low frequency modes at long wavelength: a longitudinal acoustical, a torsional mode, and two flexure modes.