A lattice dynamical treatment for the total potential energy of single-walled carbon nanotubes and its applications: relaxed equilibrium structure, elastic properties, and vibrational modes of ultra-narrow tubes

In this paper, we proposed a lattice dynamic treatment for the total potential energy for single-walled carbon nanotubes (SWCNT's) which is, apart from a parameter for the non-linear effects, extracted from the vibrational energy of the planar graphene sheet. Based upon the proposal, we investigated systematically the relaxed lattice configuration for narrow SWCNT's, the strain energy, the Young's modulus and Poisson ratio, and the lattice vibrational properties respected to the relaxed equilibrium tubule structure. Our calculated results for various physical quantities are nicely in consistency with existing experimental measurements. Particularly, we verified that the relaxation effect brings the bond length longer and the frequencies of various optical vibrational modes softer; Our calculation provides the evidence that the Young's modulus of armchair tube exceeds that of the planar graphene sheet, and the large diameter limits of the Young's modulus and Poisson ratio are in agreement with the experimental values of the graphite; The calculated radial breathing modes for the ultra narrow tubes with diameter range between 0.2 - 0.5 nm coincide the experimental results and the existing {\it ab initio} calculations with satisfaction; For narrow tubes of diameter 2 nm, the calculated frequencies of optical modes in tubule tangential plane as well as those of radial breathing modes are also in good agreement with the experimental measurement. In addition, our calculation shows that various physical quantities of relaxed SWCNT's can actually be expanded in terms of the chiral angle defined for the correspondent ideal SWCNT's.