Phonon vibrational frequencies of all single-wall carbon nanotubes at the Lambda point: Reduced matrix calculations

With a simple method-the reduced matrix method, we simplified the calculation of the phonon vibrational frequencies according to SWNTs structure and their phonon symmetric property and got the dispersion properties of all SWNTs at Γ point in Brillouin zone, whose diameters lie between 0.6 and 2.5 nm. The calculating time is shrunk about 2-4 orders. A series of the dependent relationships between the diameters of SWNTs and the frequencies of Raman and IR active modes are given. Several fine structures including “glazed tile” structures in ω ∼ d figures are found, which might predict a certain macro-quantum phenomenon of the phonons in SWNTs.