Raman characterization of 0.4 nm single-wall carbon nanotubes using the full-symmetry line group

Raman spectra of single-wall carbon nanotubes produced in the channels of zeolite AFI single crystals have been analyzed in the light of the full symmetry group, the line group. The phonon dispersion curves of the tubes (5,0), (3,3), and (4,2) are calculated based on the lattice dynamical model and the phonon branches are assigned to their quantum numbers (irreducible representations). The structures of Raman spectra of different samples are reproduced well by the density of states of relevant Raman-active phonons. The result is useful to evaluate the contents of these tubes in real crystals where the contents are not well defined before.