Full symmetry, optical activity, and potentials of single-wall and multiwall nanotubes

The full symmetry groups for all single-wall and multiwall carbon nanotubes are found. As for the single-wall tubes, the symmetries form non-Abelian nonsymorphic line groups, enlarging the groups reported in the literature. In the multiwall case, any type of line and axial point groups can be obtained, depending on single-wall constituents and their relative position. The isogonal symmetry is related to the macroscopic tensor properties of the nanotubes. The optical activity is studied within this framework, and a detailed classification of the possible nanotube-based optical devices is given. At the microscopic level, full symmetry is used to find general forms of the characteristic functions of the nanotubes. In particular, the stability of the double-wall configurations and the possibility of chiral currents are discussed. Several other consequences are discussed: quantum numbers and related selection rules, phonon spectra, etc. In particular, the bands of the zigzag and the armchair tubes are generally expected to be fourfold degenerate.