Review on the symmetry-related properties of carbon nanotubes

In this work we review the basic properties of carbon nanotubes from the standpoint of group theory. The zone folding scheme is reviewed in the light of the helical symmetry of the nanotube. The group theory for chiral and achiral nanotubes is reviewed, and the representations of the factor group of the wavevector k are obtained. The similarities and differences between the formalism of the group of the wavevector and that of line groups are addressed with respect to the irreducible representations and quantum numbers associated with linear and angular momenta. Finally, we extend the results of group theory to illuminate the electronic and vibrational properties of carbon nanotubes. Selection rules for the optical absorption and double resonance Raman scattering are discussed for the case where the electron-electron interaction is negligible (metallic nanotubes) and for the case where exciton binding energies are strong and cannot be neglected