Symmetry and selection rules in the Raman spectra of carbon nanotubes

The symmetry of achiral single-walled (n,0) and (n,n) carbon nanotubes (CNTs) was examined and the frequencies and types of vibrations allowed in the Raman spectra were calculated. The vibrational spectrum was evaluated as the eigenvalues of the dynamical matrix at the Γ-point of the Brillouin zone. The selection rules for the Raman active vibrations were estimated by the values of the matrix elements responsible for the intensities of corresponding vibrational transitions. The (n,n)-CNTs with even and odd n values are characterized by five and six allowed Raman active vibrations, respectively. The number of Raman active vibrations for (n,0)-CNTs is five if n is even and eight if n is odd. Detailed analysis of the results obtained is presented for the (10,10)-CNT as an example.