How many Raman-active vibrations do exist in carbon nanotubes?

Abstract

In this paper we present the results on Raman-active modes calculation in achiral $(n,n)$ and $(n,0)$ carbon nanotubes (CNTs). vibrational spectra are derived as the eigenvalues of corresponding dynamical matrix at the $Γ$-point of Brillouin zone. Diagonal components and $α_{xy} (=α_{yx})$ are the only non-zero components of polarization tensor. Selection rules for Raman-active modes are determined by direct estimations of matrix elements responsible for the intensities of corresponding vibrational transitions. Few $E_{2g}$ modes can be non-Raman-active because of their ``fine'' vibrational structure. We claim that numbers of Raman-active modes for achiral CNTs are: 5 ($(n,n)$ and $(n,0)$, $n$-even); 6 ($(n,n)$, $n$-odd); 8 ($(n,0)$, $n$-odd).