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

TL;DR

This paper calculates the number of Raman-active vibrational modes in achiral carbon nanotubes, providing specific counts based on their chiral indices and analyzing the selection rules for Raman activity.

Contribution

It offers a detailed theoretical analysis of Raman-active modes in achiral CNTs, including explicit counts and vibrational mode characteristics, based on eigenvalue calculations at the Brillouin zone center.

Findings

Number of Raman-active modes in achiral CNTs: 5, 6, or 8 depending on chirality and parity.

Identification of specific vibrational modes that may be non-Raman-active.

Determination of selection rules for Raman activity in achiral CNTs.

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 $\Gamma$-point of Brillouin zone. Diagonal components and $\alpha_{xy} (=\alpha_{yx})$ 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).