On the linear representations of the symmetry groups of single-wall carbon nanotubes

TL;DR

This paper introduces a new mathematical framework for describing the symmetry groups of single-wall carbon nanotubes, simplifying their geometric and physical representations using a three-axes lattice model.

Contribution

It proposes an alternative description of nanotube symmetry groups based on a three-axes honeycomb lattice, leading to simpler and more symmetric mathematical objects.

Findings

Simplified description of atomic neighbors in nanotubes.

More symmetric and manageable mathematical representations.

Illustrative results on linear representations of nanotube symmetry groups.

Abstract

The positions of atoms forming a carbon nanotube are usually described by using a system of generators of the symmetry group. Each atomic position corresponds to an element of the set Z x {0,1,...,n} x {0,1}, where n depends on the considered nanotube. We obtain an alternate rather different description by starting from a three-axes description of the honeycomb lattice. In our mathematical model, which is a factor space defined by an equivalence relation in the set {(v_0,v_1,v_2)\in Z^3 | v_0+v_1+v_2\in {0,1}}, the neighbours of an atomic position can be described in a simpler way, and the mathematical objects with geometric or physical significance have a simpler and more symmetric form. We present some results concerning the linear representations of single-wall carbon nanotubes in order to illustrate the proposed approach.