An alternate mathematical model for single-wall carbon nanotubes

TL;DR

This paper introduces a new mathematical model for single-wall carbon nanotubes based on Miller indices, simplifying the description of atomic positions and their neighbors, and providing a more symmetric and elegant framework.

Contribution

It presents an alternative to the traditional symmetry group approach by using a factor space based on Miller indices for modeling nanotube atomic arrangements.

Findings

Simplifies the description of atomic neighbors

Provides a more symmetric mathematical framework

Enhances understanding of nanotube structure

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 is a natural number depending on the considered nanotube. We obtain an alternate rather different description by starting from a description of the honeycomb lattice in terms of Miller indices. 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.