Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes

TLDR

Almost all existing TCNT construction methods are special cases of the proposed generalized classification scheme. The study develops a generalized classification scheme for toroidal and helical carbon nanotubes that incorporates both pentagons and heptagons. The scheme assigns four indices to n‑fold symmetric TCNTs, linking each to pentagon–heptagon arrangements, and derives chiral isomers or HCNTs by applying a chiral vector or dissecting the torus along longitude. The scheme generates the entire genus‑one fullerene family, from giant TCNTs to small structures with only a few hundred atoms.

Abstract

In this study, we develop a generalized classification scheme for toroidal (TCNT) and helical carbon nanotubes (HCNT) containing both pentagons and heptagons simultaneously. We show that a particular class of TCNTs with n-fold rotational symmetry and well-defined latitude coordinates can be uniquely characterized by a set of four indices, and each of the indices can be linked to the relative arrangement of pentagons and heptagons in the corresponding torus. Chiral isomers or the corresponding helical derivatives, HCNTs, can also be readily derived either by introducing a chiral vector or dissecting a distorted TCNT through certain longitude. This generalized scheme can generate the whole family of fullerenes with genus one, ranging from giant TCNTs down to small ones containing only a few hundred atoms. To the best of our knowledge, almost all of the construction methods for TCNTs in the literature belong to special cases of our generalized classification scheme.