Symmetry of arbitrary layer rolled-up nanotubes

TL;DR

This paper analyzes the symmetry properties of nanotubes formed by rolling up layers with arbitrary lattice structures, revealing how their symmetry groups depend on lattice type and chiral vector, with implications for physical properties.

Contribution

It provides a general method to determine the symmetry line groups of nanotubes from arbitrary layered lattices and chiral vectors, extending previous specific cases.

Findings

Helical axes and pure rotations are always present in nanotube symmetry.

Mirror and glide planes depend on specific chiral vectors and lattice types.

Nanotubes are generally not translationally periodic unless specific conditions are met.

Abstract

Rolling up of a layer with arbitrary lattice gives nanotube with very reach symmetry, described by a line group which is found for arbitrary diperiodic group of the layer and chiral vector. Helical axis and pure rotations are always present, while the mirror and glide planes appear only for specific chiral vectors in rhombic and rectangular lattices. Nanotubes are not translationally periodical unless layer cell satisfies very specific conditions. Physical consequences, including incommensurability of carbon nanotubes, are discussed.