Symmetry of nanotubes rolled up from arbitrary two-dimensional lattices along an arbitrary chiral vector

Abstract

The line group describing full symmetry of a nanotube rolled up from an arbitrary layer along an arbitrary chiral vector is found. A helical axis and pure rotations are always present, while mirror and glide planes appear only for special chiral vectors in rhombic or rectangular lattices. Nanotubes are not translationally periodical unless the unit cell of the layer satisfies very specific conditions. Physical consequences, including the incommensurability of carbon nanotubes, are discussed.