Symmetry of commensurate double-wall carbon nanotubes

The symmetry groups of double-wall carbon nanotubes (DWCNs) are the line and point groups for the commensurate and incommensurate walls, respectively. For the tubes with diameters between 2.8 Angstrom and 50 Angstrom all possible DWCNs are found. Among them all the 318 commensurate DWCNs are singled out and their symmetry groups are calculated. DWCNs are low symmetry objects with respect to the constituent single-wall tubes, and this symmetry reduction is described by the symmetry breaking groups. Both symmetry and symmetry breaking groups affect the physical properties of the DWCNs. While, e.g., quantum numbers and selection rules are related to the symmetry group, the low interaction between the walls is determined by the breaking group.