Helical and rotational symmetries of nanoscale graphitic tubules

We show how all extended graphitic tubules constructed by rolling up a single graphite sheet can be defined in terms of their helical and rotational symmetries. Specification of these symmetries is practically mandatory in all but the simplest calculations of tubule properties as a function of radius and structure. We also report results of a tight-binding study implemented by using these symmetries. This study shows that independent of their helicity the larger-diameter, moderate-band-gap semiconducting tubules all have band gaps given approximately by ${\mathit{E}}_{\mathit{g}}$=\ensuremath{\Vert}${\mathit{V}}_{0}$\ensuremath{\Vert}(${\mathit{d}}_{0}$/${\mathit{R}}_{\mathit{T}}$), where ${\mathit{R}}_{\mathit{T}}$ is the tubule radius and ${\mathit{V}}_{0}$ is the hopping matrix element between nearest-neighboring 2p orbitals oriented normal to the tubule surface and centered on carbon atoms separated by a distance ${\mathit{d}}_{0}$ along this surface. In addition, we show that all tubules constructed by rolling up the graphite sheet can be labeled in a fashion familiar in the description of helical chain polymers with translational symmetry.