= Solution
Rolling <graphene> into a <single-walled carbon nanotube> quantizes the circumferential <wave vector>, leaving a set of one-dimensional longitudinal subbands. At each subband edge the <group velocity> tends to zero, producing a <Van Hove singularity> in the <density of states>. For a simple gapped Dirac subband, the contribution scales as $|E|/\sqrt{E^2-\Delta_j^2}$ outside its edge. At low <temperature> the valence states below the <Fermi level> are occupied and the conduction states above it are empty. A semiconducting tube has a gap; an ideal metallic tube has gapless channels as well as higher-subband singularities.
For light polarized along the tube, the strongest allowed interband transitions connect corresponding subbands while approximately conserving axial <wave vector> and circumferential <angular momentum>. The large <joint density of states> makes distinct absorption resonances, conventionally labelled $E_{11},E_{22},\ldots$. Strong <electron>–hole interactions produce bound <excitons>, so the observed optical resonances differ from the noninteracting band separations. Defects, doping, bundling and <dielectric screening> change peak energies and <oscillator strengths>. The original schematic below depicts the subband-edge structure, not measured spectra.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2021/paper-2-nanotube-dos.png]
{title=Original schematic occupied and unoccupied nanotube densities of states and allowed optical transitions}
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