= Solution
A <single-walled carbon nanotube> confines the <electron> around its circumference while allowing motion along its axis. <Nanotube zone folding> therefore turns the <graphene> bands into one-dimensional subbands. A <semiconducting carbon nanotube> has no allowed circumferential wave vector through a <Dirac valley>; a metallic tube has a band crossing at the <Fermi energy> in the ideal model. Each massive subband edge gives a <one-dimensional van Hove singularity> in the <density of states>, and corresponding peaks appear in the <joint density of states>.
For <light> polarized along the tube, the main <electric-dipole selection rule> conserves circumferential angular momentum: transitions connect matching valence and conduction subbands. These finite-energy transitions are conventionally $S_{11},S_{22},\ldots$ in a <semiconducting carbon nanotube> and $M_{11},M_{22},\ldots$ in a metallic tube. A metallic band crossing does not make its first strong interband peak have zero energy. Perpendicular polarization admits different angular-momentum channels, often much weaker because of depolarization. <Excitons>, dielectric screening and trigonal warping modify the simple band-edge transition energies.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-2-nanotube-bands.png]
{title=Nanotube optical transitions and two example roll-up vectors}
{description=Original band and lattice schematics. Arrows indicate matching-subband transitions; the metallic crossing is separate from its finite-energy interband resonance. Lattice points represent graphene unit cells rather than individual atoms.}
{source=paper-2-nanotube-bands.py}
Experimentally, measure absorption versus photon energy; <Raman spectroscopy> becomes resonantly enhanced when the excitation matches an optical transition. A <photoluminescence excitation spectrum> of dispersed semiconducting tubes correlates excitation near $S_{22}$ with emission near $S_{11}$ after relaxation. Metallic tubes generally lack comparable bright emission because nonradiative relaxation is efficient. Combining optical energies with radial-breathing-mode <Raman spectroscopy> or <electron diffraction> assigns tube diameter and roll-up indices; a <Kataura plot> organizes those assignments.<Structure-assigned optical spectra of single-walled carbon nanotubes by Sergei M. Bachilo and others>[Excitation-emission spectroscopy and assignments to nanotube structures]{ref}
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