Chirality Distribution and Transition Energies of Carbon Nanotubes

From resonant Raman scattering on isolated nanotubes we obtained the optical transition energies, the radial breathing mode frequency, and the Raman intensity of both metallic and semiconducting tubes. We unambiguously assigned the chiral index $(n_1,n_2)$ of $\approx 50\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{u}\mathrm{b}\mathrm{e}\mathrm{s}$ based solely on a third-neighbor tight-binding Kataura plot and find ${\omega }_{\mathrm{R}\mathrm{B}\mathrm{M}}=(214.4\pm 2)\mathrm{c}{\mathrm{m}}^{-1}\mathrm{n}\mathrm{m}/d+(18.7\pm 2){\mathrm{c}\mathrm{m}}^{-1}$. In contrast to luminescence experiments we observe all chiralities including zigzag tubes. The Raman intensities have a systematic chiral-angle dependence confirming recent ab initio calculations.