Cutting lines near the Fermi energy of single-wall carbon nanotubes

Analytic expressions for the cutting line number μ of a single-wall carbon nanotube (SWNT) which gives van Hove singularities for optical transitions are found to exist around μ=N∕3 for most SWNTs, where N is the number of cutting lines of the SWNT. However, for metal-2 SWNTs, the cutting line number shows an irregular dependence on $(n,m)$. In this paper a classification is proposed for metal-2 nanotubes which are classified as either metal-$2p$ or metal-$2m$ SWNTs. The appearance of metal-1, metal-$2p$, and metal-$2m$ SWNTs in a two-dimensional $(n,m)$ plot is not regular.