Electronic properties of carbon nanotubes with distinct bond lengths

In band structure calculations commonly used to derive the electronic properties of carbon nanotubes it is generally assumed that all bond lengths are equal. However, hexagonal carbon lattices are often irregular and may contain as many as three distinct bond lengths. A regular $(n,m)$ carbon nanotube will be metallic if p=(n-m)/3 for integer p. Here we analytically derive the generalized condition for metallic irregular carbon nanotubes. This condition is particularly relevant to small radius nanotubes and nanotubes experiencing small applied strains.