Determination and mapping of diameter and helicity for single-walled carbon nanotubes using nanobeam electron diffraction

The atomic structures of 124 single-walled carbon nanotubes, described by their diameter and helicity or equivalently by the two chiral indices (u,v) that define the perimeter of each nanotube, have been determined unambiguously by nanobeam electron diffraction. A mapping of 70 possible nanotubes in the range of 1.20-1.65 nm in diameter with all possible helicities has been constructed experimentally for a carbon nanotube sample produced by arc discharge. Among the total 124 nanotubes characterized experimentally, 58 nanotubes of different structure have been identified. By examining the histogram of occurring helicities, we find that, while certain nanotubes were observed slightly more often than others, the overall feature showed a rather uniform distribution in occurrence. Basing on a nucleation-and-growth model, we suggest that the uniform distribution of helicity be originated from the weak dependence on helicity of the formation energy of carbon nanotubes, while the growth prefers slightly the structure with helicity 15 deg. -30 deg. for which the addition of carbon dimers would facilitate the growth of carbon nanotubes