On the measurement of helicity of carbon nanotubes

For a single-walled cylindrical carbon nanotube, its diameter and helicity determine the atomic structure of the nanotube except its handedness. An analysis has been given of the relationship between the true helicity of a single-walled cylindrical carbon nanotube and the apparent semi-splitting angle measured in an electron diffraction pattern from the tubule. Depending on the order of the Bessel function that dominates the diffraction intensity of the specific layer line on which the measurement was made, a difference between the two angles can be as large as 70%. This deviation is caused by the cylindrical curvature of the diffracting nanotube. The prefactors that relate these two quantities have also been deduced to account for the cylindrical effect. Using the proposed cylindrical correction factors, the indices of two carbon nanotubes were determined to be [12, 1] and [31, 13], respectively. The reciprocal space structure of cylindrical carbon nanotubes, for both single- and multi-walled, has been described as well based on the analytic expression for structure factors.