Electron diffraction and microscopy of nanotubes

Carbon nanotubes were discovered by electron microscopy in the carbon soot produced in an electric arc between graphite electrodes, as used in the production of fullerenes. Details of this microstructure have been studied mainly by the combined use of electron microscopic imaging and electron diffraction. Due to the small size of the tubes, diffraction patterns of single tubes, which are the most informative ones, can only be obtained by electron diffraction. For a complete interpretation of the observed diffraction effects a detailed theory is required. Successively more refined approximations of the theory allow us to understand the origin of the different features of the diffraction patterns. The most complete kinematical theory for the diffraction by single shell chiral straight tubes is obtained by the direct summation of the complex amplitudes of the waves scattered by the carbon atoms arranged on a helically wound graphene network. The closed form analytical expressions deduced in this way make it possible to compute the geometry and the intensity distribution of diffraction space. Diffraction patterns are computed as planar sections of this diffraction space. High-resolution electron microscopic images reveal the geometry of individual graphene sheets and their defects in multishell tubes. As well as the characteristic features of straight nanotubes those of helix shaped tubes are also discussed. It is shown how the combined use of electron diffraction and electron microscopy makes it possible to completely characterize the geometry of carbon nanotubes.