Plasma Losses to a Pair of Nanotubes from a Charged Particle

We use the RPA to calculate the inverse dielectric function for a pair of single-walled hollow cylindrical nanotubes of arbitrary radii. The axes of these nanotubes are parallel and their separation exceeds the sum of their radii. We have also derived a general formula for the energy loss of a charged particle moving parallel to the axes of the cylinders, given in terms of the inverse dielectric function and the impact parameter. In our formalism, we separate the contributions to the stopping power from the collective plasmon modes and the single-particle excitaions. Our analysis shows that these contributions depend on the impact parameter and their characteristics are determined by the dispersion relation of the plasma spectrum and the relation between their group velocity and that of the charged particle.