Energy loss of charged particles moving in cylindrical tubules

The interactions of charged particles with cylindrical tubules are studied within the framework of the dielectric theory. Elementary excitations on a tubule are modeled by an infinitesimally thin layer of free-electron gas, uniformly distributed over the surface of the tubule. The dielectric function of such a system, obtained from the random-phase approximation, exhibits a dimensional crossover from two-dimensional to one-dimensional electron gas, when the radius of the tubule decreases. Energy loss of a charged particle, moving paraxially in a tubule, can be divided into a single-particle excitation part and a resonant excitation part. It is shown that the resonant excitation modes on a tubule, which dominate the energy loss in the high-velocity regime, are quite different from those in a cylindrical cavity in a solid, described by a bulk dielectric function of the surrounding three-dimensional electron gas