Collective effects in energy loss by relativistic clusters of charged particles

A general expression for the energy loss per unit length of a relativistic cluster of particles is derived. The collective nature of the energy loss is examined using a classical dielectric description of the medium to which the particles lose energy in distant collisions. A special case, in which the particles in the cluster are assumed to have equal velocities, is also given using an extension of Fermi's method for deriving the density effect. In the nonrelativistic limit of the general result, new expressions for collective energy loss are obtained for clusters of particles moving with different velocities, as well as for particles having identical velocities. The simple dielectric function used in the nonrelativistic results can represent the frequency-dependent response function for a gas, a solid, or a plasma medium