Radial expansion of a self-pinched beam with distributed energy

Abstract

The structure and radial expansion of a relativistic particle beam are calculated in the presence of Coulomb scattering. The beam is assumed to be self-pinched and paraxial, but to have a distribution in energy. For the case in which the beam energy spread is small enough to satisfy (βγ)minβ−1γ−1≳1/2, solutions are found in which the entire beam expands self-similarly at an exponential rate proportional to β−1γ−1, where β≡v/c, γ−2≡1−β2, and the bar denotes the average over γ. The beam profile is calculated exactly in this case. If the inequality is not satisfied, low-γ particles expand faster than the main beam, at an exponential rate proportional to (2βγ)−1. Approximate time-dependent solutions, including initial transients, are presented for both cases.