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.