Thermal distribution of relativistic particle beams with space charge

The Maxwell-Boltzmann (‘‘thermal’’) distribution constitutes the natural thermodynamic equilibrium state for a charged particle beam, and knowledge of its properties is therfore of fundamental importance. The Boltzmann relation for the particle density has a nonanalytic form when the space-charge force is included. We use numerical integration to determine the transverse and longitudinal density profiles for a relativistic beam in a linear focusing system at different temperatures ${\mathit{T}}_{\mathrm{\perp }}$ and ${\mathit{T}}_{\mathrm{\parallel }}$. The calculated profiles are related to space-charge tune despression, rms width, perveance, and emittance of the beam.