Thermalization of an intense, space-charge-dominated electron beam in a long focusing channel

A continuous electron beam with a correlated emittance will eventually thermalize. Initially, the beam emittance for an intense, high-brightness, space-charge-dominated beam will oscillate, but after a sufficiently long time, it will reach an equilibrium value. The emittance oscillations are due to coherent transverse plasma oscillations in the beam and are a manifestation of periodic energy exchange between potential and kinetic energies. The beam eventually attains an equilibrium emittance, as the beam equipartitions the kinetic and potential energies. This equipartioning is reached as the beam thermalizes due to a form of Landau damping of the radial oscillations at different radial positions within the beam. Slight differences in the transverse plasma oscillation frequency for different radial positions lead to incoherence in the oscillations. In this paper, we calculate the equilibrium time scales required for equipartioning. We show that the equilibrium emittance scalings and magnitude can be predicted by conservation of energy considerations. In addition, we show that, in the space-charge dominated regime, there is a correspondence between the energy-conservation approach and the kinematic approach.