Self-consistent limited beam current in RF field

An analytical solution for a self-consistent particle equilibrium distribution in an RF field with uniform transverse focusing is applied to the problem of the high brightness beam current limit. The distribution function in phase space is determined as a stationary function of the energy integral. Particle distribution of the bright beam always tends to such a shape that the space charge beam potential is opposite to the external potential regardless of the applied field. Analytical expressions for r−z equilibrium beam profile and beam current limit in an RF field are obtained.