Self-consistent modified betatron equilibria and their adiabatic evolution

An analytic and numerical scheme is developed to determine self-consistent equilibria in modified betatrons. As the beam accelerates over a time scale of many poloidal drift times, the equilibrium develops by passing through a sequence of time dependent equilibria. At any time, the equilibria are characterized by number of particles in a drift (Pθ) surface, and the toroidal flux through a Pθ surface. One can follow the equilibria as the e beam accelerates. At some energy, the beam begins to make a transition from diamagnetic to paramagnetic poloidal drift. This transition is characterized by a change in topology of the Pθ surfaces. Depending on the change in shape at transition, the new Pθ surface can either be confined in the liner, or run into the liner. Conditions for confined orbits at the transition will be given for parameters of the NRL modified betatron experiment.