Self-consistent equilibrium and adiabatic evolution of a high-current electron ring in a modified betatron

The equations for the self-consistent equilibrium of a cold electron beam in an axisymmetic modified betatron are derived, where it is shown that the equilibrium is uniquely defined by specifying the particle energy and toroidal field as functions of the canonical angular momentum. The generalization to a slow evolution is derived, using magnetic moment, canonical angular momentum, and toroidal flux as invariants. Results pertaining to equilibrium, allowable energy spread, self-flux diffusion, compensation for flux diffusion by discrete coils, and acceleration are presented.