Cumulative beam breakup in linear accelerators with periodic beam current

An analytic formalism of cumulative beam breakup in linear accelerators is developed. This formalism is applied to both low-velocity ion accelerators and high-energy electron accelerators. It includes arbitrary velocity, acceleration, focusing, initial conditions, beam-cavity resonances, finite bunch length, and arbitrary charge distribution within the bunches, and variable cavity geometry and spacing along the accelerator. For both direct-current beams and beams comprised of δ-function bunches, both the steady-state and transient displacements of the beam are calculated, and scaling laws are determined for the transient beam breakup. The steady-state transverse displacement of particles between bunches is also calculated since, if allowed to impinge on the accelerating structures, these particles could cause activation over long periods of continuous-wave operation. The formalism is then applied to high-current ion accelerators by studying the effects of finite bunch length and arbitrary charge distribution within the bunches. The role of focusing in controlling cumulative beam breakup is quantified in each of these cases. Additionally, the effects of random initial conditions and a distribution of deflecting-mode frequencies in the cavities are also quantified.