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Effects of finite bunch length and arbitrary current distribution on steady-state cumulative beam breakup with periodic beam current

The equation of transverse motion governing cumulative beam breakup has been solved using Fourier analysis. This technique simplifies the investigation of the effect of finite bunch length and arbitrary current distribution within the bunches on beam breakup, which is an especially relevant consideration for low-velocity, high-current ion linacs. It also simplifies the calculation of the transverse dynamics of unbunched particles which constitute a diffuse longitudinal halo between bunches. If allowed to impinge on the accelerating structures, these particles could cause activation over long periods of continuous-wave operation. Simulations illustrating the salient effects of finite bunch length on steady-state cumulative beam breakup are presented here. The role of focusing in suppressing the transverse displacement of a beam which is misaligned at the accelerator entrance is also explored, and an analytic formula is derived for use in estimating the required focusing strength.