A general theory of beam breakup instabilities in linear acelerators

Abstract

We re‐examine the beam breakup instabilities (BBU) in linear accelerator with the use of a simple continuum model. The standard mode coupling approach in microwave engineering is adopted to treat the cumulative, regenerative and convective BBU in a unified manner. In addition to being able to recover all major classical solutions as limiting cases, our simplifed treatment, in several instances, provides a rather different perspective on the role of the focusing magnetic fields, spreads in betatron frequencies and spreads in the breakup mode frequencies, etc. Dimensionless parameters are constructed which provide an immediate assessment of the relative importance of these effects, as well as the anticipated BBU growth when these various factors are incorporated. We have also devised a highly accurate numerical scheme which directly integrates the governing equations. Excellent agreement between the numerical results and the analytic theory is observed. We have incorporated in our numerical study other effects such as stagger tuning and nonlinear focusing which defy analytical treatment.