Effect of space charge on bunch compression near the transition

It is shown that energy conservation in the longitudinal envelope equation can be used to derive analytic expressions to model fast bunch compression and the effect of space charge in terms of a dimensionless Coulomb parameter Σ (∝1/η, with η the slip factor). For small |η| (below transition, hence Σ≫1), the rf voltage required is nearly independent of η and dominated by space charge repulsion. The extra voltage generates the coherent momentum spread δ∝1/√|η| required to compensate the increasing space charge force gradient during compression. This sets a clear limit to the useful approach to transition. An η jump scheme is discussed to minimize this effect. Particle-in-cell computer simulation confirms the validity of our results also for more realistic beam distributions. Noticeable tails in momentum space due to the nonlinear space charge force are found for Gaussian line density bunches and Σ≫1.