Soliton-like longitudinal holes in debunched beams perpetuated by space-charge forces

Because the longitudinal space-charge forces within a local rarefaction of the particle density in phase space are focusing toward the centre of the perturbation, holes in beams might be expected to be self stabilized below transition energy. We shall report measurements made at the CERN PS Booster (PSB) and computer simulations that demonstrate the surprising longevity of such perturbations on an unbunched charged-particle beam. The negative mass instability is not an explanation for this persistent beam structure and the usual wave theory for perturbations on unbunched beams (the Keil-Schnell criterion) cannot explain their stability. The holes are solutions of the non-linear Vlasov-Poisson equation for which we have found self-consistency and stationarity conditions; these have scaling-laws similar to the Keil-Schnell criterion