Longitudinal holes in debunched particle beams in storage rings, perpetuated by space-charge forces

Stationary, self-consistent, and localized longitudinal density perturbatio ns on an unbunched charged-particle beam, which are solutions of the nonlin earized Vlasov-Poisson equation, have recently received some attention. In particular, we address the case that space charge is the dominant longitudi nal impedance and the storage ring operates below transition energy so that the negative mass instability is not an explanation for persistent beam st ructure. Under the customary assumption of a bell-shaped steady-state distr ibution, about which the expansion is made, the usual wave theory of Keil a nd Schnell for perturbations on unbunched beams predicts that self-sustaini ng perturbations are possible only (below transition) if the impedance is i nductive (or resistive) or if the bell shape is inverted. Space charge give s a capacitive impedance. Nevertheless, we report numerous experimental mea surements made at the CERN Proton Synchrotron Booster that plainly show the longevity of holelike structures in coasting beams. We shall also report o n computer simulations of boosterlike beams that provide compelling evidenc e that it is space-charge force which perpetuates the holes. We shall show that the localized solitonlike structures, i.e., holes, decouple from the s teady-state distribution and that they are simple solutions of the nonlinea rized time-independent Vlasov equation. We have derived conditions for stat ionarity of holes that satisfy the requirement of self-consistency; essenti ally, the relation between the momentum spread and depth of the holes is gi ven by the Hamiltonian-with the constraint that the phase-space density be high enough to support the solitons. The stationarity conditions have scali ng laws similar to the Keil-Schnell criteria except that the charge and mom entum spread of the hole replaces that of the beam.