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

Abstract

Stationary, self-consistent, and localized longitudinal density perturbations on an unbunched chargedparticle beam, which are solutions of the nonlinearized Vlasov-Poisson equation, have recently received some attention. In particular, we address the case that space charge is the dominant longitudinal impedance and the storage ring operates below transition energy so that the negative mass instability is not an explanation for persistent beam structure. Under the customary assumption of a bell-shaped steadystate distribution, about which the expansion is made, the usual wave theory of Keil and Schnell for perturbations on unbunched beams predicts that self-sustaining perturbations are possible only (below transition) if the impedance is inductive (or resistive) or if the bell shape is inverted. Space charge gives a capacitive impedance. Nevertheless, we report numerous experimental measurements made at the CERN Proton Synchrotron Booster that plainly show the longevity of holelike structures in coasting beams. We shall also report on computer simulations of boosterlike beams that provide compelling evidence that it is space-charge force which perpetuates the holes. We shall show that the localized solitonlike structures, i.e., holes, decouple from the steady-state distribution and that they are simple solutions of the nonlinearized time-independent Vlasov equation. We have derived conditions for stationarity of holes that satisfy the requirement of self-consistency; essentially, the relation between the momentum spread and depth of the holes is given by the Hamiltonian -with the constraint that the phase-space density be high enough to support the solitons. The stationarity conditions have scaling laws similar to the Keil-Schnell criteria except that the charge and momentum spread of the hole replaces that of the beam.

Vlasov moments, integrable systems and singular solutions

“…Interestingly, the Vlasov equation (4.1) resulting from the hodograph transformation of the Benney equation is exactly the same as the equation that regulates coasting proton beams in particle accelerators (see for example [32] where a bunching term is also included). Now, this is a very remarkable fact: the integrability of the Vlasov-Benney equation implies soliton solutions, indications for which seem to have been found experimentally at CERN [21], BNL [2], LANL [9] and FermiLab [27]. (In the last case solitons are shown to appear even when a bunching force is present.)…”

Longitudinal confinement and matching of an intense electron beam

“…Similar wave propagation was viewed on a coasting beam at the CERN Proton Synchrotron Booster by Koscielniak. 25 Simulations by Friedman 26 for Integrated Research Experiments and fusion drivers also exhibited similar waves launched from modulations in beam velocity.…”

Soliton gas in space-charge-dominated beams

“…Recently, Koscielniak et al [10] reported formation of holes in the longitudinal density profile which has been predicted by computer simulations and observed at the CERN PS Booster and at Fermilab. Although it is not straightforward to compare our results with the findings of Koscielniak et al, it is worth noting that regardless of the case (longitudinal or transverse phase space) the nonlinear interaction between density waves in space-chargedominated beams as a rule leads to the formation of holes in the density profile.…”

“…As a result, we obtain the new Hamiltonian 10) where the self-field potential U(x; θ) (φ = √ βU ) satisfies the equation…”

“…In the cold-beam limit v T → 0 (or equivalently ω → 1) the second terms on their right-hand sides can be neglected as compared with the other terms. Therefore, we can write 10) where the local gauge transformatioñ…”