Abstract
Stationary, self-consistent, and localized longitudinal density perturbations on an unbunched charged-particle 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 steady-state 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.
Highlights
Stationary, self-consistent, and localized longitudinal density perturbations on an unbunched charged-particle beam, which are solutions of the nonlinearized VlasovPoisson equation, have recently received some attention in the literature
We shall 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
Colestock [1] and Spentzouris [2] have reported some experimental observations, and we shall report on numerous measurements [3,4,5] made at the CERN Proton Synchrotron Booster (PSB)
Summary
Stationary, self-consistent, and localized longitudinal density perturbations on an unbunched charged-particle beam, which are solutions of the nonlinearized VlasovPoisson equation, have recently received some attention in the literature. For the impedance due to space charge and the induced charges in a perfectly conducting wall, the constancy of impedance divided by wave number Znvrev͞n implies the eigenfrequencies vn nv and that all waves have the same speed; one might expect to be able to construct localized, stationary perturbations. This hypothesis was put forward by Koscielniak [18]. The stationarity conditions have scaling laws similar to the Keil-Schnell criteria, and all reduce to essentially the same condition of a Hamiltonian with bound states in a potential well
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