Sheet beam model for intense space charge: Application to Debye screening and the distribution of particle oscillation frequencies in a thermal equilibrium beam

Abstract

A one-dimensional Vlasov-Poisson model for sheet beams is reviewed and extended to provide a simple framework for analysis of space-charge effects. Centroid and rms envelope equations including imagecharge effects are derived and reasonable parameter equivalences with commonly employed 2D transverse models of unbunched beams are established. This sheet-beam model is then applied to analyze several problems of fundamental interest. A sheet-beam thermal equilibrium distribution in a continuous focusing channel is constructed and shown to have analogous properties to two-and three-dimensional thermal equilibrium models in terms of the equilibrium structure and Debye screening properties. The simpler formulation for sheet beams is exploited to explicitly calculate the distribution of particle oscillation frequencies within a thermal equilibrium beam. It is shown that as space-charge intensity increases, the frequency distribution becomes broad, suggesting that beams with strong space-charge can have improved stability relative to beams with weak space-charge.

Noiseless Vlasov–Poisson simulations with linearly transformed particles

“…Next, in Section 3.3 we study a more applied test case consisting of a mismatched beam in a constant (continous) focusing channel, derived from the 1d sheet beam model developed in Ref. [22].…”

“…The tune depression σ/σ 0 -defined as the ratio between the phase advance of the particles in the presence and absence of beam charge -can be calculated [22] as…”

“…Note that a correction similar to (11) can be used here to make the deposition conservative, in the sense that ρ n h (x) dx = f n h (z) dz. In the above deposition formula (22), we observe that the evaluation of the "integrated particles" ρn h,k is not straightforward, due to the linear transformation of their shape. To compute them we have considered two methods.…”

“…to be specified below. Following the analysis carried out in [22], thermal equilibrium distributions can then be obtained as follows. Given a specific value for the positive, dimensionless parameter…”

Envelope model for passive magnetic focusing of an intense proton or ion beam propagating through thin foils

“…The choice of Gaussian b may be appropriate for a neutralized intense beam near a laser-plasma source where a short pulse laser beam typically has a Gaussian-distributed focal spot and the expanding plasma sheath appears to produce an approximately Gaussian beam [30,31]. Conversely, the choice of uniform b may be appropriate for an unneutralized intense beam emerging from a linear transport channel where Debye screening of the linear applied-focusing force leads to an approximately uniform charge-density beam [28,32,33]. The two density profiles are contrasted in Fig.…”