Kinetic description of intense beam propagation through a periodic focusing field for uniform phase-space density

Abstract

The Vlasov-Maxwell equations are used to investigate the nonlinear evolution of an intense sheet beam with distribution function f b ͑x, x 0 , s͒ propagating through a periodic focusing lattice k x ͑s 1 S͒ k x ͑s͒, where S const is the lattice period. The analysis considers the special class of distribution functions with uniform phase-space density f b ͑x, x 0 , s͒ A const inside of the simply connected boundary curves, x 0 1 ͑x, s͒ and x 0 2 ͑x, s͒, in the two-dimensional phase space ͑x, x 0 ͒. Coupled nonlinear equations are derived describing the self-consistent evolution of the boundary curves, x 2 . The resulting model is shown to be exactly equivalent to a (truncated) warm-fluid description with zero heat flow and triple-adiabatic equation of state with scalar pressure P b ͑x, s͒ const͓n b ͑x, s͔͒ 3 . Such a fluid model is amenable to direct analysis by transforming to Lagrangian variables following the motion of a fluid element. Specific examples of periodically focused beam equilibria are presented, ranging from a finite-emittance beam in which the boundary curves in phase space ͑x, x 0 ͒ correspond to a pulsating parallelogram, to a cold beam in which the number density of beam particles, n b ͑x, s͒, exhibits large-amplitude periodic oscillations. For the case of a sheet beam with uniform phase-space density, the present analysis clearly demonstrates the existence of periodically focused beam equilibria without the undesirable feature of an inverted population in phase space that is characteristic of the Kapchinskij-Vladimirskij beam distribution.

One-dimensional kinetic description of nonlinear traveling-pulse and traveling-wave disturbances in long coasting charged particle beams

“…[14], the analyses in Ref. [15] assumed that the longitudinal distribution F b ðz; p z ; tÞ corresponded to a so-called waterbag distribution [16][17][18][19], where F b ¼ const within moving boundaries in the phase space ðz; p z Þ. The weakly nonlinear analysis in Ref.…”

Analytical Solutions for the Nonlinear Longitudinal Drift Compression (Expansion) of Intense Charged Particle Beams

“…In the present analysis, we employ a one-dimensional warm-fluid model [86][87][88]91] to describe the longitudinal nonlinear beam dynamics with average electric field given by the g-factor model with e b E z = −e 2 b g∂λ/∂x [79][80][81][82][83][84][85]. For example, for a space-charge-dominated beam with flat-top density profile in the transverse plane, g 2 ln(r w /r b ) [85].…”

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

“…Analytic descriptions of collective modes in sheet beams have been derived for a continuously focused waterbag (i.e., uniform phase-space) distribution by Startsev and Davidson [12] and by Okamoto and Yokoya for approximate waterbag distributions in both continuous and periodic focusing [8,13]. Davidson et al also analyzed a waterbag distribution in periodic focusing channels in terms of the evolution of the phase-space boundary [14].…”