A core-particle model for periodically focused ion beams with intense space-charge

Abstract

A core-particle model is derived to analyze transverse orbits of test particles evolving in the presence of a core ion beam described by the KV distribution. The core beam has uniform density within an elliptical cross-section and can be applied to model both quadrupole and solenoidal focused beams in periodic or aperiodic lattices. Efficient analytical descriptions of electrostatic space-charge fields external to the beam core are derived to simplify model equations. Image charge effects are analyzed for an elliptical beam centered in a round, conducting pipe to estimate model corrections resulting from image charge nonlinearities. Transformations are employed to remove coherent flutter motion associated with oscillations of the ion beam core due to rapidly varying, linear applied focusing forces. Diagnostics for particle trajectories, Poincaré phase-space projections, and single-particle emittances based on these transformations better illustrate the effects of nonlinear forces acting on particles evolving outside the core. A numerical code has been written based on this model. Example applications illustrate model characteristics. The core-particle model described has recently been applied to identify physical processes leading to space-charge transport limits for an rms matched beam in a periodic quadrupole focusing channel [Lund and Chawla, Nuc. Instr. and Meth. A 561, 203 (2006)]. Further characteristics of these processes are presented here.

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

“…Effective relaxations of initial semi-Gaussian beam distributions resulting from phase mixing and nonlinear wave interactions are found in simulations to occur most rapidly for = 0 $ 0:5 where results here suggest that the frequency spectrum is most broad [49]. h Point 5.-Simulations and theory show that, if beam stability is defined practically in terms of limited rms emittance growth and halo generation, then a wide variety of 2D initial distributions are stable when transported in a periodic quadrupole transport channel without errors regardless of space-charge intensity-so long as the applied focusing strength is 0 & 85 per lattice period [45,46]. Large rms emittance growth results from significant numbers of near-edge particles rapidly evolving outside the statistical edge (core) of the beam and rapidly increasing in oscillation amplitude due to interaction with matched envelope oscillations of the core beam, rather than from growth of collective modes internal to the core of the beam.…”

“…(43)], but the higher-frequency breathing mode is not supported. Furthermore, scaling of space-charge forces with distance outside the envelope of an rms-equivalent core is different in the sheet-beam model (constant) relative to higher-dimensional models (falloff with distance) [46]. Together these features likely result in different characteristic halo resonances in the sheet-beam model relative to higher-dimensional models.…”

“…h Point 3.-The statistical width F w (or equivalently, the rms width F ) of the frequency distribution F for a thermal equilibrium sheet beam appears broadest for = 0 $ 0:4 but remains relatively broad over a wide range of strong space-charge-including the extreme case shown with = 0 ¼ 0:1. To the extent this broad parametric width is preserved for other relatively smooth, but nonthermal, near-equilibrium and rms-matched beam distributions, this width helps clarify why space-charge dominated beams have been observed in laboratory experiments and simulations appear to have robust stability to internal modes [44][45][46]. Although the single-particle frequencies do not simply correspond to collective mode frequencies, the spectrum of frequencies of the equilibrium beam strongly influences collective mode properties.…”

“…However, this does not imply that all KV model results are invalid. For example, lowest order linear (envelope) instability appears robust enough to extrapolate to other distributions [44][45][46]. Also, Startsev et al [52][53][54] analyzed a robust low-order transverse-longitudinal collective instability by applying the method of characteristics using (single-frequency) KV orbit equilibrium characteristics with perturbation operators for a smooth (thermal) core distribution to estimate growth rates, and found good agreement with r-z Vlasov simulation results.…”

“…However, for nontenuous halo processes leading to space-charge induced transport limits as described in Refs. [45,46], it is possible that resulting stability thresholds due to a variety of collective processes driving a significant fraction of particles well outside the beam core may be similar in sheet-beam models, though the final saturated beam states could differ due to different halo resonances supported. h…”