Generation of initial kinetic distributions for simulation of long-pulse charged particle beams with high space-charge intensity

Abstract

Self-consistent Vlasov-Poisson simulations of beams with high space-charge intensity often require specification of initial phase-space distributions that reflect properties of a beam that is well adapted to the transport channel-both in terms of low-order rms (envelope) properties as well as the higher-order phasespace structure. Here, we first review broad classes of kinetic distributions commonly in use as initial Vlasov distributions in simulations of unbunched or weakly bunched beams with intense space-charge fields including the following: the Kapchinskij-Vladimirskij (KV) equilibrium, continuous-focusing equilibria with specific detailed examples, and various nonequilibrium distributions, such as the semi-Gaussian distribution and distributions formed from specified functions of linear-field Courant-Snyder invariants. Important practical details necessary to specify these distributions in terms of standard accelerator inputs are presented in a unified format. Building on this presentation, a new class of approximate initial kinetic distributions are constructed using transformations that preserve linear focusing, single-particle Courant-Snyder invariants to map initial continuous-focusing equilibrium distributions to a form more appropriate for noncontinuous focusing channels. Self-consistent particle-in-cell simulations are employed to show that the approximate initial distributions generated in this manner are better adapted to the focusing channels for beams with high space-charge intensity. This improved capability enables simulations that more precisely probe intrinsic stability properties and machine performance.

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

“…In the fully self-consistent beam evolution, the form factor will likely evolve during the focusing cycle due to evolution in the beam charge density. In cases with applied focusing ( Þ 0) and strong spacecharge, evolution outside of self-similar form may be relatively small because strong Debye screening of the linear focusing force should result in a beam charge profile which is nearly uniform in the core out to the radial edge where the density rapidly drops to small values [28,32]. In such situations, little evolution in the form of F should be possible.…”

“…represents the effect of linear applied-focusing forces from any solenoid magnetic field applied to the system with the focusing function ðzÞ specified by Eq. (32).…”

“…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.…”

“…is the transverse rms emittance measured in the Larmor frame, which provides a statistical measure of the transverse phase-space area of the beam [28,32,36,37]. If there is no solenoidal magnetic field (B z0 ¼ 0), then the implicit Larmor-frame x-x 0 variables employed (see Appendix B) become the usual x-x 0 laboratory-frame variables and Eq.…”

“…(32). Here, we have assumed that both v and g are functions only of r and have employed the fact that the Larmor transform (B1) preserves radial distance, i.e., r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi…”