Halo formation in mismatched, space-charge-dominated beams

A semianalytic formalism was recently developed for investigating the transverse dynamics of a mismatched, space-charge-dominated beam propagating through a focusing channel. It uses the Fokker-Planck equation to account for the rapid evolution of the coarse-grained distribution function in the phase space of a single beam particle. A simple model of dynamical friction and diffusion represents the effects of turbulence resulting from charge redistribution. The initial application was to sheet beams, In this paper, the formalism is generalized to fully two-dimensional beams.<<ETX>>


INTRODUCTION
velocity space.We discuss beam evolution from the perspective of a comoving coordinate system.
According to We recently introduced a semianalytic formalism describ-the Fokker-Planck equation, the evolution of the coarseing the dynamics of transverse emittance growth and halo grained distribution function W(x,u,t), in which x, u, formation in nonrelativistic, mismatched beams arising as and t denote position, velocity, and time, respectively, is a consequence of nonlinear space-charge forces [1,2] The Fokker-Planck equation in-in which K is the net force per particle mass M in the corporates coefficients of dynamical friction and diffusion in velocity space.Turbulence excited as a consequence of comoving frame, charge redistribution enhances these coefficients and con- (2) verts free energy due to mismatch into thermal energy.
At ' 2 At If the local free-energy density is sufficiently high, mi-are the dynamical-friction vector and diffusion tensor, recroinstabilities may cause turbulent fluctuations to grow to spectively, and At is a short time during which the fluclarge amplitudes during a fraction of a plasma period These processes generate emittance ever, it should be possible to infer them by studying indigrowth and halo by injecting particles into high-amplitude vidual particle orbits in N-body simulations [6].In general, orbits.They also dissipate any fine structure present in the coefficients may be expected to be functions of posi- To solve the coupled Fokker-Planck and Poisson equations c_ tion, which says the beam has uniform "temperature", self-consistently, we decompose the distribution function is likely to be most appropriate for particles moving with into complete sets of orthogonal polynomials: velocities not much exceeding the thermal velocity, just as oo oo +oo oo it is when only binary coulomb collisions drive relaxation W= E E E _ A_"¢m(Ur)¢'_(u°)¢Pq (r)eip°' (6) [5].lt may therefore be expected to apply to "typical" m=On=Op=-oo q=0 particles comprising the central region of the beam.In actuality, the relaxation rate is slower for fast particles be-where Cm are Gauss-Hermite polynomials, cause they have less time to interact with localized field e-_"' H,.(4"au), (7) fluctuations.
Consequently, because halo particles either Cre(u)= _--_.move rapidly through the core or do not sample the core, the halo may be expected to thermalize more slowly than the core.This is seen in simulations [4].Despite this short-and ¢_qare Gauss-Laguerre polynomials, coming, our simplified coefficients should be useful both for t studying the evolution of fine structure in thebeamand ¢_q(r)= a [ q, ]2 for investigating halo generation from the core.By design, _ ([pl+q)!(ar2)_e-ar_L_Pl(ar2). ( 8) the formalism developed here can be adapted to accommodate coefficients with spatial and velocity dependencies In these polynomials, we require a = a(t) = B(t)/2D(t), once they are known, while a = a(t) is a free time-dependent variable.The dynamics we have described obviously operate in Two consideratio:ns motivate the use of these sets of three dimensions.
In what follows, we consider only the polynomials.
The first is that both sets begin with Gaustwo-dimensional dynamics in a plane orthogonal to the ac-sians at zeroth order.This is desirable because the celerator axis.The Fokker-Planck equation in cylindrical Maxwell-Boltzmann distribution of a beam in the absence coordinates (r, O, Ur,Uo = r dO/dt) is of space charge is Gaussian in both velocity and position (Gaussian ¢0 and Gaussian ¢°0).The second is that this where K is given by the superposition of the focusing force which, when integrated over r and 0, must always yield .IV'.
Upon evaluating the mean-square radius, we find (r 2} = and the space-charge force found from the coarse-grained 1 01 1/(r2), a-(1-A00 ).lt is therefore convenient to let apotentials (bI and (b8 respectively, in the manner ' so that A°lo = 0 for all t.The coefficients APmq n are complex,
After solving Poisson's equation for the space-charge potential, we find the components of acceleration to be where Q is the particle charge.
[3]. tuations modify the distribution function [5].Both the The fluctuations attenuate, however, via Landau damping distribution fimction and the net force are regarded to be on the same time scale [3], and heating can therefore oc-smoothed out.cur very rapidly.Relaxation toward Maxwell-Boltzmann The spectrum of electric-field fluctuations determines F equilibrium ensues on a time scale deternfined by weak and D [5].We do not know these coefficients a priori; howresidual turbulence.