# Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits Page: 1 of 5

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SLAC-PUB-12943

SELF-CONSISTENT COMPUTATION OF ELECTROMAGNETIC FIELDS

AND PHASE SPACE DENSITIES FOR PARTICLES ON CURVED PLANAR

ORBITS *

G. Bassi, University of Liverpool, J. A. Ellison t, K. Heinemann, University of New Mexico,

M.Venturini, LBNL, and R. Warnock, SLAC

AbstractWe discuss our progress on the self-consistent calcu-

lation of the 4D phase space density (PSD) and electro-

magnetic fields in a Vlasov-Maxwell formulation. We em-

phasize Coherent Synchrotron Radiation (CSR) from arbi-

trary curved planar orbits, with shielding from the vacuum

chamber, but space charge forces are naturally included.

Our focus on the Vlasov equation will provide simulations

with lower numerical/statistical noise than standard PIC

methods, and will allow the study of issues such as emit-

tance degradation and microbunching due to space charge

and CSR in bunch compressors. The fields excited by the

bunch are computed in the lab frame from a new double in-

tegral formula. The field formula is derived from retarded

potentials by changes of variables. It is singularity-free and

requires no computation of retarded times. Ultimately, the

Vlasov equation will be integrated in beam frame coor-

dinates using our method of local characteristics. As an

important intermediate step, we have developed a "self-

consistent Monte Carlo algorithm", and a corresponding

parallel code. This gives an accurate representation of the

source and will help in understanding the PSD support. In

addition we have (1) studied carefully a 2D phase space

Vlasov analogue and (2) derived an improved expression of

the field of a 1D charge/current distribution which accounts

for the interference of different bends and other effects usu-

ally neglected. Bunch compressors will be emphasized.

INTRODUCTION

Our basic starting point is the Vlasov-Maxwell system

in 6D, i.e., we assume collisions can be ignored and that

the N-particle bunch can be approximated by a contin-

uum. Our coordinate system, (Z, X, Y), is shown in Fig.

1. We assume an external force due to a magnetic field,

Bet(R), in the Y-direction. We define a reference or-

bit, R (s) (Z (s), XT(s)), lying in the Y 0 plane,

which is a solution of the Lorentz equation for E 0

and B Bet(R)ey. Here R := (Z, X) and s is arc

length along the reference orbit. In Fig. 1 we sketched

R (s) for a 4 dipole magnetic chicane bunch compres-

sor. We focus on the evolution of S := (Ez, Ex, By)

and take (Ey, Bz, Bx) 0. We model shielding by the

vacuum chamber by taking = 0 at Y g, where

h 2g is the height of the vacuum chamber as shown

* Work supported by DOE grants DE-FG02-99ER41104 and DE-

AC02-76SF00515

t ellisonatmath.unm.eduE =0

By=0

x

Y R ( s )

h z

h

E .0

B 0

Figure 1: Basic lab frame setup.

in Fig. 1. We let H(Y) be the fixed Y density defined for

JYJ < g, then the coupled 4D Vlasov-Maxwell system

for the field vector S(R, Y, u) and the phase space density

H(Y)3(Py)4'(R, P, u), with the shielding boundary con-

dition, takes the form:DS(R,Y,v) H(Y)S(R,v),

R +R -VR -' +PVp' =0,

8(R, Y h/2, u) = 0,

where v = ct, c is the speed of light,'= d/du, Q

andS(R,u)=ZQ (

R = P/my(P)c,

P = [Ell (R, Y,uctzp+ auJz

ctxp+ auJx

txJz - zJx) + (cR x By (R, Y, u)

(1)

(2)

(3)

= a ,

(4)

)]. (5)C

Here Zo is the free space impedance, Q is the total

charge, QH(Y)p(R, u) is the lab frame charge density

(with f HdY = f pdR = 1), QH(Y)(Jz,Jx)(R,v)

is the current density (which, of course, has no Y com-

ponent), m is the electron mass, q is the electron charge

(so that Q = Nq where N is the number of particles in

the bunch), -y is the Lorentz factor, EI = (Ez, Ex) and

By (Be6t(R) + By (R, Y, u))ey. Equations (1-2) are

incomplete without the coupling between S and T given

byp(R, u) := dP4 (R, P, u),

JI1 (R, u) := dP(P/my(P))4 (R, P, u),(6)

(7)Contributed to Particle Accelerator Conference (PAC 07), 06/25/2007--6/29/2007, Albuquerque, New Mexico

,

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Ellison, J.A.; Bassi, G.; Heinemann, K.A.; U., /New Mexico; Venturini, M.; /LBL, Berkeley et al. Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits, article, November 2, 2007; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc888527/m1/1/: accessed September 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.