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 April 26, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.