Self-Consistent Computation of Electromagnetic Fields and Phase Space Densities for Particles on Curved Planar Orbits Page: 1 of 5
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SELF-CONSISTENT COMPUTATION OF ELECTROMAGNETIC FIELDS
AND PHASE SPACE DENSITIES FOR PARTICLES ON CURVED PLANAR
G. Bassi, University of Liverpool, J. A. Ellison t, K. Heinemann, University of New Mexico,
M.Venturini, LBNL, and R. Warnock, SLAC
We 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.
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-
Y R ( s )
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:
R +R -VR -' +PVp' =0,
8(R, Y h/2, u) = 0,
where v = ct, c is the speed of light,'= d/du, Q
R = P/my(P)c,
P = [Ell (R, Y,u
txJz - zJx
) + (cR x By (R, Y, u)
= a ,
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
p(R, u) := dP4 (R, P, u),
JI1 (R, u) := dP(P/my(P))4 (R, P, u),
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.