Computation of the Longitudinal Space Charge Effect in Photoinjectors Page: 2 of 3
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tails. The number of mesh points was chosen to be of -
12 per wavelength and 1 million particles were used.
A systematic study of linearity and number of mesh points
was done for the 120MeV case for the very short
wavelengths 15 and 30 jm. Results demonstrated a good
linearity between 1% and 5% initial amplitude of
modulation. It also proved that the use of more than 5
mesh points per wavelength, generates too large
numerical noise from the scheff routine.l. LCVI
- I
.-
- simulation
/222
'Fl o /t n iTh7
-6 .
Figure 2- a-b Energy modulation / Current modulation
for a 12 MeV beam along a drift; comparison with theory
for single and multiple frequency
Theory
The LSC impedance per unit length in free space is:
Z(k)= iZ0 1 kra K kr (
where Zo = 377 Q, rb is the beam radius for a uniform
transverse distribution, K, is the modified Bessel function.
A weak transverse dependence of the LSC field is
neglected here.
The current modulation (characterized by the bunching
spectrum b(k)) for a coasting beam at the position s in a
linac is given by the integral equation
b(k,s)= b(k,0)+ik- JdtR56'(r -> s) 47rZ(k,r) b(k,r),(2)
IA 00Z
where I is the beam current, IA ~ 17 kA is the Alfven
current, and the generalized momentum compaction factor
R56' is the ratio of the path length change at s due to a
small change in yat r, given by
R56'(2 -> s) ds'3, for y>> 1 (3)
y(s')
Equation (2) generalizes the integral equation for CSR
microbunching in a bunch compressor [6,7] to an linac
with acceleration, and ignores any Landau damping in the
linac. For example, in a simple drift space, R56' = (s-
r)//=R56/y and Eq. (2) produces the well-known
solution of the space charge oscillation. If the beam size
and electron energy vary along the linac, Eq. (2) can be
solved numerically to obtain the evolution of the current
modulation.
The change in energy modulation in the linac is thenAy(k, s) = Ay(k,0) - LfdlrZ(k, r)b(k, r) .
IA 0(4)
At sufficiently high energy or large y we have R56' = 0
and the beam current modulation is frozen [1], i.e., b(k,s)= bo(k). Thus, the energy modulation is accumulated
according to
Ib(,0) s 47Z(k, r)
Ay(k,s) = Ay(k,0) dr .
IA Z
Comparison with theory
The variation of charge density due to transverse
dimensions had to be taken into account [6] in the
theoretical model. A model which included the radial
dependency of the space charge impedance could not
explain the remaining discrepancy between simulations
and theory.
The agreement is quite good for both the energy
modulation and the bunching for the 12 MeV and 120
MeV cases (see figure 2). For the 6MeV case, the theory
does not agree as well with the results from the
simulations. The theory shows that the energy modulation
amplitude cancels. This was never observed in any of the
simulation cases studied. As the beam is travelling along
the drift, the frequency spectrum of the energy modulation
is no longer a single line, but has a much broader
extension. Computed with multiple frequencies, the
theoretical curve of the energy modulation does not go
back to zero and gets closer to the simulation curve in the
bunched beam case of the PARMELA simulations. The
exact current density was not taken into account in figure
1 but should only give a few percent correction. The
spread in frequencies has been computed for a gaussian
bunch and not for an hypergaussian one. The theoretical
model does not include either the growth in uncorrelated
energy spread, which is clearly visible in simulations. It is
still unclear how much this increase in uncorrelated
energy spread, shown by the simulations, is due to
numerical noise or if it has some physical reality. Those
aspects might explain the remaining discrepancy between
theory and simulations.
Comparison theory and simulation
A 6MeV beam with the typical LCLS transverse
parameters, but starting monoenergetic, travels through a
90 cm drift and is then accelerated up to 65 MeV in a 3m
long accelerating section, and drifts 1 m before being
accelerated up to 135 MeV. The agreement between the
theory and the simulations is very good.Theory
SimulationsC -. w
i ""lul0 0 00C
F.. <Fib -
Figure 3- Simulations and theory for energy modulation
for a drift + linac+ drift +linac system;
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Emma, P.; Huang, Z.; Limborg-Deprey, C.; Welch, J. J. & Wu, J. Computation of the Longitudinal Space Charge Effect in Photoinjectors, article, May 9, 2005; Menlo Park, California. (https://digital.library.unt.edu/ark:/67531/metadc875222/m1/2/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.