Simulation of Electron-Cloud Instability in
Circular Accelerators using
Plasma Models
A.Z. Ghalam1, T. Katsouleas1, S. Lee1, W. B. Mori2, C. Huang2, V. Decyk2,
C. Ren2
1- University of Southern California (USC), Los Angeles, CA 90089
2- University of California at Los Angeles (UCLA), Los Angeles, CA 90095
Abstract. The interaction between a low density electron cloud in a circular particle accelerator with a
circulating charged particle beam is considered. The particle beam's space charge attracts the cloud,
enhancing the cloud density near the beam axis. Beam- cloud interaction is studied with a plasma wakefield
accelerator simulation code and the results are benchmarked against an existing code. The restoring force
on the off-centered beam due to the cloud's space charge is studied. The force is stronger at the tail than it
is at the head due to the cloud compression near the tail of the beam. The beam dynamics over 20Km of the
SPS ring at CERN is studied and the head-tail dephasing is observed.
INTRODUCTION
The need to understand the interaction of intense positively charged beams with the low
density electron clouds they create in circular accelerators is well documented. These
low density clouds constitute a non-neutral plasma which support wakefields of the
beam. The wakefields affect the beam propagation in a number of ways. They lead to
focusing terms that alter the tune shift of the accelerator, longitudinal terms affecting the
synchrotron motion and deflection terms that couple small offsets between head and the
tail. The latter are believed to be responsible for a head-tail instability that leads to
emittance blow-up and limits the beam current in many existing and planned circular
accelerators.
Several simulation models have been developed for the wakes and instability of beams
in electron clouds. These typically have many approximations such as neglect of the
space charge of the cloud on itself, and condensation of the effect of the cloud to a single
kick on the beam once per turn. Perhaps of even greater concern is the newness of the
models themselves. As a result there has been little opportunity to benchmark the codes
against reference codes or experimental data.
In this paper we apply some of the simulation tools we have been developing over the
past decade for the study of plasma-based accelerators to the problem of wake production
and beam propagation in electron clouds. This is an extension of work presented recently
at the ECLOUD02 Workshop in CERN [1]. The physical mechanism of wakefield
production in electron clouds of circular accelerators is nearly identical to that in plasma
wakefield accelerators driven by positron beams; namely, the rapid drawing in of plasma
electrons to the beam axis on a beam plasma frequency time-scale. In this paper, we
briefly review the primary simulation model we use -QuickPIC. Then we apply the
model to the case of electron cloud wakefields in the SPS proton storage ring at CERN.
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
224
Comparisons
Comparisons are
are made
made to
to recent
recent models
models by
by Rumolo
Rumolo and
and Zimmerman
Zimmerman [5].
[5]. We
We also
also
examine
examine the
the propagation
propagation of
of offset
offset beams
beams through
through aa significant
significant length
length of
of the
the accelerator
accelerator
(27
(27 Km
Km which
which is
is one
one turn
turn of
of SPS)
SPS) in
in their
their self-consistent
self-consistent wakefields.
wakefields. The
The effects
effects of
of the
the
cloud
cloud wake
wake and
and image
image forces
forces from
from the
the wall
wall provide
provide an
an additional
additional restoring
restoring force
force to
to the
the
offset
offset beam
beam contributing
contributing to
to the
the coherent
coherent tune
tune shift
shift of
of the
the accelerator.
accelerator. Finally,
Finally, we
we
comment
on
progress
toward
creating
a
complete
high-fidelity
PIC
model
that
comment on progress toward creating a complete high-fidelity PIC model that includes
includes
all
all of
of the
the relevant
relevant plasma
plasma physics
physics contained
contained here
here as
as well
well as
as the
the synchrotron
synchrotron motion
motion and
and
betatron
betatron motion
motion in
in the
the external
external fields
fields of
of the
the circular
circular machine.
machine. Through
Through high
high performance
performance
computing
predictions over
over
computing itit may
may be
be possible
possible to
to use
use such
such aa model
model to
to make
make accurate
accurate predictions
thousands
thousands of
of turns.
turns.
BRIEF
BRIEF DESCRIPTION
DESCRIPTION OF
OF SIMULATION
SIMULATION MODELMODEL- QUICKPIC
QUICKPIC
One
One of
of our
our main
main simulation
simulation tools
tools for
for beam-plasma
beam-plasma interactions
interactions is
is the
the particle-in-cell
particle-in-cell
(PIC)
code,
QuickPIC
[3]
and
OSIRIS
[2].
The
description
of
the
(PIC) code, QuickPIC [3] and OSIRIS [2]. The description of the two
two code
code and
and the
the
benchmarking
benchmarking against
against each
each other
other has
has been
been discussed
discussed extensively
extensively elsewhere
elsewhere [2,3].
[2,3]. We
We
choose
choose QuickPIC
QuickPIC to
to model
model the
the electron
electron cloud
cloud effects
effects on
on the
the beam
beam for
for the
the reasons
reasons which
which
become
become clear
clear later
later in
in this
this paper.
paper. Here
Here is
is aa brief
brief description
description of
of the
the QuickPIC
QuickPIC code.
code.
QuickPIC
QuickPIC is
is aa 3-D
3-D PIC
PIC code
code using
using aa quasi-static
quasi-static or
or frozen
frozen field
field approximation.
approximation. This
This
approximation
is
specifically
useful
for
studying
wakes.
It
requires
approximation is specifically useful for studying wakes. It requires that
that the
the beam
beam not
not
evolve
evolve significantly
significantly on
on the
the time
time scale
scale that
that itit takes
takes the
the plasma
plasma to
to pass
pass through
through it,
it, or
or in
in
other
other words,
words, β>>
(3» σazz.. This
This is
is typically
typically well
well satisfied.
satisfied. The
The basic
basic equations
equations for
for QuickPIC
QuickPIC
follow
in the
follow from
from the
the wave
wave equations
equations for
for A
A and
and φ0in
the Lorentz
Lorentz gauge
gauge [3]
[3] as
as illustrated
illustrated in
in box
box
in
in figure
figure 1.
1.
Maxwell
Maxwell equations
equations inin
/ 1I ∂u −
((^r^-T2
cc22 ∂t
dt2
22
1 ∂
(, I2 d 2 −
c ∂t
2
Lorentz
Lorentz gauge
gauge
XT22 \ A = 44 π# j.
∇
V )A
)A = ——J
cc
^ 22 x) φ, = 4 π ρ
∇
jJ == jh
ρ bbzˆz
b +
+ jLe ≈« jhb ==ccp
Reduced
Reduced Maxwell
Maxwell equations
equations
Quasi-static
Quasi-static approx.
approx.
_______KK
fc
"
^
φ, A = ϕ , A( z − ct )
(A
$ == $
]|
4,i>)
4Aπ<n i
22
-V
−∇
j
V ⊥A
_ LAA =- —c— J
−∇ 2⊥ φ = 4 π ρ
Local-Local- φ,Α
0,A at
at any
any z-slice
z-slice depend
depend
only
only on
on ρ,j
p,j at
at that
that slice!
slice!
Forces
Forces ::
plasma
plasma ::
Ψ = φ − A//
beam
beam ::
Fe⊥ = − e∇ ⊥ φ
Fb ⊥ = − e ∇ ⊥ Ψ
FIGURE
FIGURE 11 Quasi-static
Quasi-static or
or frozen
frozen field
field approximation
approximation used
used in
in QuickPIC
QuickPIC
The
The quasi-static
quasi-static approximation
approximation assumes
assumes that
that the
the wakes
wakes are
are functions
functions of
of z-ct
z-ct only
only and
and
leads
ϕ -- A
leads to
to equations
equations for
for the
the wake
wake potentials
potentials ϕ(p and
and Ψ
*P ==(p
A\\|| that
that involve
involve only
only solving
solving 2-D
2-D
Poisson
Poisson equations.
equations. The
The QuickPIC
QuickPIC cycle
cycle is
is illustrated
illustrated in
in figure
figure 2.
2. The
The Poisson
Poisson equations
equations
are
solved
on
a
2-D
slab
of
plasma
(using
a
well
established
bounded
are solved on a 2-D slab of plasma (using a well established bounded 2-D
2-D PIC
PIC code
code
BEPS
BEPS as
as aa subroutine)
subroutine) with
with conducting
conducting boundary
boundary conditions.
conditions.
225
2-D Plasma
Plasma Slab
Slab
2-D
Wake
Wake(3-D)
(3-D)
Beam (3-D)
(3-D)
Beam
initialize
11.. initialize
beam
solve
V\cp
22.. solve
∇
ϕ =
= p,
ρ , V\\j/
∇ 2⊥ ψ == pρe e =>
⇒ FFpp, ,\j/ψ
push plasma
plasma ,, store
y
33.. push
store ψ
step slab
slab and
and repeat
repeat 2.
44.. step
2.
5.
use
\j/
to
giant
step
beam
5 . use ψ to giant step beam
2
⊥
FIGURE2.2.QuickPIC
QuickPICcycle,
cycle, ItIt uses
uses 2-D
2-D Poisson
FIGURE
Poisson solver
solver to
to calculate
calculatepotentials
potentialsand
andupdate
updateparticles
particles
Thewakes
wakes are
are stored
stored and
and used
used to
to update
update the
The
the plasma
plasma in
in the
the slab
slab and
andthe
theslab
slabisisthen
then
pushed
back
a
small
step
through
the
beam.
After
transiting
the
beam,
the
stored
pushed back a small step through the beam. After transiting the beam, the storedvalues
values
areused
usedtoto find
find the
the force
force on
on the
the beam
ofofϕ(pare
beam (treated
(treated as
as aa 3-D
3-DPIC
PICmodel)
model)and
andititisispushed
pushed
through
a
large
step
(of
the
order
(1/30).
The
need
to
solve
for
only
a
2-D
through a large step (of the order β/30). The need to solve for only a 2-Dslab
slaband
andthe
the
largertime
timesteps
stepsof
of the
the 3-D
3-D push
push enable
enable aa time
larger
time savings
savings of
of 2-3
2-3 orders
ordersof
ofmagnitude.
magnitude. Both
Both
the 3-D outer layer and the 2-D inner layer of the code have been written in a parallel
the 3-D outer layer and the 2-D inner layer of the code have been written in a parallel
fashion to allow domain decomposition along z and y, respectively.
fashion to allow domain decomposition along z and y, respectively.
Next we apply QuickPIC to the electron cloud case. We remove the background ions
Next we apply QuickPIC to the electron cloud case. We remove the background ions
usually present in the plasma simulations and initialize a cloud and beam with the
usually
present
the plasma
simulations
and atinitialize
cloud
and beam
with in
the
parameters
usedinpreviously
[5] for
the SPS ring
CERN. aThe
parameters
are given
parameters
used
previously
[5]
for
the
SPS
ring
at
CERN.
The
parameters
are
given
in
table 1.
table 1.
TABLE 1. Parameters of SPS and KEKB
TABLE 1. Parameters
Variable of SPS and KEKB
Symbol
Bunch Variable
population [101Q]
Bunchmomentum
population [GeV/c]
[1010]
Beam
Beam
momentum [GeV/c]
Circumference
[Km]
Electron
density [ [Km]
1012 m3]
Circumference
Rms bunch
length
[mm]
Electron
density
[ 1012
m3]
Hor. Beam
size[mm]
Rms bunch
length
[mm]
RmsHor.
vert.Beam
size [mm]
Rms
Beam size[mm]
Rms vert.Beam size [mm]
SPS
SPS
10
10
26
26
6.9
1
6.9
300
1
3
300
2.3
3
2.3
Symbol
Nb
NPb
P
C
e
C
ez
X
z
xX
x
KEKB
KEKB
3.3
3.3
3.5
3.5
3.0
1
3.0
41
0.44
0.06
0.4
0.06
Figure 3 shows the initial beam and cloud density profiles in the X-Z plane. From this
Figure
3 shows
initial beam
and cloud
densitya peak
profiles
in theenhancement
X-Z plane. factor
From this
we
see that
cloudthe
electrons
are sucked
in reaching
density
of
we
cloud 1.9a
electrons
arethesucked
150see
at that
a location
behind
beam. in reaching a peak density enhancement factor of
150 at a location 1.9σ behind the beam.
226
13
25
38
50
S3
FIGURE
Initial
Beam
and
Plasma
density.
Cloud
electrons
are
suckedin
inat
at1.9
1.9a
behind the
thebeam.
beam.
FIGURE
3.3.3.
Initial
Beam
and
σσ behind
beam.
FIGURE
Initial
Beam
andPlasma
Plasmadensity.
density.Cloud
Cloudelectrons
electronsare
aresucked
sucked
in
at
1.9
behind
the
The
analytically
expected
enhancement
factor
the
center of
of the
the beam
beam is
is given
given in
inRef.
Ref.
The
analytically
given
in
Ref.
The
analyticallyexpected
expectedenhancement
enhancementfactor
factoratat
atthe
thecenter
center
of
the
beam
is
bebe
approximately
100
and
the
simulation
70.
Thecloud
cloud response
response gives
givesrise
riseto
to
66to
be
approximately
gives
rise
to
6toto
approximately100
100and
andinin
inthe
thesimulation
simulationitit
itisis
is70.
70. The
The
cloud
response
the
wakefields
shown
figure
4.The
longitudinal
wake field
field reaches
reaches aaa maximum
maximum
the
maximum
thewakefields
wakefieldsshown
showninin
infigure
figure 4.The
4.The longitudinal
longitudinal wake
wake
field
reaches
retarding
field
10
V/m
near
the
center
of
the
beam.
This compares
compares to
to the
the analytic
analytic
retarding
the
analytic
retardingfield
fieldofofof10
10V/m
V/mnear
nearthe
thecenter
centerof
of the
the beam.
beam. This
This
compares
to
expression
in
Ref.
6
which
estimates
the
field
at
the
center
about
10V/m.
expression
in
Ref.
6
which
estimates
the
field
at
the
center
about
10V/m.
expression in Ref. 6 which estimates the field at the center about 10V/m.
OuIcKPIO e-eloud wakes
Bunch
Bunch
Force
Force
(m)
FIGURE
Longitudinalforce
forceon
onthe
thebeam
beamatat
at1.9
1.9
behind
the
beam
FIGURE
Longitudinal
force
on
the
beam
1.9a
behindthe
thebeam
beam
FIGURE
4.4.4.Longitudinal
σσbehind
Alsofor
forcomparison
comparisonwe
wereproduce
reproducethe
the results
results of
of Rumolo
Rumolo and
and
Zimmermann
Also
for
comparison
we
reproduce
the
results
of
Rumolo
and Zimmermann
Zimmermann [5]
[5] in
in
Also
[5]
in
figure5.
For
identical
parameters
we
see
that
the
QuickPIC
result
and
ECLOUD
result
are
figureS.
For
identical
parameters
we
see
that
the
QuickPIC
result
and
ECLOUD
result
are
figure5. For identical parameters we see that the QuickPIC result and ECLOUD result are
quitesimilar
similarinininthe
themain
mainpart
partof
ofthe
the beam,
beam, but
but the
the ECLOUD
ECLOUD result
result
has
quite
similar
the
main
part
of
the
beam,
but
the
ECLOUD
result has
has unphysical
unphysical
quite
unphysical
divergences
at
the
extreme
head
and
tail.
divergencesatatthe
theextreme
extremehead
headand
andtail.
tail.
divergences
227
liiJnc: h
—1———0.75 >d.5 "-0-25
0.25
.
0.5
0.75
•
-
FIGURE
Longitudinal
Force
onthe
thebeam
beamfrom
from
FIGURE 5.
5. Longitudinal
Longitudinal Force
Force on
on
the
beam
from
In
figure
we
show
In
results
for
the tilted
tilted beam.
beam. The
Thebeam
beamisisisinitially
initially
In figure
figure 666 we
we show
show corresponding
corresponding results
results for
for the
the
tilted
beam.
The
beam
initially
tilted
by
over
length
tiltedby
byσσ
arrr over
over σσ
azzzlength
length of
of the
the bunch.
bunch.
tilted
13
25
30
50
13
63
25
38
50
63
FIGURE
6.
Initial
tilted
beam
beam isistilted
σ over the
bunch length
FIGURE6.
6.Initial
Initial tilted
tilted beam
beam and
and Plasma
Plasma Density.
Density. The
The
FIGURE
and
Plasma
Density.
The beam
beam is tilted
tilted aσrrrover
over the
the bunch
bunchlength
length
The
structure
of
the
cloud
density
interesting and
can be
understood as
The structure
structure of
of the
the cloud
cloud density
density in
in figure
figure 666 is
is
The
in
figure
is interesting
interesting and
and can
can be
be understood
understood as
as
follows:
A
compression
peak
is
formed
along
the
tilted
axis
of
the
beam
due to
the
follows:
A
compression
peak
is
formed
along
the
tilted
axis
of
the
beam
follows: A compression peak is formed along the tilted axis of the beam due
due toto the
the
drawing
in
of
electrons
nearest
further away,
(nearest the
pipe
drawing in
in of
of electrons
electrons nearest
nearest the
the beam.
beam. Electrons
Electrons from
from
drawing
the
beam.
Electrons
from further
further away,
away, (nearest
(nearest the
the pipe
pipe
walls)
receive
their
strongest
kick
from
the
peak
of
the
beam
current
(in
the
center
of
the
walls) receive
receive their
their strongest
strongest kick
kick from
from the
the peak
walls)
peak of
of the
the beam
beam current
current (in
(in the
the center
center of
ofthe
the
box). By
the
time
they
arrive
have
fallen
behind
creating
the
box).
By the
the time
time they
they arrive
arrive to
to the
the axis,
axis, they
they
have
fallen
behind
creating
the
box).
By
to
the
axis,
they
have
fallen
behind
creating
the
compressions
on
axis
at
the
bottom
compressions on
on axis
axis at
at the
the bottom
bottom of
of the
the figure.
figure.
compressions
of
the
figure.
There
are
33 deflecting
focusing
force
elements,
force
due
to
the space
charge
There
are
deflecting
focusing
force
elements, deflecting
deflecting
force
due
to
charge
There
are
3
deflecting
focusing
force
elements,
deflecting
force
due
to the
the space
space
charge
of
the
cloud
and
those
due
to
the
beam
and
the
cloud
images
in
conducting
walls.
The
of
the
cloud
and
those
due
to
the
beam
and
the
cloud
images
in
conducting
walls.
The
of
the
cloud
and
those
due
to
the
beam
and
the
cloud
images
in
conducting
walls.
The
two
later
elements
have
been
omitted
in
past
work
[5].
The
cloud’
s
image
contributes
aa
two
later
elements
have
been
omitted
in
past
work
[5].
The
cloud's
image
contributes
two
later
elements
have
been
omitted
in
past
work
[5].
The
cloud’
s
image
contributes
a
coherent
tune
shift
that
is
larger
and
in
the
opposite
direction
to
the
number
of
tune
shift
coherent tune
tune shift
shift that
that is
is larger
larger and
and in
in the
the opposite
opposite direction
to
the
number
of
tune
shift
coherent
direction
to
the
number
of
tune
shift
caused
by
the
image
charge
of
the
is because
of
the usual
cancellation of
causedby
by the
the image
image charge
charge of
of the
the beam
beam itself.
itself. This
This
of
usual
of
caused
beam
itself.
This is
is because
because
of the
the
usual cancellation
cancellation
of
γ2.2 In
the
the
electric
and
magnetic
forces
between
the
beam
and
its
image
to
order
of
1/
the
electric
and
magnetic
forces
between
the
beam
and
its
image
to
order
of
l/f.
In
the
γ
.
In
the
the
electric
and
magnetic
forces
between
the
beam
and
its
image
to
order
of
1/
presence of
the
electron
cloud,
shift due
to image
charges should
be
presence
of the
the electron
electron cloud,
cloud, the
the coherent
coherent tune
tune
presence
of
the
coherent
tune shift
shift due
due to
to image
image charges
charges should
should be
be
added
to
the
one
caused
by
beam
image
charges:
added
to
the
one
caused
by
beam
image
charges:
added to the one caused by beam image charges:
∆ν
=Aimage
Av=A
image
∆ν
=A
∆ν
=Aimage
ηeγ2)
image(1Av=A
image(l-rief}
∆ν=Aimage
(1-ηeγ2)
(no
(no e_cloud)
e_cloud)
(no
e_cloud)
((with
with
e_cloud)
e_cloud)
( with e_cloud)
(1)
(1)
(1)
(2)
(2)
(2)
Where
Where ηr/e is
is the
the fractional
fractional neutralization
neutralization of
of the
the beam.
beam.
Where ηee is the fractional neutralization of the beam.
228
r]e=Hen(/nb= rje(z)
(3)
ηe=Henc/nb= ηe(z)
(3)
Where
the beam
beam density.
density.
Wherennc cisisthe
thecloud
clouddensity
density before
before the
the beam
beam and
and n
nbb is
is the
Note
that
Av
varies
along
the
bunch
providing
an
additional
mechanism
for head-tail
head-tail
Note that ∆ν varies along the bunch providing an additional mechanism for
offsets
to
form
and/or
grow.
offsets to form and/or grow.
Figure
charge of
of the
the cloud
cloud as
as
Figure77shows
showsthe
thetransverse
transverse force
force on
on the
the beam
beam due
due to
to the
the space
space charge
aafunction
of
the
longitudinal
coordinates.
The
force
is
stronger
at
the
tail
than
it
is
at
the
function of the longitudinal coordinates. The force is stronger at the tail than it is at the
head
nonuniform force
force on
on the
the beam
beam
headdue
duetotothe
thecloud
cloudcompression
compression behind
behind the
the beam.
beam. This
This nonuniform
eventually
causes
head-tail
instability
after
a
long
eventually causes head-tail instability after a long
run.
3a
3Vr r
Transverse
force in
in ZZdirection
direction
Transverse force
150
100
E r (z) (V/m)
50
0
-50
-100
-150
-200
-250
-0.5
0
0.5
1
z(m)
1.5
2
2.5
FIGURE7.7.Right:
Right: Beam
Beam offset
offset by
by 3a
3σrr,, Left:
Left: Plots
Plots of
σrr
FIGURE
of ∂Ψ/∂x
3^/3% at
at x=3
x=3a
Nextwe
westudy
studythe
the self
self consistent
consistent space
space charge
charge effect
effect of
Next
of the
the cloud.
cloud. Figure
Figure 88 shows
shows the
the
transversewake
wakepotential
potential due
due to
to the
the cloud
cloud space
space charge
charge at
transverse
at different
different cloud
cloud densities.
densities. The
The
potential isis normalized
normalized to
to the
the corresponding
corresponding density.
density. As
As seen
potential
seen from
from the
the figure
figure the
the self
self
66 to 1099 per cm33 range of the cloud density, which is
consistent
effect
is
negligible
in
10
consistent effect is negligible in 10 to 10 per cm range of the cloud density, which is
thetypical
typicalrange
rangeof
ofcloud
cloud density
density formed
formed in
in circular
circular accelerators,
the
accelerators, and
and start
start having
having effect
effect
109 9per
percm
cm33and
andhigher.
higher.
atat10
0
-2
0
20
40
60
-4
-4 -6
-8
-8
-10
80
80
__
11
n
/cm33
np=10
=10n/cm
_ _
9
3
n
np=10
=109/cm
/cm3
p
p
np=1066/cm33
-10
np=10 /cm
-12
-12
-14
-14
-16
-16
X(mm)
X(mm)
FIGURE 8. Normalized Wake potential due to the cloud space charge at different cloud densities
FIGURE 8. Normalized Wake potential due to the cloud space charge at different cloud densities
229
At last
last we
we study
study the beam evolution in the wake potentials above. In these simulations,
there is no
no external
external field
field (i.e.,
(i.e., no
no lattice),
lattice), and
and the
the emittance
emittance is artificially
artificially low
low thus
thus they
there
should be taken as cartoons to illustrate (and
(and in some
some sense
sense isolate) just the wakefield
should
effects on
on propagation.
propagation. Further
Further work
work is
is underway
underway to
to include
include the
the external
external environment
environment of
of
effects
storage ring.
ring. In
In particular,
particular, we
we have
have introduced
introduced betatron
betatron and
and synchrotron
synchrotron oscillations
oscillations
the storage
of the
the beam particles. These oscillations
oscillations are due to the external fields of the magnets and
RF power in the accelerator. Under the effect of these forces, individual particles (and
(and the
as aa whole,
whole, ifif off-centered)
off-centered) execute
execute oscillations
oscillations in
in all
all 33 spatial
spatial coordinates.
coordinates. The
The
bunch as
frequency of
of the
the transverse oscillation
oscillation of
of the single
single particle has been made dependent
frequency
its longitudinal
longitudinal momentum
momentum offset
offset to
to take
take into
into account
account also
also chromatic
chromatic effects
effects that
that
upon its
seem to play an
an important
important role
role in
in the
the unstable
unstable evolution
evolution of
of the
the beam.
beam. This
This work
work under
under
seem
collaboration with
with G.
G. Rumolo
Rumolo and
and F. Zimmermann
Zimmermann from
from CERN
CERN becomes
becomes in
in the
the future
future
publication [6].
[6].
In these
these simulations
simulations the
the 3-D
3-D time
time step
step is
is 50m.
50m. Figure
Figure 99 shows
shows snapshots
snapshots of
of the
the off
off –In
centered (3σ
(3arr) beam
beam and
and cloud
cloud at
at propagation distance
distance of
of Z=0,
Z=0, 15.3,
15.3, 20.4
20.4 and
and 27
27 km
km of
of
centered
SPS ring. We see
see the
the dynamic
dynamic focusing
focusing of
of the beam by the cloud and beam oscillation
the SPS
over the center
center of the
the pipe
pipe due
due to the restoring force of the cloud. At later times aa small
small
over
oscillations in
in the
the beam
beam and
and cloud
cloud density
density is
is seen.
seen. This
This oscillation
oscillation grows
grows despite
despite the
the
tail oscillations
fact that the beam and cloud are initially symmetric
symmetric except
except for
for small numerical noise. The
does not continue
continue to
to grow
grow in
in this
this example
example and
and saturates
saturates after
after 20
20 km
km of
of
instability does
propagation.
propagation.
In summary,
summary, we
we have
have applied simulation
simulation tools developed
developed and
and benchmarked for
for plasma In
accelerator research
research to the problem of beam propagation in circular accelerators
accelerators
based accelerator
with low
low density
density electron
electron clouds
clouds present.
present. We
We find
find the
the wakefields
wakefields compare
compare well
well with
with
with
analytic estimates
estimates and
and previous
previous models
models over
over most
most conditions.
conditions. We
We also
also find
find aa new
new
contribution to the coherent tune shift
shift of the accelerator due to electron cloud image
contribution
forces not
not included
included in
in previous
previous models.
models. ItIt appears
appears that
that the
the capability
capability for
for massively
forces
parallel computation
computation with
with QuickPIC
QuickPIC would
would enable
enable modeling
modeling with
with PIC
PIC accuracy
accuracy to
to be
be
parallel
extended to
to relevant
relevant lengths
lengths (i.e.,
(i.e., several
several thousand
thousand turns).
turns).
extended
£5
FIGURE 9.
9. Snapshots
Snapshots of
of cloud
cloud and
and Beam
Beam density
density
FIGURE
230
JS
50
61
ACKNOWLEDGEMENT
We would like to thank Frank Zimmermann for introducing us to this problem, G.
Rumolo, F. Decker, Wei Jei, C. Clayton, Bob Siemann, and the E-162 collaboration for
useful discussions. Work supported by:
USDOE#DE-FG03-92ER40745,NSF-PHY-00787815,DE-FC02-01ER41192,DE-FG0392ER40727, DE-FC02-01ER41179, PHY-0078508
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
T. Katsouleas et al, ECLOUD02 Conference
R. Hemker et al, Proc. 1999, Part. Accel. Conf. (1999)
C. Choung et al ,in preparation
D. H. Whittum, Phys. Plasmas, 4,1154, 1997
G. Rumolo and F. Zimmermann, "Longitudinal Wake due to Electron Cloud ", CERN notes.
G. Rumolo, A. Ghalam ," The Tune shift from an electron cloud in a circular
accelerator "in preparation for Phys,. Rev. 2002
231