Modeling of Space Charge Induced Resonance Loss
G. Franchetti and I. Hofmann
GSI, 64291 Darmstadt, Germany
Abstract. For high-current circular machines like the SIS 100, which has been proposed as driver for a future
radioactive beam facility at GSI, an exploratory study of space charge effects on resonances during long-term
beam storage is indispensable. We discuss the effect of space charge on sextupole error resonances by a 4D
model using an analytical expression for a stationary space charge. The effect of space charge induced resonance
crossing in a bunch beam is approached by a simplified model.
INTRODUCTION
Beam loss control is an important issue for high-intensity
heavy ion rings [1]. Long term bunched beam storage,
typically 106 turns, and resolution of beam loss under
the percent level are very demanding for PIC simulation.
As pointed out in [2] numerical noise generated in PIC
solvers is responsible for an artificial emittance growth.
This effect may get controlled by increasing the number
of macroparticles used in the simulations at expenses of
larger CPU time. A study of this effect in 3D simulation
was presented in [3]. Since small beam loss is of concern,
a particle-core model provides a way to avoid numerical
noise artifacts while retrieving important aspects of the
dynamics during long term beam storage. If loss are
found to be small and collective effects may be neglected
a frozen core model may give reasonable predictions.
where () = d/ds. In Eq. 2 we use the tuneshift as a
measure of the beam intensity, while the perveance is
given by K = 2(a/C) 2 (27r) 2 (2v 0 Av - Av 2 ). Equation 2
can be integrated through a micromap + nonlinear kick
symplectic method. Dividing the ring in N integration
steps the nonlinear transport map becomes
y
2nv,
(3)
with 3* = sin(27TV0/AO, and <*f = cos(27TV0/A^). The
nonlinear kick fn is fn(x) = jj (27r)2(2v0Av - Av 2 ) (1 e~ */2)/x. The lattice nonlinearities can be added to
fn(x) at any integration step through the kick AmX™. Here
Am is the integrated strength of a m-th order nonlinear
thin element in the frame {s,x}. The relation with its
integrated strength Km in the laboratory frame is Am =
2D MODEL
The model is based on a coasting beam in a constant
focusing lattice. We consider a 2D axisymmetric beam
with frozen transverse Gaussian particle distribution of
rms radius a. The analytic space charge electric field
[4] used in the particle equation of motion provides a
particle-core model. In order to simplify the discussion
we consider initially the particle dynamics only in the xplane setting for any test particle y = yr = 0. The equation
of motion becomes
where ()' = d/ds, V0 is the ring bare tune, R is the ring
radius, K is the perveance, r =
In this case
r = x. Calling C = 2nR, and by using the coordinate
transformation x = xa, s = sC we rescale Eq. 1 to the
dimensionless equation
(2)
RESONANCES AND SPACE CHARGE
With this model we study the effect of space charge on
lattice nonlinearity induced resonances. Here the space
charge acts as a perturbation leaving unchanged the main
mechanisms of single particle resonances. We consider a
ring with V0 = 13.36, a beam with maximum tuneshift
Av = 0.2, and use 1060 integration steps to resolve the
dynamics with space charge. With this choice the resonances 3v0 = 40, and 4v0 = 53 can be met if the depressed tune scans over v = 13.16, ..., 13.36. We show in
Fig. la the nonlinear tune v for particles with initial coordinates x = 0. The calculation of v is obtained with a
FFT interpolated method (see in [5]) by using the particle coordinates in a fixed ring section over 2048 turns. As
expected, at the center of the coasting beam v = 13.16,
which moves asymptotically towards V0 = 13.36 if the
particle is taken very far from the beam center. Note that
third and fourth order resonances are crossed at x = 5 .5 cr,
and x = 2a respectively. We excite these two resonances
by using 20 equal sextupolar kicks with A2 = 0.01, and
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
260
3 q = 40
133
13.3
133
13.3
1325
13.25
1325
13.25
4q
= 53
4q=
53
132
13.2
132
13.2
px C / σ
5
2
4
6
0
x/v
x/σ
x/σ
0
e mo
600
ortail
tailthe
thecurrent
currentisisminimum.
minimum.The
Theglobal
globaleffect
effectisisthat
that
or
a
single
particle
experiences
an
oscillating
instantaneous
a single particle experiences an oscillating instantaneous
beamcurrent
currentwith
withfrequency
frequencymuch
muchslower
slowerthan
thanthe
thebetabetabeam
tron
oscillations.
We
consider
first
a
linear
current
ramp
tron oscillations. We consider first a linear current ramp
whichtakes
takesthe
thetuneshift
tuneshifttoto∆Av
0.2.InInfirst
firstorder
orderthe
the
which
ν ==0.2.
current ramp
rampcauses
causesa amigration
migrationofofislands
islandsconsistent
consistent
current
withFig.
Fig.1d.
Id.IfIfthe
thetest
testparticle
particlehas
hasinitial
initialcoordinates
coordinates
with
0 =
= 0,0,;t
1.5a
thethird
thirdorder
orderisland
islandeventually
eventuallywill
will
xy
x <<7.5
σ the
crossthe
theparticle
particlephase
phasespace
spaceorbit.
orbit.The
Thespeed
speedofofthe
theis-iscross
landatatthe
thecrossing
crossingisiscrucial
crucialfor
forthe
theeffect
effectonona aparticle.
particle.
land
In
Fig
2
we
explore
this
effect.
In
the
left
column
In Fig 2 we explore this effect. In the left column wewe
b)
c)
8
2
4
6
x/σ
d)
px C / σ
^400
400
6
200
200
x/σ
ν
ν
a)
a)
200
2
4
0
-200
-200
0
2
4
0
-400
^00
-5
-5
-2.5
-23
00
2.5
23
5
7.5
x/σ
0
-2
-200
0
0.05
0,05
0.1
0,1
0.15
0,15
0.2
∆ν
-4
FIGURE 1. a) space
space charge
charge induced
induced nonlinear
nonlinear tunes,
tunes,b)b)efeffect of resonances on nonlinear
fect
nonlinear tunes,
tunes, c)
c) Poincare’
Poincare' section
sectionfor
for
= 0.2,
0.2, d)
d) amplitude
ν.
∆ν =
Av
amplitude of
of 3rd
3rd order
order island
islandas
asfunction
function ofof∆Av.
px C / σ
-4
400
-2
0
2
4
x/σ
0
500
1000
1500
turns
2000
0
500
1000
1500
turns
2000
0
500
1000
1500
2000
1000
1500
2000
turns
x/σ
-600
-<fOO
b)
2
200
Dynamics in a Bunch
Dynamics in a Bunch
The situation may be different for the single particle
The situation
may In
be fact
different
for theoscillations
single particle
dynamics
in a bunch.
synchrotron
indynamics
in
a
bunch.
In
fact
synchrotron
oscillations
induce oscillations of the instantaneous current: when the
duce
oscillations
of
the
instantaneous
current:
when
the
particle is in the center of the bunch the local transparticle
is in is
themaximum
center ofwhereas
the bunch
the bunch
local transverse current
on the
head
verse current is maximum whereas on the bunch head
0
0
-200
-2
-400
-4
400
-2
0
2
4
6
x/σ
x/σ
-4
px C / σ
53 equal
equal octupolar
octupolar kicks
53
kicks with
with A
A33 =
= 0.005.
0.005. In
In Figs.
Figs. 1b,c
lb,c
we
show
the
nonlinear
tune
and
Poincaré
section.
we show the nonlinear tune and Poincare section. The
The
amplitude of
amplitude
of the
the island
island may
may be
be measured
measured by
by the
the size
size of
of
the flat
flat region
region in
the
in the
the nonlinear
nonlinear tune
tunewhich
whichcrosses
crossesthe
theresresonance. The
The position
onance.
position of
of the
the fixed
fixed point
point isis where
whereνV00−—∆Aνv
equals the
the harmonic
harmonic of
equals
of the
the nonlinearity
nonlinearity driving
drivingterm;
term;the
the
area of
of the
the islands
area
islands seems
seems to
to depend
depend on
on the
the gradient
gradient of
of
the nonlinear
nonlinear tune:
the
tune: the
the weaker
weaker the
the nonlinear
nonlinear tune
tune gradigradient the
the bigger
bigger is
ent
is the
the island.
island. From
From the
the example
example in
inFig.
Fig.1b
Ib
it
follows
that
large
islands
are
obtained
at
6
σ
.
Setting
it follows that large islands are obtained at 6cr. Setting
ν itit isis possible
properly the
the bare
properly
bare tune
tune νV00 and
and varying
varying ∆Av
possible
to
create
islands
at
any
position.
In
Fig.
to create islands at any position. In Fig. 1d
Id we
we plot
plot the
the
outer
and
inner
separatrix
of
the
third
order
resonance
outer and inner separatrix of the third order resonance
(at xx̃˙ =
0) as
(at
= 0)
as function
function of
of the
the space
space charge
charge intensity
intensity exexpressed
in
terms
of
∆
ν
.
The
bare
tune
is
still
pressed in terms of Av. The bare tune is still νV00 =
= 13.36.
13.36.
This picture shows consistently with the interpretation of
This
picture shows consistently with the interpretation of
Figs. 1a and b) that the third order islands move outward
Figs, la and b) that the third order islands move outward
and increase their areas as the tuneshift increases. From
and increase their areas as the tuneshift increases. From
this study we find that typically we may create big isthis study we find that typically we may create big islands far beyond 3σ , whereas the islands within 3σ are
lands far beyond 3(7, whereas the islands within 3(7 are
much smaller.
much smaller.
c)
6
4
200
0
2
-200
0
-400
-2
-4
-2
0
2
4
6
x/σ
500
turns
FIGURE 2. Phase space orbit (left) and time evolution of the
FIGURE
2. Phase
space
orbitthird
(left)turn
and(right).
time evolution of the
particle
amplitude
plotted
every
particle amplitude plotted every third turn (right).
plot pictures of the single particle orbits. In the right colplot pictures of the single particle orbits. In the right column the time evolution of the x̃ coordinate. In the case
umn the time evolution of the x coordinate. In the case
a) (first two pictures at the top) a positive linear ramp
a) (first two pictures at the top) a positive linear ramp
brings the tuneshift to ∆ν = 0.2 in 2048 turns. The parbrings
the tuneshift to Av = 0.2 in 2048 turns. The particle initial coordinates are x̃ = 4, x̃˙ = 0. At the beginticle
initial
coordinates
areelliptic
x = 4,£
= 0.the
Atfrequency
the beginning the particle
lies on an
orbit,
ning
the
particle
lies
on
an
elliptic
orbit,
the
frequency
of the oscillations (picture on the right) decreases
due to
of
the
oscillations
(picture
on
the
right)
decreases
due to
the nonlinear tune change for effect of the space charge
the
nonlinear
tune
change
for
effect
of
the
space
charge
ramp. The increasing space charge moves the third orramp. The increasing space charge moves the third or-
261
νy
x/σ
7.5
a)
5
b)
13.5
2.5
13.4
0
-2.5
13.3
-5
4
0.2
0.4
0.6
turns
0.8
13.2
1
6
x 10
c)
13.7
13.8
d)
2.5
0
0
-2
-2.5
-5
-4
0.2
0.2
0.4
0.4
0.6
0.6
turns
0.8
0.8
1
x 10
xlO
6
0
0.2
0.4
0.6
turns
0.8
we plot
plot the
the third
third order
order resonance
resonance lines
we
lines and
and the
the tune
tune footfootprint
for
test
particles
distributed
over
a
larger
area
print for test particles distributed over a larger area comcomparedwith
with the
the beam
beam size.
size. Note
Note that
pared
that the
the vicinity
vicinity of
ofthe
theline
line
2
ν
ν
−
=
41
is
evacuated,
which
shows
the
existence
y
x
2Vy — vx = 41 is evacuated, which shows the existenceof
of
stop band.
band. The
The position
position of
aa stop
of this
this resonance
resonance isis chosen
chosentoto
be near
near the
the bare
bare tunes.
tunes. According
According to
be
to the
the 2D
2D interpretainterpretation
we
expect
again
that
the
periodic
resonance
tion we expect again that the periodic resonance crosscrossing will create a diffusion, and the resonance trapping
ing
will create a diffusion, and the resonance trapping
would eventually bring the particle at the farest position.
would eventually bring the particle at the farest position.
These effects are found in Figs. 3c,d for a particle with
These effects˙ are˙ found in Figs. 3c,d for a particle with
ỹ = x̃ = 1.5, x̃ = ỹ = 0. This study shows that in a bunched
y = x=l.5,x = y = Q. This study shows that in a bunched
beam the most dangerous resonances appear to lie near
beam the most dangerous resonances appear to lie near
the bare tune and not in the most densely populated part
the bare tune and not in the most densely populated part
of the footprint corresponding to the bunch case. Conseof
the footprint corresponding to the bunch case. Consequently the single particle dynamics in a bunch is then
quently
the single
in a bunch
is then
very sensitive
withparticle
respectdynamics
to the particle
synchrotron
very
sensitive
with
respect
to
the
particle
synchrotron
amplitude as it effects the tune oscillations. In order to
amplitude
as it induced
effects the
tuneperiodic
oscillations.
In order
to
quantify losses
by the
resonance
crossquantify
losses
induced
by
the
periodic
resonance
crossing a 3D self consistent space charge calculation along
ing
self consistent
space
charge calculation
along
witha a3D
consistent
modeling
of synchrotron
oscillation
is
with
a
consistent
modeling
of
synchrotron
oscillation
is
needed.
needed.
REFERENCES
4D Tracking and Outlook
REFERENCES
262
1
6
x 10
FIGURE 3.
3. a)
a) Particle
Particle evolution
evolution in
FIGURE
in aa bunch
bunch (2D
(2D case),
case), b)
b)
resonance lines
lines and
and tune
tune footprint
footprint for
resonance
for 4D
4D case,
case, c),d)
c),d) particle
particle
evolution in
in the
the 4D
4D case.
case.
evolution
4D Tracking and Outlook
We repeat the same simulation for the 4D system (the
We repeatỹthe
for the
system
condition
= ỹ˙same
= 0 simulation
is removed).
The4D
bare
tunes(the
are
condition
y =νy ==013.58,
is removed).
The bare tuneshift
tunes are
νx0 = 13.91,
and
the
maximum
is
y0
v^
13.91,
v 00.3.
= 13.58,
and the
tuneshift
∆ν y =
We excite
nowmaximum
the harmonics
41 is
by
∆ν=x =
AV.K
= Av-y
= sextupolar
0.3. We excite
41 by
using
equal
errorsnow
withthe
A2 harmonics
= 0.01. In Fig.
3b
using equal sextupolar errors with A2 = 0.01. In Fig. 3b
νx
5
2
00
13.6
y/σ
0
x/σ
der
derislands
islandsoutward.
outward.When
Whenthe
theislands
islandsreach
reachthe
theparticle
particle
orbit
orbitthe
thethird
thirdorder
orderresonance
resonancetries
triestotocapture
capturethe
thepartiparticle.
cle.This
Thishappen
happenfor
forhalf
halfofofananisland
islandrevolution.
revolution.AfterAfterwards,
wards,the
theparticle
particleisiscloser
closerthe
theinner
innerseparatrix,
separatrix,and
anddue
due
totothe
theoutward
outwardmotion
motionofofthe
theisland
islandititcrosses
crossesthe
thesepaseparatrix
ratrixagain
againand
andgets
getsunlocked
unlockedfrom
fromthe
theresonance.
resonance. The
The
total
totaleffect
effectisisthat
thatthe
theparticle
particleorbit
orbitgets
getskicked
kicked inward
inward
bybyananamplitude
amplitudecomparable
comparablewith
withthe
theisland
islandwidth,
width,then
then
itsitsorbit
orbitamplitude
amplituderemains
remainsunaffected.
unaffected.The
Theorbit
orbitjump
jumpisis
visible
visibleafter
after1000
1000turns
turnsininthe
theleft
leftpicture.
picture. Simulations
Simulations
showed
showedalso
alsothe
theinverse
inverseeffect
effectfor
foraadecreasing
decreasingcurrent:
current:
thiscase
casethe
theorbit
orbitjump
jumpisisoutward.
outward.We
Weconclude
concludethat
that
ininthis
theresonance
resonancecrossing
crossingcauses
causesan
anorbit
orbitjump
jumpwhich
whichhas
hasaa
the
directionopposite
oppositetotothe
themotion
motionofofthe
theislands.
islands.The
Theparpardirection
tialcapture
captureprocess
processdepends
dependsstrongly
stronglyon
onthe
theparticle
particleand
and
tial
islandposition
positionininthe
thephase
phasespace
spaceand
andon
onthe
thespeed
speedofofthe
the
island
islandwhen
whenapproaching
approachingthe
theparticle.
particle.IfIfaaparticle
particleinside
inside
island
islandduring
duringhalf
halfa arevolution
revolutionaround
aroundthe
thefixed
fixedpoint
point
ananisland
movesless
lessthan
thanthe
theisland
islandsize,
size,ititmay
mayget
gettrapped
trapped and
and
moves
follow
the
fixed
point.
In
Figs.
2b,c
we
show
two
examfollow the fixed point. In Figs. 2b,c we show two examplesofofparticles
particlestrapped
trappedby
bythe
theresonance.
resonance.In
Inb)
b)the
thepartipartiples
˙
cle
with
x̃
=
3.3,
x̃
=
0
is
trapped
and
moves
outwards
as
cle with x = 3.3,^ = 0 is trapped and moves outwards as
thecurrent
currentincreases
increaseslinearly
linearlytotoreach
reachAv
∆ν==0.2.
0.2.In
Inc)
c) aa
the
particlewith
withx x̃==6,x
6, x̃˙==00gets
getstrapped
trappedand
andmoves
movesinward
inward
particle
following
the
island.
Whenthe
thetrapping
trappingcondition
condition gets
gets
following the island. When
broken,
the
orbit
gets
unlocked
and
follows
the
usual
elbroken, the orbit gets unlocked and follows the usual ellipse.
From
these
simulations
we
conclude
that
a
particle
lipse. From these simulations we conclude that a particle
a bunchmay
maybebesubject
subjecttotoperiodic
periodicresonance
resonancecrossing:
crossing:
inina bunch
the
islands
oscillate
back
and
forth
periodically,
kickthe islands oscillate back and forth periodically, kicking
the
particle
orbit
in
and
out.
The
single
particle
dying the particle orbit in and out. The single particle dynamics
has
then
a
nonlinear
resonance
induced
stochasnamics has then a nonlinear resonance induced stochasregime.Over
Overmany
manysynchrotron
synchrotronoscillations
oscillationsthe
thepartipartiticticregime.
cle may reach the trapping condition and get transported
cle may reach the trapping condition and get transported
to the farest transverse position determined by the posito the farest transverse position determined by the position of the island at the maximum intensity. Fig 3a shows
tion of the island at the maximum intensity. Fig 3a shows
the evolution of the x̃ coordinate of one particle with
the evolution
of the x coordinate
of one particle with
6 turns. The
x̃ = 1.4, x̃˙ = 0 over 10
space charge oscilx = l.4,x = 0 over 106 turns. The space charge oscillations are simulated with ∆ν = 0.2[1 + cos(2πνs n)]/2,
lations are simulated with Av = 0.2[1
+cos(2nvsn)]/2,
where synchrotron tune is νs = 10−3
3 . Note the diffusivewhere
synchrotron
tune
is
v
10~
.
Note
the diffusive5
s =
like process in the first 2 · 105 turns while above it spikes
like
process
in
the
first
2
•
10
turns
while
above
it spikes
nature
bring the particle to ∼ 7σ . Due to the reversible
bring
the
particle
to
~
la.
Due
to
the
reversible
nature
of the resonance trapping process the particle does
not
ofremain
the resonance
trapping
process
the but
particle
does not
long at the
maximum
distance
falls within
3σ
remain
longit atgets
thetrapped
maximum
distance
falls within
and later
again
and is but
brought
to 6 − 3(7
7σ .
and
laterthat
it gets
trapped
and of
is brought
6 —la.
Note
∼ 6σ
is theagain
position
the fixedtopoint
for
Note
that
~
6a
is
the
position
of
the
fixed
point
for
∆ν = 0.2 as shown in Fig 1d.
Av = 0.2 as shown in Fig Id.
1.
1.
2.
2.
3.
3.
4.
4.
C. D. R. http://www.gsi.de/GSI Future/cdr/ .
C.
D. R. http://www.gsi.de/GSI
J. Struckmeier,
Phys. Rev. E54, Future/cdr/
830 (1996). .
J.J. Struckmeier,
830 (1996).
Qiang and R.Phys.
Ryne,Rev.
theseE54,
Proceedings.
J.M.Qiang
and R. Ryne,
theseand
Proceedings.
B. Ottinger,
T. Tajima,
K. Hiramoto, 1653 , 6th
M.
B. Ottinger,
T. Accelerator
Tajima, and Conference,
K. Hiramoto,Stockholm,
1653 , 6th
European
Particle
European
Particle
Conference, Stockholm,
Sweden, 22-26
JunAccelerator
1998.
22-26
JunPart.
1998.Accel. 52, 147 (1996).
5. Sweden,
R. Bartolini
et al.,
5. R. Bartolini et a/., Part. Accel. 52, 147 (1996).