Studies of the Coherent Half-Integer Resonance
Sarah Cousineau*, Jeff Holmes1^ John Galambos1^ Robert Macek**
Alexei Fedotov* and Jie Wei*
* Indiana University, Bloomington, Indiana
^Oak Ridge National Laboratory, Oak Ridge, Tennessee
**Los Alamos National Laboratory, Los Alamos, New Mexico
^Brookhaven National Laboratory, Upton, New York
Abstract. We present studies of space-charge-induced beam profile broadening at high intensities in the Proton
Storage Ring (PSR) at Los Alamos National Laboratory. Previous work has associated the observed broadening
in the vertical direction with the coherent half integer resonance [1]. Here, we study the effect of the space
charge environment on this resonance; specifically, we investigate the strength of the resonance versus beam
intensity, longitudinal bunching factor, transverse lattice tune, and two different beam injection scenarios. For
each case, detailed particle-in-cell simulations are combined with experimental results to elucidate the behavior
and sensitivity of the beam resonance response.
INTRODUCTION
The next generation of high intensity synchrotrons will
require unprecedented minimization of beam loss in order to control radiation activation of the machine. Many
of these accelerators will operate in an energy regime
where space charge effects are a primary mechanism for
halo development and beam loss. In striving towards successively higher beam intensities, an important task is to
understand space charge from both theoretical and experimental standpoints. The Proton Storage Ring (PSR)
at Los Alamos National Laboratory can operate with
very high space charge effects and therefore provides an
ideal setting for this type study. In this paper, we analyze experimentally observed beam broadening at high
intensities in the PSR ring. Both particle core model and
particle-in-cell (PIC) simulations are employed as tools
to interpret the data. In the PIC simulations, we pay particular attention to lattice, injection, and RF settings, and
in general find very good agreement between the simulated and experimentally measured beam profiles.
The quantity of merit in space charge studies has traditionally been the incoherent space charge tune shift
limit. Recently, experimental and computational work
has shown that the coherent tune bears more consequence on resonance behavior of the beam [2,3]. A complete analysis of the half-integer coherent resonance response was performed by Sacherer in his doctoral thesis [4], and we quote his results extensively here. He
used the particle core model to demonstrate that the onset of the half-integer resonance occurs at space charge
tune shifts in excess of the incoherent tune limit. He
later extended his analysis to rms second moments of
the beam distribution, thus making it applicable to realistic space charge scenarios. We have performed extensive
work to tie Sacherer's core model together with an independently developed rms core model which includes
dispersion [5,6] and also with PIC calculations and experimental data. The emphasis here is weighted heavily
towards the experimental side.
BEAM BROADENING AT HIGH
INTENSITY AT PSR
The PSR ring is 90 meters in length an can accumulate up to about 5 x 1013 protons. Beam profile broadening is observed at the highest operating intensities
(above 3 x 1013) and coincides with an escalation of
beam losses. For the experiment presented here, 7juC of
beam was accumulated over a period of 1.16ms, or about
3214 turns, injecting 1.36 x 1010 protons per turn. The
lattice tunes were set to their nominal working values
of (VK, Vy) = (3.19,2.19), and optimal injection settings
with vertical painting were used. Horizontal and vertical profiles were measured on a wire scanner in the extraction channel after the full accumulation of beam. The
intensity of the beam was varied by chopping the linac
beam pulses before injection into the ring. The profiles
in Figure 1 show the vertical beam profile at one-fourth
of the full intensity, one-half of the full intensity, and at
the full intensity. No significant difference is observable
between the one-fourth and one half intensity profiles,
whereas the full intensity case exhibits a large amount of
beam broadening.
In order to explore the area between the half and the
full intensities, we invoke PIC model simulations of the
full experiment. The simulations neglect magnetic errors,
strip foil scattering, and chromatic effects, but include a
complete set of transverse and longitudinal space charge
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
306
Experimental Vertical Beam Profiles
Experimental
Experimental Vertical
Vertical Beam
Beam Profiles
Profiles
Y Tune
Y Tune
1/4 Intensity
Intensity
1/21/4
Intensity
Intensity
Full1/2
Intensity
Full Intensity
-40
-40
-30
-30
-20
-20
-10
-10
0
10
0
10
Y [mm]
Y [mm]
20
20
3030
30
4040
40
FIGURE
1.1.1.Vertical
beam
profiles
measured
FIGURE
Vertical
beam
profi les
measuredonon
ona awire
a wire
wirescanscanFIGURE
Vertical
beam
profiles
measured
scanner
after
extraction
ofof
the
accumulated
ner
after
extraction
of
the
accumulatedbeam,
beam,shown
shownforfor
onener
after
extraction
the
accumulated
beam,
shown
foroneonefourth,
one-half,
and
full
intensity
beams.
fourth,
one-half,
and
full
intensity
beams.
fourth,
one-half,
and
full
intensity
beams.
Incoherent Tune versus Longitudinal Position
IncoherentTune
Tuneversus
versusLongitudinal
LongitudinalPosition
Position
Incoherent
Full Intensity
Full
Intensity
Full
Intensity
Half
Intensity
Half
Halfintensity
Intensity
-3-3
-3
-2
-2
-1
-1
00
1
0
1
Phi
[rad]
Phi
[rad]
Phi [rad]
2
2
3
3
FIGURE
FIGURE
Incoherent
tune
shifts
plotted
asasaaafunction
function
ofof
FIGURE2.2.
2. Incoherent
Incoherenttune
tuneshifts
shiftsplotted
plottedas
functionof
longitudinal
longitudinal
coordinate
the
ring
for
full
and
half
intensity
longitudinalcoordinate
coordinateinin
inthe
thering
ringfor
forfull
fulland
andhalf
halfintensity
intensity
beams.
beams.
The
dashed
line
corresponds
the
half-integer
stopbeams.The
Thedashed
dashedline
linecorresponds
correspondstototothe
thehalf-integer
half-integerstopstopband.
band.
band.
a aafact
fact
which
makes
for
interesting
experiments
related
toto
factwhich
whichmakes
makesfor
forinteresting
interestingexperiments
experimentsrelated
relatedto
the
longitudinal
bunching
factor.
the
longitudinal
bunching
factor.
the longitudinal bunching factor.
1.25
1.25
1.25
Normalized Amplitude
Normalized Amplitude
interactions;
interactions;therefore,
therefore,space
spacecharge
chargeprovides
providesthe
the
only
interactions;
therefore,
space
charge
provides
theonly
only
nonlinearity.
The
details
of
the
code
are
available
in
the
nonlinearity.
The
details
of
the
code
are
available
in
nonlinearity. The details of the code are available in the
the
ORBIT
reference
manual
[7].
ORBIT
reference
manual
[1].Although
Althoughthe
thebenchmarks
benchmarks
ORBIT
reference
manual
[7].
Although
the
benchmarks
are
not
shown
here,
good
are
not
shown
here,
goodagreement
agreementisis
isreached
reachedfor
for
each
are
not
shown
here,
good
agreement
reached
foreach
each
ofof
the
profiles
shown
in
Figure
1,
as
well
as
for
the
other
of
the
profiles
shown
in
Figure
1,
as
well
as
for
the
other
the profiles shown in Figure 1, as well as for the other
experimental
profiles
experimental
profilesdiscussed
discussedbelow.
below.
experimental
profiles
discussed
below.
Two
quantities
that
are
of
importance
Two
quantities
that
are
of
importance
indiagnosing
diagnosing
Two quantities that are of importanceinin
diagnosing
space-charge-inducedeffects
effectsonon
ona aabeam
beamare
arethe
thespacespacespace-charge-induced
space-charge-induced
effects
beam
are
the
spacecharge-depressed
incoherent
tunes,
and
the
second
mocharge-depressed
incoherent
tunes,
and
the
second
charge-depressed incoherent tunes, and the secondmomoment
of
the
beam,
both
available
to
us
through
the
ment
of
the
beam,
both
available
to
us
through
the
PIC
ment of the beam, both available to us through the PIC
PIC
simulations.
Figure
2 below
below
showsthe
theformer,
former,plotted
plotted
simulations.
Figure
2 2below
shows
simulations.
Figure
shows
the
former,
plottedasas
as
afunction
functionofof
of
longitudinaldistance
distanceinin
inthe
thering,
ring,for
forboth
both
a afunction
longitudinal
longitudinal
distance
the
ring,
for
both
the half
intensity and
full intensitycases.
cases. ThePSR
PSR longithe
thehalf
halfintensity
intensityand
andfull
fullintensity
intensity cases.The
The PSRlongilongitudinal
profile
is
sharplypeaked
peakedininthe
thecenter,
center,leading
leading to
tudinal
profile
is
sharply
tudinal profile is sharply peaked in the center, leadingtoto
large tune
depressions in this
area. For
the halfsize
size beam,
large
largetune
tunedepressions
depressionsininthis
thisarea.
area.For
Forthe
thehalf
half sizebeam,
beam,
the
maximally
depressedtune
tuneshifts
shiftsslightly
slightlyexceed
exceedthe
the
the
maximally
depressed
the maximally depressed tune shifts slightly exceed the
incoherent
half-integerthreshold,
threshold,νyVy==2.0.
2.0.Recall
Recallthat
that
incoherent
half-integer
incoherent half-integer threshold, νy = 2.0. Recall that
no
beam broadeningoccurs
occurs at thisstage,
stage, implyingthat
that
nono
beam
beambroadening
broadening occursatatthis
this stage,implying
implying that
the
incoherent
threshold
can
be
crossed
without
immedithe
theincoherent
incoherentthreshold
thresholdcan
canbebecrossed
crossedwithout
withoutimmediimmediate
consequence
to
the
beam.
At
thefull
fullintensity,
intensity,where
where
ate
consequence
to
the
beam.
At
the
ate
consequence
to
the
beam.
At
the
full
intensity,
where
the
broadening
is
seen,
the
beam
has
crossed
the
incoherthe
broadening
isisseen,
has
the
the
broadening
seen,the
thebeam
beamThe
hascrossed
crossed
theincoherincoherent
limit
by
an
additional
0.1.
limit
forthe
the
onset
of
ent
limit
by
an
additional
0.1.
The
limit
for
onset
ent
limit
by
an
additional
0.1.
The
limit
for
the
onsetofof
beam
broadening
relative
to
the
tune
shift
points
strongly
beam
broadening
relative
totothe
tune
strongly
beam
broadening
relative
the
tuneshift
shiftpoints
points
strongly
towards
a coherent
response
tothethe
resonance
condition.
towards
a acoherent
response
toto
resonance
condition.
towards
coherent
response
the
resonance
condition.
Furtherevidence
evidenceofofthethecoherent
coherentresonance
resonanceisisprepreFurther
Further
evidence
of thethe
coherent
resonance
is presented
in
Figure
3,
where
secondorder
ordermoments
moments
of
sented
in
Figure
3,
where
the
second
sented
in Figure
3, where
the
secondbeam
orderenvelope,
momentsof
of
the
beam,
normalized
to
the
matched
are
the
beam,
normalized
totothe
matched
beam
envelope,
are
the
beam,
normalized
the
matched
beam
envelope,
are
plotted
over oneturn
turn of theaccelerator.
accelerator.Here,
Here,the
therms
rms
plotted
over
plotted
overone
oneexecutes
turnofofthe
the accelerator.
Here,
thea sigrms
beam
envelope
four
oscillations
per
turn,
beam
envelope
executes
four
oscillations
per
turn,
a asigbeam
envelope
executes
four
oscillations
per
turn,
signature
trait
ofthethehalf-integer
half-integercoherent
coherentresonance.
resonance.The
The
nature
trait
ofof
nature
traitare
the half-integer
coherent
resonance.
The
moments
plotted
for
particles
in
the
middle
of
the
lonmoments
are
plotted
for
particles
ininthe
middle
ofofthe
lonmoments
are
plotted
for
particles
the
middle
the
longitudinal distribution, where the density is highest. An
gitudinal
distribution,
where
the
isishighest.
An
gitudinal
distribution,
where
thedensity
density
highest.
analysis of
the moments
outside
of this range
showsAna
analysis
of
the
moments
outside
of
this
range
shows
analysis
of the moments
outside
of this range
showsa a
sharp decrease
in the moment
amplitudes,
and frequensharp
decrease
in
the
moment
amplitudes,
and
frequensharp
decreasetowards
in the moment
frequencies tending
twice theamplitudes,
bare tunes and
as we
move
cies
tending towards
twice the
bare tunes
we
move
cies
towards
tunesasas
wethe
move
awaytending
from the
center.twice
This the
is anbare
indication
that
obaway
from the
center.
This
that
the
away
theprofile
center.
Thisisisananindication
indication
that
theobobservedfrom
beam
broadening
results
mainly
from
parserved
beam
broadening
results mainly
parserved
beamprofile
profile
broadening
mainlyfrom
from
particles located
at the very
center ofresults
the longitudinal
bunch,
ticles
ticleslocated
locatedatatthe
thevery
verycenter
centerofofthe
thelongitudinal
longitudinalbunch,
bunch,
2.25
2.25
2.25
2.2
2.2
2.2
2.15
2.15
2.15
2.1
2.1
2.1
2.05
2.05
2.05
2
22
1.95
1.95
1.95
1.9
1.9
1.9
1.85
1.85
1.85
•z.
One
OneTurn
TurnVertical
VerticalBeam
BeamMoments
Moments
One Turn Vertical Beam Moments
1.2
1.2
1.2
1.15
1.15
1.15
1.1
1.1
1.05
1.05
1.05
1
1
0.95
0.95
0.95
0.9
0.9
0.9
00
0
\J .
30 40
40 50
50 60
60
70
80
90
1010 2020 30
70
80
90
10
20
30
40
50
60
70
80
90
Distance[m]
[m]
Distance
Distance [m]
FIGURE3.3. Normalized
Normalizedsecond
secondmoments
momentsof
ofthe
the full intensity
intensity
FIGURE
FIGURE
3. plotted
Normalized
second
of thefull
full intensity
vertical
beam,
overone
one
turnmoments
ofthe
thering.
ring.
vertical
beam,
plotted
over
turn
of
vertical beam, plotted over one turn of the ring.
EXPLORATIVEEXPERIMENTS
EXPERIMENTS
EXPLORATIVE
EXPLORATIVE EXPERIMENTS
Thissection
sectionisisdedicated
dedicatedtotothree
threeexperiments
experimentsperformed
performed
This
This
section
isonset
dedicated
to
three experiments
to
explore
the
of
the
resonance
in
detail.
Inperformed
the first
first
totoexplore
the
onset
ofofthe
resonance
inindetail.
In
explore
the
onset
the
resonance
detail.
Inthe
the first
experiment,
the
longitudinal
profile
shape
was
altered
experiment,
the
profile shape
was
experiment,
thelongitudinal
longitudinal
wasaltered
altered
placingaanotch
notch
inthe
thecenter
centerprofile
ofthe
theshape
longitudinal
probyby
placing
in
of
longitudinal
proby
placing
a
notch
in
the
center
of
the
longitudinal
profile,
i.e.,
no
beam
was
injected
within
15
degrees
of
the
file,
i.e.,
nonobeam
was
injected
within
1515degrees
ofofthe
file,
i.e.,
beam
was
injected
within
degrees
the
centerofofthe
thebunch
bunchtrain;
train;additional
additional beam
beam was
was injected
injected
center
center
of
the
bunch
train;
additional
beam
was
injected
outsideofofthis
thisrange
range so that
that the
the final
final intensity of
of the
the
outside
outside
ofunaffected.
this rangeso
so that
the finalintensity
intensity
ofthe
the
beam
was
Beam
eventually
diffused
into
beam
unaffected.
Beam
eventually
diffused
into
beamwas
was
unaffected.
Beam
eventually
diffused
intothe
the
notched
region,
but
the
overall
peaking
was
greatly
renotched
region,
but
peaking
renotchedand
region,
butthe
theoverall
overall
peakingwas
wasgreatly
greatly
reduced,
the bunching
factor increased
from
the nomduced,
and
the
bunching
factor
increased
from
the
nomduced,
andofthe
bunching
inal
value
about
0.3 tofactor
about increased
0.45. Thefrom
effectthe
onnomthe
inal
0.3
totoabout
0.45.
The
effect
ononthe
inalvalue
valueofprofiles
ofabout
aboutis
0.3
about
0.45.4,
The
effect
the
transverse
shown
in Figure
where
the vertitransverse
profiles
isisshown
ininFigure
4,4,where
the
vertitransverse
profiles
shown
Figure
where
the
vertical profiles after the full accumulation with and without
cal
after
accumulation with
and without
calprofiles
profiles
afterthe
thefull
full
the
longitudinal
notch
areaccumulation
plotted. Notewith
thatand
the without
longithe
longitudinal
notch
are
plotted.
Note
that
the
longithe
longitudinal
notch
are
plotted.
Note
that
tudinally notched beam is about 15% thinner. the
FulllongiPIC
tudinally
beam isisabout
15%
thinner.
Full
tudinallynotched
notched
about
15%
thinner.
FullPIC
PIC
simulations
of the beam
experiment
show
that
the emittance
simulations
ofofthe
experiment
show
that
the
emittance
simulations
the
experiment
show
that
the
emittance
growth for the highest intensity case is greatly reduced
growth
growthfor
forthe
thehighest
highestintensity
intensitycase
caseisisgreatly
greatlyreduced
reduced
307
when
the
notch isis in
withthe
thefinal
finalrms
rmsemittance
emittance
when
whenthe
thenotch
in place,
place, with
with
the
final
rms
emittance
reaching
about77Inmm
mm··•mrad
mrad
with
thenotch,
notch,compared
comparedtoto
to
reaching
ππmm
reachingabout
mradwith
withthe
compared
the
9.5
mm··•mrad
mrad
reached
without
thenotch.
Thenotch
notch
the
ππmm
the9.5
9.5nmm
mradreached
reachedwithout
withoutthe
notch.The
The
notch
was
not
observedto
makeany
anydifference
differencein
theprofiles
profiles
was
wasnot
not observed
to make
make
any
difference
ininthe
the
profiles
in
lower
intensitycases,
wherethe
thebeam
beamisisisnot
notnear
nearthe
the
in
inlower
lowerintensity
cases, where
where
the
beam
not
near
the
resonance.
resonance.
resonance.
0.02
0.02
0.02
0.015
0.015
0.015
00 0
0.025
0.025
0.025
-0.005
-0.005
-0.005
-50
-40
-30
- 2-20
0 --10
1 0-10 00 0 10
1010 20
20 20 30
30 30 40
40 40 50
50 50
-50
-50 -40
-40 -30
-30-20
Y[mm]
Y
[mm]
Y [mm]
0.02
0.02
0.02
0.015
0.015
0.015
FIGURE
5.5. Vertical
Vertical
experimental
FIGURE
after
accumulaFIGURE5.
Verticalexperimental
experimentalprofiles
profiles
after
accumulation
ofofaa a55jUC
beam
17mm
painted
to to
µ5C
and
a beam
tion
µCbeam
beampainted
paintedto to17mm
17mm
and
a beam
painted
tionof
12mm.
The
solid
painted
beam
andand
thethe
dashed
12mm.
is is
thethe
large
dashed
12mm.The
Thesolid
solidredred
large
painted
beam
dashed
blue
isisthe
the
small
blue
painted
beam.
blueis
thesmall
small
painted
beam.
0.01
0.01
0.01
0.005
0.005
0.005
00
-40
-40
-40
17mm
Injection
17mm
Injection
17mm
injection
12mm
Injection
12mm
Injection
2mm
0.005
0.005
0.005
No
NoNotch
Notch
With
WithNotch
Notch
0.03
0.03
0.03
0.025
0.025
0.025
0.01
0.01
0.01
ComparisonofofExperimental
ExperimentalVertical
Vertical
Profiles
(Full
Intensity)
Comparison
Comparison
Experimental
VerticalProfiles
Profiles(Full
(FullIntensity)
Intensity)
0.035
0.035
0.035
Vertical
Beam,
Intensity
Vertical
Beam,
Full
Intensity
Vertical
Beam,
FullFull
Intensity
0.03
0.03
0.03
-30
-30
-30
-20
-20
-20
-10
-10
-10
10
000
10
10
YY[mm]
Y[mm]
[mm]
20
20
20
30
30
30
40
40
40
FIGURE 4.
4. Vertical
Vertical experimental
experimental
beam
profi
les for
for
FIGURE
µ
CC
FIGURE
4.
Vertical
experimentalbeam
beamprofiles
profiles
foraaa77juC
7µ
beam
with
and
without
a
longitudinal
notch.
The
red
solid
line
beam
with
and
without
a
longitudinal
notch.
The
red
solid
line
beam with and without a longitudinal notch. The red solid line
thebeam
beamwithout
withoutthe
the
longitudinal
notch
(bunching
factor
«≈
is
isisthe
the
beam
without
thelongitudinal
longitudinalnotch
notch(bunching
(bunchingfactor
factor≈
.3)
and
the
blue
dashed
beam
is
the
beam
with
the
longitudinal
.3)
and
the
blue
dashed
beam
is
the
beam
with
the
longitudinal
.3) and the blue dashed beam is the beam with the longitudinal
notch (bunchingfactor
factor « .45).
notch
notch (bunching
(bunching factor≈≈.45).
.45).
In the
the second
second experiment,
experiment, the
horizontal tune
was
In
In
the
second
experiment, the
the horizontal
horizontal tune
tunewas
was
kept atat the
the fixed
fixed value
value of
of νvxx == 3.19,
3.19, while
while the
the vertical
vertical
kept
kept at the fixed value of νx = 3.19, while the vertical
tune wasincrementally
incrementally lowered from
from the nominal
nominal v =
tune
tune was
was incrementallylowered
lowered fromthe
the nominalννyy y==
2.19 toto νvyy == 2.09
2.09 in
in steps
steps of
of 0.02.
0.02. The
The result
result was aa
2.19
was a
2.19 to νy = 2.09 in steps of 0.02. The resultwas
dramatic increase
increase in
in the
the rms
rms beam
beam width
width from
from 17mm
dramatic
dramatic increase in the rms beam width from17mm
17mm
at the
the highest tune
tune value to
to 22.5mm
22.5mm at
at the
the lowest
lowest tune
tune
at
at
the highest
highest
tune value
value to
22.5mm atlosses
the lowest
tune
value.
The
corresponding
experimental
increased
value.
The
corresponding
experimental
losses
increased
value.
The
corresponding
experimental
losses
increased
linearlywith
withdecreasing
decreasing tune
tune value
value up
up until
until the
the last
last data
data
linearly
linearly
with decreasing
tune value
up until most
the last
data
point, where
where
losses jumped
jumped
substantially,
likely
point,
losses
substantially,
most likely
point,
where
losses
jumped
substantially,
most
likely
causedby
byaaviolation
violationof
ofthe
thelimiting
limiting ring
ring aperture.
aperture.
caused
caused
by abeam
violation
of the limiting
ring aperture.
Finally,
broadening
was measured
measured
as aa funcfuncFinally,
beam broadening
was
as
Finally,
beamscheme.
broadening was
measured
as a vertifunction
ofpainting
painting
beam
was painted
painted
tion
of
scheme. AA55juC
µ C beam
was
vertition
of
painting
scheme.
A
5
µ
C
beam
was
painted
verticallytototwo
twodifferent
different sizes:
sizes: one
one beam
beam was
was painted
painted with
with aa
cally
cally
two different
sizes:offset,
one beam
wasother
painted
17mmtomaximum
maximum
injection
andthe
the
beamwith
wasa
17mm
injection
offset, and
other beam
was
17mm
maximum
injection
offset,injection
and the other
beam
was
painted
with
a
12mm
maximum
offset.
Profiles
painted with a 12mm maximum injection offset. Profiles
painted
with
a
12mm
maximum
injection
offset.
Profiles
recorded
at
half
of
the
run
intensity
(2.5
/iC)
show
that
recorded at half of the run intensity (2.5 µ C) show that
recorded
at painted
half of beam
the
runisisintensity
(2.5
C) show
that
the smaller
smaller
painted
beam
smaller atat
theµend
end
of accuaccuthe
smaller
the
of
the
smaller
painted
beam
is
smaller
at
the
end
of
accumulation
than
the
larger
painted
beam.
However,
with
the
mulation than the larger painted beam. However, with the
mulation
than
thecurrent,
larger painted
beam.
However,
with
the
full55jUC
of
beam
current,
thesmall
small
andlarge
largepainted
painted
verfull
µ C of
beam
the
and
verfull
5
µ
C
of
beam
current,
the
small
and
large
painted
verticalbeam
beamprofiles
profiles atatthe
the end
end of
of accumulation
accumulation are
are nearly
nearly
tical
tical
beam
profiles
at5).
theThe
endsmaller
of accumulation
are nearly
identical
(see
Figure
5).
The
smaller
painted beam,
beam,
subidentical
(see
Figure
painted
subidentical
(see
5).
Thespace
smaller
painted
beam, subject toto aamuch
muchFigure
more intense
intense
space
charge
environment,
ject
more
charge
environment,
ject
tomore
a much
moreand
intense
charge
environment,
reacts
more
quickly
and
morespace
severely
to the
the
coherent
reacts
quickly
more
severely
to
coherent
resonance
than
does
the
large
painted
beam.
reacts
more
quickly
and
more
severely
to
the
coherent
resonance than does the large painted beam.
PIC
simulations
of
the
experiment
show
that
the
small
resonance
than
does
the
large
painted
beam.
PIC simulations of the experiment show that the small
painted
beam
reacts
to
the
resonance
very
early
in
the
PIC
simulations
of
the
experiment
show
that
the
painted beam reacts to the resonance very early insmall
the
accumulation
stage,
and
displays
beam
moment
oscillapainted
beam
reacts
to
the
resonance
very
early
in
the
accumulation stage, and displays beam moment oscillations
that
are
about
10%
larger
than
those
of
the
large
accumulation
stage,
and
displays
beam
moment
oscillations that are about 10% larger than those of the large
painted
beam.
After this
this
initial
stage
ofgrowth,
growth,
the
small
tions
that
are After
about
10%
larger
than
those
ofthe
thesmall
large
painted
beam.
initial
stage
of
painted
beam
oscillates
with
moment
amplitudes
just
painted
beam.
After
this
initial
stage
of
growth,
the
small
painted beam oscillates with moment amplitudes just
slightly
higher
than
those
of
the
large
painted
beam,
and
painted
beam
oscillates
with
moment
amplitudes
just
slightly higher than those of the large painted beam, and
the emittance
emittance
of
thethose
smallof
beam
at the
thepainted
end of
ofbeam,
accumuslightly
higherof
than
the large
and
the
the
small
beam
at
end
accumuthe emittance of the small beam at the end of accumu-
308
lation
isisabout
about
8%
large
beam.
lation
thethe
lationis
about8%
8%greater
greaterthan
thanthat
thatofof
large
beam.
This
behavior
isisconsistent
consistent
Sacherer's
claims
that
This
Thisbehavior
behavioris
consistentwith
withSacherer’s
Sacherer’s
claims
that
the
emittance
response
to to
thethe
half
the
half
theemittance
emittanceofofthe
thebeam
beamgrows
growsinin
response
half
integer
thethe
space
charge
environinteger
integerresonance
resonanceand
andweakens
weakens
space
charge
environment.
In
an
accumulation
ment.
In
an
accumulation
scenario,
this
is
an
ongoing
ment. In an accumulation scenario, this is an
ongoing
process,
and
Sacherer's
process,
analytic
equaprocess,and
andaa acomparison
comparisonofofSacherer’s
Sacherer’s
analytic
equations with
with the
the PIC simulations
simulations shows that the PSR beam
tions
tions with the PIC simulations shows that the PSR beam
is sitting
sitting continuously at the very edge of the coherent
is
is sittingcontinuously
continuously at the very edge of the coherent
resonance threshold.
threshold.
resonance
emittance
ofof
thethe
beam
is growresonance threshold.The
The
emittance
beam
is growing constantly
constantly in excess of pure painting effects
effects in oring
ing constantly in excess of pure painting effects in order to
to compensate for the resonance condition. Eventuder
der tocompensate
compensate for the resonance condition. Eventually
the beam
beam
size exceeds
exceeds the
the limiting
limiting ring
aperture
and
ally
the
ring
aperture
andand
ally
the
beamsize
size
exceedslimiting
the limiting
ring
aperture
losses
escalate,
ultimately
the
attainable
beam
losses
escalate,
ultimately
limiting
the
attainable
beam
losses escalate, ultimately limiting the attainable beam
intensity.
intensity.
intensity.
CONCLUSIONS
CONCLUSIONS
CONCLUSIONS
Strong evidence
evidence supports
supports the
the conclusion
conclusion that
Strong
that beam
beam
Strong evidence
supports
the
conclusion
that beam
broadening
at
high
intensities
in
the
PSR
accumulator
broadening at high intensities in the PSR accumulator
broadening
at of
highcoherent
intensities
in the PSR
accumulator
ring
is aa result
result
half-integer
resonance
rering
is
of aa coherent
half-integer
resonance
rering
is
a
result
of
a
coherent
half-integer
resonance
response.
PIC
simulations,
all
well-benchmarked
with
exsponse. PIC simulations, all well-benchmarked with exsponse.
PIC
simulations,
all
well-benchmarked
with
experimental data,
data, show
show that
that space-charge-depressed
space-charge-depressed tunes
perimental
tunes
perimental
that space-charge-depressed
tunes
are
in excess
excessdata,
of the
theshow
incoherent
stopband limit
limit before
before the
are
in
of
incoherent
stopband
the
are
in
excess
of
the
incoherent
stopband
limit
before
onset of
of broadening,
broadening, and
and that
that the
the beam
beam envelope
envelope executes
onset
executesthe
onset
of
broadening,
and
that
the
beam
envelope
four oscillations
oscillations per
per turn,
turn, both
both strong
strong indicators
indicators of
ofexecutes
cofour
aa cofour
oscillations
per
turn,
both
strong
indicators
of the
a coherent
resonance.
Experiments
performed
to
explore
herent resonance. Experiments performed to explore
the
herent
resonance.
Experiments
performed
to
explore
resonance
show
that
at
high
intensity,
factors
such
as
lonresonance show that at high intensity, factors such as lon-the
resonance
showshape,
that attransverse
high intensity,
factors
such as
longitudinal
profile
shape,
transverse
painting
scheme,
and
gitudinal
profile
painting
scheme,
and
gitudinal
profile
shape,
transverse
painting
scheme,
and
variation
of
bare
tune
all
affect
the
response
of
the
beam
variation of bare tune all affect the response of the beam
to
the
resonance;
at
low
intensities,
before
the
onset
of
variation
of
bare
tune
all
affect
the
response
of
the
beam
to the resonance; at low intensities, before the onset of
resonance,
no significant
significant
changes
are observed
observed
from
the of
to the resonance;
at lowchanges
intensities,
before the
onset
resonance,
no
are
from
the
modification
of
these
parameters.
resonance,
no
significant
changes
are
observed
from
the
modification of these parameters.
This
work
is
supported
by
SNS
through
UT-Battelle,
modification
of
these
parameters.
This work is supported by SNS through UT-Battelle,
LLC,
under
contract
DE-AC05-OOOR22725
for
the
This
work
is supported
by SNS throughfor
UT-Battelle,
LLC,
under
contract
DE-AC05-00OR22725
the U.S.
U.S.
DOE,
and
by contract
Indiana University
University through
through aa DOE
grant,
LLC,and
under
DE-AC05-00OR22725
for the
U.S.
DOE,
by
Indiana
DOE
grant,
DE-FG02-92ER40747.
SNS is
is aa through
partnership
of
six
DOE, and by Indiana University
a DOE
DE-FG02-92ER40747.
SNS
partnership
of grant,
six
national
laboratories: Argonne,
Argonne,
Brookhaven,
Jefferson,
DE-FG02-92ER40747.
SNS Brookhaven,
is
a partnership
of six
national
laboratories:
Jefferson,
national laboratories: Argonne, Brookhaven, Jefferson,
Lawrence Berkeley, Los Alamos, and Oak Ridge. The
Los Alamos National Laboratory is operated by the Univesity of California for the U.S. DOE under contract W7405-ENG-36.
REFERENCES
1. J.A. Holmes, V. Danilov, J. Galambos, A. V. Fedotov, and
R. L. Gluckstern, in Proceedings of the Particle Accelerator
Conference, Chicago, 2001, p. 3188.
2. A. V. Fedotov and I. Hofmann, "Half-Integer Resonance
Crossing in High-Intensity Rings," PRST-AB 5, (2002)
024202.
3. A. Uesugi, S. Machida, and Yoshiharu Mori, "Experimental
study of a half-integer resonance with space-charge effects
in a synchotron," PRST-AB 5, (2002) 044201.
4. F. J. Sacherer, "Transverse Space-Charge Effects in Circular
Accelerators," PhD Thesis, University of California,
Berkeley, (1968).
5. S.Y. Lee and H. Okamoto, Phy. Rev. Lett. 80, 124201
(1999).
6. J. Holmes, J. Galambos, D. Olsen, S.Y. Lee, in Workshop
on Space Charge Physics in High Intensity Hadron Rings,
Shelter Island, NY, 1998, p. 254.
7. J. Galambos et al., ORBIT User's Manual, SNS/ORNL/AP
Technical Note 011,1999.
309