Behaviorof
ofIntense
IntenseBeams
BeamsSimulated
with
Simpsons
Intense
Beams
Simulatedwith
withSimpsons
Simpsons
Behavior
Shinji Machida
ShinjiMachida
Machida
Shinji
KEK, Accelerator Research Laboratory
KEK,
Accelerator
Research
Laboratory
KEK,
Accelerator
Research
Laboratory
1-1 Oho, Tsukuba-shi,
Ibaraki-ken,
305-0801,
JAPAN.
([email protected])
1-1Oho,
Oho,Tsukuba-shi,
Tsukuba-shi,Ibaraki-ken,
Ibaraki-ken,305-0801,
305-0801,JAPAN.
JAPAN.([email protected])
([email protected])
1-1
Abstract. We will show some examples of space charge simulation results using the code Simpsons [1]. We also
Abstract.
We
will
show
some
examples
spacecharge
charge
simulation
resultsusing
usingthe
thecode
codeSimpsons
Simpsons[1].
[1]. We
Wealso
also
Abstract.
We
will
show
some
examples
space
simulation
results
discuss
single
and
multi
particle
behavior
asofaoffunction
of simulation
parameters.
discusssingle
singleand
andmulti
multiparticle
particlebehavior
behaviorasasa afunction
functionofofsimulation
simulationparameters.
parameters.
discuss
amplitude within which the corresponding fractions of the
INTRODUCTION
INTRODUCTION
INTRODUCTION
amplitudewithin
whichthe
thecorresponding
correspondingfractions
fractions ofof the
the
amplitude
which
particles
arewithin
enclosed.
particlesare
areenclosed.
enclosed.
particles
Space
charge
effects
in a synchrotron
are
Space charge
charge effects
effects inin a a synchrotron
synchrotron are
are
Space
characterized
with
space
charge
detuning
and
characterized
with
space
charge
detuning
and
characterized
with space
spacecharge
chargeeffects
detuning
and
resonance.
fact,
inin aa rapid
resonance.In
fact,space
spacecharge
chargeeffects
effectsin
rapid
resonance.
InInfact,
a inrapid
cycling
synchrotron
(RCS)
such
as
3
GeV
RCS
JKJ
cyclingsynchrotron
synchrotron(RCS)
(RCS)such
suchasas3 3GeV
GeVRCS
RCSininJKJ
JKJ
cycling
accelerator
complex
and
50
GeV
MR
show
quite
acceleratorcomplex
complexand
and5050 GeV
GeV MR
MR show
show quite
quite
accelerator
different
RCS,
everything
isistransient:
differentbehaviors.
behaviors.In
Inaa aRCS,
RCS,everything
everythingis
transient:
different
behaviors.
Inas
transient:
bunch
shape
as
well
transverse
emittance
is being
bunch
shape
as
well
as
transverse
emittance
being
bunch shape
asso-called
well as transverse
emittance
isisbeing
created
during
painting
process
at
injection.
created
during
so-called
painting
process
at
injection.
created
during so-called
painting process
atbefore
injection.
In
acceleration
begins
the
Inaddition,
addition, quick
quick acceleration
acceleration begins
begins before
before the
the
In
addition,
quick
particle
distribution
isissettled
down.
Here,
the
stability
particle
distribution
settled
down.
Here,
the
stability
particle distribution is settled down. Here, the stability
ofofthethebeam
issue.
InInMR,
on
beamcore
corebecomes
becomesaa aprimary
primaryissue.
issue.In
MR,on
on
of the beam
core
primary
MR,
the
other
hand,
abecomes
well-shaped
bunch
isisinjected
and
the
other
hand,
a
well-shaped
bunch
injected
and
the other hand, a well-shaped bunch is injected and
any
ofofcore
are
minimized,
atat
anycoherent
coherentoscillations
oscillationsof
coreare
areminimized,
minimized,at
any
coherent
oscillations
core
least,
a
the
beginning.
The
main
concern
of
that
type
least,a athe
thebeginning.
beginning.The
Themain
mainconcern
concernofofthat
thattype
type
least,
ofofmachine
slow
and
continuous
halo
formation
and
machineis
slowand
andcontinuous
continuoushalo
haloformation
formationand
and
of machine
isisslow
resulting
beam
loss.
InInthis
paper,
first
we
will
show
resulting
beam
loss.
this
paper,
first
we
will
show
resulting beam loss. In this paper, first we will show
an
example
space
charge
effects
ininaaaRCS.
RCS.
After
exampleof
spacecharge
chargeeffects
effectsin
RCS.After
After
ananexample
ofofspace
raising
a
question
of
single
and
multi
particle
behavior
raising
a
question
of
single
and
multi
particle
behavior
raising a question of single and multi particle behavior
in
discuss
beam
behavior
ininMR.
MR.
simulation,we
wewill
willdiscuss
discussbeam
beambehavior
behaviorin
MR.
ininsimulation,
simulation,
we
will
The
space
charge
force during
during the
the injection,
injection,
The space
space charge
charge force
force during
The
the injection,
however,
changes
the
resulting
distribution.
The
however, changes
changes the
the resulting
resulting distribution.
distribution. The
The
however,
asymmetric
emittance
at
the
beginning
induces
the
asymmetric emittance
emittance atat the
the beginning
beginning induces
induces the
the
asymmetric
coupling
through
space
charge
effects.
Figure
2
shows
couplingthrough
throughspace
spacecharge
chargeeffects.
effects.Figure
Figure22shows
shows
coupling
the
horizontal
and
vertical
emittance as
as aa function
function ofof
the
horizontal
and
vertical
emittance
the
horizontal
and vertical
emittancepainting
as a function
of
turn
number
during
anti-correlated
withspace
space
turnnumber
numberduring
duringanti-correlated
anti-correlatedpainting
paintingwith
with
turn
space
charge
effects.
We
observed five
five different
different emittance,
emittance,
chargeeffects.
effects.We
Weobserved
observed
charge
five and
different
namely
38%,
68%,
90%,
95%,
99%.emittance,
Those are
are
namely
38%,
68%,
90%,
95%,
and
99%.
Those
namely
38%,
68%,
90%, 95%,
and the
99%.
Those are
defined
as
the
amplitude
in which
which
corresponding
defined
as
the
amplitude
in
the
corresponding
defined as the amplitude in which the corresponding
fractions
are enclosed.
enclosed.
fractions of
of the
the particles
particles are
ANTI-CORRELATEDPAINTING
PAINTING
ANTI-CORRELATED
PAINTING
ANTI-CORRELATED
150
150
100
100
50
50
0
0
00
0
50
vertical amplitude (pi mm-mrad)
vertical amplitude (pi mm-mrad)
horizontal amplitude (pi mm-mrad)
horizontal amplitude (pi mm-mrad)
200
200
100 150 200 250 300 350
50
150
200
250 300
300 350
350
turn200
number
50 100
100 150
250
turn
turn number
number
150
150
100
100
50
50
0
300 350
350
50 100
100 150 200 250 300
0 00 50
0
50 100 150
200 250 300 350
turn
turn number
number
turn number
600
600
edge380_125ma
edge380_125ma
500
500
400
400
300
300
200
200
100
100
0
50 100
100 150
150 200
200 250
250 300
300 350
350
0 00 50
0
50 100 150
200
250 300 350
turnnumber
number
turn
turn number
SINGLEAND
ANDMULTI
MULTIPARTICLE
PARTICLE
SINGLE
PARTICLE
SINGLE
BEHAVIOR
BEHAVIOR
In order
order to make
make sense
multi
particle
simulations,
In
senseofof
ofmulti
multiparticle
particlesimulations,
simulations,
In
order totomake
sense
saturations of
of parameters
parameters have
to
be
checked.
In space
saturations
have
to
be
checked.
space
saturations of parameters have to be checked. InInspace
charge calculations
calculations using
using aa Particle
in
Cell
(PIC)
charge
Particle
in
Cell
(PIC)
charge calculations using a Particle in Cell (PIC)
method, parameters
parameters to be
optimized
are
number
method,
be optimized
optimizedare
arenumber
numberofofof
method,
parameters toto be
macro
particles,
grid
size,
integration
time
step,
and
macro particles,
particles, grid
grid size,
size,integration
integrationtime
timestep,
step,and
and
macro
order
of
mode
expansion
if
there
is
a
smoothing
order ofof mode
mode expansion
expansion ifif there
there isis a a smoothing
smoothing
order
process like
like Simpsons
Simpsons in
frequency
space.
Although
process
frequency space.
space.Although
Although
process
like Simpsons
ininfrequency
we
may
expect
that
smaller
grid
size,
shorter
time
step,
wemay
mayexpect
expectthat
thatsmaller
smallergrid
gridsize,
size,shorter
shortertime
timestep,
step,
we
and
higher
order
modes
always
make
a
calculation
and higher
higher order
order modes
modes always
alwaysmake
makea acalculation
calculation
and
more precise,
precise, memory
memory size
and
CPU
time
force
more
size and
and CPU
CPU time
time force
force
more
precise, memory
size
compromised solutions.
compromisedsolutions.
solutions.
compromised
edge380_0ma
edge380_0ma
edge380_0ma
99%
99%95%
95%90%
90%68%
68%38%
38%
200
200
99%
95%
99%
90%
95%
68%
90%
38%
68%
38%
99%
99%
95%
95%
90%
90%
68%
68%
38%
38%
FIGURE 2.
2. Emittance evolution during
anti-correlated
FIGURE
during anti-correlated
anti-correlated
FIGURE
2. Emittance
evolution during
painting when
when the beam current is 50
mA.
painting
50
mA.
painting when the beam current is 50 mA.
Onetypical
typicalexample
exampleof
spacecharge
chargeeffects
effectsin
One
ofofspace
space
charge
effects
ininaaa
One
typical
example
RCSemerges
emergesduring
duringphase
phasespace
spacepainting.
painting.InInorder
order toto
RCS
RCS emerges during phase space painting. In order to
make K-Vlike
like particledistribution,
distribution, anti-correlated
anti-correlated
make
make K-V
K-V like particle
particle distribution, anti-correlated
paintingwas
wasproposed.
proposed.AAbeam
beamisisinjected
injected from
from the
the
painting
painting was proposed. A beam is injected from the
centerininone
onetransverse
transverseplane
planeand
andfrom
fromthe
themaximum
maximum
center
center in one transverse plane and from the maximum
amplitude inthe
the otherplane.
plane.ItItworks
worksfine
fineasasexpected
expected
amplitude
amplitude in
in the other
other plane. It
works fine
as expected
when
thespace
spacecharge
chargeisisnot
notincluded
included(Fig.
(Fig.1).
1).
when
the
when the space charge is not included (Fig. 1).
250
250
250
250
edge380_125ma
edge380_125ma
vertical amplitude (pi mm-mrad)
vertical amplitude (pi mm-mrad)
horizontal amplitude (pi mm-mrad)
horizontal amplitude (pi mm-mrad)
fractions of the particles are enclosed.
600
edge380_0ma
600
99%
99%95%
500
95%90%
500
90%68%
400
68%38%
38%
400
300
300
200
200
100
100
0
50 100 150 200 250 300 350
0 0
50 100
100 150
150
200 250
250 300
300 350
350
turn 200
number
0
50
rumnumber
number
turn
FIGURE 1. Emittance evolution during anti-correlated
Figure 3 shows a parameter dependence of
Figure 33 shows
shows aa parameter
parameter dependence
dependence ofof
Figure
emittance vs. the number of radial grids. Saturation is
emittancevs.
vs.the
thenumber
numberofofradial
radialgrids.
grids.Saturation
Saturationisis
emittance
obtained when the number of radial grids is more than
obtained
when
the
number
of
radial
grids
is more
morethan
than
obtained
when
the
number
of
radial
grids
is
50. Similarly, the macro particles should bemore
than
50.Similarly,
Similarly,the
themacro
macroparticles
particlesshould
shouldbebemore
morethan
than
50.
FIGURE
1. Emittance and right during
during anti-correlated
anti-correlated
FIGURE
painting. Left is horizontalevolution
is vertical.
Space charge
painting.
Left
is
horizontal
is
vertical.
Space charge
charge
painting.
Left
and
right
is
vertical.
Space
is not included. There are five different emittance,
namely
isis 38%,
not
There
emittance,
namely
not included.
areand
five99%.
different
namely
68%, 90%,
95%,
Thoseemittance,
are defined
as the
38%,
99%. Those are
are defined
defined as
as the
the
38%, 68%,
68%, 90%,
90%, 95%,
95%, and 99%.
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
245
5,000,
number
is
more
than
16,
and
5,000, the
the azimuth
azimuth
mode
azimuth mode
mode number
number is
is more
more than
than 16,
16,and
and
integration
step
is
less
than
2
m
(In
fact,
Simpsons
integration step
step is
is less
less than
than 22 m
m (In
(In fact,
fact, Simpsons
Simpsons
uses
as
variable,
so
that
more
5,000,
azimuth
modevariable,
number so
is
16, and
uses time
time
as
the independent
independent
that
isis
as the
independent
variable,
somore
thatitititthan
ismore
more
appropriate
to
say
that
integration
step
time
is
less
than
integration
step
is
less
than
2
m
(In
fact,
Simpsons
appropriate to say
say that
that integration
integration step
steptime
timeisisless
lessthan
than
uses
time
as
the
independent
variable,
so
that
it
is
more
9ns.)
9ns.)
question
thenwhether
whetherthe
thecoherent
coherentmodel
modelofof
ofspace
space
question
questionisis
isthen
then
whether
the
coherent
model
space
charge
effects
helps
understand
those
small
beam
charge
charge effects
effects helps
helps understand
understandthose
thosesmall
smallbeam
beam
losses.
question is then whether the coherent model of space
losses.
losses.
charge effects helps understand those small beam
order
tostudy
study
beambehavior
behaviornear
neara aresonance,
a resonance,
resonance,
InIn
Inorder
ordertoto
studybeam
beam
behavior
near
losses.
.
100
100
0
0 0
0
,;,
n
"
i
D
300
200
100
0
2 0 0 4 02 0 6 04 0 8 60 0 1 0800
2 0 turn
4 0 number
60
80
100
turn number
turn number
5 0D
0
FD
HF
JH
4 0 L0J
L
300
300
300
200
200
200
100
100
100
0
1 0 000
0
0
D
F
H
J
L
600
600
400
400400
200
200200
tail_34.8a
tail_34.8a
tall_34Jla
tail_26.1a
tail_26.1a
tail_34.8a
tail_26.1a
amplitude (pi mm-mrad)
edge990_nrg80
38%
68%
90%
95%
99%
0
-50
50 100 150 200 250 300
0 00
50 100
100 150
150 200
200 250
250 300
300
-50
0 0 50
50turn
250
300
number
-50-50 0
100
150
200
turn
number
turn
number
turn
number
200
42 00
46 00
6800
81 00 0 1 0 0
2 0 turn
4 0number
60
80
100
turn number
turn number
FIGURE
4. Single
particle
trajectory
whenwhen
the number
of of
FIGURE
4. Single
particle
trajectory
the number
FIGURE
4.4. is
Single
particle
when
the
FIGURE
Single
particle
trajectory
when
the
radial
grids
60 (left)
and 80(right).
are five
radial
grids
60 is(left)
and trajectory
80(right).
ThereThere
arenumber
five
testoftest
radial
60
(left)
and
80(right).
There
are
five
testthe
radial grids
grids
60 name
(left)
andD,F,H,J,L.
SO(right).
There
arestart
fivefrom
particles
with
the name
of
D,F,H,J,L.
They
particles
withisisthe
of
They
start
from
the
particles
with
the
name
D,F,H,J,L.
They
particles
with
thecoordinates
nameinof
of
D,F,H,J,L.
They
start from
from the
same
initial
in radial
both radial
grids.start
same
initial
coordinates
both
grids.
same
sameinitial
initialcoordinates
coordinates in
in both
both radial
radial grids.
grids.
EMITTANCE
EVOLUTION
NEAR
EMITTANCE
EVOLUTION
NEAR
A A
EMITTANCE
EVOLUTION
EMITTANCE
EVOLUTION
NEAR
A
RESONANCE
RESONANCE
RESONANCE
We have RESONANCE
already
known that, 1) Coherent tune shift
i
amplitude
amplitude(pi
(pimm-mrad)
mm-mrad)
400
400
amplitude (pi mm-mrad)
amplitude (pi mm-mrad)
D
F
H
J
L
edge990_nrg80
edge990_nrg80
amplitude (pi mm-mrad)
DF
FH
HJ
J4L 0 0
L
500
500
vertical amplitude (pi mm-mrad)
200
200
edge990_nrg60
5D0 0
verticalamplitude
amplitude(pi(pimm-mrad)
mm-mrad)
vertical
300
300
°
(
edge990_nrg60
edge990_nrg60
500
500
400
400
1
o
vertical amplitude (pi mm-mrad)
horizontal amplitude (pi mm-mrad)
vertical amplitude
amplitude (pi
(pi mm-mrad)
mm-mrad)
appropriate to say that integration
step time is less than
550000
9ns.)
99%
99%
99%
vertical amplitude (pi mm-mrad)
vertical
verticalamplitude
amplitude(pi(pimm-mrad)
mm-mrad)
horizontal
amplitude (pi
(pi mm-mrad)
mm-mrad)
horizontal amplitude
two
bare
tunesare
arepicked
pickedup.
up.The
Thefirst
firsttune
tune
is
(6.566,
two
is is
(6.566,
twobare
baretunes
tunes
are
picked
up.
The
first
tune
(6.566,
Inthat
order
to
study
beam behavior
near
a resonance,
6.200)
that
is
near
a
structure
resonance
line
at
ν
=6.0.
6.200)
is
near
a
structure
resonance
line
at
ν
=6.0.
y
6.200)
that tunes
is near
a picked
structure
resonance
lineisat(6.566,
vyy=6.0.
•a
99%
«J9%
two bare
are
up.
firstistune
95%
99%
95%
1
95%
& 440000 5 0 0
95%
Since
the
modellattice
lattice
wetook
tookThe
here
3GeV
GeV
RCS
of
95%
90%
Since
the
model
we
here
is
3
RCS
ofof
95%
90%
90%
Since
the
model
lattice
we
took
here
is
3
GeV
RCS
440000 5 0 0
90%
68%
68%
68%
68%
6.200)
that
is near
a structure
resonance
line
at
νy=6.0.
68%
38%
68%
38%
&
38%
JKJ
with
three
fold
symmetry,
allthe
the
resonances
38%
^8% 99%
JKJ
with
three
fold
symmetry,
all
resonances
99% &
W%
JKJ
with
three
fold
symmetry,
all
the
resonances
95%
95%
Since the model lattice we took here is 3 GeV RCS of
90%
330000
330000 4 0 0
400
90%
68%
located
on
=6.0are
are
structure.The
Thesecond
secondone
oneis is
is
68%
located
on
=6.0
D
38%
located
on νvνythree
arestructure.
structure.
The
second
one
- < °
38%
yy=6.0fold
JKJ with
symmetry,
all
the
resonances
,
1 200 300
' •
1 220000 3 0 0
(6.850,
6.700)
that
is
located
just
above
ν
=6.5.
The
(6.850,
6.700)
that
is
located
just
above
ν
=6.5.
The
y
200
y
(6.850,
6.700)
that
is
located
just
above
v
=6.5.
The
located on νy=6.0 are structure. The second
y one is
I 2»»
*
«
•
resonance
non-structure
and
notexcited
excited
until
resonance
isis
resonance
is non-structure
non-structure
and
not
excited
until
(6.850, 6.700)
that is locatedand
just not
above
νy=6.5.until
The
200
110000 2 0 0
110000
quadrupole
imperfections
are
introduced.
quadrupole
imperfections
are
introduced.
quadrupole
are introduced.
resonance imperfections
is non-structure
and not excited until
100
100
00
000
quadrupole
imperfections
are
introduced.
1
»
As
different
00
22 00 4400 6600 8800 1 10 00 0
00
As
for
thebeam
beamintensity,
intensity,we
wetested
testedthree
three
different
00(
222000 444000 6600 8800 11()()
Asfor
forthe
the
beam
intensity,
we
tested
three
different
20
40
60
8 0 100
radial
number
of radial
radial grids
number ofof
radial grids
grids
number
of
0number
0
levels.
The
level
intensity
low
enough
soso
that
number
levels.
The
lowest
level
intensity
is
low
enough
so
that
forlowest
the beam
intensity,
weisis
tested
three
different
0number
2 0 of 4radial
0
6 grids
0
8 0 100
0
2of0 radial
4 0 grids
60
8 0 100
levels.As
The
lowest
level
intensity
low
enough
that
number of radial grids
number of radial grids
neither
incoherent
orlevel
quadrupole
coherent
tune
shift
neither
incoherent
or
quadrupole
coherent
tune
shift
levels.
The
lowest
intensity
is
low
enough
so
that
neither
incoherent
or
quadrupole
coherent
tune
shift
FIGURE
3.
FIGURE
number
of
radial
grids
FIGURE 3.
3. Emittance
Emittance vs.
vs. the
the number
number of
of radial
radial grids
grids inin
in
does
not
the
AtAt
second
the
neither
incoherent
or quadrupole
coherent
tune
shift
does
not
reach
theresonance.
resonance.
Atthe
the
secondlevel,
level,
the
does
notreach
reach
the
resonance.
the
second
level,
the
horizontal
(left)
vertical
horizontal
(left)
(right).
FIGURE
3. Emittance
vs. the number of radial grids in
horizontal
(left) and
and
vertical (right).
(right).
incoherent
tune
shift
is
below
the
resonance,
but
the
does
not
reach
the
resonance.
At
the
second
level,
incoherent
tune
shift
is
below
the
resonance,
but
the
incoherent tune shift is below the resonance, butthe
the
horizontal (left) and vertical (right).
is
intriguing
incoherentcoherent
tune shift
below
theAt
resonance,
but
the
quadrupole
isis
still
above.
level,
ItIt
particle
trajectory
not
quadrupole
coherent
isis
still
above.
Atthe
thethird
third
level,
It is
is intriguing
intriguing that
that single
single particle
particle trajectory
trajectory isis
isnot
not
quadrupole
coherent
still
above.
At
the
third
level,
It independent
is
intriguing that
single
particle even
trajectory
is not the
necessarily
independent
of
parameters
quadrupole
coherent
is stillsuch
above.
Atboth
the
third
level,
intensity
is
high
enough
that
incoherent
necessarily
of
the
parameters
even
though
the
intensity
is
high
enough
such
that
both
incoherent
necessarily
independent
of the
the
parameters
eventhough
though
the intensity is high enough such that both incoherent
necessarily
independent
of the
parameters
even though and
the
intensity
is high
enough
such
that both incoherent
the
emittance
defined
by
multi
particles
isis
saturated.
coherent
ones
are
resonance.
the
defined
by
multi
particles
saturated.
and
coherent
ones
arebelow
belowthe
the
resonance.
the emittance
emittance
defined
by
multi
particles
is
saturated.
and
coherent
ones
are
below
the
resonance.
the
emittance
defined
by
multi
particles
is
saturated.
and
coherent
ones
are
below
the
resonance.
An
example
is
shown
in
Fig.
4
where
the
number
of
An
isis shown
in
44 where
the
number of
38%
38%
An example
example
shown
in Fig.
Fig.
where
the the
38%
68% 38%
68%
An
example
is shown
in
Fig.
4 where
number
of
radial
grids
is
changed
from
60
to
80.
According
to
the
68%
90% 68%
1000
90%
1000
radial
is
changed
from
60
to
80.
According
to
the
1000
90%
1000
radial grids
grids
is
changed
from
60
to
80.
According
to
the
95% 90%
95%
radial
grids
is
changed
from
60
to
80.
According
to
the
95%
1000
99% 95%
99%
1000
multi particle
particle optimization,
optimization, emittance
does
not
change
99% ;
99%
;
j
;
;
99%
multi
emittance
does
not
change
800
800
multi multi
particle
optimization,
emittance
does
800
800
particle
optimization,
emittance does not change
800
much
when
it
is
more
than
50.
800
much
when
it
is
more
than
50.
much much
when when
it is more
than than
50. 50.
600
600
it is more
600
550000
JUU
600
400
400
400
400
200
200
200
i
i
i
o 3S%
• 68%
38%
68%
90%
* 95%
+
95%
99%
i v i x x x x p" x x x " x x * X x t
i«
i
i
:
;
i
0
^GOOG<SGOOO<j«GOOSOOOOOOOOOcb
-500n0 0
50 100 150 200 250 300
-50
300300
-50 000 5050
50
100150
150200200
200250250
250
300
turn100
number
-50
100
150
turn
number
turn
number
turn
number
FIGURE
5. 5.
Emittance
the
tune isis above
FIGURE
Emittanceevolution
evolutionwhen
when
the
FIGURE
5. Emittance
thetune
tune is
isabove
above
evolution
whentune
the
tune
above
non-structure
resonance.
Only
incoherent
is
below
the
non-structure
resonance.
Only
incoherent
tune
is
below
the
non-structure
incoherent
tune
is
below
the
resonance.
Only
incoherent
tune
is
below
the
resonance
(left)
or orboth
coherent
and
incoherent
tune
is
resonance
(left)
both
coherent
and
incoherent
tune
is is
resonance
(left)
tune
(left)
or
both
coherent
and
incoherent
tune
is
below
the
resonance
(right).
below the resonance (right).
below the resonance (right).
Behavior
near
Behavior
nearNon-Structure
Non-StructureResonance
Resonance
Behavior near Non-Structure Resonance
Resonance
Figure
5 showsthe
theresults
resultswhen
whenthe
the bare
bare tune
tune is
is
Figure
5 shows
Figure
5 shows
results
when the
the bare
bare
tune isis
results
when
tune
chosen
abovethe
the
non-structure
resonance.
chosen
justjust
above
the
non-structure
resonance.
chosen
above
non-structure
resonance.
just
the nothing
non-structure
Withoutany
anyerror
errorfields,
fields,
nothing
happensresonance.
for any
any
Without
happens
for
Without
any
fields,
nothing
happens
for(not
any
error
fields,
nothing
happens
for
any
intensity
except
a few
growth
99%
emittance
(not
intensity
except
a few
growth
ofof99%
emittance
intensity
except
growth
of 99%
99% errors
emittance
(not
a few
growth
of
emittance
(not
shown
here).
With
random
quadrupole
errors of
of the
the
shown
here).
With
random
quadrupole
width
of
0.02,
a significant
growth
thebeam
beam
core
as
shown
quadrupole
errors
ofas
the
random
quadrupole
errors
of
the
width
of here).
0.02,
aWith
significant
growth
ofofthe
core
well
as
the
tail
is
observed
when
the
both
tune
are
significant
growth
of
the
beam
core
width
of
0.02,
a
significant
the
beam
core
as
well as the tail is observed when the both tune areas
below
the
resonance
(Fig. 5, right).
Only
the
growth
of
tail
is
observed
when
the
both
tune
are
well
as
the
the
both
tune
are
below the resonance (Fig. 5, right). Only the growth of
99% the
emittance
is observed
when
only
thethe
incoherent
resonance
(Fig. 5,when
right).
Only
growth
below
right).
Only
growth of
of
99%
emittance
is observed
only
thethe
incoherent
tune
is
below
the
resonance
(Fig.
5,
left).
In other
only
the
99%isemittance
is resonance
observed when
the incoherent
incoherent
tune
below
the
(Fig.
5,
left).
In
other
words, core growth
is observed
only
when
the
the resonance
(Fig. 5,
left).
In other
tune
is core
belowgrowth
left).
other
words,
observed
when
quadrupole
coherent isshift
is largeonly
enough
andtheit
growthshift
is observed
only
when
words,
corecoherent
only
when
the
quadrupole
is large
enough
and itthe
satisfies the
resonance,
whereas
the 99%
tail particles
quadrupole
coherent
shift
is
large
enough
and
itit
shift
enough
and
satisfies
the
resonance,
whereas
the
99%
tail
particles
start growing even below that intensity.
resonance,
whereas
the
99%
tail
particles
satisfies
the
whereas
the
99%
tail
particles
start growing even below that intensity.
even below that intensity.
start growing
Behavior near Structure Resonance
We have already known that, 1) Coherent tune shift
ishave
a relevant
measure,
not
2) tune
Ratioshift
of the
We
have
already
known
that,incoherent.
1) Coherent
Coherent
already
known
that,
1)
is aWe
relevant
measure,
not incoherent.
2) Ratio
of shift
the
coherent
and
incoherent
tune
is
5/8
in
a
quadrupole
is aa relevant
relevant
measure,
nottune
incoherent.
iscoherent
not
incoherent.
of the
and measure,
incoherent
is 5/8 in2)a Ratio
quadrupole
mode.
The
higher the
coherent
the
smaller the
coherent
and
incoherent
tune
isorder,
5/8order,
in
quadrupole
coherent
and
incoherent
tune
is
5/8
in
aasmaller
quadrupole
mode.
The
higher
the
coherent
the
Although those results are manifest,the
there
mode.difference.
The higher
higher
thethose
coherent
order,
mode.
The
the
coherent
order,
the smaller
the
difference.
Although
results
are manifest,
are still some questions.
First, those
results arethere
valid in
difference.
Although
those
results
areinmanifest,
manifest,
there
difference.
Although
results
are
are
still
some
questions.
First,
those
results
valid
in
a coasting
beam,those
but
not
obvious
a are
bunched
beam
are
still
some
questions.
First,
those
results
are
valid
are
stillwhere
some
questions.
First,
a coasting
beam,
but nottune
obvious
in a bunched
beamin
incoherent
isthose
a function
of longitudinal
a coasting
coasting
beam,
but
not
obvious
in aaofbunched
bunched
beam
awhere
beam,
not
obvious
in
beam
incoherent
tune
isas
a function
longitudinal
position
as but
well
transverse
amplitude.
The
wheresynchrotron
incoherent
tune
istransverse
function
longitudinal
where
incoherent
tune
aa are
function
of
longitudinal
position
as welloscillations
as is
amplitude.
The
also involved. Secondly,
Behavior near Structure Resonance
position
as oscillations
well
asofare
transverse
amplitude.
position
as
well
as
transverse
amplitude.
Theis a
synchrotron
alsointensity
involved.
Secondly,
the main
concern
a high
synchrotron
Figure 6 shows
theStructure
results whenResonance
the bare tune is
near
Behavior
Resonance
synchrotron
oscillations
are
also
involved.
Secondly,
the
main
concern
of asuch
high
synchrotron
a A
synchrotron
oscillations
are
small
beam
loss
asintensity
aalso
few involved.
percents
orSecondly,
even is
less.
Figure just
6 shows
when
the bareNow,
tunethe
is
chosen
abovethe
theresults
structure
resonance.
the main
main
concern
of aas
a high
high
intensity
synchrotron
small
beam
loss such
a fewintensity
percentssynchrotron
or even less.isAa
results
when
the
concern
of
chosen
just 6above
Now,tune
theisis
Figure
showsthethestructure
results resonance.
when the
the bare
bare
tune
small beam
beam loss
loss such
such as
as aa few
few percents or even less. A
small
chosen just above the structure
structure resonance.
resonance. Now,
Now, the
the
246
corecore
growth
is observed
when
only
thethe
incoherent
isis actually
by higher
higher order
order coherent
coherentmodes
modes
actually caused
caused by
growth
is observed
when
only
incoherenttune
tune
is instead
actually
caused
bybyhigher
order
coherent
modes
core
growth
is
observed
when
only
thethe
incoherent
tune
is
actually
caused
higher
order
coherent
modes
core
growth
is
observed
when
only
incoherent
tune
is
below
the
resonance
without
random
errors.
In
fact,
of
the
incoherent
one.
instead of the incoherent one.
is below the resonance without random errors. In fact,
is actually
caused
byone.
higher
growth
is observed
when
only
the
incoherent
tune instead
of of
thethe
incoherent
is below
thethe
resonance
without
random
errors.
In In
fact,
isνcore
below
resonance
without
random
errors.
fact,
instead
incoherent
one. order coherent modes
vy=6.0
is
structure
of
all
orders
so
that
it
is
reasonable
=6.0
is
structure
of
all
orders
so
that
it
is
reasonable
y
5 10
instead
of
the
incoherent
one. 5 10
is
below
the
resonance
without
random
errors.
In
fact,
ν
is
structure
of
all
orders
so
that
it
is
reasonable
y=6.0
νto
is that
structure
ofgrowth
all orders
sodue
that
it the
isincoherent
reasonable
5 10
5 10
y=6.0
to conclude
the the
growth
is not
due
to to
the
conclude
that
is not
incoherent
5 10
5 10
νyconclude
=6.0 isthat
structure
of
allisorders
so
that
itthe
is incoherent
reasonable
to conclude
thethe
growth
not
due
to
the
incoherent
410
4 10
4410
10
to
that
growth
is
not
due
to
5 10
5 10
but
by higher
order
coherent
motions.The
Thehigher
higher
tune,tune,
but
by
higher
order
coherent
motions.
4 10
4 10
4
10
4 10
to
conclude
that
the
growth
is
not
due
to
the
incoherent
tune,
but
by
higher
order
coherent
motions.
The
higher
!
tune,
but
by
higher
order
coherent
motions.
The
higher
3
10
3
10
order
of coherent
motionbecomes,
becomes,the
thesmaller
smaller
4 10
4 10
the the
order
of bycoherent
motion
3 10 3 10
3 103 10
tune,
but
higher
order
coherent
motions.
The
higher
thethe
order
of
coherent
motion
becomes,
the
smaller
order
of
coherent
motion
becomes,
the
smaller
1 ! ! ; ii ! ! :
difference
between
incoherenttune
tuneshift
shiftand
andthe
the
!
2 10 j
2 10
difference
between
thethe
incoherent
3 10
3 10
the order
of coherent
motion becomes,
the
smaller
difference
between
the
incoherent
tune
shift
andand
thethe
2 10 2 10
2 102 10
difference
between
the
incoherent
tune
shift
coherent
tune
shift.
Practically
speaking,
the
\ \ \ l/l|,fl i i :
coherent
tunebetween
shift. thePractically
Practically
speaking,
the
1 10
1 110
10
difference
incoherent
tune
shift and
the
110
2 10
2 10
coherent
tune
shift.
speaking,
the
coherent
tune
shift.
Practically
speaking,
the
1 10 1 10
incoherent
tune
is agood
goodmeasure
measurefor
for the
the
wM 1 101 10 0
incoherent
tune
shiftshift
'%A/y^ IA,
coherenttune
tune
shift.
speaking,
the
0
incoherent
isis isaa Practically
measure
for
W
1 10 0
1 10 0
incoherent
tuneshift
shift
agood
good
measure
forthe
the
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.5
0.6
structure
resonance
although
the
incoherent
tune
shift
0 0()
0 0 C
0.1
0.2 tune 0 3
04
05
0.1 0.2 0.3 0.4 0.5 0.
structure
resonance
although
the
incoherent
tune
shift
incoherent
tune although
shift
is the
a the
good
measure
for
the
0 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5
0 0 0.1 0.10.2 0.20.3 tune
structure
resonance
incoherent
tune
shift
0.3 0.4 0.4 0.5 0.5 0.6 0.6
structure
resonance
although
incoherent
tune
shift
0
0
tunetune
tunetune
is
not
the
true
source
of
the
growth.
is
the
true
source
of the
the growth.
growth.
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
resonance
although
the incoherent tune shift
is not
not
thethe
truetrue
source
of
isstructure
not
source
of the
growth.
tune
tune
sft51.pwr_5
5
55
5
4
4
5
5
5
5
5
5
5
5
VH
5
5 5
5
5
V
4
4
4
4
44
4
4
'^CTTW'";
tail_3a
1000
tail_4a2
COHERENT OSCILLATIONS OF A
COHERENT
OSCILLATIONS
OF
COHERENT
OSCILLATIONS
AA
COHERENT
OSCILLATIONS
OF
BUNCHED
BEAM OF
COHERENT
OSCILLATIONS
OFAA
BUNCHED
BEAM
BUNCHED
BEAM
BUNCHED
BEAM
BUNCHED
BEAM
So far, the
simulation results
are all for coasting
So
far,
simulation
for
coasting
So
far,
the
simulation
results
all
forare
coasting
beams.
It the
isthe
interesting
toresults
seeare
ifare
there
coherent
So
far,
the
simulation
results
are
allall
So far,
simulation
results
are
allfor
forcoasting
coasting
beams.
Itininteresting
isa interesting
to
see
if
there
are
coherent
beams.
It
is
to
see
if
there
are
coherent
motions
bunched
beam.
As
a
first
step,
instead
of
beams.
It
is
interesting
to
see
if
there
are
coherent
beams. It is interesting to see if there are coherent
motions
inbunched
a bunched
beam.
As
a afirst
step,
instead
of
defining
coherent
motions
asfirst
whole
beam,
we
motions
in
a
beam.
As
a
step,
instead
of
motions
a bunched
beam.
a first
step,
insteadof
of
motions
in ain
bunched
beam.
AsAs
a first
step,
instead
defining
coherent
motions
whole
beam,
we
observed
the motions
quadrupole
motions
of we
each
defining
coherent
as as
a awhole
beam,
defining
coherent
motions
as
awhole
whole
beam,
we
defining
coherent
motions
as
a
beam,
we
observed
the sliced
quadrupole
motions
each
longitudinally
beam. Figure
7 shows
tune
observed
motions
of of
each
observedthe
thequadrupole
quadrupole
motions
ofthe
each
observed
the
quadrupole
motions
ofthe
each
longitudinally
sliced
beam.
Figure
7slice
shows
the
tune
spectra
of
each
slice.
As
expected,
a
of
bunch
longitudinally
sliced
beam.
Figure
7
shows
the
tune
longitudinally
sliced
beam.
Figure
7shows
showsthe
thetune
tune
longitudinally
sliced
beam.
Figure
7aand
spectra
of
each
expected,
slice
of
the
bunch
center
has
theslice.
maximum
tunea slice
tune
shift
spectra
of each
slice.
AsAs
expected,
ofthe
the
bunch
spectra
of
each
slice.
As
expected,
a
slice
of
the
bunch
spectra
of each
slice.
As
expected,
aand
slice
oftune
the
bunch
center
has
the
maximum
the
tune
shift
becomes
smaller
near
thetune
bunch
tail.
It
is
also
center
has
the
maximum
tune
and
the
shift
center
has
the
maximum
tune
and
the
tune
shift
center
has
the
maximum
tune
and
the
tune
shift
becomes
smaller
near
the
bunch
tail.
It
is
also
confirmed
that
the
shift
of
the
coherent
quadrupole
becomes
smaller
near
the
bunch
tail.
It
is
also
becomes
smaller
nearthethebunch
bunchtail.
tail.It Itis isalso
also
becomes
near
confirmed
shift
coherent
quadrupole
tune
is smaller
the
same
as
the
oneof
ofthe
acoherent
coasting
beam
with the
confirmed
thatthat
thethe
shift
of
the
quadrupole
confirmed
that
the
shift
of
the
coherent
quadrupole
confirmed
that
the
shift
of
the
coherent
quadrupole
isline
the density.
same
as the
a coasting
beam
with
same
tunetune
is the
as the
oneone
of aof
coasting
beam
with
thethe
tune
is same
the same
as the
a coasting
beam
with
the
tune
isline
the
same
as the
oneone
of aofcoasting
beam
with
the
same
line
density.
same
density.
same
line density.
Figure
8
shows
the
emittance
evolution
when
the
same line density.
Figure
8is shows
emittance
evolution
when
bare
tune
near
the non-structure
resonance.
Figure
8 shows
thethe
emittance
evolution
when
thethe
Figure
8 chosen
shows
the
emittance
evolution
when
the
tune
is
chosen
near
the
non-structure
resonance.
At
the
all
three
intensity
levels
defined
above,
Figure
8
shows
the
emittance
evolution
when
the
barebare
tune
is
chosen
near
the
non-structure
resonance.
bare tune is chosen near the non-structure resonance.
theall
all
threeintensity
levels
defined
above,
emittance
growth
is intensity
observed
and
it defined
becomes
larger
for
bare
tune
chosen
the non-structure
resonance.
At At
the
three
levels
above,
At
theis all
threenear
intensity
levels
defined
above,
emittance
growth
is
observed
and
it becomes
larger
for
the
tail.
The
clear
threshold
intensity
disappears
that
At
the
all
three
intensity
levels
defined
above,
emittance
growth
is
observed
and
it
becomes
larger
for
emittance growth is observed and it becomes larger for
The
clear
threshold
intensity
disappears
that
was
observed
in
the coasting
beam
simulation.
emittance
growth
isthreshold
observed
and
it becomes
larger
for
thethe
tail.tail.
The
clear
intensity
disappears
that
the
tail.
The
clear
threshold
intensity
disappears
that
was
observed
in
the
coasting
beam
simulation.
Qualitatively,
there
is
no
difference
whether
the
the
The
clear
intensity
disappears
that
wastail.
observed
in threshold
the the
coasting
beam
simulation.
was
observed
in
coasting
beam
simulation.
Qualitatively,
there
is
no
difference
whether
the
coherent
quadruople
tune
is
below
and
above
the
was
observed there
inthere
the
beamwhether
simulation.
Qualitatively,
is no
difference
thethe
Qualitatively,
iscoasting
no
difference
whether
coherent
quadruople
tune
isbelow
belowandand
abovethe
the
resonance.
coherent
quadruople
tune
is difference
above
Qualitatively,
there is
no
whether
coherent
quadruople
tune
is below
and
abovethe
the
resonance.
resonance.
coherent
quadruople
tune resonance,
is below and
resonance.
Near
the structure
the above
behaviortheis
resonance.
Nearthe
the
structure
resonance,
is
similar
the
coasting
beam.
Namely,
thebehavior
beam is
core
Near
structure
resonance,
thethe
behavior
Nearto the
structure
resonance,
the
behavior
is
similar
to the
coasting
beam.
Namely,
the
beam
core
starts
growing
when
the
incoherent
tune
is
below
the
similar
to
the
coasting
beam.
Namely,
the
beam
core
similarthe
to the
coastingresonance,
beam. Namely,
the
beam core
Near
structure
the
behavior
is
starts
growing
the
incoherent
tune
is the
below
the
resonance.
Itwhen
iswhen
also
reasonable
totune
say
that
growth
starts
growing
thebeam.
incoherent
is
below
the
starts
when
the
incoherent
tune
is below
the
similar
to growing
the Itcoasting
Namely,
the
beam
core
resonance.
is
also
reasonable
to
say
that
the
growth
resonance.
It
is
also
reasonable
to
say
that
the
growth
resonance.
It
is
also
reasonable
to
say
that
the
growth
starts growing when the incoherent tune is below the
tall_4a2
tail_4a2
tail_4a2
90%
38%
• 68%
68%68%
95%
amplitude (pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrac
«** \
-----ir^iii^r"---
FIGURE 8. Emittance evolution of a bunched beam when
FIGURE
8.
Emittance
evolution
ofaofbunched
a abunched
beam
when
FIGURE
Emittance
evolution
of
beam
when
the tune8.is8.
above
non-structure
resonance.
Only
incoherent
FIGURE
Emittance
evolution
bunched
beam
when
FIGURE
8.above
Emittance
evolution
of aorbunched
beam
when
the
tune
is
non-structure
resonance.
Only
incoherent
thethe
tune
is
above
non-structure
resonance.
Only
incoherent
tune
is
below
the
resonance
(left)
both
coherent
and
tune
isis above
non-structure
resonance.Only
Onlyincoherent
incoherent
the
tune
above
non-structure
resonance.
tune
below
resonance
(left)
both
coherent
and
tune
is isbelow
thethe
(left)
or or(right).
both
coherent
and
incoherent
tune
isresonance
below
the resonance
tune
below
the
resonance
(left)
or
both
coherent
and
tune isis tune
below
the
resonance
(left)
or
both
coherent
and
incoherent
tune
is
below
the
resonance
(right).
incoherent
is
below
the
resonance
(right).
38% resonance (right).
incoherent
the
38%
incoherenttune
tune is
is below
below the
resonance
(right).
68%
68%
tail_3a
90%
38% 38%
95%
o 68%
38%
68%
38%
99%
90% 90%
68%
* 95%
90%
95%
90%
x 99%
95%
99%
95%
•> 99%
99%
1000
tail_3a tail_3a
10001000
800
10001000
tail_3a
tail_3a
800 800
600
800
600 600
400
600
400 400
200
400
200 2000
200-50
200
>*** ****<
****
****.
h
amplitude (pi mm-mrad)
FIGURE 6. Emittance evolution when the tune is above
FIGURE
6. Emittance
evolution
when
the
tune
above
FIGURE
6.
Emittance
evolution
when
the
tune
is
above
FIGURE
Emittance
evolution
when
the
tune
above
structure
resonance.
Only
incoherent
tune
isisis
below
the
FIGURE
6.
Emittance
evolution
when
the
tune
istune
above
structure
resonance.
incoherent
tune
isisisbelow
the
structure
resonance.
Only
incoherent
tune
below
theis
resonance
(left)
orOnly
both
coherent
and
incoherent
structure
resonance.
Only
incoherent
tune
below
the
structure
resonance.
Only
incoherent
tune is tune
below
the
resonance
(left)
both
coherent
and
incoherent
tune
is
resonance
(left)
or
both
coherent
and
incoherent
is is
below
the
resonance
(right).
resonance
or or
both
coherent
and
incoherent
tune
resonance
(left) (right).
or (right).
both coherent and incoherent tune is
below
the
resonance
below
the
resonance
below
the
resonance
(right).
below the resonance (right).
1000
90%
• 38%
68%
68%
95%
68%
tall_3a
tail_3a tail_3a
1000
38%
38%
10001000
10001000
90% 90%
99%
99%
1000
90%
: x x ; x x *90%
x 68%
95%
x 95%
tail_3a
tail_4a2
68%
95%
95%
95%
800
800
!x* * 95%
-• 99% !
99%
1000
1000
90%
90%
99%• 99%
99%99%
95%
95%
800
800 800
800 800
99%
99%
;
x*
600
600
800
800
600 600
600 600
600
400
400
600
600
400 400
400 400
400
200
200
400
400
x«X.
:
•••!
200 200
200 200
200 ---#£; ^
*..• «M L. _ „ _ ! _ _ _ _ _
0
0
200-50 0
200-50- 0 Aooooioooopooaxbooocxi
; 300
ICGOCX
XDOOG
100
150
200 250
50 100 150 200 250 300
OGOOC
<ixxx:oc50
0 00
0 0
turn number
turn number
-50 -5
0 0050 50
100 100
150 150
200 200
250 250
300
150100
200150
250200
300300
-50
50
100
150
200
250
100
150
200
250300300 -50 -50
250 30
00
0 0-500 5005010050
number
number
number
-50 0 turn
50 turn
100
150 200 250 300
-50 0 turn
50 turn
100number
150 200 250 300
turn number
turn number
tail_4a
1000
tail_4atail_4a
10001000
800
1000
amplitude
mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi (pi
mm-mrad)
amplitude (pi mm-mrad)
-V.tv;;-.
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
-"4-"
4
FIGURE 7.
7. Spectra
Spectra of
coherent
quadrupole
tune
thethe
FIGURE
of
coherent
quadrupole
tuneatthe
atthe
FIGURE
7. 7.Spectra
of of
coherent
quadrupole
tune
FIGURE
Spectra
quadrupole
tuneat atspace
center slice
(left)
and
thecoherent
tail one
(right). Without
center
slice
(left)
and
thecoherent
tail
one
(right).
Without
space
FIGURE
7.
Spectra
quadrupole
tunespace
at
the
center
slice
and
theof
one
(right).
Without
center
slice
(left)
and
the
tail
one
(right).
Without
space
charge,
the(left)
peak
should
betail
located
at
0.44.
charge,
peak
should
be
located
at0.44.
0.44.
center
slice
(left)
and
thelocated
tail atone
(right).
Without space
charge,
thethe
peak
should
bebe
located
0.44.
charge,
the
peak
should
at
38%
charge, the peak should be38%
68%located at 0.44.
68%
38%
38%
38%
o 38%68%
38%
• 68%
90%
38%
68% 68%
95%
J
90%95%
38%
99%
90%
H
95%99%95%
68%
90%
99% 99%
95%
XXX
99%
amplitude (pi mm-mrad)
68% 68%
95%
tail_34.8a
tail_34.8a
tail_34.8a
tail_34.8a
4
4
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
1000
90%
68%38%
amplitude (pi mm-mrad)
•
38% 1000 1000
1000 1000
99%
: tail_26.1a; x 90%
95% 90%
tail_34.8a
!
: + 95%
99% 95%
;68%
800
800
90%
99% 99%
1000
1000
95%
,xxxx
800 800
800 800
99%
600
600
800
800
:^l±jJV.t:|:^tL...J
600
600 600
600 600
400
400
600
600
_____j_____|_____i____^
------400
400400
400 400
400
200
200 •i
;
400
400
200200
200 200
200
0°oioooo JOCCO ;GGOCXpC 000
0UoOGCBOOOO :OOOOSOGOCXpOOOG<p
0
200-50
200-50 0
0
50 100 150 200 250 300
50 100 150 200 250 300
n0
0n 0
0
turn number
turn number
-50
0
100
150 150
200 200
250 250
300 300
-50 -50
-50 -50
50
150
200
250
300
-50
0 0 50
150150
200200
250250
300300
0 100
50 100
0 50 100
50100100
150
200
250
300
0
0
turn
number
turn
turn
number
turn
number
turn
number
turn
number
-50 0
50 100 150 200 250 300
-50 0
50number
100
150 200 250 300
turn number
turn number
amplitude (pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi mm-mrad)
amplitude (pi mm-mrad)
tal!2«.la
tail_26.1atail_26.1a
amplitude
mm-mrad)
amplitude
(pi mm-mrad)
amplitude
(pi (pi
mm-mrad)
tail_26.1a
1000
sft51.pwr_2
4
4
4
4
5
is not the true source
of38%
the growth.
o 3S%
68%
38%
sft51.pwr_2
sft51.pwr_2
sft51.pwr_2
4
44
4
sft51.pwr_5
5
power (arb.)
power (arb.)
power
(arb.)(arb.)
power
power
(arb.)
5
4
sft51.pwr_5
sft51.pwr_5
5
power
(arb.)(arb.)
power
power
(arb.)
5
tree150
nonnnt
0
50 100
200 250 300
turn number
,*,OOOG SOOOO iOOOO )0000!
0 0
0 0 050 50
-50 n-50
100 100
150 150
200 200
250 250
300 300
number
-50 00 turn
50 turn
100
150 200
200 250
number
-50
50
100
150
250 300
300
turn number
tail_4a
tail_4a
800 800
600
800
90%
38%38%
95%
68%68%
38%
99%
n 68%
90%90%
68%
95%95%
90%
* 95%
99%99%
95%
+ 99%
99%
600 600
400
600
400 400
200
400
Z»< ****<
.*»«»;»***
200 2000
200-50
Rrff
nn^
0 f-50 100 150 200 250 300
vOOOOfflOOOOC
OOOOnumber
Soooo" turn
0 0
0n 0 0 50 50100100
-50 -50
150150
200200
250250
300300
turn
number
-50
0
50
100
150150200200250250
300 300
turn
number
-50 0 50turn100
number
turn number
FIGURE 9. Emittance evolution of a bunched
beam when
the tune9.is9.
above
structure
resonance.
incoherent
tune
FIGURE
Emittance
evolution
ofa bunched
a Only
bunched
beamwhen
when
FIGURE
Emittance
evolution
of
beam
FIGURE
Emittance
evolution
of
aaOnly
bunched
when
9.9.
Emittance
evolution
ofcoherent
bunched
beam
when
is
below
resonance
(left)
or both
andbeam
incoherent
the
tune
isthe
above
structure
resonance.
incoherent
tune
theFIGURE
tune
is above
structure
resonance.
Only
incoherent
tune
the
tune
is
above
structure
resonance.
Only
incoherent
tune
isthe
below
the resonance
(right).
istune
below
the
resonance
(left)
or
both
coherent
and
incoherent
tune
is
above
structure
resonance.
Only
incoherent
tune
is the
below
resonance
(left)
or
both
coherent
and
incoherent
is below
the resonance
(left)(right).
or both coherent and incoherent
is below
the
resonance
istune
below
thethe
resonance
(left)
or both coherent and incoherent
tune
is below
resonance
(right).
tune
is below
the
resonance
(right).
tune is belowACKNOWLEDGMENTS
the resonance (right).
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
We ACKNOWLEDGMENTS
would
like to thank Prof. Y. Mori for his
ACKNOWLEDGMENTS
We
would
like
tothank
thank support
Prof.Y.Y.
Mori
his
encouragement
and
ofMori
the
work.
WeWe
would
like
tocontinous
forfor
would like
to thankProf.
Prof. Y.
Mori
forhishis
encouragement
and
continous
support
of
the
work.
We
would
like
to
thank
Prof.
Y.
Mori
for
encouragement
andand
continous
support
of of
thethe
work.
encouragement
continous
support
work. his
encouragement and continous support of the work.
REFERENCES
REFERENCES
REFERENCES
REFERENCES
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REFERENCES
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Space
Machida,
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Machida,
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Machida,Physics
S., and
M., “Simulation
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ainSynchrotron”
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onon
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by
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T.
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edited
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American
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of
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American
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448, and
New W.
York:
American
of
by
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Physics,1998,pp.73-84.
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448, New York: American Institute of
resonance. It is also reasonable to say that the growth
247
Physics, 1998,pp.73-84.