Space Charge Studies at PSI
Space Charge Studies at PSI
Andreas Adelmann
Andreas Adelmann
Paul Scherrer Institut & Lawrence Berkeley National Laboratory
Paul Scherrer Institut & Lawrence Berkeley National Laboratory
Abstract. MAD 9 P (methodical accelerator design version 9 - parallel) is a general purpose parallel particle
tracking program
3D space
chargedesign
calculation.
First
results
on thepurpose
B870 injection
and
Abstract.
MAD9Pincluding
(methodical
accelerator
version
9 -simulation
parallel) is
a general
parallel line
particle
coastingprogram
beam studies
of the3DPSI
Injector
2 cyclotron
areFirst
presented.
Goodresults
agreement
measured
beam
tracking
including
space
charge
calculation.
simulation
on thebetween
B870 injection
line
and
profiles and
calculations
coasting
beam
studies of are
thereported.
PSI Injector 2 cyclotron are presented. Good agreement between measured beam
profiles and calculations are reported.
INTRODUCTION
INTRODUCTION
where
T denotes the step size, ^K is the map correspondwhere τ denotes the step size, Ml1 is the map corresponding
to
J^J obtained by differential algebra (DA) mething to H1 obtained by differential algebra (DA) methods
from
Hamiltonian and
and M
<^K2 is
is
ods from aa general
general relativistic
relativistic Hamiltonian
2
the
map
corresponding
to
Jf?
.
<^
is
obtained
by
disthe map corresponding to H22 . M22 is obtained by discretizing
on aa rectangular
rectangular
cretizing the
the resulting
resulting Poisson
Poisson problem
problem on
mesh
using
Fourier
techniques
to
solve
the
time
consummesh using Fourier techniques to solve the time consuming
cyclic
convolution
in
^(MlogM),
where
M
is the
the
ing cyclic convolution in O(M logM), where M is
number
of
grid
points.
Open
and
periodic
boundary
connumber of grid points. Open and periodic boundary conditions
ditions are
are available.
available.
MAD9P
(methodical accelerator design version 9 MAD 9 P (methodical accelerator design version 9 parallel)
parallel) isis aa general
general purpose
purpose parallel
parallel particle
particle tracktracking
program
including
3D
space
charge
ing program including 3D space charge calculation.
calculation.
ItIt isis based
and two frameworks:1 CLASbased on
on MAD9,
MAD 9, and two frameworks:1 CLAS SIC
(class library for accelerator system simulation
SIC (class library for accelerator system simulation
and
(parallel object oriented
and control)
control) and
and POOMA
POOMA (parallel object oriented
methods
and
applications)
[1].
methods and applications) [1]. AA detailed
detailed descripdescription
of
MAD9P
and
more
design
studies
tion of MAD 9 P and more design studies can
can be
be found
found
inin [2].
[2]. The
The canonical
canonical MAD
MAD web
web page
page isis located
located at:
at:
www.
cern . chlmadlmad9 .html.
www.cern.ch/mad/mad9.html.
MAD9P
Particle Tracker
Tracker with
with
MAD 9 P a
a general
general 3D
3D Particle
Space
Charge
Space Charge
Problem
Problem Frame
Frame
In
of writing
writing efficient
efficient paralparalIn order
order to
to ease
ease the
the task
task of
lel
applications,
we
employ
the
POOMA
framework.
lel applications, we employ the POOMA framework.
POOMA
for mathematical/physical
POOMA provides
provides abstraction
abstraction for
mathematical/physical
quantities
(particles,
fields,
meshes,
differential operators
operators
quantities (particles, fields, meshes, differential
etc.)
in
an
n-dimensional
parallel
fashion.
For
an archiarchietc.) in an n-dimensional parallel fashion. For an
tectural
see Figure
Figure 1.
1. The
The objectobjecttectural overview
overview of
of MAD9P
MAD 9 P see
oriented
complexity of
of explicit
explicit
oriented approach
approach manages
manages the
the complexity
parallel
encapsulates the
the data
data distribudistribuparallel programming;
programming; it
it encapsulates
tion
real or
or virtual
virtual procesprocestion and
and communication
communication between
between real
sors.
POOMA
and
all
the
other
components
are
implesors. POOMA and all the other components are implemented
as
a
set
of
templated
C++
classes.
When
used
mented as a set of templated C++ classes. When used
PSI's
PSI’scurrent
currentsituation
situationand
andthe
theplanned
plannedupdate
update is
is sumsummarized
in
[3].
In
order
to
achieve
the
challenging
marized in [3]. In order to achieve the challenging goals
goals
ofofreliability
reliability and
and high
high power
power operation
operation we
we need
need aa betbetter
qualitative
understanding
of
the
beam
behavior
ter qualitative understanding of the beam behavior in
in the
the
transport
transportlines
linesand
andthe
thecyclotrons.
cyclotrons.
PHYSICAL
PHYSICALAND
ANDMATHEMATICAL
MATHEMATICAL
MODEL
MODEL
MAD9P
MAD 9 P isis based
based on
on the
the Vlasov-Maxwell
Vlasov-Maxwell equations.
equations. In
In
this
thismodel,
model,particle
particlemotion
motionisisgoverned
governedby
byexternal
external fields
fields
and
andaamean-field
mean-field approach
approach for
for the
the space-charge
space-charge fields.
fields.
Particle
collisions
and
radiation
are
neglected.
Particle collisions and radiation are neglected. All
All physphysical
icalelements
elements are
are assumed
assumed to
to be
be perfectly
perfectly aligned.
aligned. The
The
total
totalHamiltonian
Hamiltonianfor
foraabeam
beamline
lineelement
elementcan
canbe
bewritten
written
asasaasum
+H
^,
sum of
of two parts, <ff
H = J#[
H1 +
2 , which correspond
totothe
theexternal
externaland
andspace
spacecharge
chargecontributions.
contributions.A
A secondsecondorder
orderintegration
integrationalgorithm
algorithm (split
(split operator)
operator) for
for aa single
single
step
stepisisthen
thengiven
givenby
by
M (τ ) = M1 (τ /2) M2 (τ ) M1 (τ /2) + O(τ 3 )
MAD
(1)
(1)
Methodical Accelerator Design
CLASSIC Class Library for Accelerator Simulation System and Control
FIGURE
MAD9P.
FIGURE 1. Architectural overview of MAD
9 P.
in
MAD9P
partiin a parallel
parallel environment, MAD
9 P partitions the particles
cles in
in a particle
particle container among the separate processors. The particle
particle spatial layout will keep a particle on
sors.
11
We
Weuse
usethe
thenotion
notionof
offramework
frameworkin
inthe
thefollowing
following sense:
sense: aa framework
framework
isisaaset
setofofco-operating
co-operatingclasses
classesininaagiven
givenproblem
problem frame.
frame.
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
316
cal
direction see
fitting
the
4-dimensional
cal
see Figure
Figure 3)
3) after
after fitting
fitting the
the 4-dimensional
4-dimensional
cal direction
direction see
Figure
3)
after
transverse
distribution
and
aa global
space-charge
neutraltransverse
distribution
and
global
space-charge
neutraltransverse distribution and a global space-charge neutralisation
factor
fee using
aa stochastic
fit
algorithm based
based on
on
isation
factor
f
using
stochastic
fit
algorithm
isation factor fe using a stochastic fit algorithm based on
0.01
0.009
B870Injection
InjectionLine
Line
B870
The
starting
point
for
all
B870
injection
line
calculaThestarting
startingpoint
pointfor
forall
allB870
B870injection
injectionline
linecalculacalculaThe
tions
is
a
4-dimensional
transverse
phase
space
distributions
is
a
4-dimensional
transverse
phase
space
distributions is a 4-dimensional transverse phase space distribution,
which
has
been
proven
be
physically
satisfactory
tion,which
whichhas
hasbeen
beenproven
proventoto
tobe
bephysically
physicallysatisfactory
satisfactory
tion,
the
daily
operation
ofthe
thebeam
beam
line.The
Thelongitudinal
longitudinal
inthe
thedaily
dailyoperation
operationof
beamline.
inin
dimensions
areuniform
uniform
spaceand
andmomenta.
momenta.The
The iniinidimensionsare
uniformininspace
dimensions
tially
DC
beam
is
modeled
by
using
a
characteristic
lontiallyDC
DCbeam
beamis modeled by using a characteristic lontially
gitudinal
beam
length
of
whereλA
thewave
wavelength
length
gitudinalbeam
beamlength
lengthof
ofβ/3A,
βλλ, ,where
λisisthe
gitudinal
of
the
rf
and
/3
the
relativistic
factor.
The
double
gap
therfrfand
andββ the
therelativistic
relativisticfactor.
factor. The
The double gap
gap
ofofthe
buncher
is
modeled
by
(analytic)
sinusoidal
momenta
buncher
is
modeled
by
(analytic)
buncher is modeled by (analytic) sinusoidal momenta
modulation
the
beam.Figure
Figure22shows
shows the
the horizontal
horizontal
modulationofofthe
thebeam.
modulation
beam
envelope
(similar
results
are
obtained
in
the vertivertibeam
envelope
(similar
results
beam envelope (similar results are obtained in the
MWP23
MWP23
0.008
MWP31
0.008
MWP31
0.007
MWP25
MWP27
MWP25
MWP29
MWP27
0.007
MWL09
/m
2σ2σ/m
x
0.006
MWL09
0.006
MWP29
0.005
MWP01
MWP01
x
0.005
MWP07
MWP07
0.004
0.004
MWP13
MWP13
MWL01
MWL01
MWL03
MWL03
0.003
0.003
0.002
MWL07
MWL07
MWP17
MWP17
0.002
MWP19
MWP19
MWP15
MWP15
MWL05
MWL05
0.001
MWP21
MWP21
0.001
0
0
0
2
0
4
2
4
6
8
6
8
s [m]
s[m]
s [m]
10
12
14
10
12
14
16
18
16
18
FIGURE
horizontal
beam profiles
profilesinin
inthe
theB870
B870
FIGURE
2.
Best
fit: horizontal
FIGURE 2.
2. Best
Best fit:
fit:
horizontal beam
beam
profiles
the
B870
line.
line.
line.
0.02
0.02
fluctuations in the measurement
fluctuations
in the measurement
measured
data
measured data with f =0.59
calculated
calculated data with fe =0.59
e
0.018
0.018
0.016
0.016
0.014
0.014
y
0.012
0.012
MWP26
MWP26
MPW28
MPW28
MPW30
MPW30
0.01
0.01
0.008
0.008
0.006
0.006
MWL02
MWL02
0.004
0.004
0.002
0.002
First
Simulation
Results
with
MAD9P
First
FirstSimulation
SimulationResults
Resultswith
withMAD
MAD99PP
the
context
feasibility
study
on
how
to
accelerInIn
Inthe
thecontext
contextofof
ofaaafeasibility
feasibilitystudy
studyon
onhow
howto
toacceleraccelerate
mA
proton
beam
with
the
PSI
cyclotron
facility,
ate
ateaaa333mA
mAproton
protonbeam
beamwith
withthe
thePSI
PSIcyclotron
cyclotronfacility,
facility,
necessary
upgrade
steps
for
the
different
system
componecessary
necessaryupgrade
upgradesteps
stepsfor
forthe
thedifferent
differentsystem
systemcompocomponents
are
under
consideration.
In
particular,
the
qualitanents
nentsare
areunder
underconsideration.
consideration.In
Inparticular,
particular,the
thequalitaqualitative
and
quantitative
knowledge
of
phase
space
transfortive
tiveand
andquantitative
quantitativeknowledge
knowledgeof
ofphase
phasespace
spacetransfortransformations
the
proton
beam
injection
into
the
PSI
Inmations
mationsofof
ofthe
theproton
protonbeam
beamatat
atinjection
injectioninto
intothe
the PSI
PSI InInjector
2
cyclotron
are
essential
for
the
successful
producjector
jector22cyclotron
cyclotronare
areessential
essentialfor
forthe
thesuccessful
successfulproducproduction
high
intensity
beams
with
low
losses.
To
obtain
tion
tionofof
ofhigh
highintensity
intensitybeams
beamswith
withlow
lowlosses.
losses. To
To obtain
obtain
proper
initial
conditions
for
the
Injector
2
cyclotron,
we
proper
initial
conditions
for
the
Injector
2
cyclotron,
proper initial conditions for the Injector 2 cyclotron,we
we
start
with
modeling
the
870
keV
injection
line
(B870)
startwith
withmodeling
modelingthe
the870
870keV
keVinjection
injection line
line (B870)
(B870)
start
from
Cockroft-Walton
pre-injector
to
the
Injector
2.
fromCockroft-Walton
Cockroft-Waltonpre-injector
pre-injectorto
tothe
theInjector
Injector2.
2.
from
fluctuations in the measurement
measured
data
fluctuations
in the measurement
calculated data with f =0.59
e
measured data
calculated data with f =0.59
e
0.01
0.009
2σ /m
/m
2σ
y
the
node which
contains the
section of
the field
in which
the
the node
node which
which contains
contains the
the section
section of
of the
the field
field in
in which
which
the
particle
isislocated.
IfIfthe
particle
moves
to
aanew
pothe
particle
located.
the
particle
moves
to
the particle is located. If the particle moves to a new
new poposition,
this
layout
will
reassign
itit toto aa new
node
when
sition,
this
layout
will
reassign
new
node
when
sition, this layout will reassign it to a new node when
necessary.
This will
maintain locality
between the
partinecessary.
necessary. This
This will
will maintain
maintain locality
locality between
between the
the partiparticles
and
any
distributed
field
and
it
will
help
keep
parcles
cles and
and any
any distributed
distributed field
field and
and itit will
will help
help keep
keep parparticles
which
are
spatially
close
to
each
other
local
to
ticles
ticles which
which are
are spatially
spatially close
close to
to each
each other
other local
local to
to
the
same
processor
as
well.
With
this
concept
(inherthe
the same
same processor
processor as
as well.
well. With
With this
this concept
concept (inher(inherited
in the
particle and
field class)
we do
not need
an
ited
ited in
in the
the particle
particle and
and field
field class)
class) we
we do
do not
not need
need an
an
explicit
and
complicated
particle
manager
class.
A
3D
explicit
explicit and
and complicated
complicated particle
particle manager
manager class.
class. A
A 3D
3D
parallel
particle-mesh solver
is implemented
on top
of
parallel
parallel particle-mesh
particle-mesh solver
solver is
is implemented
implemented on
on top
top of
of
the
POOMA
framework
and
makes
use
of
their
efficient
the
POOMA
framework
and
makes
use
of
their
efficient
the POOMA framework and makes use of their efficient
parallel
parallel
Fourier
transformation
routines.
Using
up
to
32
parallelFourier
Fouriertransformation
transformationroutines.
routines.Using
Usingup
up to
to 32
32
processors
on
a
Beowulf
cluster
at
PSI
with
flat
comprocessors
on
a
Beowulf
cluster
at
PSI
with
flat
comprocessors on a Beowulf cluster at PSI with flat communication
munication
structure,
we
obtain
87.5%
of
the
optimal
municationstructure,
structure,we
weobtain
obtain 87.5%
87.5% of
of the
the optimal
optimal
speedup.
Using
128
processors,
again
on
a
Beowulf
speedup.
Using
128
processors,
again
on
a
Beowulf
clusspeedup. Using 128 processors, again on a Beowulfcluscluster,
but
with
a
non
flat
communication
structure,
we
ter,
but
with
a
non
flat
communication
structure,
we
get
ter, but with a non flat communication structure, we get
get
still
(using
ten
million
still
37.5%
the
optimal
speedup
(using
ten
million
still37.5%
37.5%ofof
ofthe
theoptimal
optimalspeedup
speedup
(using
ten
million
3
3
particles
particles
and
mesh
with
128
points).
On
the
other
particlesand
andaaamesh
meshwith
with 128
1283 points).
points). On
On the
the other
other
hand,
it
is
well
known
that
the
numerical
noise
only
dehand,
it
is
well
known
that
the
numerical
noise
only
dehand, it is well√known
that
the
numerical
noise
only
√N when the number of particles N decreases
with
1/
creases
with
1/V^V
when
the
number
of
particles
N
creases with 1/ N when the number of particles N isis
is
increased.
allows
us
to
use
rouincreased.
The
parallel
approach
allows
us
to
use
rouincreased.The
Theparallel
parallelapproach
approach
allows
us
to
use
routinely
ofofN=
tinely
configurations
10777· •·· •·· •·10
10888 and
and
mesh
sizes
tinelyconfigurations
configurations
ofNN==10
10
10
and mesh
mesh sizes
sizes
3
2
3
2
3
2
from
128
to
128
×
2048.
MAD
9
P
runs
on
a
variety
from
128
to
128
x
2048.
MAD9P
runs
on
a
variety
of
from 128 to 128 × 2048. MAD 9 P runs on a variety of
of
UNIX
platforms
such
as
the
SGI-ORIGIN
2000,
IBMUNIX
platforms
such
as
the
SGI-ORIGIN
2000,
IBMUNIX platforms such as the SGI-ORIGIN 2000, IBMSP2
and
Linux
clusters.
Validation
of
the
code
done
SP2
SP2and
andLinux
Linuxclusters.
clusters.Validation
Validationof
ofthe
the code
code isis
is done
done
by
comparison
with
(semi)
analytic
models.
by
bycomparison
comparisonwith
with(semi)
(semi)analytic
analyticmodels.
models.
0
0
MWL08
MWL08
MWP02
MWP02
MWL04
MWL04
0
2
2
MWP32
MWP32
MWL10
MWL10
MWP14
MWP14
MWP16
MWP16
MWP18
MWP18
MWP22
MWP22
MWP06
MWP06
MWL06
MWL06
MWP20
MWP20
4
4
6
6
8
8
10
10
12
12
14
14
MWP24
MWP24
16
16
18
18
s/m
ss /m
/m
FIGURE
3.
Best
fit: vertical
FIGURE
verticla beam
beam profiles
profilesininthe
theB870
B870line.
line.
FIGURE 3.
3. Best
Best fit:
the
B870
line.
Simulated
as follows,
follows,
Simulated
Simulated Annealing.
Annealing. Define
Define F
F as
^monitors
#monitors
#monitors
F=
F
F=
=
∑
£
n=l
n=1
n=1
(XmeaM-X
(s(s
. 22.
n))
(X
Xsim
mea (sn ) − sim
n ))
2
(2)
(2)
(2)
This
function
is
measure of
of the
the degree
degree of
of conformity
conformity
This
conformity
This function
function is
is aa measure
between
simulation
and
profile
monitor
measurements,
between
measurements,
between simulation
simulation and profile monitor measurements,
where
X
mea
where X
Xmea
(snn)n)) is
rms quantity
quantity at
at
the
powhere
(s
is aa measured
measured rms
at the
the popomea(s
sition
along
the beam
beam line
line and
(s
the
corren)n ) is
sition sssnnn along
along the
Xsim
(s
is
the
corresition
and X
the
corresim
sponding
calculated
quantity obtained
obtained by
by MAD
MAD9P.
The
sponding calculated
calculated quantity
sponding
MAD99PP.. The
The
fitting
procedure
then minimizes
minimizes F
in (2).
(2). As
As shown
shown
in
fitting procedure
procedure then
F in
fitting
shown in
in
Figure
and
we
obtain good
good agreement
Figure 222 and
and 333 we
we obtain
between
meaFigure
agreement between
between meameasurement
and
simulation. The
The space-charge
space-charge neutralisaneutralisasurement and
and simulation.
simulation.
surement
neutralisation
factor
f
=
0.59
obtained
is
in
the
expected
range
tion factor
factor ffeee =
= 0.59 obtained is in the expected
tion
expected range
range
[4].
The
discrepancy
in
MWP15
is
not
fully
understood.
[4]. The discrepancy in MW P15 is not fully understood.
One possible
possible reason
reason is
is the
the large
large background
One
background which
which is
is not
not
included
by
the
model
simulation
at
the
moment.
included by the
model
moment. The
The
deviations
seen at
at MWP25
to MWP31
are related
related
to
the
deviations seen
seen
MW P25 to
MW P31 are
deviations
related to
to the
the
buncher
and
the
high
dispersive
region
in
this
part
of
the
buncher and the high dispersive region in this part of the
the
beam line.
line. More
More detailed
detailed modeling
modeling is
beam
is needed
needed in
in order
order to
to
minimize the
the gap
gap between
between theory
minimize
theory and
and observation.
observation.
317
Injector
2,
Coasting
Beam
Injector 2,
2, Coasting
CoastingBeam
Beam
Injector
Beam
Injector
2,
Coasting
OUTLOOK
OUTLOOK
OUTLOOK
OUTLOOK
AA model
of
lattice
based
on
hard-edge
of the
the Injector
Injector 2222lattice
latticebased
basedon
onhard-edge
hard-edge
of
the
Injector
based
on
hard-edge
A model
model
of
the
Injector
lattice
elements
is
used
for
various
coasting
beam
simulations.
is
used
beam
simulations.
elements
used for
for various
variouscoasting
coastingbeam
beamsimulations.
simulations.
elements
is
used
for
various
coasting
The
2D
of
Stefan
Adam
[5],
which
predict
The early
early
2D
results
Adam
[5],
which
predict
2D results
results of
of Stefan
StefanAdam
Adam[5],
[5],which
whichpredict
predict
earlyround
2D
results
of
Stefan
aaThe
stable
distribution
in
horizontallongitudinal
round
horizontallongitudinal
stable
round distribution
distribution in
in horizontalhorizontal-longitudinal
longitudinal
a
stable
round
distribution
in
configuration
space
has
been
verified
(see
Figure
and
configuration
verified
(see
Figure
and
configuration
space has
has been
beenverified
verified(see
(seeFigure
Figure4444and
and
configuration space
space
has
been
5)
with
the
full
3D
model.
The
data
shown
in
the
Figures
full
3D
model.
The
data
shown
in
the
Figures
5) with the
full
3D
model.
The
data
shown
in
the
Figures
the full 3D model. The data shown in the Figures
44 and
and
mA.
The
effect
the
are
for
MeV
and 55 are
are for
for 555 MeV
MeV and
and 111 mA.
mA. The
Theeffect
effectofof
ofthe
the
4 and 5 are for 5 MeV and 1 mA. The effect of the
0.02
0.02
0.02
0.02
0.02
0.02
10
1010
0
0
1
1 1
-0.01
-0.01
-0.02
-0.02
-0.02
0.01
0.01
0.01
Longitudinal/ m
/m
Longitudinal
Longitudinal / m
Longitudinal/ m
/m
Longitudinal
Longitudinal
0.01
0.01
-0.01
-0.01
-0.01
-0.005
0
0.005
0.01
0.01
0.01
-0.005
00
0.005
-0.005
0.005
Horizontal
/m
Horizontal
Horizontal//mm
1
1
-0.01
-0.01
-0.01
-1
-0.015
-0.015
-0.015
10
10 10
0
0 0
10 -1
-1
10
0.015 10
0.015
0.015
-0.02
-0.02
-0.02
1
-1
-0.015
-0.01
-0.005
-0.015
-0.015 -0.01
-0.01 -0.005
-0.005
0
0.005
0.01
0.015
10 -1
10 10 -1
Adding acceleration to the hard-edge model of Injector 2
Adding
acceleration
to
hard-edge
model
of Injector
2
Adding
acceleration
to the
thethe
hard-edge
model
of Injector
Injector
Adding
acceleration
to
hard-edge
model
of
22 inand
using
the
initial
conditions
obtained
from
B870
and
using
the
initial
conditions
obtained
from
B870
in- inandusing
using
the
initial
conditions
obtained
from
B870
and
the
initial
conditions
obtained
from
B870
injection
line
simulations
will
allow
to
compare
beam
jection
line
simulations
will
allow
us us
to compare
compare
beam
jectionline
line
simulations
will
allow
us
to
compare
beam
jection
simulations
will
allow
us
to
beam
profile
measured
in
the
cyclotron
[6,
7,
8]
with
calculaprofile
measured
in in
thethe
cyclotron
[6,[6,
7, 8]
8]
with
calculaprofile
measured
cyclotron
7, with
8]
with
calculaprofile
measured
in
the
cyclotron
[6,
7,
calculations.
tions.
tions.
tions.
Missing
physics
such
aa residual
gas
model,
beam
Missing
physics
such
as as
residual
gasgas
model,
beam
Missingphysics
physics
such
as
residual
model,
beam
Missing
such
as
aa residual
gas
model,
beam
collimation
including
secondary
effects,
and
neighboring
collimation
including
secondary
effects,
andand
neighboring
collimation
including
secondary
effects,
neighboring
collimation
including
secondary
effects,
and
neighboring
turns
the
cyclotron,
will
added
to
turns
inin
the
cyclotron,
will
bebe
added
to MAD
MAD
turns
in
the
cyclotron,
will
be
added
to MAD9P.
MAD
turns
in
the
cyclotron,
will
be
added
to
99PP..9 P.
Different
types
parallel
field
solver,
such
as treetreeDifferent
types
ofof
parallel
field
solver,
such
as treetreeDifferent
types
of
parallel
field
solver,
such
as
Different
types
of
parallel
field
solver,
such
as
based,
multigrid
and
hybrid
versions
are
planned
to
be aa
based,
multigrid
and
hybrid
versions
are
planned
to
be
based,
multigrid
and
hybrid
versions
planned
to aabe
based,
multigrid
and
hybrid
versions
areare
planned
to be
part
of
MAD9P.
Carefully
benchmarking
the
field
solvers
part
of
MAD
9
P
.
Carefully
benchmarking
the
field
solvers
part
of
MAD
9
P
.
Carefully
benchmarking
the
field
solvers
part of MAD 9 P. Carefully benchmarking the field solvers
against
other
codes
and
analytical
models
help
us
against
other
codes
and
analytical
models
willwill
help
us to
to
against
other
codes
and
analytical
models
will
help
us
against
other
codes
and
analytical
models
will
help
us to
to
understand
their
different
behavior.
understand
their
different
behavior.
understand
their
different
behavior.
understand
their
different
behavior.
0 0/ m
0.005
Horizontal
0.005 0.01 0.01 0.0150.015
Horizontal
/ m/ m
Horizontal
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
FIGURE
Charge density
in
a.u.:
Turn
1 and
6.
FIGURE
density
in
a.u.:
Turn
and
FIGURE 4.
4.
FIGURE
4. Charge
Chargedensity
densityin
ina.u.:
a.u.:Turn
Turn111and
and6.6.
6.
Longitudinal
Longitudinal
/m
Longitudinal
/ /mm
0.01
0.01
0.01
10
10
10
0
0
0
1
1
-0.01
-0.01
10
0.01
0.01
0.01
-0.02
-0.02
Many
thanks
toto
Martin
Humbel
andand
thethe
PSIPSI
proton
ac- acMany
thanks
Martin
Humbel
proton
Many
to
Martin
Humbel
and
the
PSI
proton
acManythanks
thanks
to
Martin
Humbel
and
the
PSI
proton
acceleration
operation
staff
for
providing
the
experimental
celeration
operation
staff
for
providing
the
experimental
celeration
operation
staff
forfor
providing
thethe
experimental
celeration
operation
staff
providing
experimental
data.
I Iwould
also
acknowledge
thethe
continuous
support
data.
also
acknowledge
continuous
support
data.
also
acknowledge
the
continuous
support
data.I would
I would
would
also
acknowledge
the
continuous
support
given
bybyand
and
thethemany
many
fruitful
discussion
with
Stefan
given
and
many
fruitful
discussion
with
Stefan
given
by
the
fruitful
discussion
with
Stefan
given by and the many fruitful discussion with Stefan
Adam,
Ralph
Eichler,
Rolf
Jeltsch,
Thomas
Stammbach
Adam,
Ralph
Eichler,
Rolf
Jeltsch,
Thomas
Stammbach
Adam,
Ralph
Eichler,
Rolf
Jeltsch,
Thomas
Stammbach
Adam,
Ralph
Eichler, Rolf Jeltsch, Thomas Stammbach
and
Robert
Ryne.
and
Robert
Ryne.
and
Robert
Ryne.
and
Robert
Ryne.
2
10
10
10
0
0
0
1
1
-0.01
-0.01
1
-0.01
10 2
10 2
0.02
0.02
0.02
Longitudinal/ /mm
Longitudinal
Longitudinal / m
10 22
10 2
10
0.02
0.02
0.02
1
-0.01
-0.02
-0.02
-0.02
-0.015
-0.015
-0.01
-0.01
-0.015
-0.01
-0.005
0
0.005
0.01
0.01
-0.005
0
0.005
Horizontal
/m
Horizontal
-0.005
0 / m
0.005
Horizontal / m
0.01
-0.02
-0.015
-0.015
0.015
0.015
0.015
-0.01
-0.01
-0.015
-0.005
0
0.005
0.01
0.015
-0.005
0
Horizontal
m0.005 0.01 0.015
Horizontal
//0 m
-0.005
0.005
0.01
0.015
Horizontal / m
FIGURE
Charge
in a.u.:
a.u.: Turn
Turn 10
10 and
and 60.
60.
FIGURE 5.
5. Charge
Charge density
density in
FIGURE
FIGURE 5.
5. Charge density
density in
in a.u.:
a.u.: Turn
Turn 10
10 and
and 60.
60.
beam
intensity
on
the
of the
the rms
rms beam
beam sizes
sizes
beamintensity
intensity on
on the
the development
development
of
beam
development
of
the
rms
beam
sizes
beam
intensity
onand
thelongitudinal
developmentdirections
of the rms
in
the
horizontal
directions
is beam
shownsizes
in
in
the
horizontal
and
longitudinal
is
shown
in
in
the
horizontal
and
longitudinal
directions
is
shown
in
in
the horizontal
and longitudinal
directions in
is shown
in
Figure
6 for 60 turns.
The strong oscillations
the
first
Figure
6
for
60
turns.
The
strong
oscillations
in
the
first
Figure
6
for
60
turns.
The
strong
oscillations
in
the
first
few turns are due to an initial ‘mismatch’ of the beam.
few
turns
are
due
to
an
'mismatch'
of the
few
turnsthat
arethe
due
tobeam
an initial
initial
‘mismatch’
the beam.
beam.
The fact
rms
size increases
withofincreasing
The
fact
that
the
rms
beam
size
increases
with
increasing
The
that the
rms beam
size that
increases
with increasing
beamfact
current
strongly
suggests
the matching
of the
beam
current
strongly
that
the matching
of
beam
current
strongly
suggests
that
of the
the
incoming
beam
has to suggests
be
adapted
to the
the matching
beam intensity
incoming
beam
has
to
be
adapted
to
the
beam
intensity
incoming
haskey
to be
to the
beam intensity
and mightbeam
be the
for adapted
a very fast
development
of
and
might
key
for
very
of
and
might be
be the
theand
keystable
for aadistribution.
very fast
fast development
development
the desired
round
This first trulyof
the
round
and
distribution.
This
the
desired
round
and stable
stable
distribution.
This first
first
truly
3D desired
simulation
suggests
again,
that the concept
oftruly
an
3D
simulation
suggests
that
concept
of
isochronous
cyclotron
is again,
well
for
intensity
3D
simulation
suggests
again,suited
that the
the high
concept
of an
an
isochronous
cyclotron
isis is
well
suited
for
intensity
operation. More
research
however
isochronous
cyclotron
well
suitedneeded
for high
high
intensity
in order
to
operation.
More
research
isis however
needed
finally make
predictions
the redesign
operation.
More
researchfor
however
needed
in order
order
to
of in
the
B870to
finally
make
predictions
for
the
redesign
of
the
line, to make
allow predictions
operation beyond
finally
for thethe
redesign
of achieved
the B870
B870
presently
line,
to
2 mA.
line,
to allow
allow operation
operation beyond
beyond the
the presently
presently achieved
achieved
2mA.
2 mA.
0.01
0.01
0.007
0.007
0.006
0.006
0.007
0.01
0.008
0.008
0.005
0.005
0.006
0.008
0.006
0.006
0.004
0.004
0.005
0.003
0.003
0.004
0.006
0.004
0.004
0.002
0.002
0.003
0.002
0.004
0.002
0.001
0.001
0.002
00
0
0.001 0
0
0
25
25
25
25
50
50
SO
50
75
75
100
100
125
125
150
150
175
175
200
200
75
799
725
759
775
299
75
100
125
150
175
200
225
225
225
225
0.002
00
00
25
25
50
50
75
75
100
100
0
0
REFERENCES
REFERENCES
REFERENCES
REFERENCES
-0.01
25
50
75
125
125
150
150
175
175
200
200
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1995.
Workshop, 1995.
225
225
I 725 759 775 299 225
100 125 150 175 200 225
s/ m
FIGURE 6. Horizontal and Longitudinal rms beam sizes at
different intensities.
FIGURE
FIGURE 6.6. Horizontal
Horizontal and
and Longitudinal
Longitudinal rms
rms beam
beam sizes
sizes at
at
different
different intensities.
intensities.
318