Simulation Studies for Inspection of the Benchmark Test
with PATRASH
Y.Shimosaki, S.Igarashi, S.Machida , M.Shirakata , K.Takayama ,
RNoda*, K.Shigaki*
*KEK, Japan
JAERI, Tokai, Ibaraki, Japan
#
Abstract. In order to delineate the halo-formation mechanisms in a typical FODO lattice, a 2-D simulation code
PATRASH (PArticle TRAcking in a Synchrotron for Halo analysis) has been developed. The electric field originating
from the space charge is calculated by the Hybrid Tree code method. Benchmark tests utilizing three simulation codes of
ACCSIM, PATRASH and SIMPSONS were carried out. These results have been confirmed to be fairly in agreement
with each other. The details of PATRASH simulation are discussed with some examples.
other macro particles outside the beam core is divided
into small square-cells. In the Tree method, the electric
field generated by PIC-style and macro particles placed
in each cell is calculated in a way of multipole
expansion. The Tree method consists of two processes,
one of that is called "upward pass" and the other
"downward pass". In the upward pass, the coefficients
of the multi-pole expansion around the center of each
cell are obtained; then, those are transformed to the new
coefficients around the center of the square-cell of
upper level, which involves the lower class four cells.
Same process is repeated until the total area is filled
with 24 cells. In the downward pass, the electric filed
exerted on a particular tracking particle is obtained by
using the above plural coefficients according to some
rule.
INTRODUCTION
One of the major issues in high-power hadron
accelerators is activation of the environment
surrounding an accelerator due to beam loss. Beam loss
must be reduced to a sufficiently low level to allow
hands-on-maintenance. In order to produce an
acceptable design, a reliable simulation code is essential
for estimating beam loss caused by space charge effects,
nonlinear components of magnetic fields and machine
imperfection. For the purpose to justify the simulation
codes in our hands now, benchmark tests utilizing three
simulation codes of ACCSIM, PATRASH and
SIMPSONS were carried out by the 3GeV ring
simulation group of the Joint Project. The machine
parameters are based on the early version of the 3GeV
Ring lattice for the Joint Project. The lattice is
characterized by (1) three-fold symmetry, (2) high yt,
(3) dispersion free in the straight sections, (4) beta
function with low symmetry and (5) half integer tune
split. The operating tune has been chosen in the region
far from the major structure resonances, where the
lattice functions are stable under perturbations and give
a relatively small beam size. In this paper, the detail of
PATRASH is described in the following subsections.
The characteristics of each code are summarized in [1].
In the next section, the preliminary results for the
inspection of the benchmark tests by using PATRASH
are presented.
BENCHMARK TEST OF CODES
As benchmark tests, 2D calculations were performed
over 1000 turns at the injection energy of 400MeV and
A/yP = 0. Saturation of simulation results against the
simulation parameters such as longitudinal step-size
and number of macro particles had been checked by
varying those parameters in the individual codes.
Sextupole fields for chromaticity correction are
included here. An initial distribution was prepared,
which is generated by the anti-correlated painting
method without space-charge forces and other
non-linear magnetic fields.
In the benchmark test, the temporal evolution of 90%,
95% and 99% Courant-Snyder invariants were
compared as a function of peak current /peak in the
bunched beam. Over the entire region of revolution,
simulation results were in good agreement with each
other within the 10% relative error as shown in Figs.l.
We believe that ACCSIM, PATARSH and SIMPSONS
The electric field originating from the space charge
is calculated by the Hybrid Tree method briefly
explained below. Charge of the macro-particles, which
are tracked in simulations, located in the beam core
region is assigned as PIC-style charge, where the beam
core is defined as a square where the length of one side
is twice the rms beam size. Then, the transverse area
including the above PIC-style particles on the grids and
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
High Intensity and High Brightness Hadron Beams, edited by W. Chou, Y. Mori, D. Neuffer, and J.-F. Ostiguy
© 2002 American Institute of Physics 0-7354-0097-0/02/$ 19.00
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FIGURE 1.
FIGURE
1. Time
Time evolution
evolution of
of the
the (a)-(c)
(a)-(c) horizontal
horizontal and
and (d)-(f)
(d)-(f) vertical
verticalCourant-Snyder
Courant-Snyderinvariant(C-S)
invariant(C-S)asasaa
function
of
I
.
Black,
blue
and
red
represent
ACCSIM,
PATRASH
and
SIMPSONS,
respectively.
peak
function of /peak. Black, blue and red represent ACCSIM, PATRASH and SIMPSONS, respectively.
particles
EMITTANCE GROWTH
particles at
at early
early stage
stage has
has been
been carefully
carefully examined
examined by
by
EMITTANCE
GROWTH AT
AT EARLY
EARLY
PATRASH
from
a
beam–dynamics
point
of
view
asasaa
STAGE
JUST
AFTER
DEPOSIT
OF
THE
PATRASH
from
a
beam-dynamics
point
of
view
STAGE JUST AFTER DEPOSIT OF THE
preliminary
preliminary study
study for
for the
the benchmark
benchmarktest.
test.
INITIAL BEAM
INITIAL
BEAM
The
The beam
beam distributions
distributions atatthe
the 4th
4th turn
turnare
areshown
showninin
Figs.2.
It
is
clearly
shown
that
the
inner
particles
Figs.2. It is clearly shown that the inner particlesatatthe
the
initial turn move to the outside in the vertical phase
initial turn move to the outside in the vertical phase
space. In the horizontal phase space, the structure can be
space. In the horizontal phase space, the structure can be
also seen around the outer edge. These structures in the
also seen around the outer edge. These structures in the
phase space were not seen at the initial turn. On the
phase space were not seen at the initial turn. On the
other hands, the distribution in the real space
other hands, the distribution in the real space
maintained the asymmetry in the vertical direction from
maintained the asymmetry in the vertical direction from
initial turn.
initial turn.
For understanding what happened at the early stage,
For understanding what happened at the early stage,
The Poincaré plot analysis was carried out. Its results
The Poincare plot analysis was carried out. Its results
As shown
shown in
in Figs.l,
Figs.1, the
horizontal emittance
As
the horizontal
emittance rapidly
rapidly
blows
up
to
240π
µmrad
at
the
early
blows up to 24071 jimrad at the early stage
stage (1-20turn)
(l-20turn)
without the dependence on the peak current in contrast
without
the dependence on the peak current in contrast
to the vertical emittance. Because this rapid emittance
to the vertical emittance. Because this rapid emittance
blow up independent of the beam intensity indicates
blow up independent of the beam intensity indicates
that there is the possibility of the rapid halo generation
that there is the possibility of the rapid halo generation
independent of the beam intensity at the early stage, the
independent of the beam intensity at the early stage, the
inspection of the benchmark tests becomes important
inspection of the benchmark tests becomes important
for making it clear that this phenomenon comes from
for making it clear that this phenomenon comes from
the physics such as the resonance or the artificial error
the physics such as the resonance or the artificial error
in these simulations. Therefore, the behavior of the
in these simulations. Therefore, the behavior of the
257
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FIGURE 3. Poincaré plot
of the characteristic 4
FIGURE
3. Poincare plot
plot of
the characteristic 4
FIGURE
particles
of the3.18Poincaré
test partices. of the characteristic 4
particles
of
the
18
test
partices.
particles of the 18 test partices.
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FIGURE 2.
2. Beam
Beam (c)
distributions
(a),
phase
and
(c)
in
the
real
space.
phase
and (c) in
real space.
FIGURE
2. theBeam
distributions (a), (b) in the
*£»
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y' [mrad]
y' [mrad]
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of the resonances are now under analyzing.
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8
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(b) (b)
vertical
vertical
40
40
20
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20
0
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0
-200
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x' [mrad]
x' [mrad]
are shown in Figs.3, where the positions of the
are shown were
in Figs.3,
the turn
positions
the
test-particles
plottedwhere
turn by
at theoffoil
test-particles
were
plotted
turn
by
turn
at
the
foil
position
until 20turn.
The plots
ofthe
the positions
red and green
are shown
in
Figs.3,
where
the
position
until
20turn.
The
plots of
the red
and ofgreen
particles
in
the
horizontal
direction
clearly
indicate
the
test-particles
were
plotted
turn
by
turn
at
the
foil
particles in
the horizontal
direction
clearly
indicate
the
2 islands.
moves
from ofthe
to green
the
position The
untilblack
20turn.
The plots
theoutside
red and
2
islands.
The
black
moves
from
the
outside
to
the
inside
in horizontal
and from the
insideclearly
to the outside
particles
in the horizontal
direction
indicatein
the
inside in horizontal and from the inside to the outside in
vertical.
The blue
is in moves
reversefrom
of the
the outside
black. to
The
2 islands.
The black
the
vertical. The blue is in reverse of the black. The
inside in
inside to seem
the outside
behaviors
ofhorizontal
the blackand
andfrom
bluetheparticles
to bein
behaviors of the black and blue particles seem to be
vertical.
The blue
is inresonances.
reverse of For
the black.
The
induced
by
some
coupling
finding
a
induced by some coupling resonances. For finding a
behaviors
of the resonance,
black and blue
particlesofseem
to be
trace
of
the
coupling
the
histories
the
each
trace
of thebycoupling
resonance,
the histories
the eacha
induced
some
coupling
resonances.
Forofinfinding
Courant-Snyder
invariant
areare
plotted
asasshown
Fig.4.
Courant-Snyder
invariant
plotted
shown
in
trace
of the coupling
resonance,
the
histories
of theFig.4.
each
TheThe
Courant-Snyder
invariants
of
the
black
and
blue
Courant-Snyder
invariants
of theasblack
and
blue
Courant-Snyder
invariant
are
plotted
shown
in
Fig.4.
particles
clearly
move
straight
lines.
The
particles
clearly
movealong
alongsome
some
straight
lines.
The Courant-Snyder
invariants
ofthat
the
black
and The
blue
canonical
perturbation
theory
tells
us
a
combination
canonical
perturbation
theory
tells
us that
a combination
particles
clearly
move
along
some
straight
lines.
The
of ofthethehorizontal
and
action
variables
horizontal
andthe
thevertical
vertical
action
variables
canonical
perturbation
theory
tells us that
a combination
become
a constant
if if
a coupling
resonance
is induced.
become
constant
a coupling
resonance
of the ahorizontal
and
the vertical
actionis induced.
variables
WeWe
suspects
that
thetheemittance
blow
upupatatthe
early
suspects
that
emittance
blow
early
become a constant if a coupling resonance isthe
induced.
stage
is
induced
by
some
coupling
resonances.
The
stage
is
induced
by
some
coupling
resonances.
The
We suspects that the emittance blow up at the early
specifications
of of
thethe
coupling
and
specifications
coupling
resonances
andthe
thesource
source
stage is induced
by
someresonances
coupling
resonances.
The
of the
resonances
arethe
now
under
analyzing.
ofspecifications
the
resonances
are
now
under
analyzing.
of
coupling
resonances
and the source
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200
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200
300
100
200
300
00Horizontal
100C-S [π
200
300
mm mrad]
Horizontal C-S
Horizontal
C-S [π
[n mm
mmmrad]
mrad]
FIGURE 4.
Temporal evolution of the
FIGURE 4.
4.
Temporal evolution
ofof the
FIGURE
Temporal
evolution
the
Courant-Snyder
invariant.
The color
corresponds
to
Courant-Snyder invariant. The color corresponds to
Courant-Snyder
invariant. The color corresponds to
Figs.3.
Figs.3.
Figs.3.
258
SUMMARY
The results of PATRASH were fairly in agreement
with ACCSIM and SIMPSONS. PATRASH can be said
as one of the reliable codes in our hands.
The rapid emittance blow up independent of the
beam intensity in the horizontal direction may be
induced by the coupling resonances. The preliminary
study is presented in this paper and a whole analysis
will be given in the coming paper.
REFERENCES
1. http://hadron.kek.jp/member/onishi/tdr/index.html
259