Tracking Studies For JHF Lattice
Alexandre Molodojentsev, Shinji Machida
KEK, High Energy Accelerator Research Organization
1-1 Oho, Tsukuba, Japan 305-0801
Abstract. Dynamic aperture of the Rapid Cycling Synchrotron and the Main Ring of the JAERI-KEK project is
simulated. The non-linear resonances, excited by the fringe fields of the magnets and the field nonlinearity of the
sextupole magnets for the chromaticity correction, are studied. The operating point for both machines is optimized to
provide maximum survived beam emittance. Finally, main misalignment errors, that introduce additional field nonlinearities, are included in the simulations.
magnets for the chromaticity correction in
combination with the fringe field effect of the dipole
magnets ('sextupole'-like effect), respectively.
INTRODUCTION
Accurate tracking study for JHF [1] (JAERI-KEK
Project) lattice should be performed to eliminate
excitation of the resonances, which can limit the
dynamic aperture. These resonances are determined by
the intrinsic non-linearities of the magnetic field even
without any magnetic field imperfections or alignment
errors. In the case of RCS (Rapid Cycling
Synchrotron) acceptance of the synchrotron and the
aspect ratio of the magnets (inner diameter over
magnet length) should be large to meet the multi-turn
injection process. Then the fringe fields of the magnets
can change the particle motion significantly. In the
case of MR (Main Ring) another source of the
magnetic field non-linearity should be taken into
account, in particular, the field nonlinearity of the
sextupole magnets used to correct the 'linear'
chromaticity. Optimization of the working point
position is performed to provide maximum available
beam emittance survived in the collimator acceptance.
The bending arc of MR consists of 8 identical
modules with the betatron phase advance of 0.75 per
module both in the horizontal and vertical phase
planes. This phase advance in the horizontal phase
plane is needed to provide the "zero-dispersion"
condition outside of the arcs. The design working
point of MR has the betatron tunes of 22.33 and 22.28
in the horizontal and vertical phase planes,
respectively. The horizontal tune is determined by the
slow extraction technique, based on the 3rd-order
horizontal resonance. In the vicinity of the design
working point the following structure resonance lines
are located: 2QH-2QV=0 and 2QH=45. The natural
linear chromaticity of MR is about -30 in the
horizontal and vertical phase planes that should be
corrected by the sextupole magnets placed in the arcs.
For the design betatron phase advance per module of
the arc all driving terms of the third-order resonances
excited by the sextupole magnets and the average
sextupole field components of the bending magnets are
cancelled [2]. The beam emittance and the collimator
acceptance are 54 and 817C-mm-mrad, respectively.
ANALYSIS OF LATTICES
The design working point on the betatron tune
diagram for RCS is 6.72 and 6.35 in the horizontal and
vertical planes, respectively. After the multi-turn
injection the beam should have the full transverse
emittances of 216 7C-mm-mrad in the both phase planes
with the momentum spread of (Ap/p) = ±0.01. The
natural 'linear' chromaticities before the correction are
-8.253 and -8.359 in the horizontal and vertical
planes, respectively. To correct the momentum tunespread two families of the sextupole are used. The
collimator acceptance of RCS is 324 7C-mm-mrad. In
the area near the design working point the following
structure resonance lines are located: 4QH=27 and
2Qv-Qn=6. Even without any magnetic field
imperfections these resonances can be excited by the
fringe fields of the quadrupole magnets ('octupole'-like
effect) and the non-linear field of the sextupole
Simulations of the dynamic aperture limitation are
performed by COSY INFINITY [3]. This code is
based on the Differential Algebraic and the symplectic
integration. To assess the long-term stability of
particles in a periodic accelerator structure in a
reasonable amount of time, the code uses an
approximation of the one turn map, and then track
particles with the map. For the accurate presentation of
the system, the maps have to be computed to a
sufficiently high order.
Linear Detuning Effect
The fringe fields of the magnets induce a variety of
amplitude/momentum dependent effects beginning
from changing of the betatron tunes. The linear
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
143
4Q
4QHH=27
=27 excited
excited by the fringe field of the quadrupole
magnets.
magnets.
amplitude
amplitude dependent
dependent detuning
detuning coefficients
coefficients calculated
calculated
for
for the
the design
design working
working points
points of
of RCS
RCS and
and MR
MR are
are
collected
collectedininTable
Table1.1.
TABLE
TABLE1.1.Linear
Linearamplitude
amplitudedetuning
detuningcoefficients
coefficients
Fringe
Sextupole
Combined:
Fringe
Sextupole
Combined:
Field
Chromaticity
FF+SCC
Field
Chromaticity
FF+SCC
Correction
Correction
RCS
RCS
-1
aahh
13.909
1.408
14.113
1.408
14.113
13.909
[m']1]
hh [m
-1 1
aa
]]
12.824
25.625
33.981
[m"
12.824
25.625
33.981
vvw[m
-1 1
[in
18.354
aahvhv[m
]]
18.354
-22.997
-6.748
-22.997
-6.748
MR
MR
-1 1
30.834
180.064
207.610
aa^
]]
30.834
180.064
207.610
[m"
hh [m
-1 1
[m"
35.813
135.128
161.758
aa
[m
]]
35.813
135.128
161.758
vvw
-1 1
30.388
171.553
198.183
[in
aahvhv[m
]]
30.388
171.553
198.183
QvV=6.35
=6.35
-Q
QV=6.30
~Q
QvV=6.27
=6.27
-Q
8
/ εBEAM
5
4QHH=27
2QV-QH=6
ε
DA
6
ΗV
7
443322-
2QV-QH=6
0.58 0.60
0.60 0.62
0.62 0.64
0.64 0.66
0.66 0.68 0.70 0.72 0.74 0.76
0.58
Fractional horizontal
horizontal tune
Fractional
FIGURE 1.
1. On-momentum
On-momentum dynamic
dynamic aperture of RCS for
FIGURE
different Q
QHH and
and Q
QVv around
around the design values.
different
For RCS
RCSthe
the detuning
detuning effect
effect isis determined
determined by
by the
the
For
combined effect
effect ofof the
the non-linear
non-linear fringe
fringe field
field of
of the
the
combined
ring magnets
magnets and
and the
the sextupole
sextupole magnets
magnets for
for the
the
ring
chromaticitycorrection.
correction.InInthe
the case
case of
of MR
MR mainly
mainly the
the
chromaticity
sextupole magnets
magnets for
for the
the chromaticity
chromaticity correction
correction
sextupole
determinethis
thiseffect.
effect.
determine
In FIG.1
FIG.l the initial beam emittances in the
In
horizontal and
and vertical phase planes are equal to each
horizontal
other. The
The symplectic
symplectic tracking was performed
other.
performed during
1000 turns.
turns. If
If the
the operation working point is QHH=6.72,
1000
Qv=6.35
the dynamic aperture is limited by
Q
the
V=6.35
£n,v/£BEAM~3.
This value can be improved about 2
εH,V/εBEAM~3. This
times
if
the
operating
working point is located in the
times if the
the
area
Q
=6.64..
.6.70
and
QVv= 6.27…6.30.
6.27.. .6.30.
area QHH=6.64…6.70 and Q
DYNAMICAPERTURE
APERTURE
DYNAMIC
The onon- and
and off-momentum
off-momentum dynamic
dynamic aperture
aperture for
for
The
both machines
machines isis analyzed
analyzed including
including the
the fringe
fringe field
field
both
effects ofofthe
thering
ring magnets
magnets and
and the
the sextupole
sextupole magnets
magnets
effects
usedfor
forthe
the 'linear'
'linear' chromaticity
chromaticity correction.
correction. For
For these
these
used
simulations the
the dynamic
dynamic aperture
aperture (or
(or the
the dynamic
dynamic
simulations
acceptance) ofof the
the machine
machine isis determined
determined as
as the
the
acceptance)
maximum
initial
beam
emittance
that
remain
maximum initial beam emittance that remain
circulating inin the
the chamber,
chamber, which
which has
has the
the horizontal
horizontal
circulating
and
vertical
sizes
much
bigger
(10
times)
than the
the
and vertical sizes much bigger (10 times) than
physical aperture
aperture atat the
the observation
observation point.
point. The
The
physical
physical limitation
limitation ofof the
the chamber
chamber size
size at
at the
the
physical
observation point
point isis determined
determined by
by the
the corresponding
corresponding
observation
Twissparameters
parametersand
andthe
thecollimator
collimator acceptance.
acceptance. Main
Main
Twiss
goalofofthis
thisstudy
studyisisoptimization
optimization of
ofthe
the working
working point
point
goal
position
to
avoid,
first
of
all,
any
limitations
of
the
position to avoid, first of all, any limitations of the
dynamic
aperture
caused
by
the
nearest
structure
dynamic aperture caused by the nearest structure
resonances. Moreover,
Moreover, the
the beam
beam survival
survival in
in the
the
resonances.
collimator
acceptance
for
both
machines
will
be
collimator acceptance for both machines will be
discussed
including
main
misalignment
errors.
discussed including main misalignment errors.
From the
the practical point of view it is necessary to
From
consider survival
survival of the beam inside of the collimator
consider
collimator
aperture. Moreover,
Moreover, the
the misalignment
misalignment errors
aperture.
errors of
of the
the
ring
magnets
can
lead
to
excitation
of
additional
ring magnets can lead to excitation of additional
resonances, which
which are
are not
not structure
structure ones.
ones. In
resonances,
In the
the case
case of
of
RCS we
we should
should study
study the
the influence
influence of
of the
the coupling
RCS
coupling
resonance Q
Qn+Qv=13,
excited by
by the
the skew
skew
resonance
excited
H+QV=13,
quadrupoles, and
and the
the 33rdrd order
order resonances
resonances 3Q
3QVV=19
quadrupoles,
=19 and
and
2QHH-Q
-Qv=7,
excited by the skew sextupoles. The axial
2Q
V=7, excited by the skew sextupoles. The axial
shift of
of the
the sextupole
sextupole magnets
magnets generates
generates an
an additional
additional
shift
source
for
the
skew
quadrupole
field
component.
source for the skew quadrupole field component.
The following
following misalignment
misalignment errors
errors are
are assumed
assumed for
The
for
this simulation
simulation that
that correspond
correspond to
to the
this
the design
design
requirements: σOTILT
= 0.5mrad and OSHIFT= 0.5mm.
requirements:
TILT = 0.5mrad and σSHIFT= 0.5mm.
The Gaussian
Gaussian distribution
distribution of
of the
the misalignment
misalignment errors
The
errors
has been
been generated
generated with
with the
the maximum
has
maximum value
value 2.5a.
2.5σ. In
In
the case
case of
of the
the shifted
shifted sextupole
sextupole magnets
magnets the
the
the
contribution to
to COD
COD is
is negligible.
negligible. In
In the
the case
contribution
case of
of the
the
design
Q
value
(Q
=6.35)
the
survival
of
the
beam
v
v
design QV value (QV=6.35) the survival of the beam is
is
limited first
first of
of all
all by
by the
the structure
structure resonance
resonance 2Q
limited
2QVV-QH=6
=6 and
and the
the 'linear'
'linear' coupling
coupling resonance
resonance Q
Qn+Qv=13.
Q
H
H+QV=13.
If
the
Q
is
changed
(Q
=6.27),
the
beam
survival
v
v
If the QV is changed (QV=6.27), the beam survival can
can
be improved
improved in
in the
the Q
QHrange
range of
of 6.64…6.70
6.64.. .6.70 (FIG.2).
(FIG.2).
be
H
Further study
study including
including the
the intrinsic
intrinsic magnet
Further
magnet field
field
nonlinearities
(systematic
and
random)
is
nonlinearities (systematic and random) is needed
needed to
to
estimate the
the reduction
reduction of
of the
the dynamic
dynamic aperture
aperture by
estimate
by the
the
high-order resonances.
resonances.
high-order
RCSDynamic
Dynamic Aperture
Aperture
RCS
For the simulation of the RCS dynamic aperture the
thFor the simulation of the RCS dynamic aperture the
order Taylor
Taylor map
map ofof the
the magnetic
magnetic field
field has
has been
been
88th order
used.
In
the
case
of
RCS
the
nearest
structure
used. In the case of RCS the nearest structure
resonanceexcited
excited by
by the
the sextupole
sextupole field
field non-linearity
non-linearity
resonance
of
the
sextupole
magnets
and
the
fringe
field of
of the
the
of the sextupole magnets and the fringe field
dipole
magnets
is
the
third-order
difference
resonance
dipole magnets is the third-order difference resonance
2Qv-Qn=6.
Another limitation of the dynamic
dynamic aperture
aperture
2Q
V-QH=6. Another limitation of the
th
th order resonance
of
RCS
is
determined
by
the
4
of RCS is determined by the 4 order resonance
144
Q =6.35
-Q
QvV=6.35
=6.35
QVV=6.27
Q
=6.35
V
-Q
QvV=6.27
=6.27
QV=6.27
1.6
1.6
1.6
COLLIMATOR
COLLI
MAJOR
COLLIMATOR
COLLIMATOR
1.4
1.41.4
1.4
1.2
BEAM
BEAM
BEAM
BEAM
1.0
1.0
1.0
0.8
0.80.8
ε
ε
MAX
/ε
MAX
HV/ ε BEAM
ε HV
BEAM
MAX
/ εBEAM
HV
1.2
1.21.2
0.8
0.6
0.60.6
0.6
0.40.4
0.4
0.62 0.64 0.66
0.68 0.70
0.72 0.74
0.76
0.58
0.60
0.58
0.60
0.76
0.58
0.600.62
0.62 0.64 0.66
0.66 0.68
0.68 0.70
0.70 0.72
0.72 0.74
0.74 0.76
0.76
horizontal
tune
Fractional
Fractional
horizontal
tune
Fractional
horizontal
tune
caseof
design
modified
operation
tunes
case
the
design
and
modified
operation
tunes
the
case
ofof
thethe
design
andand
modified
operation
tunes
the the
case
ofbeam
the design
and
modified
operation
tunes the by
MR
survival
without
any
errors
is
determined
MR
MR beam
beam survival
survival without
without any
any errors
errors isis determined
determined by
by
MR
beam
survival
without 2Q
any errors
is determined
byN
the
coupling
resonance
, where
H-2Q
V=nN
super
super
the
coupling
resonance
2Q
-2Q
,
where
N
the
coupling
resonance
2Q
er,
where
N
H-2Qv=nN
V=nN
super
super
H
sup
supe
the
coupling
resonance
2QH-2QV=nNsuperof
, where
Nsuperr =3).
isnumber
number
of
the
super-periodicity
MR
(N
super
is
of
the
super-periodicity
of
MR
(N
r=3).
is
number
the
super-periodicity
of
MR
(N
=3).
supe
super
is number of the super-periodicity of MR (Nsuper
=3).
Thisresonance
resonance
is
excited
by
'octupole'-like
fringe
This
excited
by
the
'octupole'-like
fringe
This
is excited
excited
by the
thethe
'octupole'-like
fringe
This resonance
resonance is
is
by
'octupole'-like
fringe
field
effect
of
the
quadrupole
magnets.
field
effect
of
the
quadrupole
magnets.
field
effect
of
the
quadrupole
magnets.
field effect of the quadrupole magnets.
If
errors
are
included
the
misalignment
errors
included
in the
If
the
misalignment
errors
are are
included
in the
the
If Ifthe
thethemisalignment
misalignment
errors
are
included
inin
consideration,
the
linear'
coupling
resonance
is
excited
consideration,
thethe
'linear'
coupling
resonance
excited
consideration,
'linear'
coupling
resonance
is excited
consideration,
the
'linear'
coupling
resonance
isisexcited
by
of
the
skew
quadrupole
by
the
skew
component
of the
thethe
skew
quadrupole
theskew
skewcomponent
component
of
skew
quadrupole
bybythe
the
skew
component
of
skew
quadrupole
magnets
and
the
shifted
sextupole
magnets.
magnets
thethe
shifted
sextupole
magnets.
magnets
and
shifted
sextupole
magnets.
magnets
and
the
shifted
sextupole
magnets.
2. 2.Survival
FIGURE
Survival
collimator
RCS
for
FIGURE
Survivalof
beamin
thecollimator
collimatorofof
RCS
for
FIGURE
2.
ofofbeam
beam
ininthe
the
collimator
ofRCS
RCSfor
for
different
working
different
operating
fringe
field,
differentoperating
operatingworking
workingpoints
pointsincluding
including fringe
fringe
field,
different
points
including
fringefield,
field,
field
non-linearities
sextupole
field
sextupole
field
non-linearitiesand
andmisalignment
misalignmenterrors.
errors.
sextupole
field
non-linearities
and
misalignment
errors.
QV=22.28, dp/p=0
q,=22.28,
dp/p=0
QV=22.28,
dp/p=0
......
, dp/p=+0.01
,, dp/p=+0.01
dp/p=+0.01
......
,dp/p=0
dp/p=-0.01
Q......
=22.28,
V
,, dp/p=-0.01
dp/p=-0.01
......
QV=19.28,
dp/p=0
......
,dp/p=0
dp/p=+0.01
q,=19.28,
QV=19.28, dp/p=0
......
, dp/p=-0.01
QV=19.28, dp/p=0
8
6
V
εV / εΒΕΑΜ
ε /ε
ΒΕΑΜ
εV / εΒΕΑΜ
6
6
4
4
4
2
2
2
0
COLLIMATOR
COLLIMATOR
0 COLLIMATOR
0
1
2
3
0
1
2
3
0
0
1
2
3
4
5
4
6
εH /5εΒΕΑΜ 6
ε /ε
4 H ΒΕΑΜ
5
6
εH / εΒΕΑΜ
7
7
7
8
8
8
9
9
9
FIGURE 3. On- and off-momentum dynamic aperture in
FIGURE
Onand
off-momentum
dynamic
aperture
in
FIGURE
3.of Onaperture
in
the units3.
the and
beamoff-momentum
emittance for dynamic
the design
operating
thepoint
units
emittance
the
units
offorthe
the
beam
emittance
for the
the design
design operating
operating
andof
thebeam
modified
point. for
FIGURE
3. the
Onand off-momentum
dynamic aperture in
point
and for
for
modified
point.
point and
the
modified
point.
TheofMR
aperture for
(FIG.3)
for theoperating
design
the units
thedynamic
beam emittance
the design
The
MR
dynamic
aperture
design
The
dynamic
aperture
(FIG.3)
for
theand
design
working
point
(QV=22.28)
for
the for
on-the
offpoint
and MR
for the
modified
point. (FIG.3)
MAX
1.6
AsAswas
wasstressed
stressedabove
abovethe
theMR
MRlattice
latticeisisdesigned
designedto
to
As was stressed above the
the MRlattice
latticeisisdesigned
designedtoto
suppress
suppressallalldriving
drivingterms
termsofofthe
thethird-order
third-orderresonances
resonances
resonances
suppress
allofdriving
termsworking
of the third-order
case
ofthe
thedesign
design
working
point.From
Fromresonances
theother
other
in inthethecase
point.
the
From
the
other
in the
case
of
the
design
working
point.
From
other
side,this
thisworking
workingpoint
pointisislocated
locatednear
nearthe
theDC
DCthe
'linear'
'linear'
side,
'linear'
side,
this
working
point
is
located
near
the
DC
'linear'
couplingresonance
resonanceQtrQv=0.
Forcomparison
comparisonthe
theMR
MR
H-Q
V=0.For
coupling
QQ
H-Q
V=0.
coupling
resonance
QisHcalculated
-Q
comparison
theand
MR
V=0. Forfor
dynamic
apertureis
calculated
for
the design
design
and
the
dynamic
aperture
the
design
and
rd
rd
dynamic
aperture
is
calculated
for
the
design
modified
Q
.
The
horizontal
tune
is
fixed
near
the
3rd
QVv.VThe horizontal tune is fixed
modified Q
fixed near the 3and
modified
QV. The
horizontal
tune
isfringe
fixed field
near
thethe
3rd
orderhorizontal
horizontal
resonance
line.
The
fringe
fieldof
of
the
resonance
line.
The
order
fringe
field
the
ring
magnetsand
and
thesextupole
sextupole
magnets
forthe
the'linear'
'linear'
order
horizontal
resonance
line.magnets
The
fringe
field
of the
ring
magnets
the
for
the
'linear'
chromaticity
correction
areincluded
included
infor
thethe
tracking
ring
magnets and
the sextupole
magnets
'linear'
chromaticity
correction
are
in
the
tracking
tracking
th
during1000
1000turns
turnsbybyusing
using
the
orderTaylor
map.
chromaticity
correction
arethe
included
inTaylor
the map.
tracking
during
99thth order
during 1000 turns by using the 9th order Taylor map.
8
MAX
Survival:
Survival: εε HV / εBEAM
BEAM
MAX
Survival: ε HV / εBEAM
I
MR
Dynamic
MRDynamic
DynamicAperture
Aperture
MR
Aperture
8
1.8
1.8
Q
QVvV=22.28
=22.28
Q
=22.28
QV=22.28
1000
turns
1000
turns
1000
turns
1000
turns
9th-order
map
9th-ordermap
map
9th-order
9th-order map
1.6
-FF+ChromCorr
FF+ChromCorr
FF+ChromCorr
FF+ChromCorr
-+++Skew-Q-Set#1
Skew-Q-Set#1
Skew-Q-Set#1
+ Skew-Q-Set#1
-+...+
Shift-S(H/V)-Set#1
+...+Shift-S(H/V)-Set#1
Shift-S(H/V)-Set#1
+...+
+...+
Shift-S(H/V)-Set#1
-+...+
Shift-S(H/V)-Set#2
+...+Shift-S(H/V)-Set#2
Shift-S(H/V)-Set#2
+...+
+...+ Shift-S(H/V)-Set#2
-•+...+
Shift-S(H/V)-Set#3
+...+Shift-S(H/V)-Set#3
Shift-S(H/V)-Set#3
+...+
+...+ Shift-S(H/V)-Set#3
£ 1.41.4
1.4
1.21.2
1.2
1.0
1.01.0
1.0
0.8
0.80.8
0.8
Q
QHH-Q
-QVV=0
=0
0.6
0.6
QH-QV=0
0.6
0.24
0.24
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.24
0.26 0.28
0.28 0.30
0.30 0.32
0.32 0.34
0.34 0.36
0.36 0.38
0.38 0.40
0.40 0.42
0.420.44
0.44
0.6 0.26
Fractional
0.24 0.26 0.28
0.30 Horizontal
0.32 0.34Tune,
0.36υVHυH0.38 0.40 0.42 0.44
Fractional
Horizontal
Tune,
Fractional
Horizontal
Tune,
H
Fractional Horizontal Tune, υH
FIGURE 4.
4.
Survival
FIGURE
of
beam
in
the
collimator
MR
FIGURE
4. Survival
Survival of
of beam
beam in
in the
the collimator
collimatorofof
ofMR
MR
around
the
design
working
point
including
fringe
around
the
design
working
point
including
fringe
field,
FIGURE
4.
Survival
of
beam
in
the
collimator
of MR
around the design working point including fringe field,
field,
sextupole
field
non-linearities
and
errors.
sextupole
and
misalignment
errors.
around field
the non-linearities
design working
point
including
fringe field,
sextupole
field
non-linearities
andmisalignment
misalignment
errors.
sextupole
non-linearities
and the
misalignment
errors.
Survivalfield
of
Survival
of
the
beam
in
collimator
for
Survival
of the
the beam
beam in
in the
the collimator
collimator for
for
Q
=22.28
is
shown
in
FIG.4
including
the
V Survival
Qv=22.28
is
shown
in
FIG.4
including
the
of
the
beam
in
the
collimator
QV=22.28 is shown in FIG.4 including the for
misalignment
errors
with
theinσ-value
toto the
misalignment
with
similar
QV=22.28 errors
is shown
FIG.4similar
including
misalignment
errors
with the
the a-value
σ-value
similar
to the
the the
RCS
design
parameters.
In
the
case
of
Q
=19.28
the
V
RCS
design
parameters.
In
the
case
of
Q
=19.28
the
v
RCS
design parameters.
In the
of QV=19.28
the
misalignment
with
thecase
σ-value
similar
to the
beam
survival for
forerrors
Q
=22.34…22.38
has
the same
limit
beam
survival
Q
has
same
limit
beam
for
QHHH=22.34...22.38
=22.34…22.38
has the
the
same
limit the
RCS
design
parameters.
In
the
case
of
Q
=19.28
MAX survival
V
(εMAX
HV/εBEAM ~1.2). But in this case the width of the
MAX
(8
Hv/£BEAM
~1.2).
in
width
of
(ε
~1.2).
But
in this
this case
case the
the
width
of the
thelimit
beam
survival
for
QBut
has
the same
HV/ε
BEAM
H=22.34…22.38
resonance
2QH-2Q
-2Q
V=6 becomes bigger than the width
MAX
resonance
2Q
=6
becomes
bigger
than
the
width
H
V
resonance
-2Q
=6
becomes
bigger
than
the
width
H Q~1.2).
V
But in this case the width of the
HV/εBEAM
of(εthe resonance
H-QV=0 in the case of QV=22.28.
of
the
in
of
ofresonance
the resonance
resonance
Q
=0becomes
in the
thecase
casebigger
of Q
QvV=22.28.
=22.28.
H-Q
2QHQn-Qv=0
-2Q
=6
than the width
VV
of the resonance
Q
-Q
=0
in
the
case
of
H
V
ACKNOWLEDGMENTS QV=22.28.
ACKNOWLEDGMENTS
ACKNOWLEDGMENTS
The authors would like to thank Y.Mori and
The
would
like
to
Y.Mori
and
ACKNOWLEDGMENTS
The authors
authors
would
like
to thank
thank and
Y.Mori
and
S.Ohnuma
for many
useful
discussions,
M.Berz,
S.Ohnuma
for
many
useful
discussions,
and
M.Berz,
S.Ohnuma
for
many
useful
discussions,
and
M.Berz,
K.Makino
and B.Erdelyi
theirtosupport
COSY and
The and
authors
wouldforlike
thankofY.Mori
K.Makino
K.Makino
and B.Erdelyi
B.Erdelyi for
for their
their support
support of
of COSY
COSY
simulations.
S.Ohnuma
for
many
useful
discussions,
and
M.Berz,
simulations.
simulations.
K.Makino and B.Erdelyi for their support of COSY
simulations. REFERENCES
REFERENCES
REFERENCES
1. K.Shigaki,
F.Noda,
K.Yamamoto,
S.Machida,
working
point
(Qv=22.28)
for
the
working
point
(Q
for Additionally
the onon- and
andtheoffoffmomentum
particles
is depicted.
onV=22.28)
1.
K.Shigaki,
F.Noda,
K.Yamamoto,
S.Machida,
A.Molodojentsev,
Y.Ishi,
"The
JHF
Lattice",
this
1.
K.Shigaki,
F.Noda,
K.Yamamoto,
S.Machida,
The
MR
dynamic
aperture
(FIG.3)
for
the
design
REFERENCES
momentum
is
the
momentum
particles
is depicted.
depicted.
Additionally
the ononmomentumparticles
dynamic
aperture Additionally
is shown also
for
A.Molodojentsev,
Y.Ishi,
"The
JHF
Lattice",
this
proceedings.
A.Molodojentsev,
Y.Ishi,
"The
JHF
Lattice",
this
working
point
(Q
=22.28)
for
the
onand
offmomentum
aperture
is
shown
for
V
QV=19.28. dynamic
For both
cases the
aperture
momentum
dynamic
aperture
is dynamic
shown also
also
for
1. proceedings.
K.Shigaki,
F.Noda,
K.Yamamoto,
S.Machida,
proceedings.
2.
"The
proton
driver
design
study",
FERMILAB-TMmomentum
particles
is
depicted.
Additionally
the
onQv=19.28.
For
both
cases
the
dynamic
aperture
is bigger
the aperture
physical
QVwithout
=19.28. any
For errors
both cases
the than
dynamic
A.Molodojentsev,
Y.Ishi, "The FERMILAB-TMJHF Lattice", this
2.
proton
2136,
December
2000.design
2. "The
"The
proton driver
driver
design study",
study", FERMILAB-TMwithout
anydynamic
is
bigger
than
the
momentum
also for
acceptance
oferrors
the collimator.
without
any
errors
isaperture
bigger is
thanshown
the physical
physical
proceedings.
2136,
December
2000.
December
2000. COSY INFINITY version 8,
acceptance
the
collimator.
Qacceptance
For
both
cases the dynamic aperture
3. 2136,
K.Makino
and M.Berz,
of
the
V=19.28. of
Survival
of collimator.
the beam in the collimator of MR
2. K.Makino
"The proton
driver design
study",
FERMILAB-TM3.
and
M.Berz,
COSY
INFINITY
version
Nuclear
Instruments
Methods,
A427:338-343,
without
any
errors
is
bigger
than
the
physical
3. K.Makino and M.Berz,and
COSY
INFINITY
version 8,
8,
Survivalonof
beam
in
collimator
depends
thethe
betatron
of the
particles.of
the
Survival
of
the
beam tune
in the
the
collimator
ofInMR
MR
2136, December
2000.
1999.
Nuclear
Instruments
and
Methods,
A427:338-343,
acceptance
of
the
collimator.
Nuclear Instruments and Methods, A427:338-343,
depends
on
the
betatron
tune
of
the
particles.
In
the
depends on the betatron tune of the particles. In the
1999.
3. 1999.
K.Makino and M.Berz, COSY INFINITY version 8,
Survival of the beam in the collimator of MR
Nuclear Instruments and Methods, A427:338-343,
depends on the betatron tune of the particles. In the145
1999.