Beam
BeamDynamics
Dynamics of
of FFAG
FFAG Accelerator
Accelerator
Beam Dynamics of FFAG Accelerator
Masamitsu Aiba
Masamitsu Aiba
Department of Physics, University of Tokyo
Department of
Department
of Physics, University
University of Tokyo
Abstract. Beam dynamics of FFAG Accelerator is described, focusing on the non-linearity. The tracking simulations
Abstract.
Beam
dynamics
is motion
described,
focusing
the of
non-linearity.
The
Abstract.
Beam
of FFAG the
Accelerator
focusing
onfield
The tracking
tracking simulations
simulations
were
performed,
in dynamics
order
to understand
non-linear
in the
guidingon
FFAG accelerator.
were performed,
performed, in
in order
order to
to understand the non-linear motion in the guiding field of
were
of FFAG
FFAG accelerator.
accelerator.
6
10 6
10
10%
5
10 5
10
horiaontalacceptance
acceptance
horiaontal
π
mm-mrad.)
(
( π mm-mrad.)
INTRODUCTION
INTRODUCTION
INTRODUCTION
A large acceptance of FFAG is ideal as a phase
A large
large acceptance of
of FFAG
FFAG is
is ideal
ideal as
aa phase
A
asnot
phase
rotator
and a acceptance
secondary particle
accelerator,
only
rotator
and
a
secondary
particle
accelerator,
not
only
rotator
and
a
secondary
particle
accelerator,
not
only
as a high current proton accelerator. On the other hand,
as
a
high
current
proton
accelerator.
On
the
other
hand,
as a high
current
On the other hand,
lattice
magnets
of proton
FFAG accelerator.
are full of non-linearity.
It is
lattice magnets
magnets of
of FFAG
FFAG are
are full
full of
of non-linearity.
non-linearity. It
lattice
It is
is
not obvious why FFAG has large transverse
not obvious
obvious why
why FFAG
FFAG has
has large
large transverse
transverse
not
acceptance
and
what
limits
the
acceptance.
We
started
acceptance and
and what
what limits
limits the
the acceptance.
acceptance. We
We started
started
acceptance
systematic
systematic study
study toto
to answer
answer these
these questions,
questions, both
both
systematic
study
answer
these
questions,
both
analytically
analyticallyand
andby
bytracking.
tracking.
analytically
and
by
tracking.
8
8
Number of Turn = 128turn
Number
Turn
Number
ofRadius=9.9m
Turn ==128turn
128turn
N=16,of
n_ N=16,
N=16, Radius=9.9m
Radius=9.9m
N=32,
Radius=20.5m
o_ N=32,
N=32, Radius=20.5m
4
10 4
10
3
10 33
10
io
I 1022
1.0 1.5 2.0 2.5 3.0
|100.5
0.5 1.0 1.5 2.0 2.5 3.0
0.5 phase
1.0 advance
1.5 2.0per 2.5
3.0
cell (rad.)
(rad.)
phase
phase advance
advance per
per cell
cell (rad.)
FIGURE
vs. Phase
Phase Advance
Advance
FIGURE 1.
1. Horizontal
Horizontal Acceptance
Acceptance vs.
NON-LINEAR
NON-LINEARMOTION
MOTIONIN
INFFAG
FFAG
NON-LINEAR
MOTION
IN
FFAG
InInFFAG
FFAGaccelerators,
accelerators,for
forthe
thesake
sakeof
ofachieving
achievingthe
the
Inthe
FFAG
accelerators,
for
the
sake
of
achieving
the
zero
chromaticity
condition,
the
higher
zero
the
chromaticity
condition,
the
higher order
order
zero
the
chromaticity
condition,
the
higher
order
components
componentsare
areintroduced
introducedinto
intothe
theguiding
guidingfield.
field.Then,
Then,
components
are
introduced
into
the
guiding
field.
Then,
thethebetatron
tune
has
nonodependence
upon
momentum
betatron
tune
has
dependence
upon
momentum
the betatron tune has no dependence upon momentum
but
buthas
hasa aadependence
dependenceupon
uponits
itsoscillation
oscillation amplitude.
amplitude.
but
has
dependence
upon
its
oscillation
amplitude.
Then
the
bare
tune
(the
tune
of
vanishingly
small
Then
the
bare
tune
(the
tune
of
vanishingly
Then the bare tune (the tune of vanishingly small
small
amplitude)
should
be
selected,
taking
into
account
amplitude) should
should be
be selected,
selected, taking
taking into
into account
accountthe
the
amplitude)
the
non-linear
non-linearmotion
motionatat
ata aalarge
largeamplitude.
amplitude.
non-linear
motion
large
amplitude.
HORIZONTAL
HORIZONTALMOTION
MOTION
HORIZONTAL
MOTION
Relation
Relationbetween
betweenthe
theHorizontal
Horizontal
Relation
between
the
Horizontal
Acceptance
and
the
Phase
Acceptance
and
the
Phase
Advance
Acceptance and the PhaseAdvance
Advance
The
Thetracking
trackingsimulations
simulations(Runge-Kutta
(Runge-Kuttaintegration)
integration)
The
tracking
simulations
(Runge-Kutta
integration)
using
two-dimensional
field
of
“hard
edge
model”[2]
[2]
using
two-dimensional
field
of
“hard
edge
model”
using two-dimensional field of "hard edge model"
[2]
were
performed
to
calculate
the
horizontal
acceptance.
were
performed
to
calculate
the
horizontal
acceptance.
were performed to calculate the horizontal acceptance.
Sucha aasimulation
simulationisis
isone
oneofof
ofthe
theeffective
effectivemethods
methods to
to
Such
Such
simulation
one
the
effective
methods
to
analyzethe
thenon-linear
non-linearmotion.
motion. Figure
Figure 11 shows
shows the
the
analyze
analyze the non-linear motion. Figure 1 shows the
simulationresults
resultsfor
forthe
thecases
casesofofthe
theperiodicity
periodicityofof16
16
simulation
simulation
results
for
the
cases
of the
periodicity
of 16
and3232 with
with triplet
triplet (DFD)
(DFD) lattice.
lattice. InIn Fig.1,
Fig.1, the
the
and
and 32 with triplet (DFD) lattice. In Fig.l, the
horizontal axiscorresponds
corresponds to thephase
phase advance per
per
horizontal
horizontalaxis
axis correspondstotothe
the phase advance
advance per
cell. Comparing
Comparing the
the rings
rings having
having the
the different
different
cell.
cell. Comparing the rings having the different
periodicity, it is generaland
and convenient toto use
use the
periodicity,
periodicity, ititisis general
general and convenient
convenient to use the
the
phase
advance
per
cell.
Two
common
tendencies
in
phase
phaseadvance
advanceper
percell.
cell.Two
Two common
common tendencies
tendencies inin
tworings
ringscan
canbebeseen.
seen.One
Oneisis that
that the
the acceptance
acceptance
two
two rings can be seen. One is that the acceptance
becomes smallerasasincreasing
increasing thephase
phase advance.The
The
becomes
becomessmaller
smaller as increasingthe
the phaseadvance.
advance. The
otherisisthat
thatthe
theacceptance
acceptance becomes
becomes small
small rapidly
rapidly
other
other is that the acceptance becomes small rapidly
around thestructure
structure resonancecorresponding
corresponding to the
around
aroundthe
the structureresonance
resonance corresponding toto the
the
phase advanceofof2π/3,
2π/3, 2π/4,2π/5.
2π/5.
phase
phaseadvance
advance of 2n/3,2π/4,
2n/4, 2n/5.
FIGURE 1. Horizontal Acceptance vs. Phase Advance
First
tendency
is
explained by
the large
large kk value.
value.
First
tendency
is
by
the
First
tendency
is explained
explained
by
the
large k value.
The
k
value
is
main
knob
of
the
horizontal
phase
The
is
main
knob
horizontal
phase
The kk value
value
isphase
main advance
knob of
of the
the
horizontal
phase
advance;
the
becomes
larger
as
advance;
the
phase
advance
becomes
larger
as
advance;
the
phase
advance
becomes
larger
as
increasing
the
k
value.
Equation
1
shows
the
radius
increasing
the
kk value.
Equation
11 shows
the
radius
increasing
the
value.
Equation
shows
the
radius
dependence
of
FFAG
guiding
field
and its
its expansion.
expansion.
dependence
of
FFAG
guiding
field
and
dependence
offield
FFAG
guiding
fieldradius
and its
Here,
BB00 isis the
strength
at
the
r00.. expansion.
Here,
the
field
strength
at
the
radius
r
Here, B0 is the field strength at the radius r0.
k
k ( k − 1)
rr k
kk
xx +
B 00 k ( k − 21) xx 22 ++ ⋅⋅⋅⋅⋅⋅
BBzz == BB00
== B
+B
B00 ++ B
B00
B=B\—\
=
rr00
rr00
2!!rr00 2z
r0
" 22!r
0
((Taylor
around r
= + x).). (1)
(1)
Taylor Expansion
Expansion
(Taylor
Expansion around r000,, rr == rr000 ++ xx).
(1)
As
shown
in
Eq.1,
a
large
k
value
enhances
the
As
As shown
shown in
in Eq.1,
Eq.l, a large k value enhances
enhances the
the
non-linearity.
It
seems
a
good
solution
to
take
a
small
non-linearity.
non-linearity. It
It seems
seems a good solution to take
take aa small
small
phase
advance so
that the acceptance becomes
becomes large;
phase
phase advance
advance so
so that
that the acceptance becomes large;
large;
however,
the
phase
advance,
in other word the
value,
however,
the
phase
advance,
however, the phase advance, in other word the
the kkk value,
value,
should
keep
the
orbit
should
be
kept
certain
should be
be kept
kept certain
certain large
large in
in order
order to
to keep
keep the
the orbit
orbit
excursion
in
a
feasible
value.
excursion
in
a
feasible
excursion in a feasible value.
N=16, k value=13 & 15
N=16,
N=16,kvalue=13&
5.6
5.6
5.4
5.4
5.2
5.2
phase advance =2ππ/3
phase
/3
phase advance =2c/3
O
5.0
5.0
u»
horizontal
tune
horizontalune
tune
ori
Second
separation
Second
tendency
Second tendency
tendency isis explained
explained by
by the
the separation
separation
between
the
bare
tune
resonance
between
the
bare
tune
and
the
structure
resonance
line.
between the bare tune and the structure resonance line.
line.
Figure
2
shows
the
the
cases
that
Figure
2
shows
the
tune
shifts
for
cases
that
the
Figure 2 shows the tune shifts for the cases that the
the
phase
advance
is
around
phase
advance
is
around
2π/3.
phase advance is around 2;c/3.
000
10 20
20 30
30 40
50 60
10
40
10 initial
20 ∆30
40 50
50 60
60
r (mm)
(mm)
initial
∆
r
initial Ar (mm)
FIGURE 2.
2. Horizontal Tune
Tune Shift
Shift
FIGURE
FIGURE 2. Horizontal
Horizontal
TuneAdvance
Shift of 2π/3
around Phase
Phase
around
Advance
π/3
around Phase Advance of
of 22n/3
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
204
The direction of tune shift faces to the structure
The direction
of tune shift
faces toof the
structure
resonance
line. Therefore,
“the valley
acceptance”
resonance
line.
Therefore,
"the
valley
of
acceptance"
appears and naturally the acceptance becomes zero on
appears
and resonance.
naturally the acceptance becomes zero on
the
structure
the structure resonance.
the periodicity. Therefore, using Fig.3, it is possible to
the
periodicity.
Therefore,acceptance
using Fig.3,
possible
to
estimate
the horizontal
of ita is
ring
with any
estimate
the
horizontal
acceptance
of
a
ring
with
any
parameters.
parameters.
VERTICAL MOTION
VERTICAL MOTION
Soft Edge Model of FFAG Focusing Field
Soft Edge Model of FFAG Focusing Field
Normalization of Horizontal Acceptance
Normalization of Horizontal Acceptance
In order to obtain more general result on the
In order
to obtain more
general result
on the
relation,
a normalization
of horizontal
acceptance
is
relation, a asnormalization
of horizontal acceptance is
introduced
follows.
introduced as follows.
With an approximation of the large k value, the low
an of
approximation
of the large kinto
value,
the low
orderWith
terms
Eq.1 can be transformed
Eq.2.
order terms of Eq.l can be transformed into Eq.2.
2
3
1 k
1 k
k
(2)
x + ⋅⋅⋅ .
x +
x +
Bz ≅ B0 1 +
(2)
__,,+!
1,,+
2
!
3
!
r
r
r.ji+^i
0
0
0
In order to study the vertical motion including its
In order to
study“soft
the vertical
its
non-linear
term,
edge motion
model”including
of threenon-linear
edgein model"
of threedimensionalterm,
field is "soft
introduced
the followings.
dimensional field is introduced in the followings.
First, three components of the cylindrical
First, three
components
of polynomial
the cylindrical
coordinate
are expressed
by the
of the
coordinate
are
expressed
by
the
polynomial
of the
vertical coordinate as shown in Eq.6.
vertical coordinate as shown in Eq.6.
Bi = Bi 0 (r ,θ ) + Bi1 (r ,θ ) z + Bi 2 (r ,θ )2z 2 + ⋅ ⋅ ⋅
(6)
=Bi0(r,0) + B
+ Bi2(r,0)z
(6)
i = r ,θ , z
i = r,09z
In Eq.2, the quantity kx/r0 can be treated as
In Eq.2,coordinate
the quantity
kx/rdistance.
as
0 can be
normalized
of the
Ontreated
the other
normalized
coordinate
of
the
distance.
On
the
other
hand, the emittance is expressed in Eq.3,
hand, the emittance is expressed in Eq.3,
(3)
W = x 22 / β ,
W=x /]3,
Inserting these equations into Maxwell’s Equation,
these
into Maxwell's
Equation,
the Inserting
coefficients
of equations
the polynomial
can be solved,
using
the
coefficients
of
the
polynomial
can
be
solved,
usingby
the boundary condition on the median plane shown
the
boundary
condition on the median plane shown by
Eq.7
and Eq.8,
Eq.7 and Eq.8,
(7)
Br 0 ( r , θ ) = Bθ 0 ( r , θ ) = 0 ,
(3)
where β is the beta-function. Using approximations of
where
p is the beta-function.
Using
approximations of
the
beta-function
and the betatron
tune,
the beta-function and the betatron tune,
(7)
k
k
×N = ,
2
—
N
N
β ≅ r0 / ν h , ν h ∝
(4)
k
(4)
r
B z 0 ( r , θ ) = B0
E (θ ) ,
= B0\ —r0 \ E(ff),
t0(r9ff)
(8)
(8)
where νh is the horizontal tune and N is the periodicity,
where v is the horizontal tune and N is the periodicity,
Eq.3 canh be transformed into Eq.5, which includes
Eq.3 can be transformed into Eq.5, which includes
normalized
normalized coordinate.
coordinate.
W ∝x
r
k
k
= 0
x
r0 N
kN r0
where B0 is the field strength at the radius r and E(θ),
where
B0 is the field strength at the radius r and E(0),
which changes
changes from
from 11 to
to 0,0,represents
representsthe
thedistribution
distribution
which
of
the
fringing
field.
of the fringing field.
Finally, equations
equations from
from 99 toto 11
11 are
are obtained
obtainedasas"soft
“soft
Finally,
edge
model”,
edge model",
2
2
.
(5)
(5)
Nor. acceptance * kN/r0
Then,
and that
that is
is the
the
Then, the
the remaining
remaining factor
factor is
is only
only kx/r
kxlr00 and
normalization
factor
of
the
acceptance.
As
the
result
of
normalization factor of the acceptance. As the result of
the
thenormalization,
normalization, Fig.3
Fig.3 is
is obtained.
obtained.
12000
12000
10000
r10000
−
N=16
--n—N=16
N=32
-o—N=32
8000
z
(k − 2) B 0 r
1
( Ek 2 + E ( 2 ) )
3
r0
3!
r0
Bθ (r , θ , z ) = E
B0 r
r0 r0
1.5
1.5
2.0
2.0
2.5
2.5
−
3.0
3.0
phase
phase advance
advance per cell (rad.)
"
"
205
z3 + ⋅⋅⋅
%
&
.
k −2
'
−
,(10)
,(10)
k −3
"#$
%
(
Three
the
Three parameters
parameters in
in the
the normalization factor
factor are the
main
phase
main parameters
parameters of
of the
the ring.
ring. And the horizontal phase
advance
advance isis almost
almost determined
determined only
only by the k value and
z 3 + ⋅⋅⋅
k
'
r
= EB
( r ,θ , z ) =
B2z(r,0,z)
EB0 0\B
r0
z
B r
1 (1) 2
( E k + E ( 3) ) 03
3!
r0 r0
()*
FIGURE
FIGURE 3.
3. Normalized
Normalized Horizontal
Horizontal Acceptance vs.
Phase
Phase Advance
Advance
k −1
"#$
!
1.0
1.0
k-3
(1)
, (9)
,(9)
k −3
!
§ 6000
6000
4000
£ 4000
2000
2000
0o
I 0.5
0.5
k −1
kB r
B r ( r ,θ , z ) = E 0
r0 r0
B r
1
( Ek 2 + E ( 2 ) ) 02
2!
r0 r0
()*
(
%
%
&
z2 ⋅⋅⋅
(11)
(11)
2
y
k
!"
r k
k
k ( k − 1) 2
(13)
(13)
EB
£B00 -I
=,E !" B0 + B?0 k xv+-1- BR0 k ( k −21) xv2 ++⋅ ⋅ ⋅ ,,« (13)
r =
!r02 x +⋅ ⋅ ⋅
EB0 r0 = E B0 + B0 r0 x + B0 22.r
r0
r0
2!r0
where the notation is changed; the coordinate
coordinate yy
where the the
notation
is changed;
y
represents
coordinate
z. Then the
the coordinate
natural result
represents
the
coordinate
z.
Then
the
natural
result
appears; the guiding field of FFAG accelerator can be
appears; the
field of of
FFAG
accelerator
can be
expressed
by guiding
the summation
multipole
components.
components.
expressed by the summation of multipole components.
!"
!"
2
vertical tune
vertical tune
I
^
o
2
2 3
3
4
horizontal
horizontaltune
tune 4
horizontal tune
(a)
(a) Two
Two Simulation
Simulation Conditions
Conditions in
in the
the Tune
Tune Diagram
Diagram
(a) Two Simulation Conditions in the Tune Diagram
(b)
(b) Bounded
Bounded motions
motions with
with avoiding
avoidingthe
the non-linear
non-linear coupling
coupling
(b) Bounded motions with avoiding the non-linear coupling
(A)
(A)
(A)
8
8 1
6
6 6•
4
554 45
2
".
•
^2 2
: •"""••"
0
0
0
0 0
-2
-2
•
B •"
•
"
~N -4
.4:
-5
-4
-5-5-6
m
-6 -6-10
_in , , , , , , - , , , , ,-8,-88
-10
4996
49985000
50005002
5002
5004
5006 -15-15
0 5 5510 10
499649985000500250045006
-15
-10-5 -5
-5
10
15
4996
4998
5004
5006
-10-10
0
15 15
0
r(mm)
r(mm)
z fei m )
r(mm)
r'(mrad)
r'(mrad)
10
10-,
10
vererticaltune
tune
verertical
44
33
'
-J 2
" "--V... /\
FIGURE
Non-linear
Coupling
with
“soft
edge
model”
FIGURE5.5.
5.Non-linear
Non-linear
Coupling
with
"soft
edge
model"
FIGURE
Coupling
with
“soft
edge
model”
11
0a
0
00
. " :"jft"I-"
(c)
Unbounded
motions
due
the
non-linear
coupling
(c)Unbounded
Unboundedmotions
motions
due
to
the
non-linear
coupling
(B)
(c)
due
toto
the
non-linear
coupling
(B)(B)
periodicity
periodicity=8
periodicity
==88
222
•S
o
10
10-,
10
555
000
-5
-5
8
8 666 44
4 222 00•
0 -2
-2 -4
-4 -6
-6-6 -8
-10
-10 4996
4996 4998
4998 5000
50005002
50025004
50045006
5006-8 -15
-15 -10
-10 -5
-5 00 55 10
10 15
15
5004 5006
4996 4998 5000
5002
-15 -10 -5 z(mm)
0z(mm)
5 10 15
r f
cm
)
z(mm)
Vertical Tune Shift
Shift
Vertical Tune Shift
The vertical
tune
is
mainly
occurred
by
the
tune shift
shift is
is mainly
mainly occurred
occurred by
bythe
the
The vertical
vertical tune
shift
component of normal
octupole.
The
effect
of
higher
normal
octupole.
The
effect
of
higher
component of normal octupole. The effect of higher
order components more than octupole
is
weak,
unless
octupole is
is weak,
weak,unless
unless
order components more than octupole
the
vertical
tune
is
very
close
to
the
value
affected
by
the vertical tune is very close to the value affected by
these
components.
Then,
the
direction
of
tune
shift
tune
shift
is
these components. Then, the direction of tune shift isis
one-side.
The
tracking
simulation
with
“soft
edge
simulation
with
"soft
edge
one-side. The tracking simulation with “soft edge
model”
for
various bare
bare tunes.
tunes. The
The
model"
for various
model” was
was carried
carried out
out for
results
shown
in
Fig.4
indicate
that
the
direction
of
the
Fig.4 indicate
indicate that
that the
thedirection ofofthe
the
results shown
shown in Fig.4
vertical
tune
shift
is
upward.
shift
upward.
vertical tune shift is upward.
A B
A B
z'(mrad)
z'(mrad)
°Q
3
3
r'(mrad)
r'(mrad)
Furthermore, first term of the
the radial and
and vertical
first term
of thein
and
vertical
fieldFurthermore,
can be expanded
as
Eq.12
and
Eq.13
expanded
as shown
shown
inradial
Eq.12
and
Eq.13
field can be expanded as shown in Eq.12 and Eq.13
respectively.
respectively.
k£
k£(£-1)
(k − 1)
j
y + B0 k (k − 1) xy
B0
k −1
k
r
rro0 xy
kB r B r0 y + B0
(12)
,,
(12)
E kB0 r k −1z = E 0 ro0
r
,
(12)
0
0 r
−
−
k
(
k
1
)(
k
2
)
r
E 0 0
z = E +1 B *(*-l)(*-2) xx 2,y ⋅ ⋅ ⋅ 0 k ( k − 1)(k
2 − 2) 2
r0 r0
22.r
!r002
x y ⋅ ⋅ ⋅ J
+ B0
2!r0
CONCLUSION
CONCLUSION
CONCLUSION
11
22
33
4
44
horizontal
horizontaltune
tune
FIGURE
Shift
FIGURE 4. Vertical
Vertical Tune
Tune Shift
Shift
with “soft
edge
model”
"soft edge
edge model”
model"
Then, the
the criterion
criterion of the tune
Then,
selection isis obtained;
obtained;
tune selection
the vertical
vertical tune
tune should not
be
set
just
bellow
the
the
set
just
bellow
the
not be
strong resonance
resonance line.
strong
strong
resonance
line.
Non-linear Coupling
Coupling
Coupling
Non-linear coupling
coupling
Non-linear
induced
by
normal
sextupole
Non-linear
coupling induced
induced by
by normal
normal sextupole
sextupole
is
observed
with
the
simulation
as
is
shown
in
Fig.5.
is observed
observed with
with the
the simulation
simulation as
as shown
shown in
in Fig.5.
Fig.5.
When
the
pair
of
tunes
is
very
close
to
the
non-linear
When
the
pair
of
tunes
is
very
close
to
the
non-linear
When the pair of tunes is very close to the non-linear
+2ννyy=q),
=q), the
resonance line
line ((ννxx+2
the
motions
in
each
phase
resonance
resonance
line
(vx+2Vy=q)
the motions
motions in
in each
each phase
phase
space are
are not
not bounded.
bounded.
As9 shown
in
Eq.2,
the
guiding
space
As
shown
in
Eq.2,
the
space are not bounded. As shown in Eq.2, the guiding
guiding
field has normal
normal sextupole, intrinsically.
Especially,
field
field has
has normal sextupole,
sextupole, intrinsically.
intrinsically. Especially,
Especially,
the
non-linear
structure
resonance
line
should be
the
the non-linear
non-linear structure
structure resonance
resonance line
line should
should be
be
avoided.
avoided.
avoided.
The
beam
dynamics
FFAG
accelerator
was
The
The beam
beamdynamics
dynamicsofof
ofFFAG
FFAGaccelerator
acceleratorwas
was
studied
obtain
large
acceptance.
We
defined
studied
studied toto
toobtain
obtaina aalarge
largeacceptance.
acceptance.We
Wedefined
defined
normalized
coodinates.
Using
that
horizontal
normalized
normalized coodinates.
coodinates.Using
Usingthat
thatthethe
thehorizontal
horizontal
acceptance
described
phase
advance
acceptance
phase
advance
acceptanceisisisdescribed
describedasas
asa afunction
a function
functionofof
of
phase
advance
per
cell,
independently
of
the
periodicity.
Furthermore,
per
cell,
independently
of
the
periodicity.
Furthermore,
per cell, independently of the periodicity. Furthermore,
“soft
toto
study
thethe
vertical
“soft
edge
model”
was
introduced
study
vertical
"softedge
edgemodel”
model"was
wasintroduced
introduced
to
study
the
vertical
motion.
motion.
shows
the
vertical
tune
shift
and
motion. ItItIt shows
showsthe
thevertical
verticaltune
tuneshift
shiftand
andthethe
the
nonlinear
nonlinear
coupling
FFAG.
We
found
that
nonlinear coupling
couplinginin
inFFAG.
FFAG.We
Wefound
foundthat
thatthethe
the
direction
has
toto
bebe
taken
direction
the
tune
shift
fixed
and
has
taken
directionofof
ofthe
thetune
tuneshift
shiftisisisfixed
fixedand
and
has
to
be
taken
into
account
to
choose
bare
tune.
into
into account
account to
to choose
choose bare
bare tune.
tune.
REFERENCES
REFERENCES
REFERENCES
1.1.
inin
Japan:
Based
onon
Y.
Mori,
‘Neutrino
Factory
Japan:
Based
1.Y.
Y.Mori,
Mori,‘Neutrino
'NeutrinoFactory
Factory
in
Japan:
Based
on
FFAG
Accelerator’,
Proc.
of
EPAC02,
p278-p280.
FFAG
Accelerator’,
Proc.
of
EPAC02,
p278-p280.
FFAG Accelerator', Proc. of EPAC02, p278-p280.
2.2. M.
Aiba etet al.,
‘Study ofofAcceptance
of FFAG
2. M.
M. Aiba
Aiba et al.,
al., ‘Study
'Study of Acceptance
Acceptance of
of FFAG
FFAG
Accelerator’,
Proc.
of
EPAC02,
p1226-p1228.
Accelerator’,
Accelerator', Proc.
Proc. of
of EPAC02,
EPAC02, p1226-p1228.
pl226-p!228.
206