Proceedings of the 7th Annual Meeting of Particle Accelerator Society of Japan (August 4-6, 2010, Himeji, Japan)

APPLICATIONS OF ADVANCED SCALING FFAG ACCELERATOR
JB. Lagrange∗ 1) , T. Planche1) , E. Yamakawa1) ,Y. Ishi2) , Y. Kuriyama2) , Y. Mori2) , K. Okabe2) , T. Uesugi2) ,1)
Graduate School of Engineering, Kyoto University, Kyoto, Japan
2)
Kyoto University Research Reactor Institute, Kumatori, Japan
Abstract
Until today, scaling FFAG accelerator were only designed in a ring shape. But a new criteria of the magnetic
ﬁeld conﬁguration satisfying the scaling condition even forstraight FFAG beam line is described here. Moreover, combining different types of cells can be used to imagine newlattices. Various applications using these recent developments are here examined: inprovements of the PRISM
project and the ERIT project, and a zero-chromatic carbon
gantry concept are presented.

INTRODUCTION
Recent developments in scaling FFAGs have opened new
ways for lattice design [1, 2]. Indeed, it is possible to guideparticles with no overall bend in scaling FFAGs, with an
exponential ﬁeld law. Different types of cells can be combined in the same lattice. It offers possibilities in termsof shape for rings and transport lines, but also leads to thecreation of new functions, such as dispersion suppressors.
This concept can be applied to overcome problems like in
the PRISM project, or to improve existing schemes, likethe ERIT lattice. Finally, a scaling FFAG carbon gantry
concept is brieﬂy described.

SCALING LAW
The scaling condition gathers two hypotheses:• similarity of the closed orbits,• similarity of the betatron oscillations, i.e.
chromaticity of the system.

zero-

To deal with the betatron oscillations, we use the linearized equations of motion around a closed orbit for agiven momentum:
 2
d x 1−n


x=0
 2 +
ds
ρ2
,
(1)
2


d z + nz=0ds2
ρ2with ρ the curvature radius, s the curvilinear abscissa, x
and z the perpendicular displacement off the closed orbit
respectively in horizontal and in vertical and n the ﬁeld
index deﬁned as
(
)
ρ ∂B
.
(2)
n=−
B ∂x z=0

To satisfy the scaling condition, Eq. (1) has then to be independent of momentum.
In circular elements, we start from a set of closed orbits
for each momentum lying in the median plane. In this case,similarity of the closed orbits means that each of them is aphotographic enlargement of each other. The general formof the vertical component of the magnetic ﬁeld B in the
mid-plane is[3, 4]( )kr
B = B0F(θ),(3)
r0with r and θ the polar coordinates, B0 the magnetic ﬁeld
at the radius r0 , F an arbitrary periodic function, and k thegeometrical ﬁeld index.
While in circular elements, the concept of closed orbit isobvious, the corresponding notion of reference trajectoryin straight elements has ﬁrst to be deﬁned. A reference
trajectory of a straight cell is a trajectory followed by aparticle whose coordinates (i.e. angle and position) are thesame at the entrance and at the exit of the cell.
In the same way than in circular elements, we start froma set of reference trajectories for each momentum lying in
the median plane. In this case, similarity of the reference
trajectories means that each of them is a translation of eachother. Each reference trajectory is speciﬁed by its average
abscissa X over one cell.
The betatron oscillations are invariant in momentum if
Eq. (1) is the same for every momentum, and since s isindependent of momentum, the condition to have invariant
betatron oscillations is
( )

∂
1


 ∂p ρ2 = 0
( )
.
(4)

∂
n


=0
∂p ρ2
The similarity condition of the reference trajectories validates the ﬁrst equation of the system (4), since it gives thatthe curvature radius depends only on s, so is independentof momentum. We now focus on the second equation of
the system (4).
If we differentiate the equation p = qBρ, for a particleof momentum p and charge q, with respect to x, we have:
B

∂B
1 dp
∂ρ
+ρ
=
∂x
∂x
q dx

(5)

If we introduce the ﬁeld index n in Eq. (5), we have:∗ lagrange@rri.kyoto-u.ac.jp

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n=

∂ρ ρ dp−.
∂x p dx(6)

Proceedings of the 7th Annual Meeting of Particle Accelerator Society of Japan (August 4-6, 2010, Himeji, Japan)

To express Eq. (6) in the coordinates (X, s), we now introduce the parameters η and ϵ that link dx with ds:
η = dx/dX,
ϵ = ds/dX.
Then, rewriting Eq. (6) with η and ϵ, we get()
dρ1mρ + ϵ
,
n=−
η
ds

(7)
(8)

(9)

with m a parameter given by
m=

1 dp
.
p dX

(10)

Rewriting m with the magnetic ﬁeld B, we get:
m=

1 dB
.
B dX

(11)

In consequence, we have
1
n
=−
ρ2
ηρ

(
)
ϵ dρ
m+
.
ρ ds

(12)

The similarity gives that ρ, η and ϵ depend only on s, therefore they are independent of momentum. So, starting fromthe similarity of the reference orbits, The betatron oscillations are independent of momentum if and only if m is
independent of momentum. In this case, we can integrate
Eq. (11) and with the initial condition B(X0 ) = B0 , weobtain the scaling law:in circular elements, with r the radius of the closed orbit,
and k the geometrical ﬁeld index of the cell, and( )∂a1
Dstr. (p0 ) = p0
=
(17)
∂p p0
m
in straight elements, with a the abscissa of the reference
trajectory and m the factor in the exponential law we discussed in Sec. .
In consequence, to combine a straight section and a circular section, after matching a special momentum p0 , dispersion can be matched with
k+1
,
(18)
R0
with R0 the radius of the closed orbit for the momentum
p0 at the border of the cell in the circular section. But this0matching is done only to the ﬁrst order in R−R
R0 , with R
the radius of the closed orbit for a momentum p at the border of the cell in the circular section. Higher orders effects
create a reference trajectory mismatch for momenta otherthan p0 [1]. By a proper choice of p0 , the mismatch could
be minimized for the considered momentum range. An example has been computed by simulating the insertion ofstraight sections in the 150 MeV FFAG ring of KURRI,
for kinetic energies between 20 MeV and 150 MeV. The
mismatch x as a function of kinetic energy is presented inFig. 1. This maximum mismatch is around one cm for thiscase. It would be smaller for larger rings.
m=

98m(X−X0 )B = B0 e

F(s),

7

(13)

6
x [mm]

where F is an arbitrary periodic function.
As a remark, in the linear approximation, the ﬁeld index
n can be written in the circular section asρ
n = k,
(14)
r0

5
4
3
2
1
0

20

40

60

80
100
Ekin [MeV]

120

140

and in the straight section asn = mρ.

(15)

INSERTIONS AND DISPERSION
SUPPRESSOR PRINCIPLE
Matching of reference trajectories
FFAG straight sections could be used with circular FFAG
sections, but since ﬁeld laws in each section are different,
it will occur a discontinuity of reference trajectories at theborder of these two sections. Before combining them together, it is useful to consider them separately. In scaling
FFAG, the periodic dispersion function D of a cell for agiven momentum p0 is deﬁned as
( )
r
∂r
=
(16)
Dcirc. (p0 ) = p0
∂p p0
k+1

Figure 1: Mismatch of reference trajectories between circular cells and straight cells in 150 MeV FFAG ring example.
In the same way, it is possible to match two circular
FFAG sections with different radii r1 and r2 . The dispersion matching condition is kept by adjusting the geometrical ﬁeld index:
k2 + 1
k1 + 1
=
.
(19)
r1
r2

Matching of linear parameters
Once the reference trajectories are matched, the periodic beta-functions of the cells have also to be matched tolimit the amplitude of the betatron oscillations. If a correct
matching is not achievable, then an insertion with a phaseadvance multiple of 180 deg can be done for one of the-236-Proceedings of the 7th Annual Meeting of Particle Accelerator Society of Japan (August 4-6, 2010, Himeji, Japan)

two different types of cells. This insertion becomes thustransparent, limiting the betatron oscillations.

consider the use of straight cells in the lattice and a new design is proposed (see Fig. 3). Parameters are summarizedin table 1.

Dispersion Suppressor

D2 =D1 + D3
.
2

4
2
y [m]

In FFAGs, dispersion suppressors can be useful at theend of a transport line, or to reduce excursion where rf cavities are set in FFAG rings. The effect of a dispersion suppressor would be to decrease the excursion, i.e. to bringcloser the reference trajectories around a matched one. This
excursion reduction can even be a complete suppression(see Fig. 2).
A principle of a dispersion suppressor in scaling FFAGs
is presented in Fig. 2. The components of this scheme arethree types of scaling FFAG cells, straight or circular. The
area 1 contains FFAG cells with a dispersion D1 at the border, the area 2, constituting the dispersion suppressor itself,contains FFAG cells with a dispersion D2 at the borders,
and the area 3 contains FFAG cells with a dispersion D3 atthe border. The conditions to have a dispersion suppressor
are a phase advance of 180 deg. for the cells of the area 2
and the dispersion D2 has to verify-4-10 -8 -6 -4 -2

0
2
x [m]

4

6

8

10

Figure 3: Closed orbits of 55 MeV/c, 68 MeV/c and
82 MeV/c muons µ− in the PRISM lattice with straightsections.

Table 1: Parameters of the new PRISM lattice.
Circular Straight
SectionSectionType
k-value or m-value
Radius/Length
Horizontal phase advance
Vertical phase advance
Number of cells(20)This principle is based on the linear theory, so is valid aslong as the effect of non-linearities is negligible.

Figure 2: Principle of a dispersion suppressor with scaling
FFAG cells. The upper scheme shows the case of a complete suppression of the dispersion, the lower one the caseof a remaining dispersion after the dispersion suppressor.

0
-2

FDF
2.55
2.7 m
60 deg.
90 deg.
2

FDF
1.3 m−1
1.8 m
27 deg.
97 deg.
3

Particle tracking is done with Runge-Kutta integration
in soft edge ﬁelds with linear fringe ﬁeld falloffs. Components of the ﬁeld off the mid-plane are obtained from a ﬁrst
order Taylor expansion, satisfying the Maxwell equations.
The original PRISM design has a very large dispersion
function (∼ 1.2 m) that makes difﬁcult the injection andthe extraction. The new proposal starts then from a smaller
one (∼ 0.8 m). After minimizing the mismatch of the betafunctions, the bending part of the ring is made transparent to limit the effect of the remaining mismatch on theamplitude of the betatron oscillations. The resulting betafunctions for a momentum of 68 MeV/c are presented inFig. 4. The working point is chosen in the tune diagram
so that it is far from the structural normal resonances. The
present working point has a tune of 2.9 in horizontal and6.3 in vertical.

PRISM CASE

6
5
4
β [m]

The PRISM (Phase Rotated Intense Slow Muon beam)
project aims to realize a low-energy muon beam with ahigh-intensity, narrow energy spread and high purity. For
this purpose, a scaling FFAG ring has been proposed [5].
Requirements for the FFAG ring include a large transverse and longitudinal acceptance. The original design of
the FFAG ring for PRISM is based on 10 identical DFD
triplets. If this design fulﬁlls the requirements of acceptance, the excursion is very large and the injection and extraction still remains difﬁcult. To solve this problem, we

3
2
1
0

0

2

4

6

8

10 12 14 16 18 20s [m]Figure 4: Horizontal (plain red) and vertical (dotted purple)beta-functions for half of the ring of the PRISM lattice.

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Proceedings of the 7th Annual Meeting of Particle Accelerator Society of Japan (August 4-6, 2010, Himeji, Japan)

The transverse acceptance in both planes is studied bytracking over 30 turns a particle with a displacement off
the closed orbit and a small deviation in the other transverse direction (1 mm). Collimators (± 1 m in horizontal,± 30 cm in vertical) are used to identify the lost particles.
The regions drawn by the particle with the largest initialstable amplitude in the horizontal and vertical phase spaces
are respectively presented in Fig. 5 and 6. Horizontal(∼ 24000 π.mm.mrad) and vertical (∼ 6000 π.mm.mrad)acceptances are then measured by the area of the biggest
ellipse included in this region.

The purpose of the insertion is to decrease the valueof the vertical betafunction at the target. In the existing
scheme, the vertical betafunction is 0.8 m. The length of
the insertion is settled from the RF frequency of the existing cavity (18.2 MHz). To minimize the effect of themismatch on the amplitude of the betatron oscillations, thearc is modiﬁed by changing the k-value (from 1.92 to 2.57)
to become transparent. This change in the design will havea small effect on dispersion (from 0.8 m to 0.65 m). The
parameters of the new lattice are presented in table 2.
Table 2: Parameters of the new ERIT lattice.
Circular Straight
SectionSection0.08
0.06
Xprime [rad]

0.04
0.02

Type
k-value or m-value
Radius/Length
Horizontal phase advance
Vertical phase advance
Number of cells

0

-0.02
-0.04
-0.06
-0.08
-0.5 -0.4 -0.3 -0.2 -0.1
X [m]

0

0.1 0.2 0.3

Figure 5: Horizontal phase space. Two particles with an
initial displacement of 15 cm and 29 cm are tracked in the
PRISM lattice over 30 turns. The dotted ellipse is the oneused to measure the acceptance in the middle of the straightsection.

FDF
2.57
2.35 m90 deg.
90 deg.
8

DFFD
1.52 m−1
1.4 m
41 deg.
148 deg.
2

Particle tracking is done again with Runge-Kutta integration in the same conditions than in the PRISM case. The
tune is 2.22 in horizontal and 2.82 in vertical.
3
2

0.1
1
y [m]

0.08
0.06
Z' [rad]

0.04

0
-1

0.02
0

-2

-0.02
-3
-5

-0.04
-0.06

-4

-3

-2

-1
x [m]

0

1

2

3

-0.08
-0.1
-0.1

-0.05

0
Z [m]

0.05

0.1

Figure 7: Closed orbit of 11 MeV proton in the ERIT latticewith the insertion.

Figure 6: Vertical phase space. Two particles with an initialdisplacement of 3.5 cm and 7 cm are tracked in the PRISM
lattice over 30 turns. The dotted ellipse is the one used tomeasure the acceptance in the middle of the straight section.

ERIT CASE
The ERIT (Energy/Emittance Recovery Internal Target)
project with a scaling type of FFAG proton storage ringhas been proposed as an accelerator-based intense thermal
or epithermal neutron source (ABNS) for boron neutron
capture therapy and constructed in KURRI [6]. Emittanceblow up due to multiple scattering and energy straggling in
the target is limited by ionization cooling. The results arepromising [7], but since limitation of the survival rate of
the protons comes from the emittance growth in the verticalplane, an insertion with a minimum of vertical betafunction
could improve this scheme.

Betafunctions are obtained with a small amplitude motion (see Fig. 8): at the target, the horizontal betafunctionis 3.2 m and the vertical one 0.29 m. The vertical betafunction is thus smaller by a factor 3 at the target, but thehorizontal one is bigger by a factor 2.5 compared with theoriginal lattice. Further study must be realized to checkif the horizontal acceptance (more than 10000π mm.mrad,see Fig. 9) is enough to handle the overheating due to theincrease of the betafunction.

TRANSPORT LINE AND CARBONGANTRYFFAG transport line could be useful to transport different momenta in a short time in the same line, like in hadron
therapy gantries. In zero-chromatic FFAGs, each momentum has a different reference trajectory. Dispersion suppressors are then necessary at the beginning and at the endof the gantry since all different momenta come from and ar--238-

Proceedings of the 7th Annual Meeting of Particle Accelerator Society of Japan (August 4-6, 2010, Himeji, Japan)

3.5

REFERENCES

3[1] JB. Lagrange et al, “Straight section in scaling FFAG accelerator”, PAC’09, Vancouver, FR5PFP002 (2009).

β [m]

2.5
2
1.5

[2] JB. Lagrange et al, “Zero-chromatic FFAG straight section”,
FFAG’09, Fermilab (2009).

1
0.5
0

[3] K. R. Symon et al., Phys. Rev. 103, 1840-1842 (1956).
0

1

2

3

4
5
s [m]

6

7

8

9

[4] A. A. Kolomensky and A. N. Lebedev, Theory of Cyclic Accelerators, (North-Holland, Amsterdam, 1966).

Figure 8: Horizontal (plain red) and vertical (dotted purple)beta-functions for half of the ring of the ERIT lattice.

[5] “The PRISM Project - A Muon Source of the World-Highest
Brightness by Phase Rotation -, LOI for Nuclear and Particle
Physics Experiments at the J-PARC 217 (2003).

0.08[6] Y. Mori, “Development of FFAG accelerators and their applications for intense secondary particle production”, Nucl.
Instrum. Methods Phys. Res. , Sect. A 562, 591-595 (2006).

0.06

X' [rad]

0.04
0.02
0

[7] K. Okabe et al, “Study for FFAG-ERIT neutron source”,
IPAC’10 , Kyoto (2010).

-0.02
-0.04
-0.06
-0.08
-0.15 -0.1 -0.05

0
X [m]

0.05

0.1

0.15

Figure 9: Horizontal phase space. An 11 MeV proton withan initial displacement of 13 cm is tracked in the ERIT lattice over 1000 turns.
rive at the same point. Another constraint comes from thereverse bend in the line. It induces to reverse the dispersion
between the two bends, or to use a negative-k lattice in oneof the bends. A schematic layout is presented in Fig. 10.

Figure 10: Schematic layout of a zero-chromatic FFAG
gantry. The dotted red line represents the trajectory of a
middle momentum, and the plain red line the maximum
momentum. The mixed lines represents the rotation axis.
The upper scheme shows the case of a reverse dispersion,
the lower one the case of a negative-k lattice (in yellow).

SUMMARY
To overcome the problem of injection/extraction in the
PRISM project, a new lattice using straight sections is proposed. An improvement of the ERIT scheme is then presented with a low-betafunction insertion in the ring. Finallythe concept of a zero-chromatic FFAG carbon gantry is described. if these proposals need further studies, they open
a promising way to improve lattices and schemes.
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