JAERI-Conf 95-021

PB36
Design Parameters of a Spiral Inflector for Axial Injection
Project of the CYRIC Cyclotron
M. Fujioka, T. Honma, T. Shinozuka, 0. Morishita* and T. Tachikawa*Cyclotron and Radioisotope Center, Tohoku University,
Sendai 980-77 Japan*Quantum Equipment Division, Sumitomo Heavy Industries Ltd. Niihama, 792 Japansuffix H denotes the horizontal component. For the sakeof simplicity we use the following abbreviation.

Abstract====

Cn cos(nkt) cos(nO),sn sin(nO), and ctn cot(nO).
A derivation is given of the Root's solution for the(2)beam orbit in the spiral inflector. On the basis of this solution a survey of parameters of the inflector was made, We further define E~ and E~ , the former being parallel
and appropriate design parameters were proposed con- and the latter perpendicular to vu , respectively. For
simplicity we assume E~ = 0 , thensidering the central-region geometry of the cyclotron.
1.

Eu ex: vu

Introduction=zu ,hence Ex= En·!:_and Ey = Eu·j_.
VHVH(3)A preliminary numerical study of an axial injec- From Ez = E s1 , we have Eu = E c1 and hencetion scheme was made on the basis of an off-axis injecVx = Ec1~ =(E) ct1.X and Ey =(E) ct1iJ.
tion and a mirror infl.ector. 1> Since this scheme requires Ex= Ec1 vuv Slv
vrather complex deflection electrodes within the central(4)plug through the yoke, we abandoned this and turned to On the other hand the magnetic field -Bez gives a horusing a spiral inflector 2 ) together with an on-axis injec- izontal force on the particletion.
F: = -eBiJ and F: = eB.X ,(5)
In the following we give a derivation of the Root's
34solution ) and then apply this to our CYRIC cyclotron, ) giving the equations ,of motion for x and y under theassuming an on-axis injection, which, however, requires presence of E and -Bez ;a change of the tips of the central plugs as well as minor
mx = (eEfv)ctl.X- eB iJ I my= (eEfv)ctl iJ + eB .X .
changes of the central electrodes.

(6)

2.

Derivation of the Root's &olutionv

1!

We take a coordinate system of Fig.l. In the absence of the magnetic field the central trajectory of theinjected beam at t = 0 is easily seen to be (we takev, RE, E ~ 0)
X(t) ::: (:m )= RE ( 1-z(t)with E = E (yc~s(kt) )-sin(kt)c~; ) , and 0 =kt ,(8=-Bez)

(1)

0smO

where RE is the radius of curvature under an electricfield E, kvf RE and v is the velocity. Since E · v
E · z = 0 the magnitude of the volocity v is a constantof the motion.
When a uniform magnetic field -ezB is applied, z(t)and Ez are not affected, but zu changes and Eu must
be changed accordingly for conserving E · v = 0; the==Fig.l. Deflection of central beam in the absence of B.

-350-

JAERI-Conf 95-021

Defining the radius of curvature RM under the presenceof only B,REv
mv
RM :=-,hence eB/m = v/RM = -R · -R
MEeB

=2kJ{,(7)==RE/(2RM)· From eqs.(6) andwhere k v/RE and K(7) we finally obtain the equations of motion for x andy;~~} -c~~{( : ) =k()(:)=kM(t) ( t)(15)
We note that (Ex, Ey) ex: (x, y) and that E · v = 0 .

(8)In eq.(8) the matrix M(t) can be diagonalized by a unitary transformation;

(t )

t)= ~

3.

Application to the CYRIC cyclotron

Before applying eq.(15) to our cyclotron, it is notedthat the injected beam comes down along the mechanicalaxis of the cyclotron so that the entrance point is at-i )11 (.
(x, y, z) = (0, 0, 0) and the exit point is at (p, </J, z) =W1th U = f i -i 1(g) (Pex, ell, -RE) where the beam comes out horizontally inthe median plane of the cyclotron; (p, </J, z) being theEq.(8) then becomes<Pex(x 2 + y 2) 112 and 4>
cylindrical coordinates; p
is the total rotation angle of the beam orbit in the x-yplane.
) .
) = kU+ M(t)U (((10)
From eq.(15) we note that we have two free parameters) ( X )
0_ k ( ct1 + 2iK
out of, e.g., RM, K, and RE because of the relation of'Y
ct1 - 2iK0RE/(2RM) ; we take RM and K as independentKparameters. As shown in Fig. 2, as a crucial conditioni.e., decoupled equations for X and Y . Eq.(10) can be
we require that the beam leaving the inflector electrodes
integrated as follows.
smoothly travel along a circle of radius RM and passes
perpendicularly through the slit in the radial part of thek(ct1 + 2iK)dt =In ls1l + 2ikKt, (11) grounded housing of the inflector electrodes; the slit atIn IX(t)l =a distance Rs faces the puller at Rp from the origin sothat Rs = Rp = 2.2 em for our cyclotron. For a smoothgivingof the beam at the electrode exit we requireconnection(12)= U(~ ) and also (U()==~~=1t

We take from inspection Y = 0 and from eq.(9) obtainfb.. 2ikK . eik<_~-ik< • (( ~y ) = ff
2•

/in= /out '(16)where lin is the angle between VH and eq, at the exit
point, i.e.,

1_ )
-t

[ -i(c2K+1-C2K-1)+S2K+l-S2K-1].
-i(s2K+1- S2K-d- C2K+1 + C2K-1
_1
(17).
.
. + yy)lin= tan(13)xy- yx e=,./2
Taking the real part of eq.(13) and considering the initialand final conditions, x(O) = x(O) = y(O) = y(O) = 0 and On the other hand, from the geometry of Fig. 2 we have(x 2 + y2 )(kt = () = 1rj2) = v2, we arrive finally at theat the exit point
required solution;c(xx= 2.:}2(and(X )y_X )

.

Y= 2'

S2K+1- S2K-1 )-C2K+l + C2K-1(- 2K+1 + 2K-1 - 4K2-1_-&2v (~C2K-l_ S2Ktt+ s,K_,

2K+12K-1

lout- COS2)'(14)which exactly coincides with the previous results 2· 3 ) ,considering that we take Bz = -B. Before going further,we give complete forms of solution;-1+ 2RMRs)(P~x- R~
2R MPex(18)

Satisfying this condition (eqs.(16)-(18)) leaves one freeparameter, e.g., RM . We calculated numerically under
the condition of eq.(16) various parameters as functionsof RM as shown in Fig. 3. Considering the central-region
electrodes and the results given in Fig. 3, we chose RM= 1.3 em corresponding to K = 1.1135 as shown in Table-351-JAERI-Conf 95-021

1; see also Fig. 2.
3.0

The above discussion deals with only the central trajectory of the beam. Before manufacturing more ellaborate studies are needed resorting to a higher-order optical method or a numerical one.

y(cmlTable 1. Design parameters of the spiral inflector
2.0Parametersa}

Rs=Rp
RM

K:: RE/2RMREb)

-3.0

-1.0

-2.0

Pex~if>ex==

1.0 x(cmllinCl'f3Fig. 2. Orbit projected to the x-y plane and definitionof various parameters (B # 0);
R, = 2.2 em and /{ = 1.1135.

Dlout

Design value
2.200 em
1.300 em
1.11352.895 em
2.004 em
130.42°19.99°
29.59°
49.58°
0.99 ema) See Fig. 3 and the text.
b) RE essentially defines the height of
the inflector electrodes.

1009080
1.570<i>/2(degl

K60
1.0

50
400(cml30
0.5a(degl20
10y(degl

1.10

1.20

1.30

1.40
RM (cml01.50

Fig. 3. Dependence of various parameters on RM for R,2.2 em. Note that f3 a+ 1 and RE 2/{ RM,and the arrows indicate the scale to refer.

===Fig. 4. Spiral inflector in the central region ofthe cyclotron.

References
4.

Conclusion[1] Honma T., Fujioka M., Goto A. and Shinozuka T.,
Proc. 9th Symp. Accel. Sci. Tech., Tsukuba, JapanFor the sake of understanding we gave a rather de(1993)300.
tailed derivation of the equation of the central trajectoryof an axially injected ion beam (eq.(15)), and applied it [2]Inoue M., RCNP-T-10, April (1983), in Japanese.
to the CYRIC cyclotron, and proposed a set of design
parameters compatible to the central-region structure of [3] Root L. M., Design of an inflector for the TRIUMF
cyclotron, Master's thesis, Univ. British Columbia,
the cyclotron at the cost of least change of it;.see Fig.
April, 1972.
4. We would like to stress that the results of the survey calculation (Fig. 3) should be applicable to other [4]Morita S., Fujioka M., Orihara H., et al., IEEE
cyclotrons with an appropriate scaling, i.e., a normalTrans. Nucl. Sci. NS-26 (1979)1930.
ization with respect toR, = Rp.

-352-