The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Beam Orbit Simulation in the Central Region of the RIKEN AVF Cyclotron
Dragan Toprek, Akira Gotoand Yasushige Yano

The Institute of Physical and Chemical Research (RUffiN)
Hirosawa 2-6, Wako-shi, Saitama, 351-01 Japan
Abstract
This paper describes the modification design ofthe central region for h = 2 mode of acceleration in theRIKEN AVF cyclotron. The central region is equippedwith an axial injection system. The spiral type inflector
is used for axial injection. The electric field distribution
in the inflector and in four acceleration gaps has been
numerically calculated from an electric potential map
produced by the program RELAX3D. The magnetic field
is measured. The geometry of the central region has beentested with the computations of orbits carried out bymeans of the computer code CYCLONE. The opticalproperties of the spiral inflector and the central region
are studied by using the program CASINO andCYCLONE, respectively. We have also made an effort
to ·minimize the inflector fringe field using theRELAX3D program.

measured, is stored in a polar mesh with 11(} = 1.8 °and Ar = 2em.
2. Spiral intlector
Scaling of the trajectories through the spiralinflector demands that the radius of curvature Rm,
which the ion would have if it were acted upon by amagnetic field in the absence of any electric field, isconstant. The value for Rm is given by the following
expression:

Rm =(1)

where: U ex- the extraction voltage of the ion source in
kV, B 0 - the center field value in T, 1]- the specific

1. Introduction
The basic characteristics of the RIKEN AVF
cyclotron are described in Refs. [1,2]. Here we repeatthem briefly.
The RIKEN A VF cyclotron is a compact·isochronous cyclotron with four spiral sectors. The
magnetic induction at the centre of the machine varies
from 0.5 T to~ 1.7 T. Its constant is K = 10MeV.
Extraction radius of the machine is re = 71.4 em. The
ions coming from the ECR ion source (the maximum
extraction voltage U :;'is 13 kV) are injected in theAVF cyclotron axially from up. They are introduced
into the median plane of the machine by a tilted spiralinflector. The acceleration then takes place at fourelectric gaps per turn. The angular span of the dees is85°. The vertical aperture of the dees inside the central
region is 24 mm. The maximum amplitude of the deevoltage is 50 kV and its frequency goes froth 12 MHz to24 MHz. The number of trim coils is 9, and the numberof harmonic coils is 1 in the extraction region.
In the Refs. [1,2] is shown the shape of theelectrodes of the central region, but in the former design,when the ion trajectory through the spiral inflector (withparameters: A = 2.6 em and k' = 0.0) is analyticallycalculated and the electric field distribution in the fouracceleration gaps is described by a gaussian function.
Now, to treat more realistically both the spiral inflector
and the central region, the electric fiela distributions in
the inflector and in the four accelerating gaps has been
numerically calculated from electric potential maps,
produced by RELAX3D program [4,5]. For the analysisof the ion trajectories through the spiral inflector and in
the central region, which includes both the electric andmagnetic fields data, we used the CASINO [6] andCYCLONE [7] programs, respectively. The medianplane magnetic field map B(r,O), which is thecharge of the ion, E 0 = me 2 I A -the rest energy of theion per nucleon in MeV.
The value of Rm for the representative ion(17 = 0.357,B 0 = l.ST, E 0 = 931.5MeV/n,Uex = 10 kV)

is Rm = 1.61 em. If we want to use the same spiralinflector for another kind of ion, the extra.ption voltage
U ex and the center field value B 0 for that kind of ion
can be obtained by scaling aecording to Eq. (1). We haveselected the spiral inflector with following parameters:
A = 2.2 em [3] (A is the radius of curvature which theion would have if it were acted upon by an electric fieldin the absence of any magnetic field, in fact it is theheight of the spiral inflector) and k' = 0.3. [3] (k' is aparameter which is related to the direction of the electricfield) with the tapered electrodes, trying to minimize theinflector fringe field and the horizontal cross section ofthe inflector. The electrode spacing at the inflector
entrance is d 0 = 0.8 em, the width of its electrode is 1.6
em (aspect ratio is 2) and the maximal electrodepotential is ±3.6 kV. The distance between the center oftrajectOry at the inflector exit and the center of thecyclotron (so called off-center) is p c = 1.1 em .
The electric field produced from the electric
potential has characteristic fringe field· distribution at theends of the inflector. A real inflector has fringe fields,
which increase the effective length of the inflector. We
can compensate the entrance fringe field of the inflector
by decreasing the voltage between the inflector
electrodes or by cutting a certain pieces of the inflector.
The fringe fields change the optics of the inflector andpush the particles to oscillate around the design line
(central trajectory in the case when electric field isanalytic [3]) of the inflector. This may be observed byrunning CASINO [6].

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3. Phase space at the entrance and exit of the spiralinflector

prO= 0.67 em) for different Starting phase.

The phase space diagram in two transversalplanes (u, p ~) and (h, ph) [3] at the inflector entrance

250
A(dashed and dotted line) and at its exit (full line curves
marked by letters fu and fh respectively) are presented

200k'= 0.3

Rm= 1.61 ema)

150
u

100

E50~in Fig. 1. Seven particles from the contour in the ( u, p")0and ( h, ph) planes, which correspond to the inflector
entrance emittance of e. = 140n mmmrad= 2.2 em

0QJ"' -50:;(dashedQ_-~ -100line) and e h = 130 n mm mrad (dotted line) and therelative change of the longitudinal component of themomentum of the ion at the inflector entrance is
8pv = !J.pv I p = -0.01, opv = 0.0, 8pv = +0.01, were
run through the spiral inflector using the code CASINO[3]. pis the total momentum of the ion in length units.

-150.t.\s= 0.0 mm-200·- · · · · ·- e

x = 8%

infl_ ~ntrance E, -140 '7f mm mrod

infl. entronce· t: 0 -130 Tl mm mrod

-2504---~--~---.---.----.---+-4~20246-6

coordinate [mm]For each of these three values of 8p v we have threedifferent curves at the inflector exit. Full line curves withfull triangle, full circle and full rectangular symbols at

250~--~---L--~----L---~---rA

the inflector exit correspond to Opv = -0.01, Opv = 0.0

200= 2.2 emk'= 0.3

150

and Opv = +0.01 at the inflector entrance, respectively.

c:o_··or·r ··... ,100500Due to the coupling effects we consider the motion ofthe particles through the spiral inflector in all threeplanes. Fig. 1a shows the case when the particle isinflected by voltage decreased by 8% ( !J.s = 0 mm,
X= 8 %) and in Fig. lb we compared results at theinflector exit when ~s = 0 mm , X = 8 % (full line curvewith full rectangular symbol) and when the particle isinflected by cutting off 1.5 mm piece of the inflector atits entrance ( !J.s = 1.5 mm , X =0 % - full line curve withE• .

J\..
&-"' -50:;
Q_..

-~ -100-150'I ,/fh/o0QJb)""'6p, = 0.0-200~fu'• as =0.0 mm• ll.s =1.5 mm

infL entrance E.- 140,. mm mrad··e

infL entrance jj•-1.30 ,mm mrod-2504----.---.---,----,---,---1-6-4~20246coordinate [mm]full circle symbol), but Dpv = 0.0 at inflector entrance.
From Fig. 1b we can see that the differences ofthe optical properties of our spiral inflector in the casewhen !J.s = 0 mm , X = 8 % compared to the case when!J.s = 1.5 mm , X = 0% is negligible. Because of that wewill consider further only the case when !J.s = 0 mm ,Fig. 1. Phase space diagram in two transversal planes(u, p") and (h, ph) at the inflector entrance (dashed anddotted line) and at its exit (full line curves marked byletters f. and fh respectively). The inflector entrance

x=8%.

emittance

4. Orbit studies and optical properties of the central
region4.1. Horizontal motion of the central ray
Our results prove that the energy gain
corresponding to the central ion trajectory is sufficient toclear all the obstacles successfully, Fig. 2. TheISe" = 140 n mm mrad

eh = 130nmmmrad(dashed line)

and(dotted line) and the relative changeof the longitudinal component of the momentum of theion at inflector entrance is 8pv = -0.01, 8pv = 0.0 and
8pv = +<).01. The others marks are explained in the text.

trajectories of reference ion ( 17 = 0.357, B 0 = 1.5 T ,h = 2) for three value of starting phase ¢>0 = -50° (fullline), ¢>0 = -55° (dashed line) and ¢>0 = -60° (dotted line)
during the first two turns are also presented. The deevoltage is U 0 = 50 kV . These orbits are the "centralrays" (the initial radius of the ion trajectory isr0 = 2.26cm and the initial radial momentum isFig. 3 shows the modificated shape of theelectrode in the central region of the RIKEN A VF
cyclotron. The dotted area shows the area that has been
cutting off of the previously designed electrode,accordimg to the new simulation result.

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each degree in the first two turns. The verticaldisplacements of the representative ion underconsideration for three different starting phase are shownin Fig. 4. Only those starting phase which give verticalenvelopes contained within the smallest vertical apertureof the central region are shown. From Fig. 4 we can seethat the phase acceptance of the central region is about
10 degrees.

105E'~

0>3+-----~----~------L-----+-57)

280= 0.357
1.5 T=u" = so kv-10Fig. 2. The central trajectory of the referent ion( 1J = 0.357,150

B 0 = 1.5 T, h = 2) in the central region of the350

e [deg]550

750

K70 A VF cyclotron for different value of the starting
phase¢0numericalwith the equipotential contours of thecalculatedfield.

¢ 0 =-500(fullline),¢ 0 = -55° (dashed line) and ¢ 0 = -60° (dotted line). The

equipotential contours are plotted at each 10% of the deevoltage U 0 •Fig. 4. The maximal vertical displacement of the beam in thefirst two turns for considered ion in the K70 A VF cyclotron fordifferent value of the starting phase ¢ 0 • The short lines
crossing the abscissa show the electric gap positions and thedotted lines parallel to abscissa indicate the minimal verticalaperture of the central region.

6. Conclusions

with0Our orbit studies through the spiral inflector
A= 2.2cm, k'= 0.3,
parameters:
following

Rm = 1.61 em shown that the differences of the optical~-1E~

>--2

2

3

45X [em]Fig. 3. The modificated shape of the electrode in thecentral region of the RIKEN A VF cyclotron. The dotted
area has cut off of the previously designed electrode.

properties of that spiral inflector are negligible in thecase when the inflection of the ion is reached by thevoltage decreased by 8% and in the case when theinflection of the ion is reached by cutting off a 1.5 mm
peace of the inflector at its entrance.
Our orbit studies during the acceleration in thecentral region of the cyclotron indicated that theclearances between the ion and the solid obstacles andthe vertical focusing in the first two turns are
satisfactory, but the smaller value for emittance at
inflector entrance in both u r anddesirable.
Referencesh r directionare

4.2. Vertical motion of the paraxial rays[1] - A. Goto et all, Injector A VF Cyclotron at RIKEN, Proc.

We tested the phase acceptance of the central
region of the machine by following the first two turns ofthe ions. The emittance of the beam was reconstructedfrom its emittance at the entrance of the inflector (seeFig. la). The family of curves marked by letters fu
and fh (which correspond to (u, p") and (h, ph) planeat the inflector exit) now correspond to (z, pz> and(r,p,)_plane,respectively, at the entrance of the central
region of the cyclotron.
The maximal axial displacement, defined as thebeam envelope, of the trajectories of seven particles
defining the contour at the inflector exit, was checked at12'h Int. Conf. on Cyclotrons and their Applications,Berlin, 1989, p. 51.
[2] - A. Go to et ail, Study of Beam Optics for the Injector AVF
Cyclotron at RIKEN, Proc. 12'h Int. Conf. on Cyclotrons
and their Applications, Berlin, 1989, p. 439.
[3] - L. W. Root, M.Sc. Thesis, Univ. of British Columbia,
1972.
[4]- C. J. Kost and F. W. Jones, RELAX3D User's Guide andReference Manual, TRI-CD-88-01, Vancouver, 1988.
[5] - Lj. M. Milinkovic, RELAX3D Spiral Inflectors, TR30DN-89-2l, Vancouver, 1989.
[6] - F. B. Milton and J. B. Pearson, CASINO User's Guideand Reference Manual, TRI-DN-89-19, Vancouver, 1989.
[7]- B. F. Milton, CYCLONE VERS 6.0, TRIUMF InternalReport, June 22, 1989.

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