NONLINEAR EFFECTS IN CYCLOTRON
T. Yamazaki, K. Hosono, M. Inoue and I. Miura
Research Center for Nuclear Physics, Osaka UniversityIbaraki, Osaka 567, Japan

accelerate protons up to 300 MeV. The proton beams areaccelerated through vy = 4/3 resonance. The phasep~ots across the vy = 4/3 resonance are shown inFlg. l. The phase plots show triangular shape aroundthe equilibrium orbit and show the existence ofvy = 4/3 resonance.
For the ring cyclotron we have studied the effectsof trimming coil shapes by calculating isochronous
magnetic fields with circular-shape approximation andorbital-shape approximation. The isochronous fields
are calculated by iteration method. Figs. 2 and 3 show
the phase plots of the stable region for protons atlarge radius for a spiral-sector ring cyclotron withfour magnets. Outer eight islands are related to thestable fixed points of vy = 12/8 and inner five islands
are related to the stable fixed points of Vy = 8/5.
Many small islands are related to higher order fixed
points. These islands are surrounded by random plots
in higher energy and outer region. The size of thisstochastic region is different between two types ofisochronization. But for both types of isochronization
the sizes of stable regions are large enough to accelerate particles.
Fig. 4 shows the phase plots above extractionenergy <Eex = 300 MeV). The size of the stochastic
region becomes larger above 330 MeV, and the stableregion disappears at 354.3 MeV. In figures FR means
the radial betatron frequency vy and FZ means thevertical betatron frequencies Vz , and FR and FZ values
near 90 show imaginary values. In 341.7 MeV case theradial betatron frequency v becomes 1.90 and close to
the stop band of v = 8/4. Yin the figure there are twostable fixed points of v = 8/4. Fig. 4 shows an orbit
behavior at stop band region. It may be difficult toaccelerate particles up to this energy, and the extraction energy is designed at the betatron frequency
Vy ~· L 7.
In same phase plots the large islands are
surrounded by small islands and these small islands are

ABSTRACT
By using static phase plots the nonlinear effectsof particle motion and stability limits are investigated for the RCNP AVF cyclotron and a proposed ring
cyclotron as a new facility of RCNP. For the ring
cyclotron the effects of trimming coil shapes by calculating isochronous magnetic fields with circular-shape
approximation and orbital-shape approximation arecompared.
INTRODUCTION
For the design study of RCNP ring cyclotron, thebetatron frequencies, eigen ellipses, phase plots andthe stability limit have been studied for various typesof sector magnets. 1 • 2 The iron fields used for thesecalculations were the measured field of a model magnet
and the artificial fields calculated with computer code
FIGER. 3
At extraction radius 375 em, protons can be
accelerated up to 300 MeV. The proton beams areaccelerated through vy = 4/3 resonance. The phase
ellipse is slightly deformed in crossing this resonance, and largely deformed near vy = 8/4 stop band.
The protons also cross the Vy = 8/5, 12/8 andother higher harmonic resonances. These higher harmonic resonances don't have large disturbance to theaccelerated orbits, but may change the fine structureof beam profiles and beam qualities required to highresolution studies on the nuclear structure. It isinterest to study the behavior of higher harmonic resonances.
RING CYCLOTRON
The proposed RCNP accelerator system consists oftwo ring cyclotrons and an injector cyclotron. A firstring cyclotron redesigned in the end of 1982 canA-U6.33-)!1.06-Bl-80tCMI-58.53'--L-~~-----1-

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Phase plots of the proton beams.
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Phase plots above vy = 8/5 resonance.
The orbital-shape isochronous field is used
for the calculation. Five islands around theequlibrium orbit move outward with energy.

EKIN 0-3519
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Phase plots above vy = 8/5 resonance.
The circular isochronous field is used forthe calculation. The size of the stableregion is smaller than that for the orbitalshape isochronous field.

Fig. 3

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Phase plots at large
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0-00
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well isochronized to larger radius,
OR rc. U. I
the size of the stochastic region
becomes larger and two islandsFig. 5
related to the stable fixed points
Complicated phase plots.
of v = 4/2 appear.
a.
Outer five islands around theequilibrium orbit are related to thestable fixed points of v = 8/5.
b.
A magnification of the littel box in A.
An island is surrounded by five smallislands.
c,d. Further magnifications of the small boxin B. Figs. c and d show the same phaseplots of the small island with different
starting conditions. They show verysmall islands around the small island.
~0260' :-1 ~ 00

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Fig. 6The large islands are surroundedby small islands. The ·behaviorof small islands changes
remarkably with small change ofparticle energy.

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Phase plots near extraction region. I~ the magnetic field is slightly lower than isoc1ronism, threeislands related to the stable fixed po1nts of v = 4/3 appear and radial betatron frequency (FR) decreases
with energy.

surrounded by very small islands. Fig. 5 shows a typical example of the complicated phase plot near
extraction energy. Fig. 6 show other examples of thecomplicated phase plots below extraction energy.
Fig. 7 shows the phase plots of protons at large
radius for a straight-sector ring cyclotron with four
sector magnets. The magnetic field at larger radius isslightly lower than isochronism. Three islands appear
on the boundary of stable region, they become larger
and th~ central stable region around the equilibrium
orbit becomes smaller with particle energy, and there
is Vy = 4/3 resonance near 188 MeV. This behavior ofphase plots at Vy = 4/3 is similar to that of vy = 3/3
resonance for three-sector AVF cyclotron near extrac~
tion radius.
As seen in Fig. 1, protons cross vy = 4/3
resonance at 113 MeV and the size of stable region
increases. The stable region has a triangular shape.
Three unstable fixed points are on the vertex of thistriangle. That is, these unstable fixed. points withv = 4/3 are on the boundary of stable region. Threestable fixed point orbits are far apart from theequilibrium orbit. With increasing energy many fixed
points appear at the center and move outward. Eachisland in the phase plots has stable fixed point in it.
Islands are accumulated at the boundary of the stableregion with energy. With increasing energy the betatron frequency go away from v = 4/3 and get nearer tov = 8/4, and the shape of the stable region changes
from triangle to ellipse. Moreover the accumulation of
the fixed points increases the size of stochastic
region near the boundary of the.stable region. The
phase plots in Fig. 1 are for four spiral-sector
magnets, but these behaviors are similar for four
straight-sector magnets.
Fig. 1 shows that three unstable fixed points ofv = 4/3 are on the vertex of the triangle. However as
seen in Fig. 7 three stable fixed points of v = 4/3
appear on the vertex of the triangular stable region,
and a new triangle appears around the equilibrium
point. There is an apparent discontinuance between thetriangle of low energy Vy = 4/3 region and high energy
The situation is different for theVy = 4/3 region.
InVy =3/3 resonance of three sector AVF cyclotron.
this case there is a continuation between the triangleof the central Vy = 3/3 region and high energy vy = 3/3
region near extraction. There is the same triangularstable region and no drastic change of stable region as
seen i·n Fig. 7.

RCNP AVF CYCLGrRONThe RCNP 230-cm AVF cyclotron is three sectormachine with K=l20 MeV. Radial betatron frequency vyfor 65 MeV proton acceleration is from 0.99 to 1.13.
Particles cross only high order (radial) resonances
except vy = 3/3. Fig. 8 shows the_phase plots of thestable region at large radius. By using the measured
magnetic field at the main magnet current of 600 A an
isochronous field· for 65 MeV protons was obtained, andthe particle trajectories were traced about 10000 revolutions in all. Many islands are seen in these phaseplots,. and they are related to higher order resonances.
Some examples are as follows: 23 islands at 49.45 MeV
are related to v = 24/23 stable fixed points, 11islands at 57. 40 and 62.96 MeV are related to v = 12/11
stable fixed points and 8 islands at 65.72 MeV arerelated to v = 9/8 stable fixed points.
A.

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REFERENCES

1.
2.
3.

261M. Fuki et al., Proc. 4th Sym. on Accel. Sci. andTech. p.93 (1982), RCNP Annual Report 1982, p.l48.
Y. Kadota et al., Proc. 4th Sym. on Acoel. Sci.
and Tech. p.95 (1982), RCNP Annual Report 1982,
p.l51.
I. Miura et al., Proc. 4th Sym. on Accel. Sci.
and Tech. p.97 (1982), RCNP Annual Report 1982,
p.l46.