The lith Symposium on Accelerator Science and Technology, Harima Science Garden City. 1997

Analysis of a Beam Centering Error Measured with a Radial Main Probe
Mitsuhiro FUKUDA , Susumu OKUMURA, Kazuo ARAKAWA, Ikuo ISHIBORI,Akihiko MATSUMURA and Takashi KARASAWA
Takasaki Radiation Chemistry Research Establishment, Japan Atomic Energy Research Institute,1233, Watanuki-machi, Takasaki, Gunma 370-12, JAPAN
Abstract
A method for estimating a beam centering error in acyclotron has been developed to optimize a beam orbit
during the early stages of acceleration. A series of actual beam positions measured with a single radial mainprobe was used for the estimation of the centering error
amplitude. The estimation was formed by fitting thebeam positions to the function representing radial betatron oscillation around an accelerated equilibrium orbit.
A radial betatron frequency and an energy gain per turnwere also obtained from the fitting calculation.
1 Introduction
Insufficient adjustment of cyclotron parameters fora center region, such as positions of an inflector and apuller in an axial injection system, brings an orbit center
away from a center of symmetry which corresponds to acenter of a cyclotron magnet. A first harmonic component of a magnetic field makes the orbit center move ina circle around the center of symmetry[1]. As a result of
this displacement, we have precession of the orbit either
in the direction of motion of a particle for a radial betatron oscillation frequency of Zlr < 1 or opposite to thedirection of motion of the particle for Zlr > 1. The displacement of the orbit center just before extraction fromthe cyclotron may improve an extraction efficiency, butsometimes may worsen the final beam quality.
A large first harmonic in the magnetic field drives alinear resonance at Zlr = 1 that makes the radial motion
unstable. The crucial resonance will cause an increasein amplitude of radial osciUation that results in loss ofa beam in a center region. On the other hand the resonance is helpful for increasing a turn separation justbefore the extraction, what is called precessional extraction.
For the optimum tune-up of the cyclotron beam, thebeam centering error has to be reduced during the earlystages of acceleration. The beam centering error can bemeasured precisely with multiple radial probes[2]. TheJAERI AVF cyclotron[3) is,· however, equipped with asingle radial main probe covering a full radius. The centering error of the beam in the center region was estimated from a beam density distribution measured withthe main probe only. The estimation of the centering error is based on the assumption that the centering error
amplitudes, Zlr and the energy gain per turn are identical
for the first several turns.
2 Method of the Centering Error Estimation
The radial motion of a cyclotron beam can be expressed as the betatron oscillation around the equilib-rium orbit. A radial position of the equilibrium orbit isuniquely determined by the particle energy. The shift ofthe equilibrium orbit due to the change of the particle
energy after acceleration depends on the energy gain per
turn. The radial position of the particle is expressed as

R(B) =A cos(vrB) + B sin(vrB) + Req(D.E, B),(1)

where A and B are amplitudes of the radial betatronoscillation, that is the centering error, () an azimuth integrated from the first acceleration, D..E the energy gain
per turn, Req the radial position of the equilibrium orbit. The Zlr and D..E are assumed to be constant during
the first several turns.
A relation between the particle energy and an average radius of the equilibrium orbit for a 10 MeV proton
beam is shown in Fig. 1. The particle energy in the center region is expressed by a second order polynomial of
the average radius, since a non-relativistic approach canbe applied to this energy region. Here we approximate
the average radius of the first turn as(2)where the constants b1 , b2 and b3 are determined by afitting of the energy vs. average radius curve, and E 0is an injection energy. The particle energy of the firstturn observed by the main probe of the JAERI AVF
cyclotron is E0 + D.E, since particles are stopped by themain probe after having the first energy gain.

~

350 lt"""""...........,.........-.-T"""......'r"'T.....-.........,....,I'""T'".....,-.-.........-:aE 3ooE-; 250::I'0 200~ 150~100~50Q.l~

QQ~~~Q~.2~~0L.4~~0~.6~~0~.8~~~~~1.2

Energy (MeV)

Fig. 1 The average radius of the equilibrium orbit asa function of the particle energy for a 10 MeV
proton beam.
Correlation between the radial position of the equilibrium orbit in the direction of the main probe axis andthe average radius for the 10 MeV proton is shown in
Fig.2.
Using a second order polynomial of the average radius, the radial position of the equilibrium orbit isexpressed as-300----2

Req = a1 + a2R + a3R ,(3)The 11th Symposium on Accelerator Science and Technology, H arima Science Garden City, 1997

«W~~~~~~~~~~~~~~~

350

Eaoo
§.250
E 200~ 150.

~ 100a:

50
100

200

300

R(mm)

average«W500

Fig. 2 Correlation between the radial position of theequilibrium orbit in the direction of the mainprobe axis and the average radius for the 10
MeV proton beam.

to the equation ( 1 ) is also plotted in the Fig. 4. The
centering error amplitudes obtained by the fitting were
A= -12 mm and B = +8 mm, which means that theorbit center was shifted by -12 mm in the main probe
direction and by +8 mm in the direction perpendicular
to the main probe axis.
The estimated energy gain and Vr were D.E = 0.052
MeV /turn and llr = 0.999. In the practical operation anominal acceleration voltage was 11.39 kV, and thus theenergy gain amounted to 0.045 MeV /turn. The energy
gain estimated by the fitting is larger by 14 % than thenominal value.
In this preliminary analysis, errors of the fitted values were inore than 50 %. The errors came from ambi- guity of the peak position determination. More clearly
separated peaks with a smaller width are required for
precise analysis. Evaluation of this method for estimating the beam centering error is in progress.

where the constants a 1 , a 2 and a 3 are determined by afitting.
By solving the equation of-motion, the average radius of the equilibrium orbit is given by__ -(Eo+ D.E + 3 : 0R- Ro
Eo+D.E

6.E) ~I

250
200c~ 150~ 100(4)";!

where Ro is the average radius of the first turn. Fromthe equations ( 2 ), ( 3 ) and ( 4 ), the radial positionof the equilibrium orbit can be expressed by the Vr andenergy gain.
The parameters of A, B, D.E and Vr are determinedby a fitting of actual beam positions obtained from thebeam density distribution using the equation ( 1 ).

~ so~001000

2000

3000

4000

Azimuth (degrees)Fig. 4 Radial positions( dots) of the first thirteen revolutions for the 10 MeV proton. The curve
accounts for a trace of the betatron oscillation
estimated by the fitting.

3 Centering Error of 10 MeV Proton Beam

The beam density distribution of the 10 MeV proton beam measured with the main probe is shown inFig. 3. Peaks corresponding to individual revolutions

References[1] J .R.Richardson, "Sector Focusing Cyclotrons,"
Progress in Nuclear Techniques and Instrumentation(North-Holland Physics Publishing, 1965), vol. 1, pp.
29.
[2] S. Adam and J.C. Collins, IEEE Trans. Nucl. Sci.
NS-26 (1979) 2362.

[3] K. Arakawaet al., Proc.13th Int. Conf. on Cyclotron
and Their Applications, Vancouver, Canada, (1992)
119.

Fig. 3 Beam density distribution for the 10 MeV proton beam measured with the main probe.
are clearly observed, but some peaks are duplicated. The
duplicated peaks resulted from an error in signal processing. A centroid of the duplicated peaks was obtained byaveraging the two peaks.
The beam positions of the first thirteen revolutions
are plotted in Fig. 4. The curve obtained by the fitting-301-