in a
Estimation of the Mechanical Abarration- Limited Enerqy Resolution
Model Beam Analyzing Magnet

Yasuhiko Okuma

567 Japan
Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka,
1~

Introductio n.

system ~E/E is limited
The energy resolution of a magnetic analyzer with an ideal detecting
l/2
22
by a following 2 relationshi p:(1)AE/E=[L)E/E (l'st order)+L)E/E (aberration) +AE/E (inhomogene ity))
which is determined by thewhere AE/E(l'st order) is the l'st order ion optical energy resolutionenergy resolution due tonffignet geometry, JE/E(aberra tion) is the ion optical aberration- limited
energy reso-limited
the higher order focusing and L)E/E(inhom ogeneity) is the mechanical aberration- is expressed withogeneity)lution resulting from the magnetic field inhomogene ity. The AE/E(inhoma radial direction
along
ity
inhomogene
fieldmagnetic
the image broadening caused by the absolutein the magnetic analyzer.

A digital computer program method which integrates

the equation of motion or traces the ray of charged
particle in the measured magnetic field can not discuss

only the mechanical aberration L)E/E(inhom ogeneity). A
sector type of a model beam-analyz ing magnet had been
fabricated to inVestigate the designs of new electro-

are installed to the AVF cyclotron of•RCNPt
2. Measuremen of the magnetic field inhomogene ity.

magnr~5Jwhich

The magnet has the bOmogenlze rs, the alr gap

spacers machined acru7rtely and the pole pieces sepa-. The schematic presentatio n of
rated from the yoke 'the magnet is given in fig.l. The specificatio ns andperformance s of the m11net and the power supply aresummarized in table 1 . The magnet has two. types ofpole pieces with a sharp-corner ed(SCOF) and a B-con-stant(B-con st) profiles, whose are made of a commercial

grade of two kinds of low carbon and forged iron plates(l and 2) in table l. A differentia l probe consistingof the fixed and the search Hall-elemen t probes has
been used to detect t~~ small difference of t~eLd. The fixed probe is set at astrength less than 10AA

point in the uniform field. The search probe is moved

along a radius at a given azimuth (8=1 and 3) over theuniform field on the median plane by a Zip-track system. The specificati on and performance of the analog fietd

1-difference -detecting system are presented in table 1. The difference between field strengths of two probes
had been measured in such a way as to not cause theeddy-curre~t e~f~3f and the magnetic after-effec t. in.
the pole p~eces

3. Magnetic field distributio n.
Fig.2 shows examples of the magnetic field distributions or the dependences of -the field difference AB
DesignFig.lon the radial distance r at a field strength 7.5
kG,

and at an azimuth 9=1 in the pole pieces

SCOFl (a)and at an ~yimuth.
pieces B-constl (b)Curvature :-adi us ·Deflection. angle-Gap--tni·c:It_n--ess;

8=3 in the pole

4. Absolute magnetic field inhomogene ity.
The absolute magnetic field inhomogene itiesor the absolute values of the magnetic field
inhomogene ities within a beam width w were
derived from the magnetic.f ield distributio ns
as shown in fig.2. Fig.3 gives the dependencesof the absolute field

inhomogeneityi~BI/B

onthe field strength B and on the azimuth e. The
dependences were measured at two azimuths 6=1

and 3 in the pole pieces S,COFl and SCOF2 in themagnets with and without the homogenizer s andalso in the pole pieces B-constl and B-const2

in the magnet with the homogenize rs. They were
estimated within two beam widths w=W and 2W.
The properties of the absolute field inhomogeneit ies are summarized as follows:l) They are independent of the field strengthand the azimuth.
2) They do not depend on the pole-piece iron.
3) They are independent of the field-settin gprocedure of the magnetizati on-demagne tization.

4) They have not been influenced by the existence of the homogenize rs.
5) Their values in the central part are smaller

than those in the edge parts of the pole

Table 1pieces.

-303-

Measure:::;ent

SCOFkSCOFhwith H
without Hw LW

B-const J, with H
SCQF2, withHSCOF2, withoutH
B-const2, with H
10~9=1

vm""'mI><lJT10""'•.·: .

v •"..... ..,. ..

v"' v \.
a•1Tav~ ••usFig.29=36) They depend on the beam width.
v •J, •7) They are related to the pole-piece profile. Thoseav
in the B-constant pole pieces are better than thosein the sharp-cornered pole pieces. In the B-constant pole pieces, they are limited by the effective pole-piece width.
5. Energy-resolution reduction due to the magnetic
field 1nhomogene1ty.
A formula for the calculation of the energy-resolution reduction due to the magnetic field inhomogene0510
was derived for a uniform field magngH-:: analyzer-BikG). The ability of·the magnetic analyzer to separate
particles with different momenta is represented by theFig.3
dispersion D defined as
D=S/R·p/AP(2)
where nP is the change in a moment p that produces a displacement S from the central ray as measured at the image in the magnetic analyzer with the radius of curvature R. The mechanical aberration-limited energy resolution AE/E(inhomogeneity) is, as a typical example, examined in a following
do~ble focusing uniform field ~nalyzer;. deflection angle=90°, object and ~age dista~ces=2 in theun1t of R, and entrance and ex1t rotat1·ng angles=26. 5 • The energy-resolut1on reduct1on due to themagnetic field inhomogeneity is expressed with the absolute magnetic field inhomogeneity I4BI/B
from eq.2
AE/E(inhomogeneity)=2 ap/p=2S/RD=l.6!ABI/B(3)
The whole calculations have been carried out by assuming the sharp-cornered fringing field.
The results in fig.3 are summarized in the column IABI/B in table 2. Table 2 presents themechanical aberration-limited energy resolution AE/E(inhomogeneity) deduced
from the absolutefield inhomogeneity !nBI/B ac~ding to eq.3. They were estimated within two beam widths w=W and 2W
in all pole pieces in the magnets with and without the homogenizers. It will be expected that thedata in tables 1 and 2 are useful! in the practical design of the magnetic analyzer because of thelack of this kind of the measurements.
The author wishes -to thank Pro-f:-=."'I'-'.""'K;,um:::::-a,-;b-e"" of Kyushu University'- Prof. K. Yagi of University ofTsukuba and late Dr.K.Matsuda of Institute of Physical and Chemical Research for._ their variablesuggestions and discussions throughout the work. Heexpresses his thank to Prof.J.Kokame of Institute for
Beal!ll
Nuclear Study, Prof.T.Ishimatsu of Tohoku University,--'.&IE.
WidthPieceCondition
Bomo9eDr.S.Motonaga of Institute of Physical and Chemical
ni:te:-s
Research and Prof.H.Matsuda of Osaka University for
vi.th
SCOF~
their supports th~oughout th~ work.
References.
SCOFl
l..l.7x:lo- 4
vitbout
1) Y.Okuma, K.Yagi, I.Kumabe and K.Matsuda, Nucl.Instr.
B-collstl
witb
l..G5xlo
Meth. 102(1972)317.
2) the same;-as the above; Nucl. Instr.Meth. 160 (1979) 29.
withSCOF2
l.SSz:l.O
2.48z.l.O3) the same as the above; Nucl.Instr.Meth. 190(1981)225.
without
SCOF2
2.88-x-lo- 44) the same as the above; presented in the I8fh INTERMAG,Boston (1980) and IEEE Tran. MAGNETICS MAG-17 (1981)
1234.
5) Y.Okuma, presented in the MT-7,Karlsruhe (1981) andwith
SC:OFl2.30xl0
3.68xl0
IEEE Tran. MAGNETICS ~AG-17 (1981) 1907, and Pros. ofvithollt
SCOFl
3'rd Sym?. on Accelerator Science and Technology at
RCNP, Osaka (1980) p.l79.
B-constl
5.L8xL0- 4
withl.24xLG
6) Y.Okuma, presented in the Conf. on Charged Particle
4.l4xl0- 4
2.7Lx.Lo"" 4vitb
SCOF2
Optics,. Gie~en (1980).
7)
H.A.Enge, Nucl.Instr.Meth. 28(1964)119.
SCOF2
4.16xLo- 4
2.60xl.O
8) K.Yagi, Nucl.Instr.Meth. 39Tl965)88.

...

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4.06xl0-4

Table 2-304-..