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

Tune Change due to Ripple of Magnetic Field in a Synchrot ron of Combine dFunction Lattice
Kazuo HIRAMOTO and Akira NODA *Hitachi Research Laboratory, Hitachi,Ltd. , 7-2-1 Omika-cho,Hitachi-shi, Ibaraki-ken, 313, Japan*Accelerator Research Facility, Institute for Chemical Research,
Kyoto University, Gokano-sho, Uji-shi, 606, Japan

Abstract

magnetic fields and improvemen t of beam spill ripple
may be easier than synchrotron s of the separated function lattice. In the present paper, we formulate the tunechange due to the ripple of the dipole and quadrupole
magnetic fields in combined function lattice. Furthermore, according to the obtained formula, we show thatthe tune change due to the above ripple of the magnetic_fields can be prevented by a sexupole magnetic field.

Tune change due to ripple of magnetic field in a synchrotron of combined function lattice is formulated. It
is shown that the tune change is proportional to theripple of the dipole magnetic field which is assumed tobe the same as that of quadrupole magnetic field. The
reductions of the betatron and dispersion functions are
effective for decreasing the ripple of the tune. It is also
shown that the tune change due to the ripple of thedipole magnetic field can be vanished by the sextupole
magnetic field chosen adequately, the strength of whichis equal to that for the chromaticity correction.

2 Tune Change by Ripple of Magnetic Field in
Combined Function Lattice

2.1 Formulation of Tune Change
1 Introductio nHigh energy ion beam has been applied to cancer
treatment because it shows rather sharp Bragg peak inhuman tissue. In ion synchrotron s for medical use, thetreatment has been performed by the slowly extracted
beam utilizing growth of the amplitude of the betatronoscillations caused by nonlinear resonance[!]. The nonlinear resonance of the betatron oscillations are generated when the amplitude of the betatron oscillations
reaches the separatrix of the nonlinear resonance, which
is determined by the tune, that is, the number of thebetatron oscillatioqs in a synchrotron , and the strengthof the nonlinear magnetic field such as sextupole or octupole magnetic field. If the separatrix of the nonlinear resonance varies due to the ripple of the magnetic
field, the beam extraction becomes intermittent . In thetreatment by the beam scanning, the control to irradiate
the target tissue homogeneously becomes complicated .
Then, research and development to reduce the current
ripple of the magnets have been performed to preventthe intermittent extraction, that is, beam spill ripple[3].
It has been considered that the ripple of the quadrupole
magnetic field is the main cause of the beam spill ripple. However,it has been also shown that the ripple ofthe dipole magnetic field causes the beam spill ripple
and the reduction of the dipole magnetic field further
improves the time structure of the extracted beam[3].
On the other hand, we have proposed a synchrotronof combinefd function lattice for simplification of the operation for medical use[2]. In the lattice of the combinedfunction, usually, the dipole and principal quadrupole
magnetic fields are generated by common electric currents in combined function bending magnets. Then, it
is expected that the ripple of the magnetic fields arealso common for the dipole and principal quadrupole

Figure 1 shows the coordinate system in the presentpaper. The distance along the reference orbit is denoted
as s. x is the horizontal deviation from the reference orbit and p( s), the curvature radius. Since the resonanat
extraction is usuallly performed in the horizontal plane,
the analysis in the present paper is focused on the horizontal tune change.

Fig. 1

Coordinate System

The equation for the betatron oscillations is designated as(1)
where By is the magnetic field for the y direction. Ifhigher order terms can be neglected, By is written as

Bs

Bd-B x- - x 2q2'(2)where Ed, Bq and B 8 are the dipole, quadrupole andsextupole magnetic fields,respectively. These magnetic
fields are written as-496-Bdo(l- 8d),(3)Bq0 (1 - 8q),

(4)

Bso(l - Ds),

(5)

The I lth Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

where 8d, 8q and 88 are the ripple components, that is,
temporal changing variables. Since the ripple frequencyof the magentic field is usually lower than several kHz
and the order of the revolution frequency of the beam insynchrotrons is about megahertz, the ripple componentof the magnetic field can be treated as constant in Eq.(2).
Then, Eq.(1) is written asd2xp+xe8d)- B o(1- 8 )xds2- -p2- = --[Bdo(1pq. q- B;o (1- 88 )x 2 ](1 + ~) 2 •(6)

1
In the following, the relationship !!. = - B
is applied.
PdOP
By expanding the above equation up to the second orderterms for x and expressing BBqo and BB.o as B 40 anddOPdOP
B 60 , respectively, Eq.(6) is written as follows,

2.2 Method of Reduction of Tune ChangeEq.(ll) shows that the tune change can be decreasedby the reduction of the ripple of the dipole magnetic
field. Eq.(ll) also shows that the tune change occurs
due to the sextupole field and closed orbit displacement,
the latter of which is caused by the ripple of the dipole
magnetic field. It is seen that the small sextupole magnetic field, the small betatron and dispersion fucntionsin the bending magnet of combined function lattice ·aregenerally effective for the reduction of the tune change.
On the other hand, according to Eq.(ll),. the tunechange due to the ripple of the dipole magnetic field canbe vanished by using a sextupole magnetic field. The
ripple of the dipole magnetic field causes the temporalvariation of the closed orbit, which makes the focusing or
defocusing strength change at sextupole magnetic field.
Then, by choosing adequate sextupole magnetic field,

d2 x[1-28d -B (1 _ 8 )]2 [1-28d240q +X
3d s2 + Xpp

_ 2B4 o(1- 8q) _ B6o(1- 8s)] _ 8d = O.
p2change of closed
orbit by the rippleof dipole magnetic( 7)pHere, we insert x = x + 'f}8d into the above equation andobtain the following equations:change of focusing
strength by sextupole
magnetic field.Jl. .
compensation ofthe tune change

Ocl 11 B6013

(8)

(9)

From Eq.(8), usual dispersion function is obtained.
When obtaining Eq.(9), all the terms higher than thefirst order were neglected. When the equation for the betatron oscillations is expressed as x" +(k+b.k)x = 0, thetune shift b.v due to b.k is written as b.v = 4~ b.kf3ds
where f3 is the betatron function of the unperturbed motion. Then, the tune shift b.v is expressed as follows:

sextupolemagnetJb.v=(10)

From the above equation, it is seen that the rippleof the sextupole magnetic field does not cause the tunechange in the first order approximation. The effect of
the ripple of the dipole magentic field decreases with anincrease of the curvature radius p. Assuming that theripple of the quadrupole magnetic field is equal to thatof the dipole magnetic field including phase, the aboveequation is written as follows:(11)

Fig. 2 Scheme of Compensation of the Tune
Changewe can prevent the tune change due to the ripple of thedipole magnetic field. This method is depicted schematically in Fig.2. The needed strength of the sextupole
field can be determined by calculation so that the .righthand side of Eq.(ll) may be vanished, because the distribution of the magnetic field along the reference orbit
can be obtained. This strength of the sextupole field isequal to that for the chromaticity correction. As a reference, the tune change b.v due to the momentumchange~ is expressed as follows:~Pb.v = - 47r

j (- - -p3 + - - +2p2

2'f/

4B4o'f/pB6o'f/- B4o)f3ds. (12)This equation can be obtained by the procedure similar
to the above one.
In the case of the third order resonant extraction,
the separatrix of the resonance is generated by sextupole
magnetic field. Then, the sextupole magnets for theabove correction of the tune change should be operated-497-The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

so that the separatrix for the nonlinear resonance may
not be varied. This can be accomplished by choosing thepositions, po'larities and strengths of the sextupole magnets dedicated for the tune correction. When the tune isapproximately 2 ~+ 1 for the third order resonance, theseparatrix size is determined by the difference of strengthof the setupole magnetic field if the sextupole magnets
are installed at the positions with 180 degrees rotational
symmetry. Then, if these sextupole magnets are excitedwith the same strength and polarity, the tune change
can be prevented without changing the separatrix. Furthermore, the prevention of the tune change and the adequate generation of the separatrix can be done by two
sextupole magnets.
Figure 3 shows an example of the combined function

3 Conclusion
Tune change due to ripple of magnetic field in synchrotron of combined function lattice is formulated. Thetune change is proportional with the ripple of the dipole
magnetic field assuming that the ripples of the dipoleand quadrupole magnetic field are the same both in amplitude and phase. The reductions of the betatron anddispersion functions are effective for decreasing the ripple of the tune. It was shown that the tune change dueto the ripple of the dipole magnetic field can be compensated by the sextupole magnetic field chosen adequately,
the strength of which is equal to that for the correctionof the tune change due to the momentum deviation.

References[1] K. Noda, et al., Nucl. Instrm. and Meth. A374 (1996)
269[2] A. Noda, et al., this proceedings[3] M. Kumada, et al., Proc. 5th EPAC, Barcelona,
1996,p2355[4] K. Matsuda and K. Hiramoto, Proc. lOth Symp. onAccel. Sci. and Technol.,Ibaraki.Jap an, 1995, p321

BMExtraction
BM : Combined Function Bending Magnet
ESD:Electrostatic Deflector
ESI :Electrostatic Inflector
SM1,SM2 : Septum Magnets
SXresl,SXres2 : Sextupole Magnets
Fig.3 A Combined Function Synchrotron
Using Two Sextupole Magnets
synchrotron, in which the sextupole magnets SXresland
SXres2 for the resonance excitation are installed in theabove configuration[2]. The difference of the strengthof SXresl and SXres2 determines the separatrix size.
Then, by the resonance exciter, SXresl' SXres2, thecorrection of the tune change can be done, if the fieldstrengths of SXresland SXres2 are determined so thatthey may vanishes the right hand side of Eq.(ll) withkeeping the difference of the strengths of SXresl and
SXres 2 constant for the separatrix generation[4].

-498-