JAERI-Conf 95-021

PB26
A RESONANT EXTRACI'ION TECHNIQUE FOR A COMPACTSYNCHROTRON; ASYMMETRIC DRIVING OF TWO SEXTUPOLE MAGNETS
Kooji MATSUDA and Kazuo HIRAMOTO
Hitachi Research Laboratory, Hitachi, Ltd. 7-2-1 Omika-cho, Hitachi-shi, Ibaraki-ken, 316, Japan
Abstract
A technique for third-order resonant extraction for acompact synchrotron is presented. Two sextupole
magnets are asymmetrically drived for two purposes, oneof which is to excite the third-order resonance and theother is to correct for chromaticity. The positions of these
two sextupole magnets are decided so that the outgoingbranches of the separatrices at the deflector aresuperimposed for all momenta. This arrangement iseffective to realize a compact synchrotron, which needs
less space to install magnets, and has a high efficiency of
extraction and good quality of the extracted beam.

1. Introduction
Resonant extraction methods have been widely applied
not only to physics experiments but also to medical
applications [1 ,2]. The resonant extraction utilizes anonlinear resonance of the betatron oscillations. A
nonlinear magnetic field brings about the resonance andmakes the separatrix, which is defined as the boundary
between stable and unstable betatron oscillations in thephase space. The separatrix size decreases as the deviationof the tune from the resonant condition becomes smaller.
As the tune depends on the particle momentum through afinite chromaticity, the separatrix size also depends on it.
The dependency of the separatrix size on the particle
momentum causes a change in the orbit gradient of theextracted beam at the deflector position. Any change ofbeam characteristics can lower the extraction efficiency,
which is sometimes disadvantageous for the beam user.
Usually, it is necessary to control the chromaticity during
the extraction procedure.
When a synchrotron utilizes the third-order resonant
extraction, two sets of sextupole magnets have been
installed, one of which act as the corrector of thechromaticity, while the other act as the exciter of thethird-order resonance. Two sextupole magnets must be
installed at a symmetrical position in the synchrotron as
the corrector of the chromaticity and drived of same
strengh so as to cancel the contribution to the third-orderresonance each other. As a result, at least three sextupole
magnets, one exciter and two correctors, are needed.
Frequently, four sextupole magnets, two exciters and twocorrectors, are installed to control the chromaticity andresonance independently.
In order to realize a high quality beam or flexible
control, synchrotrons such as for accelerator physics orhigh energy physics must have many components. These
synchrotrons become large in order to get enough space
for the components and operations may become rather
complicated. The size of the synchrotron should be madeas small as possible to reduce the cost, including both themachine construction and the civil engineering aspects[3]. It is necessary to develop a method which realizes ahigh quality beam with fewer components.
This paper proposes a technique for third-order resonant
extraction with the chromaticity correction for a compact
synchrotron. Two sextupole magnets for the exciters of
the third-order resonance serve simultaneously as thecorrectors of the chromaticity. In order to reduce theextracted beam emittance, the positions of these two
sextupole magnets are decided so that the outgoingbranches of the separatrices at the deflector aresuperimposed for all momenta. By combining the above
technique with the extraction scheme under a constantseparatrix [4], it is expected that a low emittance beam
can be easily extracted with a high efficiency.
2. Third-Order Resonant Extraction with the
Chromaticity Correction
At the third-order resonance, the stable areas aretriangles and the separatrices are straight lines (Fig. I). We
consider the third-order resonance driven by a single
sextupole of normalized strength Sn given by:f :;sn = ~~-1 s with s =

dl

(I)

where Bp is the magnetic rigidity of the beam and ~ isthe horizontal betatron amplitude function at the
sextupole magnet. The integral of the second horizontal
derivative of the vertical component of the magnetic field,
B", is evaluated over the length of the sextupole magnet.

Fig.l Separatrices of the third-order resonance in thenormalized phase space. Dotted lines show the separatrix
for an off momentum particle.

More generally, we have to consider the third harmonicof the sextupole magnetic field distribution in the
betatron phase around the machine. When there are m
sextupoles of strength si and normalized strength sni atphase <j>i, the resulting effect of these sextupoles has astrength S and a phase <1> with:-321-

JAERI-Conf 95-021.ts.. exp(3j41i) = tf31"S exp(3j41i)

S exp(3j41) = l•l

i•l

1

.

(2)

In this equation, phases are taken from the consideredpoinL In general, according to the difference of the phaseadvance from each sextupole 41;. the resulting phase 41
changes by each nonnalized sextupole strength S81• Somesynchrotrons utilize this character to rotate the separatrix
and change the orbit gradient of the extracted beam [5].
When the phase difference between each sex.tupole isnn/3, the resulting phase does not change for strength ofeach sextupole and then the separatrix does not rotate.
In the following study, we restrict ourselves to the caseof a ring which has superperiods of even number and two
sextupole magnets installed on opposite sides of the ring.
The phase difference between the two sextupole magnets
is nearly nn/3, because one turn phase advance is 2mt/3,
near the resonance condition. As the Twiss parameters atthese sextupole positions are equal, the separatrix size £and the chromaticity~ {defmed as {.1. v/V}/{.1-p/p)) become:

E=Ms~5sJz

parallel to the Y axis in the nonnalized phase plane at theposition of these sextupoles. Two other branches incline
from Y axis for 2n/3 and 4n/3 radians. So theinclinations of these three branches are tan{n/2),
tan(7n/6), and tan(lln/6).
The separatrix rotates with the phase advance fromthese sextupole magnets. So the inclinations of the
branches are tan{n/2-41 ), tan{7n/6"41 ), tan{lln/6-41 ), where
41 is the phase advance from one of the sextupole magnets
to the considered point.
Generally, a synchrotron has finite values for thedispersion function and its derivative. So the location of
the separatrix in the phase plane varies with the particle
momentum. When the chromaticity is corrected and theseparatrices for all momenta are congt'\lent, the condition
to superimpose the separatrices around the deflector for all. momenta is that one of the branches of the separatrix isparallel to the shift of the separatrix with the momentum
through the dispersion function (Fig.2). The equation of
the condition is:(3)

11' tan(0-41)-a,.. =~~=~nat+ 4ivf311(S1 + SJ
(4)
where sl and s2. are the strengths of these two sextupole
magnets, f3 and 11 are the betatron and dispersion functions(defmed as .1.x.1.p/p) at these sextupole magnets. v andare the tune and its deviation from the resonant condition.
~aat is the part of the chromaticity without the sextupole
magnets contribution, natural chromaticity. In Eq. (3),the plus (minus) sign is taken when the number n of thephase difference between the two sextupole magnets,
nn/3, is even (odd). When this sign is minus, theseparatrix size and the chromaticity can be tuned
independently by selecting sl and s2. carefully.
The difference between the strengths of these two
sextupole magnets becomes large when a rather large
chromaticity change is needed. Then the strength of oneof the sextupole magnets becomes strong. In this case,trajectories of the extracted particles curve in the phasespace due to the higher order term of the nonlinear
resonance. Trajectories curved like this are not suited tothe extraction. So this technique is not proper for a ringof large chromaticity.

o

with

a=

+'It
4 +1

(5)

where 41 is the .phase advance from one of the
sextupoles to .the deflector, a and f3 are the Twiss
parameters at the deflector, 11 and 11' are the dispersion
function and its derivative at the deflector. In the above
equation, m is an integer which satisfies the condition
41-n/2<0<41+7t/2. If the position of the deflector is fixed,
we can superimpose the separatrix by selecting theposition of two sextupole magnets according to Eq. (5).
yshift of thecenter orbit

outgoing branchof the separatrix

X

3. Superimposing the Separatrices at. the Deflector
In order to reduce the extracted beam emittance, i.e., thephase space area of the extracted particles at the deflector
position, the outgoing branches of the separatrices aresuperimposed. This condition is derived for the ring offinite chromaticity [6]. In this study, the chromaticity iscorrected by the technique shown above.
The normalized horizontal phase plane (X,Y) is definedas:x
ax+ f3x'
X= .jJf and Y:.jJf
where x and x' are the real horizontal position and orbit
gradient respectively, and f3 and a are the Twiss
parameters at the considered point.
When the phase difference between the two sextupole
magnets is nn/3, one of the branches of the separatrix isdeflector
Fig.2 Separatrices with the chromaticity corrected.
The outgoing branches of the separatrices are superimposed
for all momenta.

4. Application to a Compact SynchrotronThis technique has been studied for a lattice of acompact synchrotron dedicated to medical use [7]. In orderto reduce the size of the synchrotron and simplify its
control, this synchrotron employs a combined function
lattice. The circumference is 22.9m. The bending magnets
have n-indices for focusing and defocusing the beam. The
horizontal tune is 1.75 and the vertical tune is 0.85. In
the extraction period, two trim quadrupole magnets
change the horizontal tune to nearly 1.67. The third-order-322-

JAERI-Conf 95-021

resonant extraction is raised by a transverse perturbationof the radio frequency and the separatrix is kept constant
The effects of the present technique are analyzed throughcomputer simulations.
The two sextupole magnets are treated as thin lenses.
The momentum spread is assumed as dp/p=±0.2%.
Emittance of the beam is assumed as 247tmm·mrad. The
positions of the two sextupole magnets are selectedaccording to Eq. (5). Horizontal tune is selected as 1.67,
nearly 107t/3, in order to get proper tum separation; here
the tum separation is defined as an increment of thebetatron amplitude at the deflector position. Twiss
parameters at the deflector and sextupole magnets areshown in Table 1.
Table. 1 Twiss parameters-1.55
1.55 m0.39

1.48 m-0.32

The strengths of the two sextupole magnets are·0.95(m-1 and 12.05(m·1. These are selected so that thechromaticity of the synchrotron vanishe and the separatrix
size becomes 247tmm·mrad.
Results of the simulation show that the separatrices for
all momenta are almost congruent and the trajectories ofall extracted particles are superimposed (Fig.3).
20
15
10

I

A technique for third-order resonant extractio~ wasgiven. Two sextupole magnets were drived
asymmetrically, and served the two purposes, one ofwhich is to excite the third-order resonance and the otheris to correct for chromaticity. The positions of these two
sextupole magnets are decided so that the outgoingbranches of the separatrices at the deflector aresuperimposed for all momenta. This technique wasapplied in a compact synchrotron. Computer simulations
showed that the separatrices for all momenta were almost
congruent and that the trajectories of all extracted particles
were superimposed. High extraction efficiency and low
extracted beam emittance would be expected.
Acknowledgement

1.42m
0.83

2.96m

5. Conclusion

50

The authors would like to express their thanks to
Profs. Makoto Inoue, Akira Noda and Dr. Yoshihisa!washita of Kyoto University for their invaluable
discussions.
References[1] A. Itano et al., Proc. 7th Symp. on Accel. Sci. andTechnol., Osaka, Japan, 1989, p. 42.
[2] M. Yoshizawa et al., Proc. 1991 Part. Accel. Conf.
San Francisco, USA, 1991, p. 875.
[3] K. Endo et al., Proc. EPAC, Nice, 1990, p.l784.
[4] K. Hiramoto and M. Nishi, Nuclear Instruments andMethods in Physics Research A322(1992) p. 154.
[5] M. Takanaka and A. Itano, Proc. 5th Symp. on Accel.
Sci. and Technol., Tokyo, Japan, 1984, p. 271.
[6] W. H~dt, CERN, PS/DL/LEAR Note 81-6.
[7] K. Hiramoto et al., Bull. Inst. Chern. Res., Kyoto
Univ. Vol. 73, No.I (1995) p. 11.

-~

-5
-10
-15
-20

-40

-20

20

0

x (mm)
Fig.3 Results of the computer simulations.
Emittances are 247tmm·mrad for all particles of Ll.p/p=±0.2%

Values of the tum separation at the deflector are listed inTable 2. The particles are extracted by increasing theamplitude of the betatron oscillation through theperturbation. Since the separatrices are constant during theextraction, extracted beam parameters such as the tum
separations and orbit gradients, do not change.
Table. 2 Tum separation at 30mm
tlp/p (%)
tum separation
(mm)

-0.2

0

+{).2

9.9

8.3

6.6

-323-