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

Beam Optics and Dynamics of the JHF 50 Ge V Main Ring
Shinji MACHIDA, Yoshiharu MORI, Yoshihiro ISHI*, Shinji SHIBUYA**Accelerator Laboratory, High Energy Accelerator Research Organization
3-2-1 Midori-cho, Tanashi, Tokyo 188-0002, JAPAN(*also Mitsubishi Electric Co., Ltd., **also Sumitomo Heavy Industries, Ltd.)

Abstract
The 50 Ge V main ring is characterized by high intensity andlow beam loss. Several considerations have been taken intoaccount in the lattice design to achieve those two primary goals.
Although there are strict boundary conditions, such as the total circumference, which is limited because of the availablesite enclosed by the KEKB ring, a well-optimized lattice isobtained.

1 Overview of the Lattice
The dispersion function is adjusted so that the momentum compaction factor (MCF) becomes negative or, in otherwords, the transition energy becomes imaginary. Althoughmany synchrotrons now under operation can conquer problems at the transition crossing, the best way to minimize beamloss during acceleration is not to have a transition energy.
In order to obtain a negative MCF, three FODO cells,
starting from a defocusing quadrupole, are taken as the minimum unit of an arc. We refer to the unit as "module". In amodule, four bending magnets are put in each half cell, except for two half cells on both sides of a focusing quadrupole
magnet at the center. Due to those arrangements of the lattice,
the dispersion function is modulated in a module and has apeak at the center. In the half cells at both ends, where thebending magnets are located, the dispersion function becomes
negative. Therefore, the MCF, which can be obtained as theintegral of the disper~tion function through the finite curvature, becomes negative [1]. The beta functions in both thehorizontal and vertical directions are not very much perturbed.
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functions in the arc. In that way, a negative MCF is preserved
to be somewhat independent of the configuration of the insertion.
An abort system is essential for a high intensity proton
synchrotron. There are two abort channels: one for beams atthe injection energy, and the other for higher energy beams,of around 30 to 50 GeV. For beams at the injection energy,
about one third of the insertion space is dedicated to the fastkickers for the extraction of beams. Although the kicker
strength is not sufficient to extract higher energy beams, theaperture of that abort channel is large enough to extract a beamfilled in the vacuum chamber. For aborting higher energy
beams, the fast extraction channel is used. Since the apertureof the fast extraction channel is designed based on the adiabatically damped beams, the lowest energy that the fast extraction channel can handle is limited. Still, the core part ofthe beam should be dumped into the fast extraction channel if
it is necessary to abort rather low energy beams whose size isbigger lthan the aperture. The extracted beam eventually hits atarget for neutrino production. The fast extraction channel
should be ready whenever the machine is on.
Although the lattice is fourfold symmetry in shape, it
has a higher symmetry as far as the phase advance is concerned. Namely, the phase advance of the insertion is 2n: forthe horizontal plane and 1t for the vertical. In such a way, insertion becomes transparent for a beam and the periodicity
becomes higher. For example, the total phase advances of thesextupole resonance, 3v =hand v +2v =h (his an integer), in
the insertion are 61t and x41t, respectively. Since the sextupole
magnets are installed only in the arc, insertion is transparent
for those resonances, and the periodicity becomes 24, whichis the number of modules.
Since space charge effects are another concern, thephase advance of the lattice is carefully chosen. Although thebeam emittance is quite large, 54lrt mm-mrad at the injection(220 n: mm-mrad, normalized), the tune spread, including
image currents effects, may reach to slightly less than -0.1. A
resonance free region in the tune space should be prepared,
which can accommodate that magnitude of the tune spread.
In addition, resonances excited by the space charge force, itself, must be taken into account. According to a recent study,a space charge induced resonance significantly appears whenit couples to a local phase advance of 90 degrees [2].

2 Stability of the Linear Lattice
Figure 1: Lattice functions of "module" in the arc.
The lattice functions of the module are depicted in Fig. 1. Thearc of the ring consists of six modules.
Between the arcs, four insertions are prepared: one foracceleration, one for fast extraction, one for slow extraction,and the last one for beam abort. The ring, therefore, has fourfold symmetry. Insertion has to be sufficiently long.
A long straight insertion consists of four FODO cells.
Since the horizontal phase advance through the insertion ischosen to be 2n: and the vertical is 1t, the beta and alpha functions in both directions and the dispersion function are allmatched at both ends of the insertion. In other words, the insertion acts like a phase shifter without perturbing the lattice

In order to assure stable operation and a search for good
operating points, the stability of the linear lattice has beeninvestigated. Since the upper limit is due to the hardware design, the strengths of the quadrupole magnets are adjusted so
as to explore transverse tune space. The maximum values ofthe lattice functions in the arc and the insertion are moderateover a wide range of tune space.
The lattice functions are, then, examined as a functionof the MCF. When the operating point is fixed at (21.85, 15.40),
the MCF can be varied from -2.3 X 10•3 to 2.5 X 10-3. In termsof the transition gamma, it corresponds to 20i to 20, respectively. As shown in Fig. 2, the horizontal and vertical beta
functions are similar, and are around 31 m in the arc when the-502-

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0-1-2-3momentum compaction factor (MCF) (x 10 "3 )

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-2

-1

momentum compaction factor (MCF) (x 10"3)

Figure 2: Maximum beta functions in the arc (left), and in theinsertion (right) when the MCF is varied.
MCF is around zero. Those in the insertion also have a similarvalue of 38 m in both directions when the MCF is almostzero.
If one chooses a small magnitude for the MCF, thehigher order effects of the MCF or the coefficient of the quadratic term of Llp/p may become significant. The particle
motion may be unstable if those higher terms are large, anddifferent momentum particles see different rf bucket and tran-.0.056-0.057-0.059-0.06/v-0.0011v/-0.0012----- :---· -----""""~~ -0.0013"'

i-0.0014-0.0015-0.0016-0.01~-0.0050.005

momentum spread

O.Ql

-0.01

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momentum spread

Figure 3: Slippage factor as a function of momentum spreadat the injection energy (left) and the extraction energy (right).
sition energies. In order to examine those effects, often called
the effects of a 1, a 2 , and so on, the slippage factor TJ as afunction of Llp/p was calculated (Fig. 3). The momentum dependence within the range of +-1 % is sufficiently small and,even at the top energy, the variation of the slippage factor issmall.

tice components is determined by the sum of three parts: betatron amplitude, momentum spread with the dispersion function, and COD. Although the core size of the beam can be thesquare root of the quadratic sum of the frrst two, we define thefull beam size as simply the arithmetic sum of those three.
The maximum betatron amplitude (or emittance, asdiscussed before) is 54 1t mm-mrad. The COD can be lessthan 1 mm at the injection where the beam size is maximum.
As for the momentum spread, +-1 % is assumed for injection
with barrier buckets in future. As a result, the beam size becomes maximum at the middle of the module in the horizontal direction. In the vertical, one of the two defocusing magnets in insertion has the maximum beam size. In order to accommodate those maximum beam sizes, the beam pipe radius was determined to be 120 mm in all of the quadrupole
magnets and 120 mm (hor.) x 100 mm (ver. )in all of the bending magnets.

4 Dynamic Aperture
According to the normal injection scheme, the firstbatch from the booster stays at an injection energy for 120
ms. The sextupole magnets for chromaticity correction will
be a major source to limit the dynamic aperture. Other
nonlinearities, such as themultipole components in the normal conducting lattice magnets, are negligible.
The strength of the sextupole magnets tends to be highcompared with a normal FODO lattice for two reasons. Oneis that the number of sextupole magnets is less. The other isthat the magnitude of horizontal and vertical beta functions issimilar at the position of the focusing sextupole, even though
the dispersion function is high there.
The dynamic apertures were estimated by particle tracking codes: SAD [3] and Simpsons [4]. It is defined in the following way in those codes. Several particles are launched withdifferent amplitudes and tracked for 2,000 turns. Initially, eachparticle has equal horizontal and vertical amplitudes. We regard the beam as having been lost when its amplitude becomes
more than 1 m. After tracking for 2,000 turns, the maximum
initial amplitude among the surviving particles is defined as
the dynamic aperture.
First of all, the dynamic apertures were surveyed overa rather wide transverse tune range using the code SAD. At
interval of 0.05, tune is varied in 2D space, totally at 60 x 60
points. Since the momentum spread is assumed to be zero, the3 Emittance and AcceptanceThe beam emittance from the booster is expected to beless than 54 rc mm-mrad (the normalized emittance is 220 rcmm-mrad). The main ring acceptance has been determined to
be 54 rc mm-mrad also. It seems that no margin is reserved foradditional emittance growth in the main ring.
At the moment, however, we assume that the margin isalready included at booster injection. Since we do not inject abeam in a full acceptance of the booster, there may be only atail of the beam near to the edge of the booster acceptance. Ifthe emittance growth mainly occurs at injection, although thebeam into the main ring may have a full emittance of 54 rcmm-mrad, the core of the beam is still smaller. Thus, thequestion is what fraction of the beam is contained in the beamedge, which will be scraped by the main ring acceptance. We
have no accurate idea. If it turns out not to be negligible, wewill consider using a scraper and collector system either inthe transport line between the booster and the main ring or in
the main ring. An alternative way is to enlarge the acceptance.
In that case the hardware design should be revised.
The necessary aperture of the magnets and other lat-Figure 4: Dynamic aperture as a function ofthe transverse
two dimensional tune. In the case of full number of sextupoles(left) and one missing one (right). The darkest grey means
zero aperture and the lightest grey corresponds to four times
as much as the physical aperture in size. A chromaticity correction sextupole is excited to make the chromaticity zero.
tracking is done in 4D phase space. The strength of thesextupole magnets is adjusted to make the chromaticity zero.
As shown in Fig. 4 (left), a fairly large dynamic aper--503-

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

ture of around 200 1t mm-mrad is obtained above and below
the third integer resonance: v +2v =56. It is about twice asmuch as the size of the physic'al aperture and therefore fourtimes that of the acceptance.
To the first -order, the periodicity of the sextupole is 24
because the phase advance in the insertion is 27t and 1t for thehorizontal and vertical planes, respectively. The insertion istransparent as far as the sextupole phase is concerned. The
resonance, v +2v =56, actually has harmonics of 48 after subtracting the offsdt in the four insertions, which is an integer
multiplied with a periodicity of 24. On the other hand, otherthird integer resonances having a periodicity of an integer multiple of the superperiodicity of 4 are relatively weak, or none
in principle, because of the higher periodicity.
In reality, the symmetry of the sextupole magnets may
be broken for various reasons. The phase advance in the insertion may, for example, differ from what we designed. Nevertheless, one expects that those perturbations are still small.
The other source which one cannot treat as a perturbation is a
missing sextupole at the injection section, we assume at themoment, where kicker and septa magnets are installed in placeof the sextupole magnet. If we fix the design to inject beams
in the middle of the arc, instead of taking them to the nearest
insertion where sufficient space is available for injection without affecting other lattice elements, we must remove one
sextupole and break periodicity.
The dynamic aperture in the same region as shown inFig. 4 (right) is estimated with one missing focusing sextupole.
Although the strengths of the sextupole magnets are similar
to make the chromaticity zero, the dynamic aperture is considerably reduced. It is at most the same as the physical aperture. Because.iof the symmetry breakage of the sextupoles, thethird integer resonances of all the harmonics appear. Still, thewide valley by v +2v =56 is quite noticeable.
Although we ~sume zero chromaticity until this point,
and looked at the dynamic aperture, one may ask if it is reallynecessary to have zero chromaticity for the main ring. In fact,
the chromaticity do~ not have to be zero, because it is operated under the "transition energy" all of the time. Moreover, afinite negative chromaticity is even preferable from the beam
instability point of view.
Using the code Simpsons, the dynamic apertures areobtained as a function of the sextupole strength. The tune is
fixed at (21.85, 15.40), which is given the lower-left comer ofthe previous figures. Th~ synchrotron oscillations are included
1000

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Figure 5: Dynamic apertures as a function of the sextupole
strength. The sextupole strength of 1 makes the chromaticity
zero. The left hand figure shows them in the high periodicity(24) lattice, and the right hand figure shows them in the one
missing sextupole lattice. Synchrotron oscillations at the injection energy are included.
in that calculation. Two initial momentum spread, 0 and 0.5%are compared. As shown in Fig. 5, the smaller is the strengthof the sextupole magnets, the larger is the dynamic aperture

obtained, as expected. Also, similar magnitude of the dynamic
aperture in a high periodicity lattice is retrieved in the latticewith one missing sextupole if the sextupole strength is decreased to 70-60 %. A particle with a momentum spread of
0.5 % has the same tendency as a 0 % spread but is somewhat
less sensitive to the strength.

5 Tune SliliJft
As another measure of the nonlinearity, we have lookedat the tune shift as a function of the momentum spread andbetatron oscillation amplitudes. The higher periodicity lattice
is examined and the operating tune is fixed at (21.85, 15.40).
The sextupole magnets are excited to make the chromaticity
zero.
Since there are only two families of sextupole magnets to correct the chromaticity, no negligible higher ordermomentum dependence of the tune is observed. It is around0.005 under nominal operation, where the maximum momentum spread is 0.5 %. If the number of sextupole families isincreased, the higher order dependence is more suppressed.
The defocusing sextupole magnets are divided into two families, and the vertical tune shift becomes much smaller. Weadjusted the strength of the three families of sextupole so as
to obtain the smallest tune shift in the momentum range of+0.1 %. Increasing the number of sextupole families means anincrease in the power supply units and more complex control.
The advantage which we gain due to more sextupole families
is marginal.
The transverse tune is a function of the betatron amplitude because of the higher order effects of the sextupole mag0.1

0.025 rr:::==r=:-:-~1

0.08

"'~

0.06

----------r:: ,'--------------·--------------~--~----------::--------··r-------------,', ..i':,:g:,' :~ 0.04 ---------·--·-r···-;7··--··-:··: ,,e O.Dl5 -----------·-·t··------------·L;'~-----~.... ":a o.ot ---------------:--------... ~---+----~ ,'':..) '~,0.005 ---~-~~-;-;'·1--50transverse amplitude (pi mm-mrad)

100

150

transverse amplitude (pi mm-mrad)

Figure 6: Amplitude dependence of the tune shift. With a fullcorrection of the chromaticity (left) and half correction (right).
nets. When the sextupole is fully excited, the tune shift at an
amplitude of 54 1t mm-mrad, which is the physical aperture,
becomes 0.02 in the horizontal and 0.03 in the vertical, asshown in Fig. 6 (left). The tune shift depends almost linearlyon the amplitude, obviously due to the leading term of theoctupole. The deviation from a linear relation at large amplitude, however, indicates that higher order effects than octupole
are also noticeable. When the strength of the sextupoles ishalved, the slope becomes almost one fourth. At the same time,higher order effects than octupole are reduced, as shown inFig. 6 (right).
References[1] N. I. Golubeva, A. I. Iliev, Yu. V. Senichev, Proceedingsof International Seminar on Intermediate Energy Physics,
1989, Moscow, USSR, pp.290-297.
[2] S. Machida, Nucl. Instr. and Meth., A 384, 316 (1997).
[3] K. Oide and H. Koiso, Phys. Rev., E49, 4474 (1994).
[4] S. Machida, Nucl. Instr. and Meth., A373, 309 (1996).

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