JAERI-Conf 95-021

3a1-2
BEAM DYNAMICS ISSUES IN
THE JAPANESE HADRON PROJECT CIRCULARACCELERATORSS. Machida and JHP synchrotron design group
National Laboratory for High Energy Physics (KEK)
1-1 Oho, Tsukuba-shi, Ibaraki, 305 Japan
andUniversity of Tokyo (INS)Study,Nuclearfor
Institute3-2-1 Midori-cho, Tanashi-shi, Tokyo, 188 Japan
Abstract
Beam dynamics issues in the JHP circular accelerators aredescribed. The booster ring, which is a 3 GeV proton
synchrotron, accelerates 5x 10 13 protons per pulse (ppp)
and the main ring, which is a 50 GeV machine, accelerates 4x 10 14 ppp. Problems associated with high intensity
beams are examined, especially space charge effects anddynamic aperture.

In this paper, we will pick up some of those beam dynamics issues in the JHP synchrotrons and describe our present
understanding. For those issues which are not studied yet,
we will show our future direction of the study. As for thelattice design itself, a separate paper describes the detail[3].

II. Space Charge Effects
One of figures which describes space charge effects in acircular machine is Laslett tune shift.

I. Introduction
The proposed accelerator complex of the Japanese
Hadron Project (JHP) [1] consists of a 3 GeV fast cycling
booster and a 50 GeV slow cycling main ring. In order toobtain high average beam current, which is 200 p,A in thebooster and 10 p,A in the main ring, with technically feasible repetition rate, the number of protons per pulse becomes quite large. That is, 5x 10 13 for the 3 GeV boosterand 4 x 1014 for the 50 Ge V main ring. In terms of circulating current, they are 7 A and 14 A, respectively [2].
Obviously, beam loading effects in the longitudinal direction would be one of main beam dynamics issues in bothmachines.
Beam dynamics issues associated with the large number of protons per pulse exists in both longitudinal andtransverse directions. One could easily imagine that spacecharge effects right after the injection until beams get accelerated up to sufficient energy may cause problems such as
beam emittance growth and beam loss. The machines havequite little margin for those because the machine apertureis limited by size of each lattice components which are essentially determined by the total cost of the rings. Even asmall fraction of beam loss, say 1%, is no more negligible
because it is almost equal to the entire beam loss of theKEK PS, for example, and radio activation of the facility
becomes serious problem.
In order to keep the space charge tune shift moderatemagnitude, the beam emittance is taken relatively large
from the beginning. At the same time, to squeeze thebeam size small, a rather strong focusing lattice is adopted.
Chromaticity correction by sextupoles, therefore, introduces strong nonlinearity and dynamic aperture becomes
one of primary concerns.

7r,B2.,3cy(l + ~)B'

where l::..vy is the vertical tune shift, rp is the classical proton radius, n 1 is the total number of protons, ,8 and 1 are
Lorentz factors, c., and fy are unnormalized horizontal andvertical emittance, and B is the bunching factor. Laslett
tune shift includes, not only direct electro-magnetic field
in a free space, but the field due to image charge on vacuum chambers and magnets, which is represented by theform factor F. In fact, at the injection energy of the 3 GeV
booster, namely 200 MeV, the tune shift mostly comes froma direct space charge field, but at that of the 50 GeV mainring, the field due to image charge accounts for not a small
fraction of the total tune shift. In any case, we choose lattice parameters such that Laslett tune shift should be lessthan -0.25.
Since our design ·of machine acceptance is 32011" · mm ·mrad (unnormalized) for the 3 GeV booster and 547r·mm·
mrad for the 50 Ge V main ring, incoherent space charge
limits are 5.6 x 10 13 for the 3 GeV booster and 4.7 x 10 14for the 50 GeV main ring. Coherent space charge limit for
the 50 Ge V main ring is 4.2 x 10 14 . Bunching factor isassumed to be 0.3 in aU cases.
In addition to the fact that the space charge force shifts
the transverse tune down, the space charge force itself creates resonances. Unless the beam distribution has a uniform shape, the space charge force has nonlinear dependence of its transverse coordinates. The Hamiltonian including space charge terms, therefore, becomes-407-H1 21 2

1 [ 1 {}By22= 2P:z: + 2Py + Bp 2! ox (x - y )JAERI-Conf 95-021
le-021 {)2 By 3
2+ 3! {)z2 (:x - 3:xy )·1a3 B11 4
224+ 4! {)z3 (:x - 6x y + y ) + ...1rpA

1le-03 -..c:J.J

1z2 _y2+ R2oy3B [ (JJU~~: U~~: + Uy )Uy ( U~~: + Uy )2u~~: + u 11
142 2+ 12Ull:3(U~~: + Uy )2 :X + 2U~~:Uy ( U~~: + Uy )2 X y]
2u11 + u~~:4+ 12u 113( U~~: + Uy )2y + ....

'tl

We assumed that the beam distribution is Gaussian andU~~: and u 11 are horizontal and vertical rms beam size, respectively, and A is a line charge density. The first square
bracket shows the magnetic force terms in lattice elements.
They are essentially zero if the lattice is constructed in aperfect manner except a quadrupole term which provides
focusing force and a sextupole one which is introduced
intentionally to correct chromaticity. The second square
bracket shows the space charge te.rms of both linear andnonlinear contributions. Those exist even if the -lattice isperfectly constructed.
The driving term of one dimensional fourth integer reso~ance excited by space charge force (4v11 = N) is, for example,- 2JyJa2

12uy + U~~:JJy f32-y3 B 12u:( u~~: + O"y )2 e-i·4~w

ds,

where 2J11 is the Courant-Synder invariant, {311 is the vertical amplitude function and ¢111 is the vertical phase of betatron oscillations. One can see that only harmonics which isinteger times the number ofsuperperiod (N = integerxn,)
are non zero because the harmonics of rms beam size u~~:,yfollows lattice amplitude functions, P~~:,y· In reality, there
are small modulation in amplitude functions and harmonics other than the above number has small magnitude in
driving term. Figure 1 shows the strength of driving termsof space charge induced fourth integer resonances in theKEK PS. Note that the KEK PS has fourfold symmetry(n, = 4).
When the vertical tune is 7, the strength of the spacecharge induced resonance becomes maximum. This happens because the KEK PS has more or less a 28 normal
FODO cell structure and therefore phase advance per cellbecomes 90 degree. At the same time, the primary beamenvelope modulation, namely space charge force, has 28
harmonic components. We name it space charge excited"super structure" resonance.
Since the 3 GeV booster has 24 FODO cells and the50 GeV man ring has 88 FODO cells, the super structure
resonance is located at the tune of 6 and 22, respectively.
The operating point near that tune should be avoided.

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vertical tune

Figure. 1. Resonance width of space charge induced driving harmonics in the KEK PS. Because small amplitude
function modulation is included, structure (filled box) as
well as nonstructure resonance (empty box) is excited.
Note that the KEK PS has fourfold symmetry. Horizontal
tune is fixed to 7.12.
chromaticity of the 3 GeV booster ring is around -7 to 6 and we plan to start commissioning of the 3 GeV ring
without chromaticity correction until it is required by somereasons.
On the other hand, the natural chromaticity of the 50
GeV ring is around -30. With momentum spread of ±0.5%
in a beam, transverse tune oscillates ±0.15%, that is notallowed. Chromaticity correction is essential in the 50 GeVring.
Two families of sextupoles are installed in the missing
bend regions, which are located one for every three FODO
cells. Typical strength of the sextupoles to make the chromaticity zero is 29 T /m for focusing sextupoles and -59
T /m for defocusing ones. Those numbers include the sextupole length in it, that is 1 m. With those sextupoles
excited, we investigated the dynamic aperture as follows.

A. Time dependence
To obtain a rough idea of the dynamic aperture of thering, we did tens of particle tracking runs taking transverse
operating point as a .parameter. The simulation code SAD[4] was used without synchrotron oscillations or momentum
offset of a particle. Transverse tune area of ± 1 near thenominal operating point (24.25, 20.70) was searched. A
dynamic aperture turns out 380 1r • mm · mrad at one point
as shown in Figure 2, which is much larger than the physicalaperture, that is 54 1r • mm · mrad.

III. Dynamic Aperture Both rings are operated under transition energy withentire energy region. From collective instability point ofview, such as zeroth mode of head-tail instability, beams
are stable without chromaticity correction. In fact, naturalB. Momentum dependence
Momentum dependence of the dynamic aperture wasstudied using the simulation code Simpsons [5], which
tracks particles in 6D phase space with synchrotron os--408-JAERI-Conf 95-021

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YxFigure. 2. Dynamic aperture of the 50 GeV ring at several
operating points.
cillations. Operating point was chosen at (22.75, 22.65)
and sextupoles are excited such that chromaticity becomeszero for both planes. The rf voltage of 200 kV was appliedso that the rf bucket high is ±0.43 % in the tl.pjp axis.
The longitudinal tune at the injection energy is 0.004. As
shown in Figure 3, no strong momentum dependence was
300 .---------------- ---------------- ,Figure. 4. Dynamic aperture of the 50 Ge V ring withdifferent chromaticity correction strength. Amplitude ofbeam momentum is fixed to 0.2 %.
exist and dynamic aperture is small. When stronger correction was applied, the dynamic aperture becomes large at
one point (chromaticity= -15). The further increase of thestrength of sextupoles, and therefore smaller chromaticity,reduces the dynamic aperture gradually.
At this moment, we are not sure if we should correct thechromaticity to zero or not.

IV. Summary[J[J[J[JBased on the proposed lattice of the JHP circular accelerators, we studied space charge effects and dynamicaperture. In fact, combining those two effects is necessary, namely tracking study of dynamic aperture with spacecharge effects included. We plan to do it in near future.

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References

1-<~ 100{\j[1][2][3][4].....ue{\j>::€0 ~--r--.--~-.--,--,--~-,r--r~
-0.0

0.1

0. 3

0. 2

0.4

[5]

0.5

dp/pFigure. 3. Dynamic aperture of the 50 GeV ring withdifferent momentum amplitude.
observed.

C. Chromaticity dependence
The last figure (Figure 4) shows the dynamic aperture ofthe 50 Ge V ring with different chromaticity setting. The
simulation code Simpsons [5] was used. Natural chromaticity is around -30. With a small correction of chromaticity(chromaticity = -25 and -20), still large tune oscillations-409-

Y. Mori, et al, in this proceedings.
C. Ohmori, et a/, in this proceedings .
Y. Ishi, et al, in this proceedings.
SAD stands for "Strategic Accelerator Design" code.
It is developed by K. Oide and colleagues at KEK.
S. Machida, ''The Simpsons Program", AlP Conference Proceedings 297, p.459, 1993.