The II th Symposium on Accelerator Science and Technology, Harima Science Garden City,,l997

Simulation of Beam Bunching using Electron Cooling for the Ion Storage Ringof RIKEN RI-Beam Factory Project
Masao TAKANAKA and Takeshi KATAYAMA*
Cyclotron Laboratory, RIKEN, 2-1 Hirosawa, Wako-shi, 351-01 Japan*Center for Nuclear Study, School of Science, University of Tokyo,
3-2-1 Midoricho, Tanashi-shi, 188 Japan
2 BUNCIDNG PROCEDURE

Abstract

Beam bunching using electron cooling is planned to be
carried out under the control which keeps the momentum
spread constant. The simulation of the bunching for a
150 MeV/u U~~t beam in DSR has been done on whichspace-charge effects including the transverse dipole effects
are taken into account. Results of the simulation show thatthe dipole effect interferes with the decrease of the transverse emittance due to the cooling and results in helping
the decrease of the bunch length.

1 INTRODUCTION
Collision experiments are planned with ion-ion beams andion-electron ones in Double Storage Rings (DSR) which
is currently being designed. The luminosity of the beams· can become high when the beams are bunched. Ion beams
are bunched by using the electron cooling together with RFapplication. As the bunching goes, not only the longitudinal space-charge force but also the transverse one, which
are dominant over forces induced through the coupling impedances between beams and the vacuum chamber in theenergy region 100 MeV/u to 1.5 GeV/u of the ring, become large. The former is known to disturb shortening thebunch length below the transition energy. The dipole component of the latter is known to make betatron-tune shifts
coherently which has possibility to make the beam size ofthe equilibrium state large. Both the phenomena are notgood for getting the high luminosity. The simulation of thebunching has the aim to reveal whether there is any obstacle
against getting the high luminosity.
The bunching of ion beams is planned to be carried out
under the momentum-spread control. The previous simulation of the bunching, on which the dipole componentof the transverse force was not taken into account, showedthat non-linearity of the lattice does not make a problem
even when the incoherent betatron-tune spread reaches 0.5,
the bunching process is stable, and the transverse emittanceturns to increase near the equilibrium [1]. The simulation
has been improved as follows in order that the behavior of
the transverse emittance is understood better;A beam is injected to DSR, after it is accumulated as acoasting beam and cooled down to a given momentum
spread and to such a transverse emittance in the Accumulator and Cooler Ring (ACR) that it does not meet resonances. The beam is cooled again there by electron cooling while RF is applied under the momentum spread control which increases the RF voltage so that the momentum
spread stays constant. The beam becomes bunched as thevoltage increases. The smaller the spread is, the lower isthe threshold of the beam line density due to the microwave
instability [2]. The momentum-spread control is necessary
to a high line-density beam. The simulation has been done
just for beams in DSR.

3 SIMPLIFICATION
The following simplification has been done on the simulation mainly in order that one reduces CPU-time load andkeeps statistic accuracy.
i) Except for the Twiss parameters during the cooling
section, j3 function along the ring has been equal to theaverage one 7J = R/ v, where R is the mean radius of thering and v the betatron tune. a function has been 0.
ii) The current concerning is just about the beam behavior under space-charge effects, or the effects through thebroad-band impedances. Although the simulation is for thecase where an ion beam occupies all RF buckets, it does
not deal with coupled-bunch phenomena.
iii) The longitudinal space-charge force is induced bybeam's space charge through a perfectly conducting vacuum chamber of the inner radius b. The ions of charge qe
at the distance r from the central orbit get the energy ll.Esp
from the force during time ll.t, when the transverse charge
distribution is round and Gaussian with one-dimensional
standard deviation rr which is much smaller than the radiusb;di ll.t,-v -dll.Esp = zqe n sp 21r s• refinement of the longitudinal space-charge impedance,• taking the dipole component of the transverse spacecharge force into account.

. [z"][ z"]

n spg(r) ~--291-ao
2x-..r:..._-g (0)The simulation results of the behavior are showed here.

l=ig(r)z2/3"! 2 0'(b)(.~rr) ,-x_e-dx+2log
X2a2~ 0.577 + 2log-r,

The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

where the beam of velocity v has the current distribution
I along the longitudinal direction s, and f3 and 1 are therelativistic constants of v, and Z 0 impedance of free space.
Longitudinal variation of the transverse charge distribution
has been neglected in evaluating the impedance. The deviation (T has been replaced by the horizontal rms beam size
(T h and the vertical one (Tv;iv) As the transverse electric field due to the spacecharge, just the monopole component and the dipole one
have been taken into account. They have been estimated asif they are in the ultra-relativistic case, and the monopole
field has been let be that in free space. The impedance dueto the dipole has been estimated as a ring type;1 1). ZoR (ZJ.. = Z (32,,p g(Th(Tv - b2
45

RESULTS

Two kinds of simulation results are shown, the one being
about the longitudinal space-charge impedances, and theother about the transverse space-charge impedance.

5.1

Longitudinal space-charge impedance

The longitudinal space-charge impedance is dependent on
the transverse charge distribution of a beam, or not only on
the beam size but also on the position within the beam. As
described in Section 3, it has been approximated to be thatfor the Gaussian distribution. Figure 1 shows the difference
between g ( r) taking the position dependence into accountand non-taking g ( 0). The results are ones of the simulation
on which the transverse space-charge impedance has notbeen taken into account yet. The equilibrium state seems
to be realized after bunching of 60 ms .

.

PARAMETERS OJF' THE RING AND A BEAMThe simulation has been done just for a 150 MeV/u U~~tbeam of current 3.4 rnA. In Table 1, are shown parameters of the ring, a coasting beam, and the electron cooling,which have been used as input data of the simulation. The
initial distribution of a coasting beam for the simulation has
been flat along the RF phase axis, and Gaussian along themomentum axis, in the horizontal phase space and in thevertical one, respectively. The average current of 3.4 rnA
is the threshold due to the longitudinal microwave instability in the case where the beam bunches in 6xrms length of
0.4 m, of 6xrms momentum spread of w- 3 , and of rms
transverse emittance of w- 6 7!' rnrad.
Table 1: Parameters of the ring, a coasting beam, and theelectron cooling.
Ring
Circumference
Momentum compaction factor
Betatron tune (vh lvv )Natural chromaticity Ceh )Twiss parameters at the cooling section
a'j,c = a~c
f3'J.c = (3~cRF harmonics
Inner radius of the vacuum chamber b
Coasting beam
Momentum spread (6xrms)
Rms transverse emittance (ch = fv)
Electron coolingElectron current
Cathode temperature kTc
Length of the cooling section
Electron-beam radius at the section
Longitudinal magnetic field at the section

260m
0.03772
7.38/5.8
-35
07m
87
4cm

5A
0.1 eV
3m
25mm
1 kG

Figure I: Transition of the bunching. The solid curves arefor the position-dependent longitudinal impedance, and thedotted ones for the position-independent one.

5.2

Transverse space-charge impedance

The transverse space-charge impedance has been taken into
account just in the horizontal direction on the simulationof which results are shown in Figure 2. The dipole effectthrough the impedance is seen to interfere with the decreaseof the horizontal emittance due to the cooling. As the results, it helps the decrease of the bunch length due to thecooling. The bunch length near the equilibrium becomes
0.5 m, or half that in the previous case. The RF voltage of
300 kV is required for the short bunch.
Figure 3 shows the bunch shape, which is approximately
represented by using a Gaussian curve and a parabolic one,after the bunching of 30 ms. The parabolic curve represents
the distribution of the group of cooled ions which do hardly
synchrotron movement around the center of the bunch. Figure 4 shows the dipole moment, that is, the collective displacement times the instantaneous beam current. The amplitude of the collective horizontal displacement is seen to
be 0.1 mm around the bunch center and larger at both thesides. The horizontal emittance along the bunch is shownin Figure 5.

-292-The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

The effect of the dipole may have been overestimated,because on the simulation the beta function along the ringhas been let to be constant except for the cooling section
and just a ring type of the dipole component has beentaken into account as the dipole component of the transverse field.

8 10"6
6 10"64 10"6<*§2 10"60';< -2 10"66

1.1 10"'-4 10"6....
110"3 ~

§9 10-4 e .e.
,eo>.,. *e 4~ e\0....

-8 10"6-1.5s" --g0\0~g, :;; 38 10"4 ~~~7 10"c.: g=ill

101520Time(ms)

2530

~

~

410"6~ 3 10"6"g-~ 2 10"

6"

~~

"§70~

110"6!I:

0-1.5

60

e':::: 50"~

-1-0.500.5
Longitudinal position (m)1.5

Figure 5: Horizonatal emittance along the bunch after
bunching of 30 ms.

= 40ue""' 30.0.,

8= 20

7 REFERENCES~"£l 10.s
0-1.5

1.5

5 10"6l5 10"4

Figure 2: Transition of the bunching. The transverse spacecharge impedance has been taken into account just in thehorizontal direction.

<-0;50.5
0Longitudinal position (m)" ~.,>~ 0~~E-<
5-1

Figure 4: Dipole moment after bunching of 30 ms.

·e e~4Emit-v

0 ">u....c 200-6 10"6-g *-1) \-0.5
00.5
Longitudinal position (m)[1] M.Takanaka, T.Katayama, Proc. of PAC 97 (1997).
[2] A.Hofmann, CERN 77-13, 139 (1977).
1.5

Figure 3: Bunch shape after bunching of 30 ms.

6 CONCLUSION_The simulation of the bunching of a 150 MeV/u U~~t beamof 3.4 rnA under the momentum-spread control at the electron density of 2.5 kNm2 at the cooling section has shown
the following characteristic of the bunching.
• The process of the bunching is stable, just when thelongitudinal space-charge effect and the transverse
monopole one are taken into account on the simulation.
• The transverse dipole space-charge effect interfereswith the decrease of the transverse emittance due tothe cooling, which helps the decrease of the bunch
length.
• The beam near the equilibrium is bunched in 6 x rms
length of 0.5 m, and with the transverse rms emittanceof 10- 6 1r mrad under RF voltage of0.3 MV.

-293-