The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

CHARACTERISTICS OF ELECTRON-COOLED BUNCHED BEAMSPREDICTED BY A SIMULATION
M. Takanaka, RIKEN, Japan
Abstract
Characteristics of electron-cooled bunched ion beams
have been studied using the particle-tracking simulationin which the space-charge force and the force induced
through the transverse broadband impedance are taken intoaccount. The beams become space-charge dominated during the bunching of coasting beams. The simulation results
show that beams of beam current of 4 rnA are stabilized bya transverse feedback to be bunched by the bunch length of
16 em for the bunch spacing of 3 m at the third harmonic
RF voltage of 290 kV. The characteristics of the bunchedbeams are predicted by analyzing the simulation results.

1 INTRODUCTION
The Radioisotope Beam Factory (RIBF) [1] will have anelectron-RI beam collider (e-RI Collider). We have studied ion-beam bunching at low energies for collision experiments, using a particle-tracking simulation. In the previouspapers [2] we have shown that during the electron-coolingbunching the beams meet the transverse dipole-mode instability which can be cured by a transverse feedback andthat the seed of the instability originates in the toroid fieldsof the electron-cooling (EC) section, and have referred tosome characteristics of the stabilized beam.
In the particle-tracking simulation, we have taken thesame sources of force acting on particles into account asprevious, and have used the same beam parameters, too.
Using 40,000 macro-particles, we have simulated U~~t ion
beams of 4 rnA, or 5.4x 10 6 ions per bunch at 150 MeV/u.
Since our interest is a single bunch phenomenon, we haveassumed that every bunch behaves the same along the azimuthal direction. The simulation has showed that the longitudinal and transverse electron-cooling times are 6 ms
and 18 ms for a faint-current beam, respectively.
From the next section, using the simulation results wegive a more detailed description of characteristics of thestabilized beam than previously.

2 BEAM BUNCHING
We have the RF system which supplies a fundamental
RF voltage and a third harmonic RF voltage. The system
is located at the azimuth 1r from the EC section since themomentum spread can be the smallest at the EC sectionin this location case. The evolution of the beam bunchingis shown in Fig. 1. At first the coasting beam is cooled
to the sixfold-rms momentum spread 5 x 10 - 4 . Then, thefundamental RF voltage is increased in such a way that themomentum spread at the EC section is maintained at 5 x
w- 4 . When the sixfold-rms bunch length reaches 1 m,
the increase stops. When the momentum spread reaches
1.9 x w- 4 or most of the beam stays within the one-thirdbunch spacing, the third harmonic RF voltage is increased
up to 290 kV holding the momentum spread.

The toroid fields at the both ends of the EC section makeresonances vh + vy = i (i=integer) with the half resonance width of0.01 on the vh + vy scale near the operation
point. The resonances are seeds of the transverse instability for electron-cooled bunched beams. In order to curethe transverse instability, we have used a following transverse feedback after 30 ms. Picked up at the end of theEC section where the dispersion is zero, the horizontal andvertical beam-position signals along a bunch are fed to thebunch downstream the phase advance 1r /2 with feedback
gains 0.8/ f3h and 0.8/ f3v, respectively, and bandwidth 6GHz. Around 60 ms 0.5 percent of the beam is lost aroundthe bunch center (z=-10 mm to 10 mm). After 80 ms, there
is no beam loss and the instantaneous beam current at thebunch center is 120 rnA.
We show the characteristics of the beam at 135 ms in thenext section.

3CHARACTERISTICS OF THE
STABILIZED BEAM

Transverse symmetry of the ion distributions shown inFig. 2 means that the feedback to cure the transverse instability works well. Since the weaker the effective longitudinal space-charge impedance seems to the ions withthe larger betatron amplitudes in a space-charge-dominatedbeam, the weaker the RF-wave-form distortion seems to theions. Therefore, when the momentum spread of the beam isconsiderably small compared with the RF separatrix height,
the more the ions with the larger amplitudes are centralizedon the bunch profile as shown in Fig. 2.
Figure 3 shows the longitudinal phase-space distribution
and the projections at the EC section and just after the RF
section. Just before and after the RF section the distribution
is one with the opposite slop in the phase space. The momentum spread just after the RF section becomes five times
larger than at the EC section. The bunch length becomesthe longest at the RF section and the shortest at the EC
section. The longitudinal space-charge effect makes sucha longitudinal quasi-elastic motion of the bunch at the frequency tune 1 instead of a synchrotron oscillation. At thesame time, as we have assumed the same dispersion along
the ring except for the EC section, the bunch has a horizontal oscillation with the head-tail mode number 1 andthe frequency tune 1 due to the momentum deviation. Thebunch profile has an excess around the center comparedwith the parabolic profile [3]. The microwave structure of
the phase-space distribution is a reflection of the longitudinal space-charge impedance that smears fluctuations within
the impedance bandwidth in bunch profiles. But, the structure seems to have a little lower frequency than the microwave structure of the bunch profile, which is referred tojust below.
Figure 4 shows the spectrum of the longitudinalmonopole moment which has following features:
1. Along D/wo = kh (k <rv 200, h =RF harmonic- 89-

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

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4 rnA. The momentum spread is one at the EC section. The
transverse feedback works after 30 ms.
number) a series of signals due to the parabolic bunch profile.
2. Along the two-spread-wings frame a series of signals due to the reflection of the longitudinal space-charge
impedance. The position of the signals can be described by
the following equation which is derived by applying theVlasov equation to the case of bun~hed beams with little momentum spread under the dominance of the spacecharge impedance;n- nwo = ±nwo
where I is the instantaneous beam current averaged overthe bunch, and z~! f the effective longitudinal spacecharge impedance. The dynamics of the signal prqduction
is equal to that of splitting Schottky spectra of cooled coasting beams. As the longitudinal impedance has been as--80-60

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Figure 2: Ion distributions at 135 ms.

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Figure 3: Longitudinal phase-space distribution at 135 ms.
In the upper, the slanted distribution is one just after the RFsystem, and the non-slanted distribution is one at the endof the EC section. In the lower right, the projection with afitted parabolic curve is one at the end of the EC section.
sumed to be cut off over k = 232, the difference In - nw 0 I
decreases over the k.
3. Along fljw 0 = kh (k >rv200) a series of signals
which have the largest peak around k = 450. We guessthat the signals come from an internal oscillation inducedby the longitudinal quasi-elastic motion of the bunch.
4. Along fl/wo = kh ± i (i=integer not being 0) a seriesof signals due to the quasi-elastic motion.
Figure 5 shows the spectrum of the horizontal dipole moment which has following features:
1. Around I Dfwo - kh I= 4.4 a series of signals dueto the rigid-bunch mode of coherent betatron oscillation
which the feedback can not completely damp.

90-The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001coherent tunes are nearly 0 except for those for some ofions around the bunch center. This means that the beam
synchrotron oscillates hardly because of the RF-wave-form
distortion, too .
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1020y(mm)700-40Figure 4: Spectrum of the longitudinal monopole moment
at 135 ms. For the sake of clear sight of main signals, thespectrum normalized by the DC component is cut away below the three levels 10- 3 ,2.5 X 10- 4 , and 10- 4 .
2. At I Djw 0 - kh I= i (i=positive integer) a seriesof signals due to the head-tail mode 1 of coherent betatronoscillation with a tune of 1 which was referred to above.
The spectrum of the vertical dipole moment does nothave the second feature because of no vertical dispersion.

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ms.

4 CONCLUSIONS

0

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Figure 5: Spectrum of the horizontal dipole moment at 135
ms. The spectrum is cut away below 3 X w- 9 .
Figure 6 shows incoherent tune distributions. Whilebetatron-tune shifts are little for ions at the bunch center,large tune shifts occur for ions on the head and tail sides.
But, the large shifts for the horizontal direction are disturbed by the head-tail mode 1 of coherent betatron oscillation. Although the bare incoherent synchrotron tune is0.09 for the third harmonic RF voltage 290 kV, most of in-We have carried out particle-tracking simulations tostudy characteristics of electron-cooled bunched ion beams
and have reached following conclusions.
Although the incoherent betatron-tune shifts become
over 1 during the electron-cooling bunching, the beam of
4 mA can be stabilized by a transverse feedback to bebunched by the bunching factor of 19, by the bunch lengthof 16 em, or at the peak current of 120 mA under the third
harmonic RF voltage of 290 kV. As the phenomena due tothe RF-wave-form distortion,
1. Incoherent synchrotron tune is suppressed to nearly 0
from the bare tune of 0.09.
2. The bunch motion is a quasi-elastic one with the tuneof 1 instead of synchrotron oscillation in the longitudinal
direction, which is automatically accompanied with a headtail mode 1 of horizontal dipole oscillation because of thehorizontal non-zero dispersion.
3. The beam size can be extremely large at the bunch
center. The main bunch profile consists of a parabolic
shape and some excess around the bunch center.
There is a microwave structure on the bunch profile
beyond the bandwidth of the longitudinal phase-space
impedance.

5 REFERENCES[1] Y.Yano eta!, RIKEN Ace. Progress Report (2000) 301.
[2] M.Takanaka, PAC2001, FPAH093.
M.Takanaka, RIKEN-AF-AC-29[3] S.Hansen eta!, IEEE Trans. On Nucl. Sci. 22 (1975) 1381.

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