JAERI-Conf

95-021

PB25
Beam Current Limitations due to Single-Beam Collective Effects
in the Ion Storage Ring of RIKEN RI Beam Factory Project
M. Takanaka
Cyclotron Laboratory, RIKEN (the Institute of Physical and Chemical Research),
2-1 Hirosawa, Wako-shi, Saitama 351-01, Japan
T. Katayama
Institute for Nuclear Study, University of Tokyo,
3-2-1 Midori-cho, Tanashi-shi, Tokyo 188, Japan
Abstract
Single-beam collective effects limit the beam current whenbeams are served for colliding experiments in the presently
designed ion ring. For the some hundreds MeV/u ion beammaking the betatron tune shift -0.2, longitudinal instabilities
occur with growth times of 0.02 s, and the transverse resistive
wall instability with growth time of 1 s. The electron cooling withcooling time of 0.01 s at the full momentum spread 10·' of theion beams can damp the instabilities. Then, the luminosity ismuch less than 1 x 1027 em·' s·' ort the heavy ion side. A proposal
about the lattice of the ring is made to improve the luminosity.

Here, current limitations of ion beams and electron beams are
estimated which are due to beam-beam effects during thecollision, the direct space charge effects, and beam instabilities in
the presently designed ion ring, but not in the electron ring.
Instabilities are treated of just in the injection energies, because
the lower the beam energy, the more easily instabilities occur.
Both the lightest ion and the heaviest ion are treaded of in orderto watch the dependence on ion species. The luminosity isevaluated for the threshold beam currents. Finally, a proposal ismade to improve the luminosity.

1.
Introduction
The RIKEN Accelerator Research Facility group has beenproposing "RIKEN RI Beam Factory" as a next facility-expanding
project [1]. The factory takes the aim at providing RI beams of thewhole mass range in a wide energy range up to several hundreds
MeV/u. Ion beams from the existing RRC are accelerated and, orqualified as a cooled beam, a high current beam, and, or an RI
beam through a superconducting ring cyclotron, an accumulatorbooster-cooler ring, and, or an RI beam generator. The beams are
injected to the one of a twin storage ring that is called DoubleStorage Rings (DSR). Electron beams from a 300 MeV linac areaccumulated and accelerated in the other one. The colliding
experiments are carried out at two interaction sections (IS) inDSR. On the other hand, ion beams are injected to both the rings
for ion-ion merging experiments. Both the lattices of the rings
have been designed to be the same. Table 1 shows theparameters of DSR.

Table 1: Parameters of DSR.
Circumference of a ring
Max. magnetic rigidity
Radius of curvature
Beam energy (injection--+ top)ion(ZIA=0.5)
ion(Z/A=0.387)
electron
Betatron tune ( v J v,)
Momentum compaction factor
Natural chromaticity (~.I~')
Beta function at IS ( p;t P;>
Dispersion at IS ( qJ '1. )Max. stored electron
beam current
Number ofbunches (ion/electron)
Electron rms emittance ( e, ~ e,)
at0.3 GeV
at2.5 GeV

178.694 m
12.76 Tm
8.506 m2. Electron Beam Current Limitations
In DSR a 15-bunched electron beam circulating in the one ring
collides with a 30-buncbed ion beam circulating in the other ringat two IS's with zero colliding angle. The linear tune shift of .theion due to the beam-beam effect is described as follows under
the assumption where the charge distribution of the electronround beam is Gaussian with the rms a,. in the radius direction[2];~· = (1 + fJJ3,)fJ:N..r, Z/A'4tra,./f3:ri(l)'where /J, and fJ, are velocities of the ion and the electron beams
the beta
normalized by the light velocity c, respectively,
function at IS's, N8 , the number of electrons per bunch, ·,., theclassical proton radius, Z/ A the. charge to mass ratio of the ion,and r, the ion mass normalized by the rest mass. The stability of
the lattice of the ion ring requires

P:

icosJL- 2~r~, sinJLI:S: 1,(2)where JL is half the phase advance of the ring. One has thefollowing requirement, reserving a little safety margin;

0.4 --+ 1.2 GeV/u
0.15 --+0.82 GeV/u
0.3 --+ 2.5 GeV
6.350/5.763
0.0424-10.81/-7.43
0.60/0.60 m
0.0/0.0 m~.:s:o.os.

(3)With substitution of a,.' = eRMs.P: to Eq. (l) ( the beta function of
the electron ring at IS is designed to be equal to that of the ionring), the threshold number of the electrons per bunch isevaluated as shown in Table 2.
Table 2: The threshold number of electrons per bunch due to thebeam-beam effectIon xelectron
2.5 GeV e·
300 MeV e3xlO'2x10"
400 MeV/u 4He2 +2xlO'
1 xlO"
150 MeV/u 2~92+

0.3 A
30/15

8 x1o-' mnrad
530 x1o·• mnrad.

Colliding experiments require beams of high luminosity andsmall momentum spreads. For this, ion beams are intensified andcooled down so that the full momentum spread may become atmost 1o·• and the transverse emittances may become
1 xlO.. mnrad, before the injection to the ring. They are bunchedwith the full bunch length of 40 em in the ring. The qualification
always does not increase the luminosity, because it makes singlebeam collective effects strong and the effects limit beam
currents.

On the other hand, the stored current of the electron beam isexpected to be at least 300 mA, when the beam of 300 MeV is
injected in the ring. Then, the number N., becomes 7 x 1010 • The
electron bunch population N8 , shown in Table 2 is limited by thevalue further.
3. Ion Beam Current Limitations
3,1 Incoherent betatron tune shift due to direct space charge
For a round beam whose charge distribution is assumed to be318 -JAERI-Conf 95-021

Gaussian with the rms a,, in the radius direction and parabolicwith the full bunch length e in the longitudinal direction, theincoherent betatron tune shift of ions with small betatron
amplitude due to direct space charge is described by [3](4)where {3" is the average beta function of the ion ring, N., thebunch population of ions, and B1 = e1 ( 3 ~rR J the bunching factor,R being the average radius of the ring. Here, one properly uses
the equality 4a,,' = ej3" on treating of ion beams whose charge
distribution is not Gaussian. When resonance compensation iscarried out well, the shift of ILl vi$; 0. 2 is tolerable. Then, thebunch population N 8 , is limited as follows;
N. < 1x 109 for 400 MeV lu 'He''
8 ' -7 x 10'!50 MeV lu "'U"'(7)
where e is electronic charge, 11 the slippage factor, and Lip/ p thefwhm momentum spread. The equation (7) shows<1.4xl09
107

N

8' - ] . 6 X

ion bunch population making the betatron tune shift ILl vi$; 0. 2.
When a bunched beam circulates with the angular frequency
co 0 in a ring, it can have following longitudinal frequency
components without perturbation;(9)

co,= co0 (pB + m + nv,),

where B is the number of bunches, p = 0, ±1, ±2, ... etc., v, thesynchrotron tune, m (=O,I,· .. ,B-1) the coupled-bunch mode,and n ( = 0, ±1, ±2, ... etc.) the synchrotron mode. Withperturbation, the frequency is coherently shifted, in theSacherer-Zotter formalism [13], by

1.1 %Io 'I
pr co'n
L1com.n = _j .....r:.L
lnl +I 3(£J2n:R)' hV'[Z ]"'"__!:..

n(I 0)•ff(11)

wherehis the RF harmonic number,Vthe peak RF-voltage, andh, ( ro) the spectral power density of the n-th synchrotron mode.

The above equations are applicable to the single bunch, also.
Landau damping around the cut-off frequency region isexpected if the shift of high-order modes induced through thebroad band impedances is smaller than the synchrotron
frequency spread coming from a nonlinear sinusoidal RF bucket
16 /2TrR co.v,,
Ll co.v, ) --~(he/)'(Table 3: The imaginary parts of the longitudinal impedancesIm( ZL In)[ .a J of the ion ring of DSR at the low frequency.
400 MeV/u
150 MeV/u
Injection energy
860
1900
Space charge
-2.0
-1.4
Bellows-0.05-0.04
Flanges-0.2-0.1
Transition sections-0.003-0.002
Slits of vacuum ports
-0.21
-0.14
Clearing electrodes
-1.5
-1.5
RF equivalent BB-0.07-0.05
Strip-lines
Table 4: The imaginary parts of the transverse impedancesIm( Z" )[MD I m] of the ion ring of DSR at the low frequency.
150 MeV/u
6600-0.2(8)(6)

where a round vacuum chamber is assumed with inner radiusb =4 em. It is known that the space charge impedance isdominant in the energy region of DSR.

Injection energy
Space charge
Broad band

400 MeV lu 'He''
for !50 MeV I u 238U 92 ' •It is known that the microwave instability does not occur for the(5)3 2 Impedance budget
The lattice of DSR has been being optimized nowadays. The
structure of the vacuum chamber and numbers of the elements
have not yet determined. Here, the beam coupling impedanceswith the chamber are estimated under a rough assumption ofthem.
Longitudinal broad band impedances of bellows and flanges,transition sections, slits of vacuum ports, and clearing electrodesof disk type are evaluated with the theoretically derived
equations [4], [5], [6], and [7], respectively. Longitudinal
impedances of strip-line monitors are estimated with theequation [8].
A candidate of the RF cavity for the ion ring is a AJ4 coaxial
cavity with the use of a perpendicular bias field on the ferrite [9].
The higher order cavity modes are strongly suppressed with thedamper [1 0]. The longitudinal impedance of the RF cavity for
DSR is assumed to be like that of the 50 MHz 150 kV prototype
synchrotron cavity [11], but the frequencies of HOM areassumed to be just integer times as large as the fundamental
frequency. The RF voltage of 700 kV is given with seven cells ofcavity in order that the bunch length is reduced to 40 em.
The longitudinal impedances except the resistive wall one areshown for the injection energies of 150 Me V/u of the heaviest
ions and 400 MeV/u of the lightest ions in Table 3. The
transverse broad band impedances shown in Table 4 areevaluated with the relation]'''1n[zL ]'' =l:.!!_[z{3,b' L

3 3 Longjtndinal instabilities
The microwave instability is undesirable, because it induces
bunch lengthening when the bunch population goes beyond athreshold, and the lengthening decreases the luminosity. The
threshold peak current I,, is given with Keil Schnell criterion
under the parabolic momentum distribution [12];(12)Then, the bunch population is limited as follows;N••

400 MeVIu 'He''$;5xl06
5 x 104 for 150 MeV I u 238U 92 ' '(13)

Therefore, single-bunch and multi-bunch instabilities may be
induced for the bunch population making the tune shift
IL1vi=0.2. The growth times of the most unstable mode each n-th
mode are shown in Table 5 and Table 6, which are calculatedwith the program ZAP [ 14] which has been modified to be
applicable for ion beams by the authors.
Table 5: The growth times [s] of the longitudinal single-bunch
instabilities
150 MeV/u
400 MeV/u
Energyz3surn+
4Hez+
Ion
Mode n=1
Mode n=2
Mode n=3
Mode n=4

400 MeV/u
2300
-0.2

-319-

0.7

11-0.31030-4-0.8

JAERI-Conf 95-021

Table 6: The growth times [s] of the most unstable mode(m,n) ofth e lontzitud'mal mu Iti-bunch instabilities each n-th mode.
238u92+4He2+400 MeV/u
150 MeV/uMode(O,l)Mode(28,1)
0.02
0.07
Mode(27,2)Mode(0,2)
0.4
0.05Mode(0,3)
Mode(29,3)0.3
0.3
Mode(28,4)Mode(0,4)
0.9
0.6

3 5 Electron cooling
The electron cooling is applied to damp the longitudinal andtransverse instabilities. The cooling time is required to be less thanhalf the growth time [16];

3 4 Transyerse jnstabilitjes
As in the longitudinal direction, a bunched beam can havefollowing transverse frequency components without
perturbation;

4.
Luminosity
The length of the colliding region is two-third of the ion fullbunch length, because of the normalized velocity {3, = o. 5corresponding to the lowest beam energy. When the detectable
collision region is longer than it, the luminosity is described by
[ 17]

( 14)

1,,,[ 5: 0. 01

L=

With perturbation, the frequency is coherently shifted, in the
Sacherer-Zotter formalism [13], by~-·_-_ZL( ro,)h,( ro,- ro1)

(16)

~h.<ro,- ro1 )p=--where E is the beam energy per nucleon, and ro1 the chromatic
shift ro1 =~ro0 /7j, ~ being the chromaticity. The above equations
are applicable to the single bunch, also.
Transverse single-bunch instabilities occur when the coherent
frequencies of two neighboring low-order modes shift to become
equal, for example at IL1ro';' 0 1= ro0 v, [15]. They are called modecoupling instabilities induced through the broad band
impedances. The threshold peak current is described by(17)

v, =Then, the bunch population is limited as follows;
N. < 4x 108 for 400 MeV /u 4He" .
•·- 5x10'
150 MeV lu '"u"''(18)

Even for the bunch population making the tune shift
however, the coherent tune shift is.

0. 06
400 MeV I u HeL1roL/ro•=o.07 for 150 MeV/u '"U"'''
4N.,N.JREVIB
2n(4eRMs. + e,)/3:'

(20)

(21)

where f REV• is the revolution frequency of the ion beam. Whenthe electron cooling with cooling time of 0.01 s is used to damp
the instabilities, the bunch population making the tune shift
IL1vi=0.2 is available. Then, the luminosity is shown in Table 8.

(15)[zLr;· -=-=----[ s ].

ILl vi= 0. 2 ,2'(1. 9 )which is less than the incoherent betatron tune shift. Landaudamping is expected to suppress the instabilities.
Transverse coupled bunch instabilities are induced through theresistive wall impedances and RF narrow band ones. Here, as one
has no .data about the latter, the growth times of the instabilities
just through the former are evaluated. The resistive wall
impedance contributes mainly the mode n =0 because of its
sharp peak at the zero frequency. Without the chromaticity, themost unstable mode is known to be the mode(23, 0) or themode(24, 0) because of 30 bunches and the betatron tunes on
the two sides of 6. For the bunch population making the tune'shift IL1vi=0.2, the growth times are estimated as shown in Table
7. The avoidance of the instabilities is dependent on thechromaticity.
Table 7: The growth times [s] of the most unstable mode(m,n) ofbunch instabilities.
th e trans verse COU!lledIEnergy
400 MeV/u
150 MeV/u
4He2+''"lf92'
Ion
Mode(23,0) orI
Mode(24,0)3

Table 8: The luminosity [ em-' s-1 ] for the bunch population
making the tune shift IL1 vi= 0. 2 .
Ion xelectron
300 MeV e2.5 GeV e4He2+
27
2 X 1028
3 X 10400 MeV/u9x1024
l X 1026
150 MeV/u ''"U"'5.
Conclusion
In the presently designed ion ring of DSR, the longitudinal
single-bunch and multi-bunch instabilities are induced for thebunch population making the betatron tune shift l4vi=0.2 andthe bunch length of 40 em. The transverse multi-bunch
instabilities may be induced, which is dependent on thechromaticity. The electron cooling with the cooling time of 0.01 s
can damp the instabilities. The luminosity for 150 MeV/u "SU•"is much less than 1 x 1027 cm-'r'. In order to improve theluminosity, the design of DSR lattice should be modified asfollows; 1) the beta function at IS is reduced by one order, 2) theoptics of the lattice of the electron ring is variable enough toincrease the emittance of the 300 MeV electron beam up toaround 0. 5 x10_. n mrad, 3) the number of bunches of both thebeams are increased by a few times.
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