JAERI-Conf95-021

PB38
STUDY OF INCOHERENT BEAM- BEAM EFFECTS AT RADIOACTIVE
ISOTOPE BEAM FACTORY
Y.Batygin and T.Katayama *The Institute of Physical and Chemical Research (RII<EN), Saitama 351-01, Japan*Institute for Nuclear Study, University of Tokyo, Tokyo 188, Japan
Abstract
Merging ion-ion collisions is an important featureof the proposed RIKEN Radioactive Isotope Beam
Factoryl). In the merging collision case, the value ofluminosity 1026 l/cm2sec is several orders of magnitudeless than for head-on collisions because both beams havealmost the same vector of velocity and merging angle israther small ( 1-1 00) even when the stored number of ionsis close to the space charge limit of 1012 particles. In thepresent paper, the beam-beam effects are studied for thecoasting merging beam collisions using particle-in-cell(PIC) model in 4D phase space. Tolerable incoherent
beam - beam tune shift and beam disruption effect suchas emittance growth have been evaluated from high ordernonlinear resonances study. Beam luminosity and beam
life time due to beam-beam effects are estimated as afunction of main collider parameters.

yhx

1. Introduction

Proposed Radioactive Isotope Beam Factory is
aimed to be used for wide range of experiments withunstable nuclear beams. Among variety of planning
experiments the ion-ion merging collisions are of themost importance. Merged beam technique is very useful
method for the study of nuclear fusion processes.
Merging two RI lbeams deliver low energy collisions justabove the Coulomb barrier threshold that is difficult tobe realized in other experimental methods. The mostimportant collider parameter is luminosity which islimited among other reasons by physics of beam-beaminteraction. In this paper we analyze the luminosity
constraints originating from a beam-beam interaction as
a function of main parameters of storage ring.

Fig.1 Merge beam- beam interaction.

function of collision angle a, number of particles per
beam N 1, N2 , ring circumference 27tR and effective sizeof the beam heff at the interaction point
L = Ypr + pl- 2p1p2cosa- ~rpisin2a c Nt N2(21tR) 2sina herr

Let us note that for merge coasting beams the luminosity
is inversely proportional to beam height and does notdepend on beam width.

2. Luminosity of merge beam - beam interaction

3. Particle-in-Cell Model of Beam-Beam Interaction

We consider two coasting merge ion beams withparticle densities n1, n2 and beam velocities ;I = c~t ,-;2 = c~2 colliding with angle a (see fig. 1). Luminosity
L is defined as a ratio of interaction rate to cross section
for particle interaction L=l/cr dN/dt. Using expression
for invariant cross section2) the number of collisions dNduring the time dt isf=dt!!.N._(J- - 2(vt-V2)- [-Vt X -V2]2 nt n2 dV
c2(2)

(1)

v

where integration is performed over the volume ofinteraction. It is convenient to express luminosity as a

Beam-beam interaction was studied by combinationof particle-in-cell treatment of space charge problem atthe crossing point and transfer matrix for particle
revolution in storage ring. Self- consistent considerationof the problem is based on calculation of beam-beaminteraction arising from the intrinsic space charge field of
the beams. Beams are represented as a collection of large
number (lo3 - lo4) modeling particles. Real number ofparticles in the beams is much larger (1012). To reduce
the numerical fluctuation arisen from over evaluatedparticle-particle collisions, the smoothing particle-in cell
technique is used. For merging ion-ion collision we use
the assumption that two beams are identical therefore wecan consider behavior of only one beam (the second-356-JAERI-Conf 95-021

beam is the same). At the point of interaction influenceof one beam on another one is simulated as strong-strong
interaction of two identical beams with crossing angle.
Simulation starts with random number generation of
initial particle positions x,y and reduced particle
momentum Px= Px'Y· Py= Py'Y in four-dimensional phasespace. Initial distribution of the beam in transversal
phase space coordinates gives the elliptical phase space
projections described by root-mean-square {RMS) beam
ellipse aox 2+2boXpx+copi = £ where the RMS beamemittance e is defined as

is a nonlinear function of transverse coordinate due tononlinear space charge field of the beam. Gaussian beamwith N particles and transverse RMS size CJ delivers
space charge field:
Er=L[I-exp(- S>J= 2 ~ r+ ...
Nq81t
2o4x2 R eo r

(9)From eqs. (7), (9) the linear approximation to beambeam kick is as follows:
(10)

(3)

analogously for y, Py· Simulation of each turn of particle
with longitudinal momentum Ps = Ps'Y consists of transfer
mapping of one revolution of particle in storage ring ofradius R with betatron tunes Qx. Qy and nonlinear kick~Px. ~Py due to beam-beam interaction:(4)

Multiplication of matrix of one turn and linear beambeam kick gives the value of linear betatron tune shift formerge beam-beam collisions:(ll)where ro = q2/ 41t£o mc2 being the value of classicalradius of particle.

5. Results of simulation
analogous for yn+l, p~+l . Self-consistent beam-beamkicks ~Px. ~Py are calculated from space charge problem
with instantaneous distribution of particles :NX~Px =

qhx
~ Exi(Y)mc 2 tga~.f i=l~Py =qhx~ Eyi(Y)2mc sina~.f i=l(6)NX(7)

where mc2 /q is a rest energy divided by charge ofparticle, Exi (y), Eyi (y) are space charge field of thebeam calculated at spatial grid points, NX is a number of
equidistant mesh points along x - axis located with steph x (see fig. I). Space charge field of the beam is
calculated from the Poisson's equation in Cartesian
coordinates(8)with Dirichlet boundary conditions for potential U on
the surface of conducting pipe using Fast Fourier
Transforms.
4. Beam-Beam Tune Shift
In coasting merge beams the interaction is not x-y
symmetric. In the median plane of the beams (x-directionin fig. 1) particles experience much smaller kick than in
the vertical plane (y-direction). As a result, initiallyround beams · become prolonged in the direction
perpendicular to median plane. Realistic beam-beam kick

RI beam factory is supposed to have· 2 colliding
points, therefore the values of tune are half of betatrontunes in the ring: Qx/2 = 2.8815; Qy /2 =3.175. For twocoasting merge beams (see fig. 1) only y - direction isresponsible for degradation of beam luminosity due tocompensation of beam-beam kick in x-direction. Theclosest resonance value mQy = n to betatron tune value
3.175 is 6Qy = 19 , i.e. 6th order resonance is achievedfor Qy = 19/6 = 3.16666. Max tune shift from working
point to resonance is ~ = 3.175 - 3.1666 = 0.00833. At
fig. 2,3 and in Table the results of calculations for ~ =0.005 (non-resonance case) and ~ = 0.027 (resonancecase) are presented. Different initial nonlinear particle
distributions were treated in calculations: Gaussian
distribution p(r) =Po exp (- c2(2a2) and "parabolic"distribution p(r) = Po (1 - rl!R~) 2 • Evolution of RMS
beam envelope

2CJx = 2 {<.X£> =vf_l_N i=lxr(12)and RMS beam emittance e (see eq.3) were controlledduring the simulations as statistical averaged values overlarge number of modeling particles. From results ofsimulations it follows that below resonance threshold~<0.008 beams are stable, i. e. no envelope and RMS
emittance growth were observed. Under resonanceconditions ~ > 0.00833 beam-beam instability wasobserved. The Gaussian distribution is more unstable
than the parabolic distribution. Under the Gaussian
distribution the phase space portrait of the beam istypical for high order resonance (see fig.2 ) while forparabolic distribution the phase space projections are
ellipses as for non-resonance case .

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JAERI-Conf 95-021

Table. Results of PIC simulation of beam-beam effects(mesh NX x NY= 64 x 64).

model of beam-beam interaction. Max number ofparticles estimated from beam-beam interactions exceeds
the limit defined by incoherent space charge tune shift
for circulated beam. Further study is required for
estimation of synchro-betatron resonances on beam
parameters due to finite length of the interacted bunches.

BeamTune Turns Number Envelope EmittanceDistributionof particles Growth Growth(per 104 turns)

8. References
Gaussian

0.005 4·Io4Gaussian 0.027
Parabolic 0.027

Io3

1o4 5·1o3
2·1o4 5·103

1.01.0002(no growth)
1.03
1.01

1. Y.Yano, T.Katayama, Present Status and Future Planof RUffiN Accelerator Research Facility, Proceedings oftheEPAC94, London (1994), 515.
2. L.Landau and E.Lifshitz, The Classical Theory ofFields, Pergamon Press, 1975.

1.05
1.025

6. Limitation of Luminosity and Beam LifetimeThe calculation performed shows the limitation in max
tune shift due to beam-beam effects ~< 0.008 which istypical for ion-ion collisions. Assuming ~max = 0.008,
cry= 10-3m, a =1o, 13Y = 1.7, r0 = 1.5 xo-18m (proton) thelimited number of particles due to beam-beam interactionis (see eq.ll):

0.00050'~ax Qy cry sina 132 T
ro= 6. 1013.'r......

0.00025:. . :;~-r~.

=····

:.:.

>-Q._

N < 2 il.:: -~·· •;. .....

:i--~

·.i

0.00000.:.

:.
'"';'

... ... ,.;' ~

•,\ ~

(13)

-0.00025.·.:~~1\ ,·=:....f~J,fo.

:;:.

-0.00050.·:..... ·:;..

::.

This value is larger than the space charge limited numberof particles in the ring of radius R=28m due toincoherent space charge tune shift (Laslett tune shift)
L\Vmax= 0.25:.~.·~- .·~- ...... ,•.

?'

(14)
Taking limited number of particles Nmax = 3·1012, a=100, heff = 4·1Q-3m and assuming 131 = 132 ==1, thelimitation in luminosity is (see eq. 2):

L<Fig.2. Phase space trajectories of Gaussian beam near
nonlinear resonance of 6th order (~y = 0.027).

2

Nmax c tg !X.. =2 x1026
1(21tR) 2 herr
2sec cm 2(15)From results of simulation we observe theincrease of transverse beam size 1-3% per 10000 turns at
the resonance conditions. If we assume that for serious
degradation of luminosity the beam size should expanded
twice, the upper limit of beam lifetime in resonance is::,;
Q

0.105

>="

0.100

Nmax = 1.QQ%_ 10000 = 5 x 105
2%turns.

f-L---+--- +---r----r l(16)0.095 !:=i:::;:::;::::;:~~:::::;::~::2=::;::::i::::S~~::w~
05000
10000
15000
20000
TURN

If the resonances are avoided, the beam life time islimited by other reasons .
7. Conclusions
Particle-in-cell simulation shows significance of beambeam interaction on beam parameters. Beam lifetime andluminosity constraints were studied from strong-strong

Fig. 3. Envelope growth of the beam with parabolic
distribution, ~ = 0.027.

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