The-11th Symposium on Accelerator Science and Technology, H arima ·Science Garden City, 1997

Diffusion Instability in Noisy Beam-Beam Interaction
Yuri BATYGIN and Takeshi KATAYAMA*The Institute of Physical and Chemical Research (RII<EN), Wako-shi, Saitama 351-01; JAPAN*center for Nuclear Study, School of Science, University of Tokyo, Tanashi 188, JAPANa-Abstract
Beam-beam instability in the presence of random
fluctuations in opposite beam size is studied. Specificfeature of noise beam-beam instability is that this instability
can appear apart from excitation of nonlinear resonances .
Diffusion coefficient of noise beam-beam instability
exhibits quadratic dependence on the values of beam-beam
parameter and noise amplitude and sharp dependence onnormalized beam size. In the vicinity of low-orderresonances beam size become larger, which results in much
faster diffusion. Diffusion coefficient of instability as afunction of beam-bean parameter, amplitude of-noise andbetatron tune value is presented.

0.06

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I " ......_

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426x /crb1 Introduction

1.2

One of the main problems in ion-ion circular colliders isa small value of achievable beam-beam tune shift ~=0.005.
Physical reason for beam-beam limitation is usually
attributed to the excitatjon of a set of nonlinear resonancesdue- to a periodic nonlinear kick in linear system.
Overlapping of nonlinear resonances is an universal
mechanism of stochastic particle instability in nonlinear
systems. Another mechanism of unstable particle motion isa diffusion, created by a noise [ 1-6]. This noise can exist,
for example, due to mismatch of the beam with the channel.
In this paper we study the noise which appears in an
incoherent beam-beam interaction. As it is shown below,
such a noise always induces diffusion instability if beambeam kick is a nonlinear function of coordinate. Due to thediffusion character, noisy beam-beam instability does nothave a threshold character and can exists under any value ofbeam-beam tune shift.

1.15
0

1.1

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1.05

w
.._w

11...111.
..-...

l,.,

J...

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....

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0.9

0 20 40 60 80 100 120 140160 180 200103 nFig. 1. Stable particle motion with the values of beam-beamtune shift~ =0.005 and betatron tune Q=3.168:a) beam-beam kick, b) rms beam emittance.

2 Noise Beam-Beam Instability
Noise beam-beam instability can be easily demonstrated
via computer simulations. Let us consider for simplicity a
one-dimensional model. Results of this study are valid for
multi-dimensional problem as well. Particle motion is
described in coordinates (x, p =~ dx/dz), where ~ is a valueof beta-function of collider. Between subsequent collisions
particle experience linear matrix transformation withbetatron angle 9 =21tQ. Beam-beam interaction is treated asa thin lens with nonlinear beam-beam kick .1pn:( Xn+l) ( COS 9
Pn+l = -sin 9

sin9 )( Xn)
COS 9
Pn + .1pn(1)Beam-beam kick .1p is expressed by a Gaussianfunctionwhere x is a particle position and CJ is a standard deviationin opposite beam size. Parameter ~ is a beam-beam
parameter, which characterizes the strength of interaction:(3)
where N is a number of particles per bunch,ro q 2/(47reomc 2) is a classical particle radius, and y is aparticle energy.
Nonlinear kick (2) induces a set of nonlinear
resonances, which are stable, until islands do not overlap
each other. Isolated nonlinear resonance of the order pis
described by Hamiltonian [7]:-505~=

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

a0.06.--..

lJJ'~0.04

I

0.02~ ,....__

1/-.::!"'-'r i'-.

small beam mismatch with the channel. The value of noise
amplitude u=0.05 is chosen arbitrary to demonstrateappearance of diffusion-type instability in the presence ofsmall random perturbation of beam-beam kick. As shown inFig. 2, this noise destroys the stability. In contrast with Fig.
1, beam emittance expand with time.
Important feature of the noise regime is that this kind of
instability can exist apart from the excitation of nonlinear
resonances. Noise beam-beam instability appears if two
conditions are met:
- beam-beam kick is a nonlinear function of coordinate;- parameter of beam-beam kick (beam standard deviation <r)
is a subject of noise.

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20

64x /crb3 Analytical Treatment of Effective Beam Emittance
Growth1.2
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0

1.1c1.05.

w

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w

1 P"'""""'~ ..,....

,Idillr

Let us provide analytical estimations of emittance
growth under noise beam-beam interaction. Transfer matrix
after n turns with arbitrary momentum kick at every turn,
Api , is given by the expression [5]:~n-1

Xn =a cos (n8 + \{1)+0.95

L Api sin (n-i) e 'i=O(6)n-10.90Pn =- a sin (n8+ \{1) +20 40 60 80 100 120 140 160 180 200

i=O103 nFig.2. Beam-beam instability under 5% noise in theparameter <r during particle interaction with ~ = 0.005 and
Q=3.168: a) beam-beam kick, b) rms beam emittance
growth.

H (I,'Jf) =I·( Q- m ) + ~ U(I) + ~ V(I) cos p'Jf ,pwhere 'P is the initial phase of oscillations. Random beambeam kick (Api ) can be expressed as a function ofunperturbed trajectory. It gives an approximate treatment of
the problem, valid for small values of perturbation.
Suppose, beam-beam kick is a nonlinear function ofunperturbed trajectory:
(7)

(4)

We study evolution of the root-mean-square (rms) beam

where (I, 'If) are action-angle variables, Q is a betatrontune, m and p are integers, U(I) is a detuning factor and V(I)is a resonance strength function.
In Fig. 1 an example of stable beam-beam interactionin the vicinity of the 6th order resonance is presented. In
simulations the beam is presented as a collection of 3000
particles. Simulations are done for the value of betatrontune Q = 3.168, close to the 6th order resonance value
3.16666 (or 6Q=19). From the simulations it is clear thatbeam emittance is stable.
Another picture is observed if the parameter <r of beambeam kick (2) is a subject of noise. In the calculations,presented in Fig. 2, parameters of the process are chosen to
be the same as those in Fig.l, but standard deviation <r ischanged from turn to turn according to the expression:(J(n) = <r(o) (1 ±!!. Un) 'I Api cos (n-i) e 'emittance En = 4 V<xfi> <pfi> ~· <xn Pn>2, where brackets
mean averaging on initial phases of particles. Calculation ofrms beam emittance gives:

4 n-1

6 n-1

- [R I, & sin(2n8-2i8)- ~

22i=O
4 n-1

2L & cos(2n8-2i8)] 2i=O
6 n-1

2

- [R I, Oi cos(2n8-2i8) +~I, Of sin(2n8-2i8)] .
2 i=O
2 i=O

(8)

Effective beam emittance growth is then as follows:

(5)

(9)

2
where u is a noise amplitude and Un is a random noisefunction within the interval (0, 1). It corresponds to the noisein the size of the opposite beam, which can exist due to-506-~ (84) = 25 a8n-1n-1i=O

k=OI,. I, olo~ [ 1- cos(2i8 -2k8)] . (10)The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Eq.(lO) indicates emittance growth in presence ofnonlinearity in beam-beam kick due to nonvanishing termxlO

o2.

0.4
0.35

Comparison of eqs. (2), (7) and (9) gives an expression
for a diffusion coefficient D in beam emittance growthunder noise regime En/Eo = Y1 + D n :70.3(11)0.2/0.1

Diffusion coefficient exhibits quadratic dependence onnoise amplitude u and on value of beam-beam parameter S
as well as fourth-power dependence on normalized beam
size (a/2cr0 ). It indicates, that noise instability in beambeam interaction appears under arbitrary small values of u
and I;.

v

~

0.15

20"ov

~!70.25

0
D = Tt2 (l;u)2 (__1!__)4.

-5/~/

0.05
0~
00.02

0.12

0.1

0.08

0.06

0.04uFig. 3. Diffusion coefficient D as a function of noise
amplitude u in parameter cr for 1;=0.005: solid line calculated from eq. (11), * -numerical values.

4. Numerical Study of Diffusion Coefficient
In Figs. 3-5 results of numerical evaluation of diffusion
coefficients are presented. Working point in simulations
given in Figs. 3, 4 was chosen far enough from low-ordernonlinear resonances, so the beam size was equal to that of
the opposite beam a=2cr 0 . As it follows from Figs. 3, 4,
numerical values of the diffusion coefficient are close to theanalytical estimation ofEq. (11).
Excitation of high order nonlinear resonances results inincrease of beam size. Without noise in beam-beam kick thenonlinear resonances are stable. In the vicinity of nonlinear
resonances, where beam size become larger, diffusion has
to be faster, according to the expression (11). Diffusion
coefficient as a function of betatron tune due to different
beam size is summarized in Fig. 5. As seen, in the vicinityof low-order resonances diffusion goes much faster. The
diffusion coefficient has a peak values near resonances of
the 8-th order (Q=3.1285), 6-th order (Q=3.1666), 4-thorder (Q=3.25) and 3rd order (Q=3.333). These values ofbetatron tune correspond to the largest values of diffusion
coefficient.

xlO

0-5

0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0""
~/v

///v

~./

0~
0.002 0.004 0.006 0.008 0.01 0.012~
Fig. 4. Diffusion coefficient D as a function of beam-beam
parameters for the value of noise amplitude u = 0.05:
solid line- calculated from eq. (11), *-numerical values.

References[1]. Y.Batygin and T.Katayama, "Noise instability innonlinear beam-beam interaction", RIKEN AF-AC-3,
March 1997.
[2] K.Cornelis, Tail Production due to the beam-beam,
Proceedings of the Third LEP Performance Workshop,ed. by J.Poole, CERN SL/93-19, 1993, p.123.
[3] J.K.Koga and T.Tajima, Phys. Rev. Lett. 72, 2025
(1994).
[4] T. Sen and J.A.Ellison, Phys. Rev. Lett. 77, 1051
(1996).
[5] V.Lebedev et al, Part. Accel. 44, 147 (1994).
[6] Y.Batygin and T.Katayama, In Proceedings of the Fifth
European Accelerator Conference, Barcelona, 1996,
edited by S.Myers, A.Pacheco, R.Pascual, Ch.Petit-JeanGenaz and J.Poole (Institute of Physics Publishing,1996), p.1170, p.l173.
[7JL.R.Evans, CERN 84-15,319 (1984).

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3 3.05 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5

QFig. 5. Diffusion coefficient as a function of betatron tunein presence of noise with amplitude u=0.05.

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