BUNCH DIFFUSION IN A STORAGE RING
Y. Mizumachi

National Laboratory for High Energy Physics
1. Introduction

In the proton bunch storage experiment at the CERN SPS machine, twotypical longitudinal phenomena have been observed. One is the turbulent
coherent oscillation, mainly of quadrupole mode, in the early period of thestorage. This oscillation cause the emittance blow-up. The other is observed
after the turbulent oscillation has stabilized itself and the bucket is more
fully occupied. One sees gradual decrease of bunch intensity with the lostparticles still coasting in the ring. Therefore, it is called a diffusion
p-rocess.
Soon it was recognized that the diffusion is closely related with
the noise in the RF system; the reduction of the band width of the. radial
position feedback loop has brought about the remarkable improvement of beamlife. In this report the basic procedures of the computer simulation of thebunch diffusion process are described.
2. Principle of Simulation
In the stationary condition, the equation of phase oscillation iswritten in (~,~~X) coordinate as

1\.S/_7iAr>v\(3¥'We will consider here phase noise.
of 9' in (2) as

cp~\~/"'/

It is represented by. small disturbance'f +Ll (-t)

where n(t) is a small random time variable.
taking normal distribution for Ll ( t) •We can introduce white noise by

In order to see how the RF noise gives rise to bunch diffusion, we takemany test particles ( ~ 200) in the RF bucket. Then we trace the phasemotion of those particles and observe how the distribution varies from theinitial one. However, even if we assume a very localized distribution at
first, it is soon elongated due to the nonlinearity of phase oscillation.
We can eliminate such apparent diffusion by locating the particles on amatched trajectory. All the particles remain on the trajectory as long as
there is no noise and the noise effect is demonstrated by the deviations ofparticles from the trajectory.
From (1) and (2) we have the following constant of motion in thestationary condition without noise;
D2= ~2..£ + TC./?JEo(L:lfJJ')2.
2Yev

2(3)

This means that n2 is constant on the trajectory and the trajectory istransformed into a circle in the normalized coordinate system (X,Y) ofFig.l, where
X = ~ :f.
y == 2(-eV
7f "/Eo ( <:1f3 y )( 4-)
2In this coordinate system, D is the distance of the particle from the or1g1n
and the bucket boundary is transformed into a unit circle. Therefore, thediffusion by noise can be demonstrated by the growth of the variance of D.
1of the particles, V(Di) which is defined as follows;

ffm

V(Pi) =I

l'i-N _2- (P(- Di)N

2.

(. ~~Di =N

2: D;
t-==1-311-(5"")

3. Some Results
An example of simulation is shown in Fig.2 for large amplitude
oscillation. We see linear growth of V(D.) which does not appear insmall amplitude oscillation: This shows ·tfie effect of nonlinearity.

yX
distribution circle

Fig. 1

Distribution of test
particles in thenormalized coordinate)~·V(Di) Sinusoidal 170°a.

Uniform mode-.lOE-o·S'o--'--L----'---'-------'--'---''----'-~'--___,__.....___.

-.o.2.3.4.sTIME
Fig. 2_.s_.___..__.__..______._-7.a

Growth of particle distribution spreadfor large amplitude oscillation-312-_.____,.___,,g