Resistive Wall and the Beam Breakup Multibunch Instabilities in Beams withLong Gaps
H. Fukuma, KEK, Tsukuba, Ibaraki 305, Japan,D.V. Pestrikov, Budker INP, 630090 Novosibirsk, Russian FederationAbstract
In circular accelerators and storage rings weak, but longrange wake fields close the feedback coupling of multibunch oscillations in a beam which has a long gap containing no bunches. Even if the leading perturbation of thecoherent oscillations occurs due to a strong wake with thememory shorter than the gap, the effects of these weak,
but long-range wakes eliminate a possibility of the beam
breakup multibunch instabilities in circular machines because the oscillations are expressed by well defined eigenfunctions which are exponential functions of time. We discuss main features of resulting multibunch modes and theirrelation to conventional beam breakup instabilities.

1INTRODUCTION

The high luminosity performance in electron-positron factories is achieved in the multibunch operation modes. For
several reasons, the beam filling patterns in these rings havethe gap, which is empty of bunches. For example, in KEKB
and PEP-II the length of this gap is approximately 10% of
the circumference of the closed orbit. Severe limitations on
the high current operation performance in such machines
may pose coherent instabilities of transverse oscillations ofbunches due to the fast ion instability (FII; e.g. in [1]) in theelectron ring and due to the photo-electron instability (PEI;[2]) in the positron ring. The length of the gap in the bunchtrain is usually chosen to clean-up the orbit from the secondary charges. Therefore, the wakes due to these charges
completely disappear during the gap length. We shall callsuch wakes as the short-range ones (WSR). Without otherssuch wakes result in the beam breakup-type (BBU) multibunch instability of the beam. One of the most importantfeature of these instabilities is an absence of the eigenfrequency spectrum of the coherent oscillations (e.g. in [3]).
Even in the case where the oscillations are damped artificially, the amplitudes of BBU oscillations grow till somefinite level and then decay. Due to specific for the BBU instabilities amplification of oscillations along the train it canlimit the high current operational performance of the beam.
In real machines effects of the BBU producing wakes
are accompanied by weak, but long-range wakes (WLR)
providing the multiturn interactions of bunches and closing the feedback of coherent oscillations of bunches along
the beam. In beams with a uniform filling pattern effectsof WLR gives tiny corrections to the oscillation eigenfrequencies and, generally, can be neglected. However, in thecase of the BBU instabilities effects of WLR qualitatively
changes the spectra of coherent oscillations producing theireigenfrequency spectra. In the case of artificial damping,

all such eigenmodes will decay provided that the maximum
damping decrement exceeds the maximum instability increment. Although as was discussed in [4], the amplitudes
of coherent oscillations can grow along the train, due todamping they all may stay in a safe region.

2THE MODEL

We consider a simplified model where the WLR is presented by the resistive wall wake, while the WSR is a wakedescribing the interactions only of two neighbor bunches
in the beam. Describing (the vertical) dipole coherent oscillations of bunches using their centroids in the phasea= ẏa − imy ωy ya (where my = ±1, and a =space ξm
1, 2, . . . , Mb , Mb is the number of bunches in the beam),
assuming that wakes only slightly perturb coherent oscillations during the revolution period (T 0 = 2π/ω0 ) as wellas during unperturbed oscillation periods (2π/ω y ) and zerobunch lengths, we write(b)(b)+ imy ωy ξm= imy Aξ˙m∞(a)ξm (t − [b − a]Tb )√b−aa=b−1(b−1)− imy W ξm(t − Tb ).

(1)

The first term in the rhs of Eq.(1) describes effect of theresistive wall wake [5]. We assume that identical M bbunches uniformly fill the train with the period 2π/M . If
M − Mb > 1, the second describes the BBU interactionsof bunches in the beam. The strength of this wake is givenby the parameter W . If N is the number of particles in thebunch, d is the pipe radius and σ is the wall conductivity,
the parameter A reads
A=πN e2√,3γme d3 ωy σTbTb =T0.
M(2)For example, for storage rings in KEKB [6] we have
N/γ = 4.4 × 106 , Ts  2 ns, σ = 5.8 × 1017 1/s [copper], d = 5 cm, νy = ωy /ω0  45 and correspondingly,A  1.8×10−5ω0 . If the taken WSR simulates PEI (e.g. in[7]) a typical value of W can be estimated as W  0.01ω 0.
Using substitutions, u = (ω − ωm )/(my W ), ωm =
ω0 νm , νm = my νy ,w = W/A, and
2πiaνm(a)ξm− iωt ,(t) = Xa expMwe transform Eq.(1) to the following eigenvalue problem:
uXb =Mb
a=1

Pb,a Xa ,(3)where (b > 1)

(7)

4 OSCILLATIONS IN THE TIMEDOMAINThe amplitudes of the bunch oscillations are calculated using eigenvectors X a,q and eigenvalues u q of Eq.(3)Xa (t) =Mb −1

Xa,q Cq exp(−imy W uq t),(8)

Xa,q Cq .

(9)

q=0

Xa (0) =M
b −1

q=0

Without artificial damping the bunch oscillation amplitudes
grow with increments of the order of W . A singlebunch

0,00,51,0Re(uq)Figure 1: The mode map. Open circles – eigenvalues,crosses – approximation, harmonic number M = 200,Mb = 180, w = 100, νm = 0.2123.

10210110010-13060

M od

This equation shows that the eigenvalues have a branch
point when (1/w) → 0. As seen from Fig.2, the amplitudesof the eigenvectors have substantial amplification along thetrain.

-0,5180150
120906030r

which results in (q = 0, 1, . . . , Mb − 1)2πi[νm + q]
1expuq =  √.
1/MbMbw M − Mb

(6)

-1,0bee√,uMb w M − Mb

-1,0m

2πiνm-0,590e nu

120m bernu

Although typical values of w are large Eq.(3) cannot be
solved using the perturbation theory i.e. expanding solutions in the power series of 1/w. The reason is that in
the case of strong BBU interaction both eigenfrequencies
and eigenvectors of Eq.(3) are non-analytic functions of
1/w. Examples of numerical solutions of this equations areshown in Fig.1 (the eigenvalues) and in Fig.2 (the eigenvectors). If the BBU interaction dominates, the shape of themode map reminds very strongly the mode map calculated
for the beam containing M b bunches with a uniform filling
pattern and neglecting the resistive wall wake. For large
number of bunches in the beam and w  1 a reasonableapproximation to the calculated spectrum is obtained, if
we assume that inside the train the bunches mainly interact
due to WSR (XMb (u)  (1/u)Mb e2πiνm X1 (u), w  1),
while the resistive wall wake contributes only in the excitation of the first bunch in the train by the last one. In thiscase the (approximate) dispersion equation is obtained using X1 (u) = K(u)X1 (u),150

h3 EIGENMODES

0,0180ncand θ(x) = 1, if x ≥ 0, otherwise θ(x) = 0.

K(u) 0,5(5)Bu∞exp (2πiνm l)√,w
lM + 1 − al=11,0-Im(uq)P1,a = −∞θ(b − a)  exp (2πiνm l)√√−, (4)
w b − a l=1 w lM + b − a

Amplification

Pb,a = δb,b−1 −Figure 2: Amplification coefficients to eigenvectors|Xa,q /X1,q | versus the bunch and mode numbers. Sameconditions as in Fig.1.
damping with decrements slightly exceeding the maximum
instability increment (Γ > W Im(u q )max ) is necessary tocure this instability. Exactly at the threshold of instability(Γ = W Im(uq )max ) the oscillation amplitudes still growup due to BBU interactions of bunches (Fig.3). Generally,
the distributions of amplitudes along the bunches and intime depend on the nature of WSR. For example, similarcalculations with wakes corresponding to the FII, taken e.g.
from [1], and at the threshold of instability yield substantial amplitudes of bunch oscillations in the tail of the train(Fig.4), which appear after time intervals substantially ex-Oscillation amplitude [Arb. u.]

ceeding the risetime of the most unstable mode. This effectis specific for the BBU interactions which couple manybunches in the train (e.g. in [1]). For this particular instability the amplitudes of bunch oscillations never exceed
initials, if the damping decrement about twice exceeds themaximum increment.

10100-110-210-3

10180150120-4T im30e inte rv

60

a l [13090/(2 w60Bu)]90ncuhnmberFigure 3: Amplitudes of the bunch oscillations versus thebunch number and time. Initially all bunches have unit displacements, Γ = Im(uq )max , other conditions are same as
in Fig.1.

5 CONCLUSIONSIn a storage ring containing the beam with a long gap nobeam breakup instability can occur because of effects of
the weak but long-range wakes. In any case, such a wakecan be due to the finite resistivity of the vacuum chamber walls. Even in the cases where the WSR dominates,
WLR close the feedback of bunch oscillations along thebeam with the long gap resulting in the eigenmode spectra of multibunch oscillations in these conditions. Since noeigensolutions appear, if the long-range wakes are ignored,
effects of WLR on a multibunch instability in a storage ringcannot be calculated using the perturbation theory.
If the number of bunches in the train is large, the values of increments of multibunch modes very weakly depend on the strength of the weak wake provided that theBBU interaction dominates. In such a case, the detectionof the eigenfrequency spectrum is the main indications of
the effect of WLR. Unstable multibunch eigenmodes canbe damped using conventional singlebunch feedback systems. Due to BBU behavior of the modes the damping
decrement should exceed by some amount the maximum
mode increment. However, the oscillation amplitudes canbe kept in a safe region provided that the damping is strong
enough.

6REFERENCES[1] T.O. Raubenheimer and F. Zimmermann, Phys. Rev. E 52,5487, 1995.
[2] K. Ohmi, Phys. Rev. Lett. 75(8), p.1526, 1995.

Oscillaton amplitude [Arb. u.][3] N.S. Dikansky, D.V. Pestrikov, Physics of Intense Beams andStorage Rings. AIP PRESS, New York, 1994.

10[4] H. Fukuma, D. Pestrikov, Transverse Multibunch Coherent
Modes of a Beam with a Gap, in these Proceedings, Report
P1hlf11.

2

10[5] N.S. Dikansky, A.N. Skrinsky, Atomnaya Energiya, 21, p.
176, 1966.

1[6] KEKB B-Factory Design Report. KEK Report 95-7, 1995.

10

10[7] M.A. Furman, G.R. Lambertson, In Proceedings of MBI97,p. 170, KEK Proceedings 97-17, 1997; see also in S.A.
Heifets, In Proceedings of CEIBA95, p. 295, KEK Proceedings 96-6, 1996.

0-1

10-2

20T im e

40in t e r v

60a l [0 .

6080

1τ ]
min

90

180
150
120

30100

Bc
unhnumber

Figure 4: Same as in Fig.3 but for WSR due to FII. Initially
all bunches have unit displacements, Γ = Im(u q )max ,M = 200, Mb = 180, νm = 0.2123, time intervals aremeasured in units of the inverse maximum increment.