Transverse Multibunch Coherent Modes of a Beam with a Gap
H. Fukuma, KEK, Tsukuba, Ibaraki 305, Japan,D.V. Pestrikov, Budker INP, 630090 Novosibirsk, Russian Federation
Abstract
We study transverse multibunch instabilities in a beam
where a uniformly filled train of bunches is followed bythe gap empty of bunches. Contrary to cases of uniformly
filled beams, the amplitudes of multibunch eigenmodes insuch a beam grow towards the end of the train. This amplification of oscillation amplitudes along the beam can limit
the beam performance in high current operation of accelerators especially near thresholds of instabilities.

ya , where a = 1, 2, . . . , Mb is the bunch number. If effects of the beam induced fields are described in terms ofthe transverse coupling impedance and the bunch wakes
are weak enough to disable the coupling of betatron modes(a)ξm = ẏa − imy ωy ya , where my = ±1, the Fourier am(a)(a)
plitudes ξm (t) = ξm (ω) exp(−imy ωy t − i∆ωm t) arefound by solving the system of equations (see, e.g. in [2])(a)uξm(ω) =Mb(b)ξm(ω)Ba,b ,b=1

1

INTRODUCTION

The high-current operations of particle accelerators andstorage rings can be limited by multibunch instabilities
of coherent oscillations. In beams with uniform filling
patterns the amplitudes of multibunch modes are constantalong the beam. In modern storage rings and colliders due
to tight spacing of bunches in the beam and for many technical reasons the beam filling pattern should contain a gap(or gaps), which is empty of bunches. In these conditions
the singleturn interactions of bunches result in the beam
breakup instability (BBI) of the beam (see, e.g. [1]) withspecific amplifications of the bunch oscillation amplitudes
along the train. One of the most important feature of theBBI is that coherent oscillations of the beam have no eigenmodes (see, e.g. in [2]).
In storage rings, the bunches move along the closed orbit. The resistivity of electrodes surrounding the beam may
result in multiturn interactions of bunches, which couple
the oscillations of all bunches in the beam. Contrary to thecase of BBI, such oscillations always have the eigenmodespectra. This means that, generally, in a storage ring no
BBI can occur. However, since the gap in the beam makes
an obstacle for the propagation of the multibunch eigenmodes along the train, the energy of coherent oscillations
is accumulated at the tail of the train. Correspondingly, inthe beam containing the gap coherent oscillations may havethe properties which are specific for both conventional andfor the beam breakup instabilities. In particular, the amplitudes of multibunch eigenmodes will be amplified along
the beam.

2THE MODEL

For simplicity we make calculations for vertical dipole
coherent oscillations of identical M b bunches with zero
length which uniformly fill the train with the period T b =T0 /M , where T0 is the revolution period of particles along
the closed orbit. The beam has a gap, if M > M b . We
neglect the singlebunch interactions in the train. Coherent oscillations of bunches are described by their centroids

Here,

∆ωm =

my N e 2u. (1)
2pνy
∞1 
2πikνmexp
Z ⊥ (kTb )MMk=1M−1
2πin[b − a + k]
exp×,M
n=0Ba,b =(2)N e is the bunch charge, p is the momentum of the reference particle, ν y is the betatron tune, ν m = my νy ,Z ⊥ (t) is the transverse coupling impedance in the time domain. Separating in Eq.(2) summations over bunches in thetrain from that over the revolution periods and defining as(b)Xb = ξm (ω) exp (−2πibνm /M ), we arrive at the eigenvalue problem typical for the multibunch instabilities:
uXa =a−1
Xb Z ⊥ ([a − b]Tb )b=1+MbXb

b=1∞e2πikνm Z ⊥ (kT0 + [a − b]Tb ).

(3)

k=1

For the beam with a gap the first line in Eq.(3) describes
the coupling of bunches during the current turn in the ring
while the second describes the multiturn interactions ofbunches. Without multiturn interactions the eigenvalue
problem has no solutions except for trivial, so that only
BBI becomes possible in the train. These instabilities occur in linacs. In a storage ring, the part describing themultiturn interaction, generally, never vanishes. So thatEq.(3) always has nontrivial solutions. If the numbers
q = 0, 1, . . . , Mb − 1 mark the eigensolutions of Eq.(3),(a)the bunch centroids in the train are found using ξ m (t) =exp(−imy ωy t + 2πiaνm /M )Xa (t),Xa (t) =M
b −1
Cq Xa,q exp(−i∆ωm,q t),(4)q=0and initial conditions


Mb −1

2πiaνm(a)exp −(0) =Xa,q Cq .
ξmM
q=0(5)

the amplitudes
 Aq can be found from the normalization
condition a |ξa,q |2 = 1. Using these expressions we findthe solution to an initial value problemMb
1 (a)(b)ξm(t) =Ga,b (t)Ka−b (Mg )ξm(0),Mb(9)b=1bunches, and
2πiq[b − a]exp −i∆ωm,q t +. (10)
Mb
q=010

2,5|ξ|

2,0
1,5
1
1,00,520

40

60

80

Eigenvector amplitude

3,00

4
20-2-6-10-8-60,0100

Bunch number

Figure 1: Dependences of the bunch amplitude (solid lines)
and of the eigevector amplitudes (dashed line) on the bunch
number. M = 100, M b = 90, λT0 = 10, for all bunches(a)ξm (0) = 1, solid lines from top to bottom t = 5τ , 3τ andτ, where 1/τ is growthrate of the most unstable mode.
ence of the multibunch oscillations in the beam with a gap
from that in the beam with a uniform filling pattern. According to Eq.(7) the amplitudes of eigenvectors of multibunch modes increase towards the end of the train (Fig.1).
This behavior of |K a | increases contributions to the amplitude of oscillations of a given bunch in the train from-4-2

02Re(uq)

Mb −1
In these equations, the factor K a (Mg ) describes the differ-0,16-4(b)where the values ξm (0) are defined by initial offsets ofGa,b (t) =the preceding bunches of the train and decreases the contributions to this amplitude from subsequent bunches in thetrain. As a result, initial deflections of bunches are pushed
by the instability from the head to the tail part of the train(Fig.1). Similar amplifications of the oscillation amplitudes
along the train are specific for BBI. In a storage ring, theamplifications of eigenvectors are stronger for longer gapsand for shorter interaction radii of the beam bunches. According to Eq.(8), the eigenfrequency spectrum of multibunch modes in the beam with a gap (Fig.2) does not differ very strongly from that for a beam with a uniform filling pattern. Solving Eqs.(3) and (5) numerically, we findIm(uq)Generally, Eqs.(3) and (5) should be solved numerically.
For simplicity we consider a special case with Z(t) =−Z0 e−λt where Eq.(3) can be solved analytically. Varying λ we can change in this model the number of interacting bunches as well as the strength of the multiturn effects. Defining as ∆ωm,q = −my N e2 Z0 uq /(2pνy ), wefind (Mg = M − Mb )
−2πiqaξa,q = Aq exp(6)
Ka ,Mb
aMg
Ka (Mg ) = exp [λT0 − 2πiaνm ],
(7)
M Mb


2πi[νm + q] − λT0F, F = exp
uq =, (8)1−FMbFigure 2: Mode map. Same conditions as in Fig.1.
that described features of the multibunch oscillations in abeam with the gap hold for other multibunch wakes withthe bunch to bunch memory longer than the length of thegap.

3DAMPINGSince about a half of the multibunch modes are unstable,
the multibunch operations demand relevant damping mechanism to stabilize the beam oscillations. If unstable modes
are damped by a singlebunch feedback system providing a
damping decrement Γ, then Eq.(9) reads
M(a)(t) =ξmbe−Γt (b)Ga,b (t)Ka−b (Mg )ξm(0).
Mb(11)

b=1

As seen from Fig.1, without additional damping even forequal initial displacements of bunches along the beam theinstability pushes a maximum bunch deflection along thebeam towards end of the train. If 1/τ is the increment of themost unstable mode and if ∆t = τ is used as the time scale,
we see that during many of τ the distribution of bunch deflections along the train differs from that corresponding to
the most unstable mode. As was mentioned, this feature issimilar to BBI. As seen in Fig.3, for a 10 % gap a singlebunch damping with Γ = 1/τ results in a general decay ofbunch oscillations. So that a usual requirement for the feedback gain to provide the damping decrement slightly exceeding the maximum increment is quite safe for this case.

However, if we increase the length of the gap three times
|ξ|

0,16
3τ

0,40,12
τ

0,3

9τ

0,08|ξ|

15τ

0,2

0,04
3τ

5τ
0,00

0,1

-10

0

10

20

30

40

50

60

70

Bunch number

0,0
0

20

40

60

80

100

Bunch number

Figure 5: Same as in Fig.4, but Γ = 1.5/τ .

Figure 3: Dependences of the bunch amplitude on thebunch number. Same conditions as in Fig.1, characters nearcurves show the time interval.

4 CONCLUSIONSand decrease λ in Z(t) four times, we arrive at the oscillation history shown in Fig.4. In this case a 30% excess of|ξ|24τ

1,51,015τ

0,530τ

9τ

3τ

0,0-100102030

40

5060

70

Bunch number

Figure 4: Dependences of the bunch amplitude on thebunch number. M = 100, M b = 70, λT0 = 40, for all(a)bunches ξm (0) = 1, Γ = 1.3/τ , characters near curves
show the time interval.
Γ over the threshold value does not prevent the oscillation
amplitudes from the region where the amplitudes exceed
their initial values. According to Eq.(8), τ ∝ λT 0 . So thatthe short-range interaction does not pose additional limitations provided that Z 0 is given. The required additional
strength of the damping system becomes important, if thestrengths of the short-range wakes strongly exceed that of
the long-range wakes (see, e.g. [3]). As seen from Fig.4,
during the damping and after a rather long time interval
the amplitudes decay after passing their maximum. Thisbehavior is also specific for BBI. For this particular beamgeometry no blowup in the beam oscillation amplitudes occurs so that the oscillation amplitudes never exceed theirinitial values, if Γ = 1.5/τ (e.g. in Fig.5).

The multiturn memory of the bunch wakes prevents themultibunch BBI in the beams with gaps in storage rings.
The oscillations of bunches in these beams are presented byrelevant combinations of the multibunch eigenmodes. Nevertheless, since the gap makes an obstacle for propagationof the eigenmodes along the beam, the amplitudes of themultibunch eigenmodes increase along the train. Withoutartificial damping, the amplitudes of the bunch oscillationsat large time intervals also increase along the train.
If the multibunch modes are damped using a singlebunch
feedback system and if the number of the damped modes
is large, the bunch oscillation amplitudes will not exceed
their initial values provided that the gap is short enough. In
this case, the required power of the damping system can be
estimated assuming a uniform filling pattern in the beam.
With the short-range interaction between bunches thedamping decrement should exceed the maximum instability increment by the amount, which depends on the lengthof the gap and on the interaction radius of bunches in thebeam. A strong damping (e.g. Γ = 2τ ) eliminates initial
blowup of the bunch oscillation amplitudes, which is specific for BBI. However, near the instability thresholds theoscillation amplitudes of the tail bunches may decay after
passing the maxima exceeding their initial values and thus
can limit the high current beam performance.

5REFERENCES[1] T.O. Raubenheimer and F. Zimmermann, Phys. Rev. E 52,5487, 1995.
[2] N.S. Dikansky, D.V. Pestrikov, Physics of Intense Beams andStorage Rings. AIP Press, New York, 1994.
[3] H. Fukuma, D. Pestrikov, Resistive Wall and the Beam
Breakup Multibunch Instabilities in Beams with Long Gaps,in these Proceedings, Report P1hlf09.