QUADRUPOLE WAKE-FIELD EFFECTS IN THE KEKB LINAC
Y. Ogawa, KEK, Tsukuba, Japan

Abstract
Quadrupole wake-field effects were observed for thefirst time in a single-bunch, high-intensity primaryelectron beam for positron production at the KEKB linac.
The observed effects, which emerged as x-y coupling intransverse motion, were in good agreement with a simple
calculation based on a two-particle model. The
observational results and the calculation are presentedwith special emphasis on the parametric-excitation natureof the phenomena.

1 INTRODUCTION
Wake-field instability concerning a high-intensity
electron linac is considered to be one of the most crucial
issues for stably operating the existing linacs, such as theKEKB linac [1], as well as of future linear colliders. In
this connection, some experimental investigations of thelinac instability have so far been carried out, while
focusing mainly on a transverse dipole wake-field effectand its cure. The quadrupole wake-field effects in linacs
have been considered to be negligibly small, simply
because they are essentially second-order phenomena. Abeam of increasing intensity, such as the 8-nano Coulomb
single-bunched beam in the primary high-intensity
electron linac for positron production at the KEKB linac,
however, may allow the quadrupole wake-filed effects toemerge as observable x-y coupling phenomena ofparametric-excitation nature. Simple calculations basedon the two-particle model as well as the firstexperimental results obtained for a high-intensity singlebunched electron beam at the KEKB linac are presented.

2 TWO-PARTICLE MODEL
2.1 Equation of Motion
Consider a round single-bunch electron beam intransverse space, which is accelerating under a constantgradient ( g) with an initial energy of γ0 in the presence ofan external smooth focusing system:2• γ = E / mc = γ0 + g s,d  dx γ
 + γ k 2x = F ,•ds  ds 
where E is energy, m the electron mass, c the speed oflight, k the betatron wave number, F the external force
due to a transverse wake field, and s is the distance down
the linac.
To simplify the effects of the transverse wake in asingle-bunch beam, let us introduce a rigid two-particle

model [2]. Half of the charge is placed into each macroparticle, while two particles are separated by a distanceof about 2σz of the longitudinal bunch distribution, whichis assumed to be rigid in the relativistic region. The wakefields at the trailing particle generated by the leading
particle are thus fixed; the dipole wake, W1(2σz) and thequadrupole wake, W2(2σz) are constant in this model.
The round-shaped beam has quadrupole moments (Qn,Qs) as well as dipole moments (Dx, Dy) due to atransverse offset of (x1, y1) of the leading particle:• D x = x 1, D y = y 1,22
• Q n = x 1 – y 1 , Q s = 2 x 1 y 1,
while the transverse wake-field effects at the trailing
particle (x2, y2) are characterised as the external-force
terms:• Fx = –Nr0/2 (W1 Dx + 2W2Qn x2 + 2W2Qs y2),
• Fy = –Nr0/2 (W1 Dy – 2W2Qn y2 + 2W2Qs x2),
where N is the total number of particles in a bunch, and r0-15
is the classical electron radius (r0 = 2.818 10 m).
The equations of motion for the two particles arewritten as follows:d  dx1 γ
 + γ k 2 x1 = 0,ds  ds d  dy1 γ
 + γ k 2 y1 = 0,ds  ds d  dx2 γ
 + γ k 2 x 2 = Fx ,ds  ds d  dy 2 γ
 + γ k 2 y 2 = Fy .
ds  ds 

2.2 x-y Coupling and Parametric Excitation
The equation of motion indicates two distinct features
for quadrupole wake-field effects:• x-y coupling phenomena mainly relevant to the Qsterms,• parametric excitation nature explicitly related tothe Qn terms.
Although the effects are mutually dependent so that,rigorously, they can not be separated from each other, let
us consider two extreme cases for numerical simulations
in order to verify that the quadrupole wake-field effects
at the KEKB linac are really observable:
22• Case I : x1 = y1 , that is, Qn = 0,22
• Case II: x1 >> y1 , that is, Qs ≅ 0.

2.3 Numerical Simulation
The linac parameters used for numerical simulations
relevant to the KEKB linac operation configuration at thefirst two sectors, A and B [3], are summarized in Table I:
Table 1: Linac parameters
Parameter
SymbolValue
10N
Number of Particles
5 10 (8 nC)
1.5 mmBunch Lengthσzk
Betatron Wave Number
2π/46 1/m
Initial Energy
230/0.511
γ0g
Energy Gain
20/0.511 1/m
53
4.5 10 1/m
Dipole Wake
W1(2σz)95
5.0 10 1/m
Quadrupole Wake
W2(2σz)In this case, the initial conditions are set so that thecoupling from the vertical to horizontal direction is smallenough to evaluate any horizontal effect of parametricexcitation nature:• x1 = y1 = 0, x1’ = 0.5 mrad, y1’ = 0.01 mrad (s =25.4 m).
It is shown in figure 2 that the horizontal oscillation
increases by about 15% due to parametric excitation,
while the vertical oscillation also increases by about 13%
through x-y coupling from a large horizontal oscillation.

Case I: Qn = 0
To clarify the quadrupole wake-field effects, twosimulation results with/without W2 are compared for thesame initial conditions:•x1 = y1 = 0, x1’ = y1’ = 0.3 mrad (s = 25.4 m).
Figure 1 clearly shows mutual x-y coupling effects dueto the quadrupole wake. The relative amount of about
12% increase in the orbit oscillation is attributed to thequadrupole wake-field effect.
(a)

(a)

(b)

Fig. 1: Simulations for x-y coupling phenomena dueto the quadrupole wake-field effects in linacs. The
betatron oscillations were calculated for the KEKB
linac by the simulation based on the two-particle
model using the linac parameters given in Table I.
The initial conditions are set to x1 = y1 = 0, and x1’ =y1’ = 0.3 mrad (s = 25.4 m), corresponding to Case I(Qn = 0). (a) W2 = 0, and (b) W2 ≠ 0. The orbit
oscillation in (b) is larger than that in (a) by about
12%, which is attributed to the quadrupole wake-field
effects. The dashed line shows a simulation without
any wake fields.
Case II: Qs ≅ 0(b)Fig. 2: Simulations for the parametric-excitation
nature of the quadrupole wake-field effects in linacs.
The betatron oscillations were calculated for theKEKB linac by a simulation based on the twoparticle model using the linac parameters given in
Table I. The initial conditions are set to x1 = y1 = 0,x1’ = 0.5 mrad, y1’ = 0.01 mrad (s = 25.4 m),corresponding to the Case II (Qs ≅ 0). (a) W2 = 0, and(b) W2 ≠ 0. The horizontal oscillation increases dueto the parametric excitation by about 15% comparedto (a), while the vertical oscillation also increases by
about 13% through x-y coupling from a large
horizontal oscillation. The dashed line shows asimulation without any wake fields.
Through these two simulation cases, it is verified thatthe quadrupole wake-field effects could emerge as
observable phenomena at the KEKB linac.

3 OBSERVATIONThe situation in the simulation conditions, however,should not be realistic for a direct observation of thequadrupole wake-field effect at the KEKB linac, since ina real machine the quadrupole wake field (W2) could not
be eliminated as in simulations, and any comparison
with/without W2 could be totally impossible.

Measurements were therefore carried out so that ahorizontal kick was applied to the beam at the initialpoint of an energy of 230 MeV in the middle of sector A;
the resulting vertical betatron oscillation was observed
down the linac in sectors A and B by a beam position
monitor (BPM) system. Figure 3 shows the x-y coupling
effects observed in the vertical orbit when a horizontal
kick of about 0.3 mrad was set to the 8-nC beam at theinitial point. The solid line indicates a simulation using
the parameters listed in Table I.

a single kick was estimated to be smaller than 0.03%. In
the case of a high-current beam, however, this coupling
could be enhanced through the dipole wake-field term
(W1Dy ) in the equation of motion, although the above
estimation assures that the term could not be as large as
the quadrupole wake-field terms.
Therefore, theobserved results concerning the x-y coupling might
comprise both effects originating from dipole andquadrupole wake fields, regardless of the extent of theircontribution.
Since the separation of these two effects in theobservation is in principle difficult, we rely on theagreement between the observation and the simulation:
figure 3 shows that the agreement is quite good within
the experimental errors.
We may conclude that
quadrupole wake-field effects were surely observed forthe first time at the KEKB linac.

4 CONCLUSIONSThe quadrupole wake-field effects were calculated for
the KEKB linac using a rigid two-particle model. The
simulation showed that the effects could be observable
for an 8-nC beam at the KEKB linac if the initialconditions are appropriately chosen. Taking into accountthe recipe obtained by the simulation, measurements for
x-y coupling effects were carried out. The obtained
results are in good agreement with the simulation,suggesting that quadrupole wake-field effects wereobserved for the first time at the KEKB linac. Further
investigation on the quadrupole wake-field effects will becontinued in order to clarify the parametric-excitation
nature as well as x-y coupling phenomena.

Fig. 3: The x-y coupling effects observed in thevertical orbit when a horizontal kick of about 0.3
mrad was set to the 8-nC beam at the initial point ofan energy of 230 MeV (s = 25.4 m). The solid line
indicates the simulation results using the parameterslisted in Table I.
In order to check the results, the same horizontal kick
was applied to a low-current beam. No vertical betatronoscillation due to x-y coupling was observed within theprecision of the BPM system (< 0.1 mm). This signifies
that the observed x-y coupling for an 8-nC beam isinherent in the high-intensity beam and could be of wakefield origin. On the other hand, the x-y coupling causedby a rotational misalignment of the steering coil used forThe author wishes to thank K. Oide for his warm
encouragement and valuable discussions. He also thanks
the members of the Linac Commissioning Group (LCG)
for their help in operating the high-intensity electronlinac.

REFERENCES[1] Y. Ogawa, “Commissioning Status of the KEKB
Linac”, PAC’99, p.2984, New York City, March1999.
[2] P. B. Wilson, “High Energy Electron Linacs:
Applications to Storage Ring RF Systems and Linear
Colliders”, in Physics of High Energy Particle
Accelerators, AIP Conference Proceedings No.87,p.450, AIP, New York, 1982.
[3] Y. Ogawa, “Performanceof the Upgraded Injector+ Linac for KEKB”, e e Factories’99, p.64, KEK,Tuskuba, September, 1999.