The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

ESTIMATION OF ELECTRON-CLOUD DENSITY BY MEANS OF
MEASURING BETATRON TUNE AT KEKB LOW ENERGY RING
T. Ieiri#, H. Fukuma, T. Kawamoto and S. S. Win
KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract
The KEKB Low Energy Ring suffers from blowup in
the vertical size of positron bunches due to an electron
cloud. Solenoids are installed around the ring to confine
the electrons to the vicinity of the chamber. The densityof the electron cloud can be estimated from a change inthe betatron tune of an individual bunch. Tune-shiftmeasurements were performed while the solenoid field,
the bunch spacing and the bunch current changed.

1. INTRODUCTIONKEKB [1,2] is a multi-bunch, high-current,electron/positron collider for B meson physics. The
collider consists of two storage rings: the Low Energy
Ring (LER) for a 3.5-GeV positron beam and the HighEnergy Ring (HER) for 8-GeV electrons. Both rings store
more than 1,000 bunches, where the harmonic number is5120 with an RF frequency of 509 MHz.
The LER has suffered from blowup in the vertical beam
size since the early stage of commissioning [3]. The
blowup is caused by an electron cloud. In order to confine
electrons to the vicinity of the chamber wall, solenoids
were installed in the arc sections in September 2000, andit was confirmed that they were effective to weaken theblowup. Thus, we added solenoids in the straight sections.
The total length covered by the solenoids, including thepermanent magnets, is about 2300 m [4], where thecircumference of the LER is 3016 m.
Though the solenoids have contributed to increasing theluminosity, blowup in the vertical beam size is stillobserved at high beam currents and for narrow bunch
spacing. It is required to investigate the properties of theelectron cloud and the effects of the solenoids. We haveestimated the electron cloud acting on positron bunches
by measuring the coherent betatron tune of individual
positron bunches.

2. TUNE SHIFT DUE TO ELECTRON
CLOUD
Since the effect of photoelectrons was first identified atthe Photon Factory [5,6], many efforts have been made toclarify the mechanism of the instability. Analytical workand simulations have provided an image of an electron
cloud. A model showed that the blowup in the verticalbeam size observed in the LER was caused by a strong
head-tail instability [7]. Using a two-particle model, thethreshold of the electron density in the cloud for theblowup is given by# Email: takao.ieiri@kek.jp

ρ th =2γν s

π re L β x ,y,(1)

where γ is the relativistic factor, ν s is the synchrotron
tune, re is the classical electron radius, β x,y is theaverage value of the horizontal or vertical betatronfunction and L is the longitudinal length of the cloud.
Experimentally, the density of electrons in a cloud can beestimated from a shift of the betatron tune under thehypothesis that the electric fields due to the electron cloud
result in a shifting of the coherent betatron tune ofpositron bunches. The tune shift due to the cloud is givenby [8]r
∆ν x,y = e ⋅ β x, y ρ 0 ⋅ L ,(2)2γ
under the condition that the majority of the electrons in
the cloud are not disturbed during the passage of apositron bunch. Here, ρ0 is the average density of theelectron cloud. Combining Eqs. (1) and (2), the thresholdfor the blowup in the tune shift is simply given byνs.
(3)πThis expression shows that the threshold is determined by
the synchrotron tune alone. This condition is similar tothat for the strong head-tail instability in a single bunch.
A synchrotron tune of 0.025 results in a threshold of0.008 in the tune shift.
∆ν th ≈

3. MEASUREMENT OF TUNE SHIFTThe betatron tune is measured based on a swept
frequency method. A bunch signal picked up by a button
electrode is selected by a gate module [9] and its peak isdetected. The noise level on a detector is less than the rms
vertical beam size at the pickup electrode. The resolutionof the tune measurement is mainly determined by thebandwidth of an analyser and is estimated to be ±0.0004
in the present configuration [10].
The measurement was performed without any
collisions, where bunches were placed with an equal
spacing in a train and each bunch in a train had almost thesame intensity. Since each train was separated sufficiently,a leading bunch in a train would not be affected by thecloud. Thus, we regarded the tune of the leading bunch asa standard. The synchrotron tune was 0.0249 and thechromaticity had slightly positive values, ξx =0.7 andξy =2.2. The average values of the betatron function were
10.5 m horizontally and 11.0 m vertically.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

off

on
0 100 200 300 400 500 600 700 800
Bucket(b)offVertical Tune Shift

0.02
0.015
0.01
0.005
0-0.005

(a)

0.006
0.004
0.002
0-0.002
0 100 200 300 400 500 600 700 800
Bucket0 100 200 300 400 500 600 700 800
BucketFigure 1: Horizontal (a) and vertical (b) tunes along atrain, the dots are tunes with solenoid field and thetriangles are without a solenoid field. The bunch current
is 0.5 mA and the bunch spacing is 8 ns.

3.2 Bunch Spacing and Current Effects
The effect of the bunch spacing was measured under aconstant solenoid field of 45 Gauss. Figure 2 shows thetune shift with different bunch spacing of 4, 6 and 8 ns ata bunch current of 0.5 mA. The horizontal and verticaltune shifts behave similarly at 6 ns and 8 ns spacing.
However, the tune shift is extremely large at 4 ns spacing
in both directions. The result suggests that the current
solenoids system is insufficient to repel the electron cloudat 4 ns bunch spacing.
Next, the current dependence of the tune shift wasmeasured under a constant bunch spacing of 8 ns. Figure
3-(a) shows the tune shift as a function of the bunch
current with the solenoids off. The horizontal and verticaltune shifts linearly increase with the bunch current.
According to a beam size measurement using aninterferometer [11], the threshold bunch current for
blowup in the vertical beam size was 0.35 mA, whichcorresponded to a tune shift of 0.011. This tune shift for

(b)

0.015
0.01
0.005
0-0.005

0 100 200 300 400 500 600 700 800
Bucketon

Figure 2: Horizontal (a) and vertical (b) tunes along atrain. The dots are tunes with a bunch spacing of 8 ns, thesquares are with 6 ns and the crosses are with 4 ns. Thebunch current is almost constant, 0.5 mA.

Tune Shift

0.01
0.008
0.006
0.004
0.002
0-0.002

0.025
0.02
0.015
0.01
0.005
0(a) offVH00.2
0.4
0.6
Bunch current (mA)

0.8(b) on0.015
Tune Shift

Vertical Tune Shift

Horizontal Tune Shift

Under the current solenoid system, the effect ofsolenoids was investigated. Note that some permanent
magnets are active regardless of the solenoids being
on/off. The betatron tune increases along a train and tends
to saturate at the rear part of a train, as shown in Fig. 1.
The saturated tune shift is about 0.009 horizontally and0.015 vertically without any solenoid fields. When allsolenoids are active, the tune shift is reduced to 0.002
horizontally and to 0.006 vertically. Thus, the reduction in
the tune shift by the solenoids is about 0.007 horizontallyand 0.009 vertically. The measured tune shifts aredifferent between the horizontal and vertical directions.
This result suggests a different distribution and/ormechanism in producing the cloud between two directions.

blowup is higher than the expected value of 0.008 using
Eq.(3). Figure 3-(b) shows the current dependence withthe solenoids being on. The horizontal tune shift issuppressed by the solenoids. However, the vertical tune
shift increases with the bunch current, even if thesolenoids are active. We found a kink in the curve, wherea vertical coherent oscillation of bunches was observed at
the rear part of a train. The threshold bunch current for
blowup is 1.4 mA, which corresponds to a tune shift ofhigher than 0.012. These results are roughly consistent
with the expectation, though the measured tune shift isslightly higher than the expectation.
(a)
0.008
Horizontal Tune Shift

3.1 Effect of Solenoids

V0.01
0.005

H00

0.2 0.4 0.6 0.8 1 1.2 1.4
Bunch Current (mA)

Figure 3: Difference in the tune between the leading andlast bunches in a train as a function of the bunch current,(a) with the solenoid Off and (b) with the solenoid On. The
dots are horizontal tune and the squares are vertical one.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

4. DISCUSSION
4.1 Comparison with a Simulation

Tune Shift

The electron cloud density can be estimated from thetune shift using Eq. (2). The cloud builds up at the front
part in a train and tends to saturate at the rear part, asshown in Fig. 4. The build-up time with a linear
approximation is 70 to 80 ns in both axes. However, adifference between the horizontal and vertical axes seems
to enlarge at the rear part in a train. The horizontal tune
shift is 0.009, which corresponds to a cloud density of
12−3
1.3 × 10 (m ) , assuming L =3000 m. On the other hand,the vertical one is 0.015 at the tail. A slow increase in thevertical tune is observed at the saturation region.
A simulation can evaluate the cloud density, assuming
the yield of photoelectrons, (Y) and the reflection
coefficient, (R) of the synchrotron light. Assuming
Y=0.02 and R=0.4 [12], the central cloud density linearly
increases along a train and tends to saturate, where thebeam conditions between the simulation and themeasurement are almost equal. The linear build-up time isabout 70 ns and the saturated density is evaluated to be12−31.2 ×10 (m ) , which corresponds to a tune shift of0.0084 with L =3000 m. The simulation agrees with thetune shift on the horizontal axis.
0.02
V0.0150.01

H0.0050-0.005

0100 200 300 400 500 600
BucketFigure 4: Tune shift along a train with the solenoid field
Off. The dots are the horizontal tune and the squares are
the vertical one. The bunch current is about 0.5 mA andthe spacing is 8 ns.

4.2 Where is the Cloud?
Though the horizontal tune shift was almost suppressedby the solenoids, about one third of the vertical tune shift
remained. Where was the remaining cloud? The existenceof a cloud controlled by the solenoids might beinvestigated by partially switching off the solenoids
wound around the ring. The LER can be simply divided
into 8 parts, 4 arc and 4 straight sections. The arc sectionsare 2.5 times longer than the straight sections. The tune
shift was measured, when one fourth of the solenoids at
the arc sections were On to Off and when the solenoids at
all straight sections were On to Off. Each change in thetune shift indicates how much the cloud was suppressedby the solenoids located there. The results are
summarised in Table 1, assuming that contributions fromfour arc sections were the same. While the contribution
from the arc sections is isotropic, that from the straightsections is not. For the vertical tune shift, about half ofthe effect comes from the straight sections. The
asymmetric tune shift due to the remaining cloud mightcome from the asymmetric contribution of the cloud atthe straight sections.
Table 1 Contribution of the tune shift, measured with abunch spacing of 6 ns and the bunch current of 0.5 mA.
The remains mean the tune shifts with all solenoids on.
Arc Sections
Straight Sections
Remains
Total (estimation)

Horizontal
0.006 ~ 0.008
0.00350.001 ~ 0.002
0.011 ~ 0.014

Vertical
0.006 ~ 0.008
0.008
0.00550.020 ~ 0.022

5. SUMMARY• The horizontal and the vertical tune shifts areasymmetrical at the KEKB LER, i.e. the verticaltune shift is larger than the horizontal one, except
for the leading part in a train.
• The horizontal tune shift and its build-up time agreewith a simulation.
• The vertical tune shift at the blowup is roughly
consistent with the threshold for blowup with thehead-tail model.
• Half of the vertical tune shift suppressed by thesolenoids comes from the straight sections.
• The current solenoid system is insufficient at a bunch
spacing of 4 ns in both directions.

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[5] K. Ohmi, Phys. Rev. Lett. 75, 1526 (1995).
[6] M. Izawa et al., Phys. Rev. Lett. 74, 5144 (1995).
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[8] K. Ohmi et al., Proceedings of APAC’01, Beijing,445 (2001).
[9] T. Ieiri et al., Phys. Rev. ST Accel. Beams, 5, 094402(2002).
[10] T. Ieiri et al., HEAC2001, Tsukuba, (2001).
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