MEASUREMENT OF BETATRON TUNE ALONG BUNCH TRAIN IN
THE KEKB LOW ENERGY RING
T. Ieiri*, H. Fukuma, T. Kawamoto and M. Tobiyama
KEK, Oho1-1, Tsukuba, Ibaraki 305-0801, Japan
Abstract
The bunch-by-bunch betatron tune along a bunch train has
been measured in the KEKB low-energy positron ring.
Electron cloud density can be estimated from tune shift
under the hypothesis that electrons produced by photoemission and/or secondary emission are attracted bysuccessive positron bunches and result in shifting thecoherent betatron tune of positron bunches. Build-up anddecay of the cloud density were measured. The effect ofsolenoid fields was also measured. The electron density
estimated from the tune shift is compared with thethreshold density for vertical size enlargement.

1 INTRODUCTIONKEKB[1] is a high intensity multi-bunch collider. The
collider consists of two storage rings, the Low Energy
Ring (LER) for 3.5 GeV positron beams and the HighEnergy Ring (HER) for 8 GeV electrons. The target
luminosity is 1 × 10 34 cm −2 sec −1 , when all rf buckets
are filled with harmonic number 5120. However, theachieved luminosity is only around one third of the target
value so far. It was observed in single beam mode thatthe vertical beam size increased above a threshold current
for the LER [2]. The increase of the size is caused by aphoto-electron instability and is briefly summarized asfollows [3,4,5]:• The threshold of the size blowup is roughly determinedby the charge density (bunch current)/(bunch spacing);• The blowup starts after about 6 bunches from the headof a bunch train, which has typically 60 bunches with 4rf-bucket (8 ns) spacing [6];• The blowup is weakened by increasing verticalchromaticity.
Simulation study [7] suggests that the blowup is caused
by a single-bunch head-tail instability and the thresholdshould be determined by the electron-cloud density near
the beam. In order to investigate the properties of theelectron cloud, a special electrode was installed on achamber wall [8,9]. Energy distribution of the electrons
and their accumulation along bunch trains were measured
by giving the electrode a bias voltage. The electroncurrent obtained by those measurements, however, is notelectrons that directly act on positron bunches. We have
tried to estimate electron cloud density acting on positronbunches by measuring the coherent betatron tunes of*Email: takao.ieiri@kek.jp

individual positron bunches.

2 GATED TUNE METER
The space charge of the electron cloud makes an electricfield which should cause a tune shift in the positronbunches. The vertical tune shift of a positron bunchusing a one dimensional model is given byr0 β y L 1 ∆y∆ν y =⋅(1)∫ ρ ( y ′)dy ′ ,γ∆y 0where r0 is the classical radius of an electron, β y theaverage value of the betatron function, L is thelongitudinal length of the cloud along the ring, ∆y thevertical displacement, ρ ( y) the cloud density and γ therelativistic factor. Assuming a homogeneous density ofthe electron cloud near a positron bunch, the incoherent
tune shift within a bunch would be equal to a coherent
one. When the density is not constant, however, tune
spread occurs, which may cause a variation of tune
spectrum. Thus we can estimate the cloud density fromthe tune shift and its spectrum.
It is required to measure bunch-by-bunch tune toinvestigate the photo-electron instability. Recent highspeed electronics has enabled us to detect the bunch-bybunch signal. Figure 1 shows the betatron tune
measurement system using a switch. The tune ismeasured for a gated bunch with a swept frequency method
using a tracking analyzer (Anritsu, MS420K). The
output signal of the tracking analyzer is swept infrequency corresponding to a fractional part of the betatrontune and is modulated by a pulse synchronized with therevolution frequency. The amplitude-modulated pulse
with width about 50 ns is combined with a feedback
signal [10] and both signals are guided to deflector
electrodes after amplification. The deflector consists offour stripline electrodes mounted in a chamber. Timingof the deflection pulse is adjusted using a delay modulewith a step of 2 ns. The bunch signal is picked up by abutton electrode installed in a vacuum chamber with adiameter of 64 mm. One bunch is gated from a bunchtrain in the switch. The amplitude of the gated bunch
signal is sampled and detected by a self-triggered pulse.
Only the oscillatory part of the gated pulse is detectedowing to a feedback gain control formed in a detector
circuit. Thus the oscillation amplitude is independent of
input bunch intensity. The resolution of the tune
measurement is mainly determined by the bandwidth of

the tracking analyzer, which is estimated to be ±0.0004
in the present configuration.
Deflection
Pulse

Tracking
Generator

Stripline
Deflector

beamPower Amp.

outputBOD

Tune Shift

input

SwitchGate Pulse

increases up to the 10th bunch and tends to saturate
behind it. The measured tune shift is caused by afocusing force for all modes, which suggests aninteraction between a positron bunch and electron cloud.
The behavior of the tune shift agrees with the clouddensity obtained by simulation [7].
0.012m=10.01m=00.008
m=-1
0.006
0.004
0.00200
10203040

Button Pickup

BOD :Bunch Oscillation Detector

Bunch Number

Figure 1: Scheme of gated tune measurement system.

3 MEASUREMENT OF TUNE SHIFTThe bunch-by-bunch tune was measured without collision
along a bunch train which has 60 bunches with 4-bucket
spacing, where each bunch had almost the same intensity.
Three peaks in the vertical tune spectrum of the leading
bunch were observed on the tracking scope. The central
peak corresponded a pure betatron frequency represented by
the m=0 dipole mode. The two sideband peaks indicatedbetatron oscillation modulated by synchrotron motion
represented by the m=1 and m=-1 modes. As themeasurement proceeded to rearward bunches, those threepeaks were similarly shifted towards a higher tune, while
the peaks widened. Figure 2 shows the tune shift as afunction of bunch number from the leading bunch, wherethe original tune is defined by that of the leading bunch inthis measurement. Because there are many empty buckets
before the leading bunch, its tune is expected to be thesame as that of a single bunch. The tune shift linearly

Figure 2: Betatron tune shift along bunch train for threemodes of betatron oscillation, where the bunch current isIb = 0.21 mA and the synchrotron tune ν s = 0.015 . The
measurement was performed in July 2000 withoutmagnetic fields to trap photo-electrons.
In order to investigate the properties of the electron
cloud, an additional bunch was injected behind a bunchtrain. The tune of the additional bunch was measured as
its longitudinal position was shifted, while the train waskept constant. We find the tune shift exponentiallydecays with a decay constant of 14 buckets or 28 ns asshown in Fig. 3. Though the measurement wasperformed with the C-yoke magnets described later, themeasured decay constant agrees with the simulation [7]using a field free condition.
0.01
0.008
0.006
0.004
0.002
005
10 1520 25 30 35
Bucket Number from Last Bunch of Train
Tune Shift

Since the deflection pulse has a width of 50 ns and arise time of 20 ns, a few forward bunches are also kicked.
This perturbation may disturb the electron cloud being
measured. In order to investigate this effect, the tune of
the gated bunch was monitored while the timing of thedeflection pulse was shifted with a constant oscillation
amplitude of about 50 µm , less than the rms verticalsize, where noise level displayed on the tracking analyzer
corresponded to an amplitude of a few µm . It wasconfirmed that the variation of the measured tune stayed
within measurement errors of ±0.0004 . Therefore, we
found the perturbation of the forward bunches had anegligible effect. An effect of excitation amplitude wasalso investigated. When the excitation amplitude wasincreased over a range of more than 20 dB, the change ofthe tune was found to be within measurement error.
Though the result suggests that a nonlinear effect for theexcitation is small, the tune measurement was performed
with an excitation amplitude of less than the rms beam
size so as not to disturb the cloud.

Figure 3: Variation of the vertical tune of an additional
bunch as a function of the distance from the last bunch ofa bunch train which contains 32 bunches with 4-bucket
spacing. The bunch currents are all 0.8 mA.

4 DISCUSSION
The simulation results predict that the threshold for thephoto-electron instability is ρth = 5 ~ 10 × 1011 m −3 ,when chromaticity is zero and synchrotron tune
ν s = 0.015 [7]. Assuming a constant cloud density near
the beam, the tune shift becomes ∆ν y = 0.004 to 0.009
at the threshold, using β y = 10.9m and L = 2000 m .
On the other hand, the measured tune shift varies along
the train as shown in Fig. 2 and is also changed by thebunch current and by the bunch spacing. Figure 4 shows
the tune shift between the 1st and 6th bunches of a train
plotted for various charge densities. The 6th bunch was

Tune Shiftselected as the threshold bunch for the blowup [6]. Thetune shift almost linearly increases with the chargedensity. It was observed that the vertical size enlargement
took place above a charge density of 0.08 mA/bucket
using an interferometer [11]. The threshold for theinstability is estimated to be around ∆ν y =0.0045 fromthe tune shift measurement, assuming a constant clouddensity. Though an extremely simplified condition isused, the measured tune shift is roughly consistent withthe predicted value from the simulation.
0.006
0.005
0.004
0.003
0.002
0.001
000.02 0.04 0.06 0.08 0.1 0.12
Charge Density (mA/bucket)

Tune Shift

Figure 4: The tune shift of 6th bunch in a train as afunction of the charge density, where ν s = 0.011 .
0.01
0.008

w/o C-yoke

0.006
with C-yoke

0.004

Table 1: Linear tune shift per bunch along a train.
condition
tune shift ×10 −4
Non-correction
9.0 ± 1.0C-yoke
5.5 ± 0.5
Solenoid
2.8 ± 0.5

5 SUMMARY

0.002
0tune shift along a bunch train was compared with andwithout the C-yoke magnets. A small difference in thetune shift is observed in only the leading part of a bunchtrain as shown in Fig. 5. On the other hand, the size
measurement showed that great improvement was notobserved for the threshold, though onset and growth of
blowup were somewhat retarded in the bunch train [6].
Second, solenoid fields were applied in the arc sections of
the LER after all the permanents magnets were removed.
The maximum longitudinal field is 45 gauss and the total
length is about 800 m. The tune shift was measured andcompared with and without solenoid fields as shown in
Fig. 6. A clear difference of the tune shift was observedin the leading part of a train. Remarkable improvement
for the threshold was observed in the size measurement[4]. However, we can see that the tune gradually increases
along a train, even when the solenoids are fully excited.
Table 1 shows linearized tune shifts per bunch in theleading part of a train for each configuration. Comparing
between the C-yokes and the solenoid, the solenoid ismore effective than the C-yokes.

010 20 30 40 50 60
Bunch Number along Train

Vertical Tune Shift

Figure 5: Tune shift along bunch train, with and without
C-yoke permanent magnets, measured with Ib = 0.21mA
and 4-bucket spacing.
0.008
0.007
Solenoid OFF
0.006
0.005
0.004
0.003
Solenoid ON
0.002
0.00100
1020304050
Bunch Number
Figure 6: Tune shift along bunch train, with and withoutsolenoid fields, measured with Ib = 0.25mA and 4-bucketspacing.
Intensive efforts have continued to suppress vertical
blowup since the early stages of the commissioning.
First, permanent magnets attached to C-shaped iron yokes
were placed on the chamber surface in the arc sections.
The longitudinal length covered by the C-yokes was about
750 m, where the total length of the arcs is 2200 m. The

A fast gate module enables us to measure bunch-by-bunch
betatron tune in multi-bunch machine such as KEKB.
The tune shift measurement is useful to estimate thevariation of electron cloud density along a train of
positron bunches. The measured build-up and decay timesof the cloud are consistent with the simulation. It wasobserved that the solenoid field was more effective thanthe C-shaped magnets, which supported the size
measurement. The authors would like to thank Prof. K.
Oide for his support, and the colleagues for installing agreat number of C-shaped magnets and solenoid windings.

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[4] K. Oide, LHC2001 (2001).
[5] H. Fukuma et al., in these proceedings.
[6] J. W. Flanagan et al., EPAC 2000, Vienna, (2000).
[7] K. Ohmi and F. Zimmermann, KEK Preprint 200019 (2000) and Phys. Rev. Lett. 85-18 (2000)3821.
[8] K.C. Harkay et al., PAC 99 (1999)1641.
[9] Y. Ohnishi et al., in these proceedings.
[10] M. Tobiyama and E. Kikutani, KEK Proceedings
99-24, (2000)120.
[11] T. Mitsuhashi et al., KEK Proceedings 99-24,(2000)134.