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

MEASUREMENT OF LONGITUDINAL IMPEDANCE AT KEKB
T. Ieiri# and H. Koiso, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Abstract
The current dependence of the synchronous phase andthe bunch length was measured to estimate thelongitudinal coupling impedance of KEKB. The bunch
length was reduced to 4 mm owing to a negative
momentum compaction lattice. The loss factor wasmeasured at such a short bunch length. The bunch
lengthening was observed even with a negative
compaction in a high-current region, which suggests apossible microwave instability in the low energy ring.

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.
Bunches are squeezed at an interaction point (IP) togain the luminosity. The bunch length should be shorter
than the beta function at the IP to avoid the hourglass
effect. However, the bunch length increases with thebunch current due to inductive impedance under thelattice with positive momentum compaction. On the otherhand, when the lattice is switched to negative momentum
compaction, bunch shortening is expected due to thenegative impedance. When the bunch length is short,however, it is expected to increase the loss factor.
Reducing the loss factor is an important issue for KEKB.
The impedance had already been measured in the earlystage of commissioning [3]. Some RF cavities were added
and some vacuum components were modified after theprevious measurement. Measuring the impedance isnecessary not only to understand the current machine, butalso to design a future machine, such as SuperKEKB [4],where a bunch length of 3 mm is expected.

2. LONGITUDINAL BEAM BEHAVIOR IN
POSITIVE AND NEGATIVE MOMENTUM
COMPACTIONKEKB adopts a unique lattice composed of thenoninterleaved 2.5 π cell structure [1]. The lattice has awide dynamic aperture and flexibility for adjusting theparameters of the emittance and the momentum
compaction. The momentum compaction, defined byα ≡ (∆C /C ) /(∆p/ p) , where C is the circumference andp is the momentum, is determined by integrating thedispersion function at the bending magnets. The current
lattice uses a positive momentum compaction factor of
α = 3.4 × 10−4 for both LER and HER. The lattice wassmoothly switched to use a negative compaction factor# Email: takao.ieiri@kek.jp

without changing the absolute value in both rings. Thus,
the natural bunch length and the synchrotron tune would
not be changed by switching.
Let us consider the longitudinal behaviour of a bunch.
Accelerating cavities in a storage ring give energy to acirculating bunch to compensate for the radiation loss andparasitic loss. The synchronous phase of a bunch isdetermined by the balance between the cavity voltage andthe total energy loss. The parasitic loss is caused by theresistive impedance of the environment, represented by
the loss factor k (σ ) , depending on the bunch length. The
shift in the synchronous phase due to the parasitic loss
∆ϕ s is given byUVc sin(ϕ s0 + ∆ ϕ s ) = 0 + k(σ )T 0 I b ,ewhere Vc is the cavity voltage, ϕs 0 is the synchronous
phase of a zero-current limit, U0 is the radiation loss perturn, and T 0 is the revolution period. When the phaseshift ∆ϕ s is small enough compared with unity, it isgiven by∆ϕ sT0≈ k(σ ).
Ib
Vc cos ϕ s0
As illustrated in Fig. 1, a bunch rides on the descending
shoulder of an RF wave in the case of α > 0 . On the otherhand, the synchronous phase moves to the rising shoulder
in the case of α < 0 , to maintain the phase stability. Thesynchronous phase shifts to compensate the additionalenergy loss due to the parasitic loss. Thus, the beam phaseadvances in the case of α > 0 and lags in the α < 0 on theRF wave, as the bunch current increases. The rates of thephase shift are proportional to the loss factor.
RF Voltage
α<0α>0Phase Delayφs
Figure 1: Schematic of the synchronous phase of a bunchin positive and negative momentum compaction.
The natural bunch length is determined by the lattice
parameters at a zero current. However, the bunch shape isdetermined by its own potential-well field at finite bunch
current. Inductive or capacitive impedance of theenvironment changes the bunch length due to thepotential-well distortion. When a purely inductive
impedance is assumed in the case of α > 0 , bunch
lengthening is analytically derived [3], and given bye αI bR 3 Z i (ω )σσ 3)=( ) −(() ().
2σ l0
σl0n4 π γ sE σ l0

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 20033.1 Beam Phase

Relative Phase (degrees)

The beam phase was measured turn by turn with afour-dimensional beam-position monitor [5]. Two
amplitudes of an I/Q (in-phase and quadrature phase)
demodulator working at an RF frequency give a phasedifference between the beam and the RF phase. Assumingthat the RF phase is constant, we can obtain a relativebeam phase. We confirmed that an average phase wassettled within ±0.1 ° under a stable beam.
The measurement was performed with a single bunch.
Figure 2 shows variations of the measured beam phase asa function of the bunch current in the LER. The beamphase is advancing, as the bunch current increases, in thecase of α > 0 , and it is lagging in the case of α < 0 .
Assuming that the RF phase is constant, we can estimate
the synchronous phase, ϕs 0 from an extrapolated phaseto zero current. The estimated ϕs 0 is 10.3 ° at the LER
and 16.7 ° at the HER, respectively. On the other hand, thecalculated ϕs 0 is 11.8 ° at the LER and 15.5 ° at the HER,
the calculation is consistent with the estimation. We findthat the slope in α < 0 is steeper than that in α > 0 at
both rings. The result suggests that the loss factor in thecase of α < 0 is larger than that of α > 0.
(b)
(a)
114
-91
113
112
111
110
109
108

00.5
11.5
Bunch Current (mA)

-92
-93
-94
-95
-96
-97

00.5
11.5
Bunch Current (mA)

Figure 2: Beam-phase vs. bunch current in the LER, (a)α > 0 and (b) α < 0.

3.2 Bunch Length
The bunch length was measured using a method basedon the beam spectrum. A bunch-length monitor detecting

Bunch Length (mm)3. MEASUREMENT

two frequency components in the beam spectrum under
the condition of ω d σ t < 1 , where ω d is the detectedangular frequency and σ t is the rms bunch length in time,
indicates the rms length of a bunch in real-time, even if
the bunch shape changes [6]. The monitor detects twofrequency components of 1.02 GHz and 2.54 GHz andhas a resolution of ±0.2 mm at a bunch length of 4 mm.
Since the gains between two channels are not always thesame, we need to calibrate the detector in order to obtainan absolute value of the bunch length. The calibration isperformed based on the assumption that an extrapolated
bunch length to zero current approaches the natural bunch
length. The monitor detects an average value of the bunch
length, since the monitor does not have a function toselect a specific bunch.
The measurement was performed with a multi-bunchedbeam, where bunches with an equal intensity areseparated by 320 ns. Since bunches are well separated, theeffect of a coupled bunch would be negligible. Figure 3(a) and (b) show the average bunch length as a function of
the bunch current with the two cases of α > 0 and α < 0.
The bunch length increases with the bunch current in
α > 0 and the bunch length in α < 0 is shorter than thatin α > 0 at both rings. A bunch length of around 4 mm is
obtained at a low-current region at the LER; however, it
increases as well as the case of α > 0, when the current isabove 0.4 mA. At the HER, the bunch length keeps
around 5 mm in α < 0 , even at a high-current region.
Thus, a difference in the bunch length between α > 0 and
α < 0 expands, as the bunch current increases.
(a)8Bunch Length (mm)Here, R is the average radius of a ring, Z i (ω) is theinductive impedance and n is an integer given byn = ω / ω 0 , where ω 0 is the angular revolution frequency.
In the case of α < 0 , the negative inductance causes
bunch shortening. However, when the bunch length is too
short, bunch lengthening might occur, because an effectof negative capacitance appears due to the extended
spectrum of a short bunch. Since two effects oflengthening and shortening are actually mixed, variationsof the bunch length might be small. When the bunch
current increases further, a microwave or turbulent
instability takes place above the threshold in both cases of
α > 0 and α < 0 , which results in increasing the energy
spread as well as the bunch lengthening.

76
54
3

0

0.2

0.4
0.6
0.8
Bunch Current (mA)

1(b)

8
7654
3

00.2
0.4
0.6
0.8
Bunch Current (mA)1Figure 3: Bunch length as a function of the average bunch
current, (a) dots in the cases of α > 0 and squares α < 0at the LER, (b) dots in the cases of α > 0 and squares
α < 0 at the HER. The arrows indicate the natural bunch
length, 4.74 mm at the LER and 5.22 mm at the HER.

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

Normalized Bunch Length

As shown in Fig. 3-(a), unexpected bunch lengthening
was observed in the case of α < 0 at the LER, comparedwith that in the HER. A microwave instability issuspected in the LER. First, The Boussard criterion isconsidered. The threshold current is estimated to be 0.26
mA at the LER, using an impedance of Z i / n =0.07 Ω [3].
However, the equation might be underestimated [7], since
it is based on the coasting beam theory. Secondly, A.
Chao and J. Gareyte have derived a scaling low to explain
the microwave instability [8]. The bunch length satisfies
the scaling property, σ l = F (ξ ) , withαIξ = 2b .
νsEWe find that the parameter ξ does not depend on α . On
the other hand, an experimental result revealed that themeasured bunch lengths for two values of α were nearly
equal at a high bunch current. Thirdly, the measured
bunch length in the case of α > 0 was fitted by thecalculated bunch length with purely inductive impedance,
as shown in Fig. 4. We find that the bunch length deviatesfrom the calculated bunch length above a bunch current of
0.6 mA. However, we do not yet have measured increaseof the energy spread. No coherent oscillation wasobserved even above a bunch current of 1.0 mA.
1.8806040200345678Bunch Length (mm)910(b)

10080
6040200345678Bunch Length (mm)910Figure 5: Loss factor as a function of the bunch length, (a)LER and (b) HER. The crosses are the values measured in
1999 and the dots are in 2003.

1.6
1.4
1.2

Table 1: Inductive impedance.
19992003
Impedance

10.8

Loss Factor (V/pC)

4.1 Possibility of the Microwave Instability

conducting cavities has increased to store a high current
beam, which may contribute to increasing the loss factor.
We find that the loss factor slightly increases in the HER.
The movable collimators have been improved [9], which
has reduced the impedance. The effects of eachmodification, however, are not reflected in themeasurement.
(a)100

Loss Factor (V/pC)

4. DISCUSSION

0

0.2
0.4
0.6
0.8Bunch Current (mA)

1

Figure 4: Measured bunch length indicated by dots in thecase of α > 0. The line is the calculated bunch length as afunction of the bunch current at the LER, assumingZ i / n = 0.06Ω .

4.2 Impedance
The loss factor obtained from the slope in the beamphase is plotted in Fig. 5, together with the data measured
in 1999. The inductive impedance estimated from thebunch lengthening is represented in Table 1. Comparing
the data obtained in 2003 with those in 1999, the loss
factor does not indicate a large change. It is noticed that aloss factor increases and tends to reach 100 V/pC at abunch length of 4 mm. Though the inductive impedance
tends to slightly reduce, the inductance at the LER is still
4-times larger than the design value [1]. During 4 years,
many components of both rings have been modified.
Among the modifications, the number of normal

LER Z i / n ( Ω )HER Z i / n ( Ω )0.072 ± 0.011

0.076 ± 0.006

0.060 ± 0.01

0.065 ± 0.006

REFERENCES[1] KEKB Design Report, KEK Report 95-7 (1995).
[2] K. Akai, et al., Nucl. Instrum. Methods A499 191(2003).
[3] T. Ieiri et al., Proceedings of the 12th Symp. on Accel.
Sci. and Tech., Wako, 409 (1999).
[4] http://kcgsrv1.kek.jp/SuperKEKB/home.html/.
[5] T. Ieiri and T. Kawamoto, Nucl. Instrum. Methods
A440, 670 (2000).
[6] T. Ieiri, Proc. of the EPAC 2000, Vienna, 1735
(2000).
[7] K. Oide and K. Yokoya, KEK Preprint 90-10 (1990).
[8] A. Chao and J. Gareyte, SPEAR-197/PEP-224,
SLAC, (1976).
[9] Y. Suetsugu et al., Rev. Sci. Instrum. 74, 7, 3297
(2003).