HEAD-TAIL INSTABILITIES OBSERVED AT J-PARC MR AND
THEIR SUPPRESSION USING A FEEDBACK SYSTEM
Y.H. Chin, Y. Kurimoto, T. Obina, M. Okada, K. Takata, M. Tobiyama, T. Toyama, KEK, Japan,
Y. Shobuda, JAEA, Japan
Abstract
This talk presents analysis of head-tail instabilities
observed at the J-PARC MR (Main Ring) and theirsuppression using a feedback system. At the MR, we havebeen observing serious instabilities during the injection
and at the onset of acceleration. Without the feed-backsystem on, nearly one third of particles are lost due to theinstabilities at the beam power of 120kW. At the injection,
the instabilities take place on the horizontal plane byinjection errors resulting in a large displacement of
injected bunches and due to collateral kicks of alreadyinjected bunches by the kicker fields. On the other hand,
the instabilities at the onset of acceleration develop on thevertical plane from the noise level. Measurements of thehorizontal oscillation initially suggested higher-orderhead-tail modes excited during the injection. However,the detailed analysis unveils that a simple dipole mode
can have temporal appearance of higher-order head-tailmodes due to a large chromaticity. The comparison
between the measurements and the theoretical prediction
shows that they have very similar behaviours.

INTRODUCTION
J-PARC consists of three proton accelerators: a 400
MeV linear accelerator (currently operating at 180 MeV),
a 3 GeV Rapid-Cycling Synchrotron (RCS) and a 50 GeV(currently 30 GeV) Main Ring (MR). The main
parameters of the MR are summarized in Table 1.
Table 1: Main parameters of the MR ring.
Circumference
1568 mInjection Energy
3 GeV
Extraction Energy
30 GeV
Repetition Period
3 sec
RF Frequency
1.67-1.72 MHz
Number of bunches
8Synchrotron tune
0.002-0.0001
Betatron tune
22.4, 20.77
In the normal operation mode, over 90% of the protons
accelerated in the RCS are directed to the muon andneutron production targets in the Materials and Life
Science Experimental Facility (MLF). The remaining
protons are transported to the MR for further acceleration
before being extracted via one of two (fast and slow) MR
extraction ports.
The MR impedance is dominated by the transverse
resistive-wall impedance of the stain-less steel chamber
and the horizontal kicker impedance [1]. The kicker
impedance has sharp peaks at about 1MHz and 10MHz. It
is almost halved after four injection kickers were replacedby parasitic-impedance-free new models with HOM
dampers (we still use five old-type kickers). The
transverse resistive-wall impedance is exceptionally high,since the first betatron unstable line appears at around
40kHz on the vertical plane where the skin-depth iscomparable to the wall thickness and the transverse
impedance is proportional to the inverse of the frequency,
not the square root of the frequency [2]. The beam
instability evaluation before the commissioning of MR
started predicted that beam instabilities would start to hurt
beam operation from the beam power of 100-200kW.
Although, the necessity of a damper system has beenrecognized, the actual development has been delayed due
to financial constraints. After the commissioning of MR
started, the careful inventory of the existing hardware
revealed that we already have some key components
necessary to compose the damper system: several unused
Stripline Position Monitors (SPM) and the exciters fortune measurements that can be exploited as damper
kickers. It was found that only remaining component tobe developed is a signal processing circuit to filter andprocess SPM data and then to produce kick signals. The
whole project costs only less that 100k USD.
Figure 2 sketches the transverse bunch by bunch
feedback system. The beam position signals are sampled
at an RF frequency times 64 rate (108.8MHz at 3GeV).
The signal processing and digital filtering circuits consistof two LLRF4 boards with four 14-bit ADCs and two 14bit DACs. They extract the betatron oscillation signals
using 8-tap FIR filters. Its system integration andfirmware development including EPICS interface wasdone by the Dimtel Inc. The kick signals are sent to thestripline damper kickers through the power amplifiers(two 500W/ 10kHz-250MHz ones for the horizontal plane
and two 1kW/100kHz-8MHz ones for the vertical plane)
to provide a single kick per bunch per passage for theboth directions.
PickupsHor. Ver.

Ver.

Hor.

Beam

Signal sampling at
RF frequency x64
Processing
circuitamp. or att.)
DigitalSingle kick per bunch filter Hybrid
operating at RF frequency 0-180

Dummy
100k-8MHz, 1kW load

10k-250MHz, 500W
RF power amplifier

Figure 2: Schematic view of the transverse bunch-bybunch feedback system.

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Figures 3 (a) and (b) show the beam oscillation signals
from the SPM recorded at a 500MHz sampling rate forthe first 128ms period covering the injection of the lastbatch of bunches from RCS (K4) and the onset ofacceleration (P2) where all instabilities take place. The
top figure (a) shows the signals with the feedback (FB)
off, while the bottom figure (b) shows one with the FB on.
The strong vertical oscillations can be seen just after the
onset of acceleration when the FB is off. The large
horizontal oscillation can be also seen just after the
injection of the last batch of bunches. They are basicallygone when the FB is tuned on. The DCCT monitor showsthat the particle losses at the injection and at the onset ofacceleration are about 500W and 4,800W, respectively,
far larger than the collimator tolerance of 400W. They are
reduced to 100W and 25W, respectively, when the FB is
on. The FB is now a must in daily operation of MR.
(a)

FB:Off

20ms

H:
K4

P2

Injection of thelast batch

Acceleration starts

V:

(b)

FB:On
FB:On

H:

V:

Figure 3: The beam oscillation signals from SPM for the
first 128ms (a) when FB is off and (b) when FB is on.
(a)Figures 4 (a) and (b) show the beam intensity measured
by the DCCT monitors without and with FB, respectively.
It can be seen that the beam loses almost 30% of particles
due to the beam instabilities when FB is off.

HEAD-TAIL INSTABILITY THEORY
Let us revisit the head-tail instability theory [3]. The
chromaticity is defined by
dω β
(1)
ξ=
/δ ,
ωβ
where ωβis the angular betatron frequency andδ = Δp / p is the relative momentum deviation. In otherwords, the betatron frequency depends on the momentum
deviation as(2)
ω β (δ ) = ω β 0 (1 + ξδ ) .
When a particle moves along the ring, the accumulated
betatron phase advance is given bydss ξ= ωβ 0 ( − τ )φ β ( s ) = ∫ ω β (δ )βcβc η ,(3)s≡ ωβ 0− ωξ τ
βc
where τ is the arrival time difference of the particle at theposition s (positive toward the head of bunch), βc is thevelocity of the particle, and η is the slippage factor. Thus,
the betatron phase advance varies linearly along thebunch and attains its minimum (maximum) at the head
(tail) of the bunch. Next, let us check how the differenceof phase advance between the head and the tail of thebunch changes over one synchrotron oscillation period,
using the four particle model in the synchrotron phasespace as illustrated in Fig. 5. We assume that the arrival
time is oscillating asτ = τˆ cos(2πν s k ),(4)where ν s is the synchrotron oscillation tune and k is therevolution turn. We also assume that the chromaticity is
negative and the ring is operated below the transitionenergy (η < 0 ). Thus particles move clockwise.

FB:Off

(a)

φβ

δ

(b)

φβ

δ

0
(b)

0

ωξτˆ − ωξτˆ

FB:On

τ

ωξτˆ − ωξτˆ0ωξτˆFigure 4: The beam intensity measured by the DCCT
monitors (a) when FB is off and (b) when FB is on.

0

τ

0

− ωξτˆωξτˆ0− ωξτˆ

Figure 5: (a) The initial phase advance setting of the four
particles. (b)The phase advance after a quarter period of
the full synchrotron oscillation.

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In principle, the betatron phase advance slows down
(quickens up) by ω ξ τˆ for every quarter period ofsynchrotron oscillation as the particle moves forward(backward) along the synchrotron orbit, respectively. As
can be seen in Fig.5, the initial phase relationship along
the bunch is preserved after the quarter period ofsynchrotron oscillation. One can quickly check that thephase pattern remains stationary over the full period of
the synchrotron oscillation. In other words, the differenceof the phase advance between the head and the tail of thebunch is constant, and we denote this constant as χ :ξχ = φβ Tail − φβ Head = 2ωξτˆ = 2 ω β 0τˆ = constant(5)ηLet us assume that the dipole moment observed at asingle point in the ring has the following standing wave
pattern:t⎧m = 0,2,4,...
⎪⎪cos(m + 1)π τ,(6)L
p m (t ) = ⎨t⎪ sin(m + 1)πm = 1,3,5,...
⎪⎩
τL
where τ L = 2τˆ is the total bunch length. The transverse
pick-up signal observed at that point on the k-th
revolution turn is given by(7)
I m (t ) ∝ p m (t ) exp(iω ξ t + i 2πkν β 0 ) ,where ν β 0 is the betatron tune. The effect of thetravelling-wave component over the standingwave p m (t ) is to shift the bunch spectrum by ωξ . Figure 6
shows an example with the transverse resistive-wall
impedance and the narrow-band kicker impedance. The
spectra are drawn for the head-tail modes 0 and 1 withpositive phase difference χ . In this example, the mode 0
is stabilized, while the mode 1 becomes unstable by thetransverse resistive-wall impedance at low frequency.
Transverse
ReZT(ω)stableresistive-wall
impedance
F (ω − ω )0ξF1 (ω − ωξ )the right peak of the bunch spectrum of the mode 1
implies that this part of the mode now oscillates faster. In
summary, the mode m=1 is degenerated at zero
chromaticity, but are now split to slower and faster
oscillating parts by non-zero chromaticity. These split
modes are equally excited due to the standing-wave
condition of head-tail modes, and thus have a node at thecentre just like the m=1 mode at zero chromaticity.

BEAM SIGNAL MEASUREMENT AT MR
We have done several beam studies to investigate beam
oscillation signals from the SPM in more depth. The
studies were done with 8 bunches with the beam power of120-140kW. The typical chromaticity pattern is shown in
Fig. 7.
ξx=-1.42ξy=-1.47

P2
ξx=-4.53
ξy=-4.7

K4

50ms
Injection oflast batch

200ms

ξx=-4.53
ξy=-4.7

50ms

Acceleration starts

Figure 7: The typical chromaticity pattern.

Vertical Instability at the Onset of Acceleration
The vertical instability at the onset of acceleration (seeFig. 3(a)) is a text book case of the head-tail instability:
noise with the right phase relationship as shown in Fig.5
is picked up and then grows. Let us count head-tail modes
that could be excited at the onset of acceleration by the
low frequency transverse resistive-wall impedance andthe 10MHz kicker impedance. The phase difference χ forthe chromaticity ξ = − 1 .45 , the total bunch lengthof τ L = 2τˆ = 150 ns and the slippage factor of 0.058 is 4.4.
That means that the low frequency impedance can excite
the m=0 or m=1 mode. The 10MHz kicker impedance canproduce χ = 13 .8 and thus the m=3 mode is likely to be
excited. Figures 8 show the maximum amplitude of thevertical oscillation when the FB is turned off (left) and isturned on (right), respectively.

ωωξNarrow-band
kicker impedance

unstable
Figure 6: An example of the bunch spectrum shift by nonzero chromaticity.

Figure 8: The time evolution of the maximum amplitude
of vertical oscillation for the FB off (left) and on (right).

The shift of the left peak of the bunch spectrum of themode 1 means that a part of the mode now oscillates
slowly. The head and the tail of the bunch will movealmost in phase, not out of phase, to synchronize with the
low frequency impedance. On the other hand, the shift of

From these figures, the growth time can be estimated as
about 1.2ms. The huge damping effect by the FB is
recognizable in the right figure of Fig. 8. Figure 9 shows
the vertical oscillation signals superimposed on 10
consecutive turns (when FB is off). There is no node

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visible, suggesting that the vertical oscillation is just a
dipole mode. The BBU approach for the growth time
estimate delivers the growth time of 1.5ms, in areasonable agreement with the measurement of 1.2ms [4].

Figure 9: The vertical oscillation signals superimposed on
10 consecutive turns when FB is off.
Another interesting finding is the pulsational behaviourof the oscillation amplitude. At the onset of acceleration,
instability develops from noise level and then theoscillation amplitude starts to grow. Eventually, someparts of the beam hit the wall and get lost. The instability
then loses its mechanism of growth and starts to decay.
After while, the instability picks up steam again and starts
to grow. The above process is repeated until the beam
loses enough particles that the instability ceases for good.

Horizontal Instability at the Injection
The horizontal instabilities during the injection havevery different features. They start, not from noise, butfrom a large transverse displacement due to injection
errors or ones by kicks by the mismatching field of theinjection kicker magnets. There is no betatron phasedifference along the bunch at the moment of the injectionor the kicks by the mismatching field. Nevertheless, if westill apply the Sacherer’s text book analysis to this case,the head count of head-tail modes that could be excited at
140kW is the m=3 mode by the low frequency resistivewall impedance and the m=6 mode by the 10MHz kicker
impedance. Figures 10 show the time evolution of the
maximum amplitude of horizontal oscillation of the firstbunch at the injection of the last batch when the FB is off(left) and is on (right), respectively. Figures 11 show themeasured horizontal oscillations of the same bunch for
the FB on (left) and FB off (right) at the 1st (top), 250-th(middle) and 501-th (bottom) turns, respectively.

Figure 10: The time evolution of the maximum amplitudeof horizontal oscillation of the first bunch at the injectionof the last batch (K4) for the FB off (left) and on (right),
respectively.

Figure 11: The measured horizontal oscillations of thefirst bunch at the injection of the last batch (K4) for the
FB off (left) and FB on (right) at the 1st (top), 250-th(middle) and 501-th (bottom) turns, respectively.
One can see that main growing mode with the FB off isthe dipole mode and it is very quickly and effectively
damped by the FB system. However, the complicatedintra-bunch oscillations occasionally emerge and thenfade out. It looks like that a lower-order head-tail mode
appears first and then it is converted to a higher-ordermode (m=6 or 7), and then converted back to the lowerorder mode again after one full synchrotron oscillation
period (∼500 turns). Although these seeming head-tailmodes impose no harm on the MR operation at present, if
they are really head-tail modes excited by the kicker
impedance and will grow more rapidly at higher beam
power, they may pose significant threat on the MR
operation. For the present narrow-band FB system cannot
damp intra-bunch oscillations. This possibility raised asignificant concern over the future operation of MR. To
defuse this threat, first we started to develop two types of
intra-bunch feedback systems. At the same time, we mademore in-depth investigation of the measured intra-bunch
oscillations to find their true identity.
Figures 12 show the horizontal oscillation signals
superimposed on 10 consecutive turns starting from 251th turn. No clear node is visible.

Figure 12: The horizontal oscillation signals
superimposed on 10 consecutive turns starting 251-th turnfor the FB off (left) and the FB on (right), respectively.

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One possible explanation is that the kicker 10MHz
impedance is not strong enough to sustain higher-orderhead-tail modes alive for long period of time by locking
the phase synchronisation with them. Another possible
explanation is that the simple application of Sacherer’ text
book analysis is not inadequate in the present case.
Let us re-investigate how the betatron phase difference
between the head and the tail of a bunch evolves over afull synchrotron oscillation period using the four particle
model (see Fig. 13). All the particles have the same
betatron phase at the beginning. At every a quarter periodof the full synchrotron oscillation, the phase difference
between the head and the tail increases by 2ωξτˆ till thehorizontal oscillation signals and found that the low
frequency signal (lower than ∼5MHz) is dominating andthere is no clear peak at high frequency. This result also
indicates that the simple dipole mode with the oscillating
betatron phase difference between the head and the tail ofthe bunch can create temporal deceptive appearance as
higher-order head-tail modes.

half of the synchrotron oscillation. Then, the phasedifference starts to decrease and come back to zero at thefull synchrotron oscillation (namely, the picture next to(d) comes back to (a)). More detailed analysis shows thatthe phase difference oscillates likeξ(8)
χ = φβ Tail − φβ Head = ωβ 0τ L ⋅ (1 − cos(2πν s k ))
η
(a)

φβ

δ

(b)

φβ

0

− ωξτˆ

τ0δωξτˆ − ωξτˆ

0

(c)

0

0

φβ

δ

ωξτˆ

0

(d)

δ

φβ

0

2ωξ τˆ − 2ωξ τˆτ0− 2ωξ τˆωξτˆ − ωξτˆωξτˆ

SUMMARY− ωξτˆωξ τˆ

02ωξ τˆ

Figure 14: The simulated horizontal oscillations and theirsuperposition on 10 consecutive turns at the 1st (top), 251th (middle) and 501-th (bottom) turns, respectively.

ωξτˆ

0
0

τ

τ

− ωξτˆ0− ωξτˆ

Figure 13: (a) The initial zero phase difference. (b-d)
Change in phase advance after every quarter period of thefull synchrotron oscillation. After (d), the phase difference
comes back to (a).
Let us see how a simple dipole mode changes its
oscillation appearance when the phase difference between
the head and the tail oscillates as specified in Eq. (8).
Figures 14 show the simulated horizontal oscillations andtheir superposition on 10 consecutive turns at the 1st (top),
251-th (middle) and 501-th (bottom) turns, respectively.
They are strikingly similar to the right figures of Figs. 11
and Fig. 12. We also analysed the frequency spectra of theIt has been demonstrated the present feedback system isquite effective to suppress beam instabilities at the MR.
The initial analysis of the oscillation signals of thehorizontal instabilities during the injection hinted
excitation of higher-order head-tail modes and it raised aconcern over the future operation of MR. However, themore detailed analysis revealed that the instabilities
during the injection are, despite their appearance, merely
the dipole oscillations with the oscillating betatron phasedifference between the head and the tail of the bunch overa synchrotron period. The kicker impedance is now
halved and the concern over higher-order head-tailinstabilities proved unfounded at least up to ∼300kW
beam power.

REFERENCES[1] Y.H. Chin et al., “Impedance and Beam Instability
Issues at J-PARC Rings”, HB2008, Nashville,
August 2008, WGA01, p.40 (2008).
[2] A. Burov and V. Levedev., “Transverse ResistiveWall Impedance for Multi-Layer Round Chambers,”
EPAC’02, Paris, June 2002, p. 1452 (2002).
[3] F.J Sacherer, “Transverse Bunched Beam
Instabilities”, CERN/PS/BR76-21 (1976).
[4] Y. Shobuda, private communications.

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