The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Estimation of Beam Instabilities
and Its Cure in the S Pring-8 Storage Ring
Takeshi NAKAMURA
SPring-8, Kamigori, Ako-gun, Hyogo, Japan

1 Introduction
Single-bunch and multi-bunch instabilities in theSPring-8 storage ring were estimated and the possible
scheme to cure them were studied using the simulation code
SISR[1] and CISR[2], respectively. SISR can simulate
single-bunch instabilities driven by broad-band impedance
and CISR can cb multi-bunc;h instabilities driven bynarrow-band impedance.

2 The S Pring-8 Storage Ring
The parameters of the ring is shown in Table 1.
T a ble 1 The parameters o f the SPnng. 8 storage nng.
Parameter
Value
Unit

Figure 1 shows the dependence of the bunch length andthe energy spread crEIE on the bunch current ~· Because thisring is rather inductive compared to collirers which have alot of cavities, the potential-well distortion lengthen thebunch length and the threshold of microwave instabilities
can not be seen until the threshold current of the transverse
instabilities which is shown later.
Transverse Instabilities
Figure 2 shows the bunch current increase vs. time usedin the simulation. Figure 3 and Figure 4 show the amplitude
of betatron motion of the bunch vs. time and the spectrumofEnergy
8GeVEnRevolution Frequency
T__il
208.77 kHz
Energy Loss per Turn
9.2
MeVUna 1.46xl0-4
Momentum Compaction Factor
Radiation Damping Time (Long.)
4.1
ms
Radiation Damping Time (Trans.) 't, ,'t
8.3
ms
Betatron Tune (horizontal)
51.22
Yv
Betatron Tune (vertical)
v,
16.16
Broad-band impedance of the ring[3] are estimated using
the core MAFIA and the estimated values of thelongitudinal broad-band impedance, the broad-band
transverse impedance of small discontinuities and localized
transverse broad-band impedance of cavities arer::rlati~ci:y S~. ;s~rY·.<::

g0"'"

5101\
20
Time [xlO- s]
Figure 2. bunch current shape vs. time~

051015Time [xl0-3 s]

20 2mAZ 11 = -9.68xlo- 8 m i + 400 + 1.49x10 8 J;i;f [QJ , (28)Figure 3. Amplitude of the betatron motion vs time (~=0).

(B BI7~8 ) = 17.3x(-2.13x10 5+5.98x10 14 &) [Q](29)

10- 3 -r--~----~--~~~----~----~~

1 o-4
10-5

and(B Cav z~av) = 10.0 X 4.2x10 19 ~(29)
, where Bss = 17 .3m and is the averaged value of the beta
functions at the small discontinuities and 13cav is the value ofthe beta function at the cavities. Transverse instabilities are
estimated only for y direction which has larger transverse
broad-band impedance caused by smaller aperture of thebeam pipe.
[Q)

3 Single-Bunch· InstabilitiesLongitudinal Instabilities

81 o-63031323334
Betatron Frequency [10 3 Hz]
Figure 4. Spectrum of the betatron motion (~= 0) .
From Figure 4, the shift of the frequency of m=O mode
is comparable to the synchrotron frequency, 1.5kHz at Ib=3mA and 2x 1. 5kHz at Ib=7mA. 1hese instabilities seems
to be a more-coupling instability of m=D and m=l at Ib=3mA and that of m=O and m=2 occurs at Ib=7mA
In Figure 5 and Figure 6, which are for ~=4, No
instabilities occurs near Ib=3mA, but the instabilities with
m=O and m=2 mode-coupling growths up at Ib= 1OmA. The
difference between 1;=0 and 1;=4 seems to be from the effectof the head-tail damping, which can be seen at the beginningof the bunch current increase at time - 5ms in Figure 5 and
it is much faster than radiation damping seen at Ib=OmA.

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The II th Symposium on Accelerator Science and Technology, Harima Science Garden City, I997

5 Acceleration Voltage Modulation
Partial-Filling

10-2
10-4

10-6
10-8 -,.--~--~--~---+--..r-20
3 1510Time [xlO- s]
Figure 5. Amplitude of the betatron motion vs. time(~= 4).
1o- 2 --r-~-=-~--"'--"'---'---'-__.~-5

0

10-4
10-6
10-8272833
29 30 31 3t
Betatron Frequency [10 Hz]

34

Figure 6. Spectrum of the betatron motion (~= 4).

4 Multi-Bunch Instabilities
Maximum allowable values of impedance to get theaverage current 1OOmA, which is the nominal current of theSPring-8 storage ring, are !noM R 11 < 1.4 [MQ ·GHz] forlongitudinal and~ R.l < 92 [MQ] for transverse, wherefHOM are frequency of impedance and ~ is beta function atimpedance. The sources of the multi-bunch instabilities inthe SPring-8 storage ring are expected to be resonator
impedance of higher order modes of acceleration cavities andresistive wall impedance of the narrow-gap undulators. In
this report, the instabilities caused by resonator impedance
is focused.
The calculated impedance of the single-cell cavities
installed in the SPring-8 storage ring is -1.5MQ at 900
MHz for longitudinal and 14M!l/m at beta function lOrn
for transverse. The parameters in Table 2 are model
impedance used in simulation of multi-bunch instabilities
and are almost 1.5 and 2 times larger values of those of asingle cavity for longitudinal and transverse, respectively.
Table 2. Model impedance for simulation with average
current 1OOmALongitudinal Resonator hnpedance

nonRIQ

4000
1018 MHz0
Frequency

Transverse Resonator hnpedance
2308 Q/m
RIQ
13000
_Q
1006 MHz
Frequency_lOrn
Beta Function at impedance
Acceleration ModeRIO

0
Frequency
Voltage driven by External Generator
Acceleration Voltage at Iave=lOOmA

1855 n15880
508.58 MHz
17 MY/turn
14 MY/turnbyWhile a bunch train passes acceleration cavities, RFvoltage decreases by beam-loading. After the bunch train
passes, RF power supplied to the cavity re-fills RF energy ·in the cavity until the bunch train comes again. The bunches
at the head of the train feel larger amplitude of acceleration
voltage and execute faster synchrotron oscillation and thebunches in the tail feel lower amplitude of acceleration
voltage and oo slower oscillation. This causes bunch-bybunch spread of synchrotron frequency and lead to decoherence of the oscillation of instabilities and cease it. This
scheme was successfully applied to the ESRF with 1/3
filling. 115 filling is easily achieved in the SPring-8 storage
ring because extracted beam from the SPring-8 booster
synchrotron has 1 micro second duration which is one fifthof the revolution period of the storage ring.
The modulation of the amplitude of acceleration voltage
by beam-loading of bunch train is obtained by thesimulation code CISR and is shown in Figure 7. Even in
112 filling, the amplitude of morulation is more than
0.5MV and resulting synchrotron frecp.Iency spread is a f. 70Hz as is shown in Figure 8. The damping time by thisspread is 't -l/(21t M.) = 2.2ms which is twice faster than thelongitudinal radiation damping time, 4.1ms.
15.0 -J----'-,-L--....__. ....__.......__...,_

>

"'o~

14.0~

13.0~~>

50
40
30
20
Bunch No.
Figure 7. Amplitude of acceleration voltage vs. bunch
number. The number of bunch per tum used in thesimulation is 59 and first 1/5, 1/3 and 1/2 bunches filledwith electrons.

u010.:;:

t

1600~---T--~---,--~~~~~~

50403020Bunch No.
Figure 8. Synchrotron Frequency vs. Bunch Number fromsimulation result.
The simulation result, Figure 9-11, shows that nolongitudinal instabilities occurs in 112, 1/3, 1/5 filling andthis amplitude morulation is enough ·to cure theinstabilities di.ven by the model impedance. For 111.
filling( e<pal filling) shown in Figure 10 and Figure 11,
instabilities saturates at sotne amplitude of bunch motion
and the bunch length become longer then. This is because of
the filamentation caused by the nonlinearity of theacceleration potential and resulting tune spread in the bunch,which was observed and analyzed in ALS and ELETTRA.
Vl-491-010

The II th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997~

I:~V: ::::::I~00.000.04. 0.08Q)

-~

03

1.0§
0z 0.50)(;J":;::0.12-5 0.0Ui
Time [s]Figure 9. Increase of the current to their maximum values
used in simulation

12681014
Chromaticity
Figure 12. Growth rate of transverse multi-bunch
instabilities by the model impedance vs chromaticity.

20024

7 Chromaticity Modulation
4·u

0~

0~---r--~--~--~--~~--~~....bO

1:l

0.00

0.04

0.08

0.12

Time [s]Figure 10. Simulation result for relative energy oscillation
amplitude of a bunch.
~S!

30~

20

c::

10'o,....,-5bOQ)....l..c::
uc::0.00;:I

0.04

0.08

0.12

Time [s]Figure 11. Simulation result for a bunch length(rms). The
growth of the instabilities was saturated as increase of thebunch length.
ill

Chromaticity moch.ilation[5] by synchrotron frequency
produces intra-bunch betatron tune spread and cause recoherence of the electrons in a bunch and cure transverse
instabilities. Because this scheme can produce tune spreadinsire of a bunch, this scheme is also effective to singlebunch instabilities.

8 Bunch Current Dependent Tune Shift
Betatron frequency of a bunch repends on its bunch
current by wake field produced by impedance and this shift isestimated to be 0.4-0.5 kHz/( rnA/bunch) as shown in Sec.1.
And single-bunch more-coupling. instabilities occurs andthe bunch is lost where this shift is comparable tosynchrotron
tune
1.5kHz-2kHz
at
3mA/bunch-5mA/bunch. By distributing the bunch with different
bunch current, inter-bunch tune spread can be obtained tocure transverse multi-bunch instabilities and the spread of
1mA/bunch is enough to cure it for the model impedance.

6 Chromaticity Control

9 Coclusion

head-tail damping
As the bunch current increases with positive
chromaticity, transverse head-tail damping becomes strong
and overcomes instabilities. This scheme was applied toseveral machines such as KEK TRISTAN[4]. In theSPring-8, this damping is estimated by the simulation by
SISR and is comparable to radiation damping at 0.20.3mA/bunch for chromaticity is 4.
tune modulation effect
Each electron with non-zero amplitucb of synchrotron
oscillation executes non-harmonic betatron oscillation.
This is caused by the betatron frequency modulation bychromaticity and the energy oscillation by synchrotron
motion. This reduce the effect of harmonic force of wakefield which impedance source produces on the electrons. Thereduction factor is1000lo82-ro;-11 + ms~)-)8 a;/1 e- -2cri2 -od82(((1)HOM

where 1;, <J8 , W 5, and hare chromaticity, energy spread,synchrotron frequency and momentum compaction factor,respectively. This factor reduces the effect of the force onbunch and affects the motion like electron energy increase
because this increase the rigidity of the bunch to coherent
excitation as shown in Figure 12.

Using simulation codes, the strength and characteristicsof the instabilities of the storage ring is estimated and theeffects of the possible scheme to cure of them were studied.
The result shows that the storage ring can store design valueof their current, average current 1OOmA and 5mA/bunch.
Referenct~s[1,2,3,5] Refer SPring-8 Annual Report 1995 and 1996.
[2] T. Nakamura, "The Simulation Study of the SingleBunch Instabilities in the SPring-8 Storage Ring," TheFifth European Particle Accelerator Conference,Sitges(Barcelona), June 1996, pll02.
[3] T. Nakamura, "Broad-Band Impedance of the SPring-8
Storage Ring," The Fifth European Particle Accelerator
Conference, Sitges(Barcelona), June 1996, pl099.
[4] K.Akasaka, S.Kamada, "Experimental Study of HeadTail Damping and Nonlinear Filamentation Based on
Precise Analysis of Coherent Betatron Oscillations inElectron Storage Rings," Proc. of The Fifth European
Particle Accelerator Conference, Sitges(Barcelona),
June 1996, pll41.
[5] T. Nakamura, "Cure of Transverse Instabilities by
Chromaticity morulation," Proc. of the 1995 Particle
Accelerator Conference, Dallas 1995, p3100.

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