The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Single-Bunch Instabilities in the SPring-S Storage Ring:
Comparison with Simulation based on Estimated Impedance
Takeshi NAKAMURA
SPring-S
1-1-1 Kouto, Mikazuki-cho, Sayo-gun, Hyogo 679-519S, Japan

1 Introduction
The comparison between the observation of the singlebunch instabilities at the SPring-S storage ring reported inthis conference[l] and the result of the simulation based on
estimated impedance was shown in this paper. The· resultsof the observation in ref.[l] is not shown here but they areused for the comparison with the simulation results.

simulation are shown in Fig. 1, Fig. 2 and Fig. 3,respectively. In the simulation results, the bunch length israther shorter and the bunch shape do not have long tailcompared with the observation[!].
The energy spread obtained by the simulation is constantand keep natural value.

Bunch Shape:::)<...... O.S

2 Estimated Impedance.€

Impedance of the beam-pipe components are estimatedwith a simulation code MAFIA[2] using model impedance.
The estimated value of the longitudinal impedance isz 11 =- iroL+R + Zc l~iwhere L =5.22xl0· 8 H, R = S75 Q and Zc = 1.15xl08 Q s- 1•
For transverse impedance, the estimated value is(13z_L)=-i(13Lj_)+(13~_]_) +(13z~) ~;cJ"'

0.6

00.4

1::Q)
Q)

00

til
..c::

u

0.2
0-400

-350

-300

-250

-200

Time/ ps
Fig. 2 Bunch shape vs. bunch current.

where (13 L_!_) = l.Sx108 Q, (13 R_!_) =1.7x10S Q s- 1, and(13 z~) = 2.1x1020 Q s- 3' 2 • 13 is the beta function and<> isthe averaged value in one revolution of the ring.
The estimation of the impedance above was performed
using the MAFIA results for the bunch length around 5mm,
which means that for the longer bunch, the error of theestimation of the impedance and the difference of thesimulation and the observation may increases.
The source of the 90% of resistance R and R.L , which are
the main source of transverse instabilities, is the bellows.

3. Longitudinal Motion
The code SISR (Single-bunch Instabilities in Storage
Rings)[2] are used to simulate the instability using theestimated impedance.

10080"'~ 60~ 40- 200Bunch Length. ---.--...
.... --------:------·
''
''''
'

-------~······t=f····

In frequency response of the betatron motion with finite
chromaticity, we can see several peaks whose frequency is
f=Av f0 ± m f8 where Av is the fractional part of the tune andf0 and f. is the revolution frequency and synchrotron
frequency respectively. From instability theories[3], we may
expect that the head-tail instability of m=O peak may occur
at negative chromaticity and at non-negative chromaticity,mode-coupling instability or head-tail instability of lml =1
may occur. At head-tail instabilities in negative
chromaticity, the growth rate of m=O peak gradually
increases as bunch current increase and when it exceeds theradiation damping rate, the betatron motion of the centre ofmass of the bunch is driven and the instability arises. At
non-negative chromaticity, m=O peak of betatron frequency
decreases as bunch current increases and if m=O peakmerges with m=-1 peak, mode-coupling instability occurs.
Until the merge of two peaks, no growth rate of modecoupling instability is exist.

1~ o. ~ f: ;_: ~_: : C': ·- ·- -· - ·!. ·- -· - ·- ·- -~i- :- ·- -: - ·- ~l- •- J
7~ • 0.5••••••• mimmmo!uuuummo:mm••--------•o:------------··t··------------~------------·-·-t··------------·j···--------

Bunch Current I mA
Fig. I''''--~---------------~-------- -------~----

3. Transverse Motion

u 0.25..s:::00u§

Bunch length vs. Bunch current

mThe bunch current dependence of the bunch length, thebunch shape and the synchronous phase obtained by the------- _____ ,_________________ ,_________________ ,__________________,____ _
.

00.01

.

.

.

0.02

0.03

0.04

Time/sFig. 3 The increase of the bunch current in the simulation.

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The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Chromaticity=- 4The threshold current is 0.4mA/bunch. The frequency
response of the betatron motion is shown in Fig. 4. The m=O
peak ( central peak ) become higher as the bunch current
increases. The increase of the betatron amplitude of thebunch vs. time is shown in Fig. 5 in which we can see thatthe gradual increase of growth rate of the instability as theincrease of the bunch current. These property of theinstability shows that this is m=O head-tail instability.

6 and we can see the merge of m=O peak and m=-1 peak at
the threshold current. The instability should be the modecoupling.

Frequency Response of
Betatron Motion

Frequency Response ofBetatron Motion

100.1

10.01

0.1

0.001

10.10 •0 1

LLLl..L.LLJ..i..LL.L.L.LJ..i..LLLLLJ..i..LL.L.LLl..i..LL .L.L.L.J

0.0001

3.2 104 3.3 104 3.4 104 3.5 104
Frequency I Hz
Fig. 6 Frequency response of the vertical betatron motion
at chromaticity 0.23.

Betatron Amplitude
Chromaticity= 0.24r-r~0 .0 1 FrT----rr-,--,-,--,--,-r-r-,---i",-r,--,--r-Frequency I Hz
Fig. 4 Frequency response of the vertical betatron motion
at chromaticity -4.

0.001

Betatron Amplitude
Chromaticity = -4

0. 0 1 FT>--r-r-r-r-r-r-,-,--rr-r-,--r-r-r-r-r-~

0.0001'0.6mAw-5

0.001
0.0001

w-5
w-6''"-~~~,:,;:_;_?
O.OlmA----- ---------~--

2"iA

0.01

"'v\ \

0.02 0.03 0.04
Time Is------------ -·---

Fig. 7 Betatron amplitude growth vs. bunch current. The
bunch current is increased as in Fig. 3. The growthrate of the instability is suddenly increased atthreshold bunch current. At low bunch current, thehead-tail damping can be seen.

Time Is
Fig. 5 Betatron amplitude growth vs. bunch current. The
bunch current is increased as in Fig. 3. The growthrate of the instability is gradually increased as thebunch current.

Chromaticity = 3. 7

Chromaticity = 0.24
The threshold current is 4mA/bunch in the simulation. The
frequency response of the betatron motion is shown in Fig.

The frequency response of the vertical betatron motion in
the simulation is shown in Fig 8 and 9. Instability arises atthe bunch current above 12 rnA/bunch but the bunch length
is more than twice of natural value and the simulation
results at this bunch length may differ from the observation.
With almost the same operation parameters, we had the-544-

The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

bunch current more than 15mA!bunch without instabilities
in the SPring-8 storage ring.

Betatron Amplitude
0 .00 1 c:-r-r--.C-'-r-h.-r-,--o-.-m,_a,_t,.,ic---,-i---rt"-r--r-,---,---3,_.7"'---.:J

Frequency Response ofBetatron Motion
0.0001
0.1

0.01

w-s
0.01

0.001

w-6
0.001

0.000

w- 7

L.J_j-'----'----'--L--'--.l_-'-'-L ..L___l---'--L--'--.l_-'-'-lll..J .._JjiJ

00.01 0.02 0.03 0.04
Time IsFrequency I Hz
Fig. S Frequency response of the vertical betatron motion
at chromaticity 3.7

Fig. 10 Betatron amplitude growth vs. bunch current. The
bunch current is increased as in Fig. 3.
4 Conclusion

Frequency Response ofBetatron Motion

By the simulation based on the estimated impedance byMAFIA, we reproduced the result of the observation and wecould estimate the type of instabilities. The estimatedimpedance is for rather short bunch length, which is enoughto estimate the phenomena a't a few rnA/bunch, but and at
longer bunch at high bunch current, the discrepancy becomelarger such as bunch shape or instability threshold above
1OmA. One possibility to extend such case is the direct
usage of the wake-function obtained by MAFIA, whichtakes lots of CPU power.

0.01

0.001

0. 000 13 104

References

L.l_l_--'--'--1--'-'--J....L_J_--'- -'--1--'-'--J....L_J_--'-L...J--'-'- -.L..J

3.4 104
Frequency I Hz

Fig. 9 Frequency response of the vertical betatron motion
at chromaticity 3.7.

[1] T. Nakamura et. al., "Observation ofthe Single-Bunch
Instabilities in the SPring-S Storage Ring", thisproceedings[2] SPring-S Annual Report 1996
[3] For example, A W. Chao, "Physics of Collective Beam
Instabilities in High Energy Accelerators", John Wiley& Sons, 1993

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