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

Resistive Wall Instability in the KEK 12GeV PS
T. Toyama, D. Arakawa, S. Igarashi, J. Kishiro, K. Takayama
High Energy Accelerator Research Organization (KEK)
1-1 Oho, Tsukuba, lbaraki 305 Japan
Abstract

Investigation of the beam loss at the beginning of acceleration clarified the loss mechanism due to the horizontal head-tail instability. The instability occurs due
to a large change in the chromaticity produced by thesextupole field induced in the beam-pipe of the dipolemagnets. The horizontal chromaticity increases up toa positive value just around 80 ms after the beginning of acceleration. The longitudinal mode 0, 1 and2 have been observed. The impedance model explains
the instability very well.

1Figure 1: Horizontal coherent oscillaltion.

INTRODUCTION

The operating point, (vx, vy), has been changed from(7.12, 7.21) to (7.12, 5.21) since January 1996, expecting a reduction of the space-charge induced effects[1).
After the modification, the beam loss emerges around80 ms after the beginning of acceleration ( P2 ). Chromaticity control using one of the two sextupole families
has been the most effective cure to suppress the beamloss. The observation using the position monitor revealed that the instability is caused by a horizontal
coherent dipole motion, which was identified as the!!ead-Iail Instability ( HTI ) of longitudinal mode,
mainly, £ = 0. The beam loss around 80 ms after P2
has almost vanished by properly exciting sextupoles.
In this paper observed phenomena and its analysis
using the Sacherer's formula are presented.

Figure 2: Growth of the coherent oscillation. Upper
trace: beam intensity (10 11 protonsfdiv), lower trace:
coherent oscillation, abscissa: 5 k samples/div.
tupoles ( ~~x sext ), a remnant of the correction sextupoles ( ~~x remnant ), a sextupole component of thedipole and quadrupole magnets ( ~~x di , ~~x quad ),and an eddy current in the dipole beam-pipe ( ~~x eddy). Taking the energy dependence of the chromaticity
into account as~x(t) = ~xnatural + ~~xsext + ~~xdi + ~~xquad +~2EXPERIMETAL RESULTS

The typical profile of the horizontal coherent dipole oscillation is shown in Fig.l. In Fig.2 the growth of theoscillation is shown in the lower trace and the degradation of a beam intensity in the upper trace. Thesignals are sampled at the timing of the bunch center,5 k samples per division. The start point of the trace
is 50 ms after P2. The instabilities occur repeatedly
even decreasing beam intensity. Around 80 ms fromP2 the mode £ = 0 was· mostly observed. On the other
hand the mode £ = 1, 2 or some mixtures were observed at the neighbour of the above time region.
It is well known that the head-tail motion of mode
£ = 0 is unstable in the positive chromaticity region under the transition energy and has no instability
threshold in the beam intensity[2)[3].
Near injection energy, there are some contributions
to the chromaticity other than natural chromaticity,[4): an excitation of two families of correction sex-~xremnant

B(t)j B(t)~B(Oms)B(t) + ~x eddy B(100ms)j B(lOOms)

we obtain the time evolution of the chromaticity, whichis shown with the measured chromaticity in Fig.3. The
solid line and the filled circles are the calculated andmeasured results, respectively, for the operation inwhich the beam loss occurs .due to the uncorrected
chromaticity at (7.12, 5.21). The calculation including the eddy current effect well agrees with the measured result and well explains the beam loss around 80
ms from P2, when the chromaticity takes the positive
value around 80 ms from P2.
The natural chromaticity is ~x = -9.22 for the previous operating point, (7.12, 7.21). After reducing thevertical tune, the natural chromaticity increased by,...., 1.3, became closer to zero. This caused zero crossing of the horizontal chromaticity around 80 ms afterP2 and forced the horizontal coherent motion, mainly,of mode 0 unstable and resulted in the beam loss.
Since the incoherent space charge tune shift has no-297-

The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997
2~-··T---,,.

·-··r-··~r-··0.00004

0•

•
I

•

i~

0.00002I..c::+>

0.00001+>-1
X

Ul'·-2

•II

I

-3~

00~

l9 -0.00001

I-40I.I
I-0.00002-5 L_-=-~-~~---__j-20 020 40 60 80 1 00 120

Time from P2 (ms)Figure 3: Horizontal cliromaticity after the acceleration start.

i0.00003<I>--___. - ...•~
I2II
46810Particle number [ x 10A11 ]Figure 5: Growth rate as a function of bunch intensity.
Table 1: Beam and machine parameters

effect for the transverse dipole mode, we need someother stabilizing effects. The cure has been achievedby the defocusing sextupole family (S D) because ofthe power supply polarity. Corresponding chromaticities are plotted in Fig.3 by the dashed line (calculatedresult) and the open circles (measured result).
In order to fully understand the feature of the instability, including modeling of coupling impedances,and to investigate most proper cure, the growth rate
was measured with various conditions.
Measured growth rate as a function of the cluomaticity with single bunch is depicted in Fig.4. Because the measurement was made during acceleration,
the growth rate is normalized by the revolution frequency w0 .

E271" x 7.67 kHz
271" X 4 kHz
5.8 x 10 11 protons/bunch
7.1575 ns
700 + 938 MeV

TJ-0.22

w0WsNb
VxTb

The growth rate normalized by w 0 isIm[.:lvf],
where0.0001<I>

0.00008+>
nlp::

0.00006..c::

0.00004~00.00002+>~l9

0-0.00002---1/v
0v/
1/,...--- -- --2

3

4

5

Chromaticity
Figure 4: Growth rate as a function of horizontal cluomaticity.
The measured growth rate as a function of the bunch
intensity was depicted in Fig.5.

3ANALYSISThe qualitative feature of the observed HTI was well
explained by a multi-particle simulation code, although a constant wake was used.[4]In order to obtain more quantitative feature, thegrowth rate is calculated using the Sacherer's formula[2][3][5] along with an impedance model.

p=integer, £ is the azimuthal mode number, w 0 therevolution frequency, w 8 the synchrotron frequency, e
the electron charge, c the light speed, Nb the bunch
intensity, Vx the horizontal tune, Tb the total bunch
length, E the total energy, w€ = ~/TJ the chromatic
frequency, and TJ the slippage factor. The parametersused for the calculation are summarized in Table 1.
For coupling impedances a resistive wall impedance,a kicker impedance and a broad band impedance areassumed. The resistive wall is considered to comprise
two parts, one from almost rectangular beam pipes[6]in the bends and quads ( zk~ 1.) and one from almost
cylindrical beam pipes in straight sections (Z~~ 1.),I.e.

-298-(m). R(m) 71"2ZRW.L ~ (sgn(w)- z) b(m) 24v{2P~'The lith Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Table 4: Broad band impedance

Table 2: Parameter s for the resistive wall impedance
R(m)
R(s)

b(m)
b( s)
p200m
140m
25.5 mm
60mm
9 x 10- 7 nm

RT3 MO/m

Wr27r X 1.4 GHz

Q1

The form factor, hm(w) for parabolic distributio n is
Table 3: Kicker parameter s (for one module)
Zs
Zoab
Vkicker I C
Vcable/c£(k)
£(k2

377 n
25 n25 mm
62.5 mm
0.028
0.67
250mm
55 m

The calculated results are depicted in Fig.4 andFig.5. In Fig.4 the growth rate for the mode £ = 0, 1
and 2 are plotted by a solid line, a dashed line anda dotted line, respectively. These results agree withthe measured values very well. It should be notedthat three impedance s contribute almost equally to thegrowth rate.

4and.) R(s) [;r&p( )((s)~ sgn w - z -b3 -~-~'zRW
Eo w
(s)
j_

where 21f'Rcml and 21f'R(s) are the total length of themagnet section and the straight section, respectively, p
the vacuum chamber resistivity, b(m) the half height ofthe beam pipe and b(s) the pipe radius. The relevant
parameter s are summarize d in Table 2.
There are five identical kickers in the 12Ge V PS. Its
impedance (real part) is[7]Re[zi"'ll =ZTo [sin(k£k) + tan(k'£Uc os(k£k)- tan(k'£U]2
1 + tan 2 (k'£Uk£kwhere a and b are the half height and the half width of
the kicker aperture, ZTo = Zs£kl(4ab ), k = wlvkicker, k' = wI Vcable, £k and £~ the length of the kicker
and cable, respectively, Z 8 is the vacuum impedance ,Vkicker = Zoal(pob) velocity in the kicker, Vcablevelocity in the cable and Zo is the characteri stic
impedance of the kicker. Each kicker is matched at
one end and connected with a cable at the other end
which is approxima tely open at the end of the cable.
The parameter s used in the calculation are summarized in Table 3.
For the broad band impedance
Z_ Wr
BBJ_-W 1

RT+ Z"Q( Wr I W - W I Wr )'SUMMA RY

The observed feature of the HTI is well explainedby the Sacherer's theory and the present impedancemodel.
We are further investigati ng the HTI, which comprises single- and multi-bun ch effects, and will optimize the cure.

5ACKNO WLEDG EMENTS

The authors wish to thank KEK-PS members fortheir interest and continual support through this work.
They wish to thank Prof. I. Yamane and Prof. H.
Sato for calling their attention to the present problem.
They also wish to thank Dr. M. Shirakata, Dr. M.
Uota and Miss K. Koba for helping in the experimen tand giving usefull comments.

6

REFERE NCES

[1] S. Machida and Y. Shoji, AlP Conf. Proc. 377 (1995)
p.160, S. Machida, Nucl. Instr. Meth. A 384 (1997) 316.
and Y. Shoji et al., submitted to the 1997 Part. Accel.
Conf., May 12-16, 1997, Vancouver, BC, Canada.
[2] B. Zotter and F. Sacherer, CERN 77-13, p.175.
[3] A. Chao, "Physics of collective beam instabilities inhigh energy accelerator s", John Wiley & Sons, Inc.,
1993 and the references there in.
[4] T. Toyama et al., submitted to the 1997 Part. Accel.
Con£., May 12-16, 1997, Vancouver, BC, Canada.
[5] R. Cappi, Part. Accel. 50 (1995) 117.
[6] K. Y. Ng, Part. Accel. 16 (1984) 63.
[7] G. Nassibian, CERN/PS 85-68 (BR), 1986.

the following parameter s are assumed.[5]-299-