Theory and Simulation of the Electron Cloud Instability
K. Ohmi
KEK, Tsukuba, Ibaraki, 305-0801 Japan
Abstract. We discuss two-stream instabilities of positron and proton beams caused by an electron cloud from the
viewpoint of theory and simulation. The instabilities and their related phenomena are called the electron cloud
effect. Build-up of an electron cloud and instabilities caused by it are discussed for various positron and proton
rings.
INTRODUCTION
The instability caused by electron cloud has been discussed for a long time. The first work was done for
CERN-ISR [1, 2], and the two-stream type of instability started to be discussed in the accelerator physics. After that the work was succeeded to ion trapping instability in electron storage rings [3, 4, 5], and then was
succeeded to electron cloud instability in positron storage rings [6, 7, 8]. In the B factories of KEK (KEKB) and SLAC (PEP-II), solenoid magnets which wipes
off an electron cloud played an important role to realize higher luminosity [9, 10, 11]. Meanwhile, an instability has been observed for bunched proton beam in a
proton synchrotron ring at Los Alamos National Laboratory (LANL-PSR). They reported that the instability was
caused by an electron cloud [12].
We discuss the electron cloud instabilities of positron
and proton beams. Electrons are produced by synchrotron light for the most part in positron ring. The
number of photon and the conversion efficiency to electron are well understood. On the other hand there are
some candidates of electron source in proton ringii. e.,
ionization, proton loss, H~ injection etc. It is not clear
which mechanism is important in proton rings. Anyway
the initial yield is far less than that of positron ring, therefore Multipacting is essential to be built-up the electron
cloud. We discuss the electron cloud build-up in Sec.2,
and a single bunch instability driven by the electron
cloud for positron and proton beam in Sec.3. The bunch
structure is very different for positron and proton beams.
Positron bunch has a typical size of ax x ay x az ~ 1 x
0.1 x 10mm3, while proton bunch has 1cmx 1cmx 100m
in high intensity proton rings as are used for neutron
sources.
ELECTRON CLOUD BUILD-UP
We discuss electron production mechanism at first. For
positron beam, photo-emission at the chamber surface
due to synchrotron radiation is dominant. The number
of photon hitting the chamber wall is given by
5n a
_ _
r
(1)
for a positron in a meter, where a and y are the fine
structure constant 1/137 and the relativistic factor, respectively. The number of photo-electron produced by a
positron at the chamber is given by
nej = >
(2)
The direct photo-emission rate was estimated to be Yr =
0.1 for an electron incidence of a shallow angle at the
copper chamber. This value was consistent with an in
situ measurement of electron current using button electrode in KEKB [13]. We installed a test ante-chamber
in KEKB-LER to study density and yield of the electron
cloud. The value for the ante-chamber was obtained to be
1/5 for the cylindrical chamber, namely, Yj = 0.02. These
rates are somewhat larger for aluminum chambers.
For proton beam, it is not known clear electron
sources. Many possibilities for primary electron production can be considered. Ionization of residual gas
due to proton beam is a candidate. Proton beam ionizes residual gas with the result that electrons and positive ions are created. Ionization cross-section for CO
and H2 is estimated as a(CO) = 1.3 x 10~22m~2 and
a(H2) = 0.3 x 10~22m~2 using Bethe formula [14].
The molecular density dm is related to the partial pressure in nPa by the relation at 20°C, dm(m~3) = 2.4 x
lOnPm(nPd). The electron production rate is 7.7 x
W~9e~/(m -p) at 2timesW~7Pa. The production rate is 7
orders smaller than that of photoelectron in KEKB. The
positive charged ions are repulsed by the proton beam,
and hit the chamber wall. Electrons are created on the
occasion of the ion absorption at the chamber surface
[15, 16]. The number is known as 5 ~ 10. The yield is
5 x 10-V/(m - p) at 2 x 10-7Pa.
Electrons are created by proton absorption at the beam
chamber surface. M. Furman et.al. [17] use an electron
production rate Yl = 4 x I0~6e~/(m • p) at the chamber
surface. Proton loss of 4 x 10~6 has been observed in a
CP642, High Intensity and High Brightness Hadron Beams: 20th ICFA Advanced Beam Dynamics Workshop on
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360
revolution at PSR (L=90m). They assume that a proton
creates 100 electrons at its loss.
H~ injection is a direct electron source. In JHF-RCS,
since protons are injected during 500 turns, corresponding electron yield is estimated as Yl = 2 x W~5e~ /(m •
p). Most of the electrons are trapped in the magnetic field
of the injection magnets, therefore the effective yield is
smaller than the value.
Secondary electron production plays an important role
for electron cloud build-up, especially quantity of primary electrons are much less than neutralization level
and beam intensity is too high to accelerate the electrons
over lOOeV. Secondary yield [18], which is the number
of electrons created by an electron incidence with an energy, is approximated by a formula as follows,
1.44
U.44'
(3)
field,
(5)
where A+ and a / x are the beam line density in a bunch
and transverse beam sizes, respectively, and re and c
are the electron classical radius and the speed of light,
respectively.
The impedance corresponding to the wake field is
expressed by
cR?
fi
(6)
_
v
CO
C0e
RS/Q, which characterizes the coupling of beam and
electron cloud, is given by
C0e
In recent studies, elastic reflection [19] of electrons is
taken into account by Y2(E = 0) = 0.5 ~ 0.8. In this
paper, we use the original formula, Eq.(3).
We calculate electron density by the simulation code
PEI [7]. Space charge force of electron cloud is taken into
account. The beam chamber is assumed to be cylindrical
and electrons are produced uniformly along azimuthal
angle in the simulation. Electrons are produced at se and
are tracked in two dimensional x — y plane. The equation
of motion of electrons is expressed by
C
Q~
(7)
c '
Q which is called quality factor characterizes a correlation length of the wake field, and is °o in a simple linear
theory.
ke is the line density of electron cloud which contributes to the interaction with the beam. The electrons
distribute uniformly whole of the chamber at the start of
the interaction with the beam. The positron beam gathers electrons in the cloud during the interaction. The
positron
beam size (~lmm) is much smaller than cloud
d2x(t) _ 2kp(se-vt)rec2
- — , ( 4 ) (chamber) size (~ 5cm). We consider the line density as
~
*
he = K x 2npeaxay [20], where K is an enhancement
factor due to pinching of electrons and contributions of
where the force FG(x) is expressed by the Bassettielectrons near by the beam.
Erskine formula normalized so that F ,
asx On the other hand, the proton beam size (~lcm) is
(~5cm), and the bunch length is a)ef/c is much larger
Figure 1 shows variation of electron cloud density for
than 1 (~100). Electrons are gathered at the position of
the low energy ring (LER) of KEKB and the 3GeV rapid
the proton beam as soon as starting the interaction with
cycle synchrotron (RCS) of JHF as a function of bunch
the beam, therefore ke is considered to be the line density
l
passage . We note the bunch length at KEKB-LER is
of electrons in the whole chamber [21].
crz =5mm in the standard deviation while that at RCS
The single bunch instability is estimated from the
is lp =90m in total length. The density increases and
wake field. We use the coasting beam model to evalusaturates at a certain density.
ate the instability, because of coeaz/c > 1. The threshold
of the instability is expressed by [22]
SINGLE BUNCH INSTABILITY CAUSED
BY THE ELECTRON CLOUD
Z0
Z0
(8)
A single bunch instability is caused by a wake field induced by the electron cloud. We focus on the vertical instability in this paper. The wake field is represented by
a resonator model. The resonator frequency (coe) corresponds to oscillation frequency of electrons in the beam
For U > 1, the beam is unstable.
We estimated the threshold value of electron cloud
density for various positron and proton rings. The results
are shown in Table 1 and 2.
The wake field approximated by the resonator model
permits us to study the instability with simple analytic
methods. However the estimation of the threshold includes somewhat ambiguous factors: i.e., for example,
1
Space charge force has not been included in the calculation of 3GeV
RCS yet
361
1 .£«+ \£.
150
1e+12 " /
& 8e+11 - + Near beam
+
1
T3
6e+11
^g&sssssas^^
"S0
o 4e+11 - g
Average
°
$
J?
2e+11 _o
°
0
100
+
50
_
\HH.
20
40
Bunch
60
80
500
10
1500
1000
s(m)
FIGURE 1. Electron cloud density as a function of bunch passage. The left and right pictures depict those of positron (KEKB)
and proton (JHF-3GeV-RCS) beams, respectively.
TABLE 1. Single bunch electron cloud instability in positron
storage rings. The enhancement factor is chosen to be K = 3.
The impedance is evaluated at pe = 1012w~3.
variable
E(GeV)
L(m)
7V+(1010)
KEKB I PEP-II I DAFNE I JLC-DR
V,
(^(jum)
ov(jUm)
az(mm)
G)C(7Z/C
Z/ZQ(m~1}
P^(1Ql2™~3)
TABLE 2.
3.5
3016
3.3
0.018
420
60
5
2.5
2877
0.54
3.1
2200
6
0.025
700
120
12
3.2
3363
1.2
0.51
97.7
4
0.012
2000
63
24
3.2
511
1.9
1.98
398
0.75
0.01
84
7.1
5
9.1
2184
4.4
Wake fi eld and stability for electron cloud instability.
JHF
variable
L(m)
7
A^(xl0 13 )
(7r(cm)
^(m)
aE/E(%)
r\
Ys
beam pipe radius R(cm)
Z(coe)L JQ (MQ/m)
Z((0e} /Q(MQ/m)
®etp/C
Ur
U
H
3GeV
50GeV
ext.
mj.
ext.
mj.
348.3
1.4
4.15
1.9
110
0.6
-0.48
0.0058
12.5
0.29
0.61
133
0.07
0.15
348.3
4.2
4.15
1.2
82
0.7
-0.047
0.0005
12.5
0.24
0.83
182
0.23
0.78
1567.5
4.2
4.15
1.1
82
0.7
-0.058
0.0026
6.5
0.68
9.7
199
0.11
1.6
362
1567.5
54.
4.15
0.5
16
0.25
-0.0013
0.0001
6.5
0.019
0.96
276
0.02
1.2
PSR
ISIS
AGS
90
1.85
3
1.0
65
0.25
-0.187
0.0003
5
0.46
0.90
166
1.6
3.2
163
1.07
1.25
3.8
60
800
3.0
1.2
0.7
68
0.28
-0.146
0.0017
5
0.024
0.37
153
0.004
0.06
0.0003
8
0.0051
0.0085
27
0.09
0.14
how to choose K and Q. To remove the ambiguity, we
have to do tracking simulations [11, 23, 24, 25].
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