EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN-SL-Note-2000-061 AP
More Electron Cloud Studies for KEKB:
Long-Term Evolution, Solenoid Patterns, and
Fast Blow Up
F. Zimmermann, H. Fukuma, and K. Ohmi
Abstract
We report recent electron-cloud studies for the Low Energy Ring (LER) of
KEKB, continuing earlier investigations [1, 2]. Here we simulate the build up and
decay time of the cloud for various solenoid and C-yoke configurations, discuss
the electron density enhancement during the bunch passage [3], and estimate the
instability threshold using the Ruth-Wang theory of fast blow up [4, 5].
Geneva, Switzerland
December 5, 2000
1
Introduction
This report is structured as follows. In Section 2 we describe simulations of electron cloud build
up and decay for a series of solenoids with equal or alternating polarity. Among other aspects,
we discuss the effect of varying period lengths, magnet strength, and changes in the solenoid
field model, considering, e.g., Fourier expansions beyond the paraxial approximation [6]. We
then briefly describe simulations for symmetric or antisymmetric C yoke quadrupoles with
10-cm and 5-cm distance between magnets, using improved parametrizations of the magnetic
field. Section 3 discusses the ‘pinch’ of the electrons during the bunch passage and the resulting
enhancement of the incoherent tune spread along the bunch. Section 4 estimates the singlebunch current threshold for the post head-tail single-bunch instability. The report closes with
a short summary and outlook.
2
Various Simulations of Electron-Cloud Build Up
The build up of the electron cloud and the electron density at the center of the chamber are
studied using a computer simulation, where electrons are represented by macroparticles. The
simulation includes the electric (and sometimes also the magnetic) field of passing bunches,
external magnetic fields, electron-cloud space charge field and, optionally, image charges of
both beam and electrons. Details of this program have been described elsewhere [7, 8, 9, 10].
Table 1 compiles parameters assumed in our simulations for KEKB. Typical results for
a short train of 40 bunches followed by a gap are displayed in Fig. 1. The figure shows the
evolution of the electron line density (number of electrons per meter length) as a function of
time for a field-free region and a solenoid. Both build-up and decay time are longer in the
presence of a solenoid field. In a field-free region the decay time is of the order of 15 ns (Fig. 1
bottom right) whereas in a solenoid it increases to about 100 ns (Fig. 1 bottom left). Also
compared in the figure are results for two different photoelectron distributions. In one case
the photoelectrons are emitted uniformly distributed around the chamber circumference; in
the other case they are confined to a narrow opening angle with rms value 11.25◦ . These two
situations are denoted by R = 1 (indicating the extreme of 100% photon reflectivity) and
R = 0 (no reflectivity). The total number of accumulated electrons for the two cases differs
by about 20%–30%.
Figure 2 depicts the central cloud density, i.e., the density at the center of the chamber, for
the same four cases as shown in Fig. 1. Again, the decay is faster in the field-free region, but
there is almost no difference between the results for R = 1 and for R = 0. In the remainder
of this report we only consider R = 1.
It had been suspected that small changes in the beam size due to beam-beam collisions
could change the region from where photoelectrons are emitted, which in turn might be responsible for the observed blow up in collision. Given the results in Figs. 1 and 2, and the
rather small difference in central electron density for R = 0 and R = 1, this seems to be an
unlikely explanation.
Figure 3 depicts the energy spectrum of electrons lost to the wall, on a linear scale. The
spectrum is concentrated at rather low energies, extending only to about 30 eV. We recall
that energy spectra plotted on a logarithmic scale in Ref. [2] revealed a peculiar fine structure
at higher energies.
Recent measurements [12, 13] indicate the existence of a large component of elastically
scattered or rediffused electrons in addition to the true secondaries. For the latter the secondary emission yield at perpendicular incidence is described by the Seiler formula [11]. The
2
variable
chamber radius
rms bunch length
rms horizontal beam size
rms vertical beam size
rf bucket length
bunch spacing
bunch population
ring circumference
average vertical beta function
photon reflectivity
photoelectron yield
center of photoelectron energy distr.
photoelectron rms energy spread
maximum secondary emission yield
energy of max. sec. emission
symbol
rp
σz
σx
σy
srf
sb
Nb
C
βy
R
Ype
Epe,0
σpe
δmax (SEY)
max
value
47 mm
4 mm
600 µm
60 µm
0.6 m
4, 8, rf buckets
3.3 × 1010
3000 m
15–20 m
100%
0.15/e+ /meter
7 eV
5 eV
1.8
300 eV
Table 1: Parameters assumed for the electron-cloud simulations.
1.6e+10
2e+10
charge/m, solenoid, R=1
charge/m , no field, R=1
charge/m, solenoid, R=0
charge/m , no field, R=0
1.8e+10
1.4e+10
1.6e+10
1.2e+10
1.4e+10
1e+10
1.2e+10
8e+09
1e+10
8e+09
6e+09
6e+09
4e+09
4e+09
2e+09
2e+09
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
0
2e+10
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
2e+10
charge/m, solenoid, R=0
charge/m , solenoid, R=1
charge/m, no field, R=0
charge/m, no field, R=1
1.8e+10
1.8e+10
1.6e+10
1.6e+10
1.4e+10
1.4e+10
1.2e+10
1.2e+10
1e+10
1e+10
8e+09
8e+09
6e+09
6e+09
4e+09
4e+09
2e+09
2e+09
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
Figure 1: Simulated electron-cloud line density in units of electrons per meter as a function
of time in seconds along a 40-bunch train with 4 bucket bunch spacing and a subsequent long
gap, comparing various cases: (first) photoelectron distribution confined to an rms opening
angle of 11.25◦ with and without solenoid field; (second) uniform photoelectron distribution
with and without solenoid field; (third) confined and uniform photoelectron distribution in
solenoid field; (last) confined and uniform photoelectron distribution in field-free region.
3
6e+12
9e+12
center density, solenoid, R=0
center density, no field, R=0
center density, solenoid, R=1
center density, no field, R=1
8e+12
5e+12
7e+12
4e+12
6e+12
5e+12
3e+12
4e+12
2e+12
3e+12
2e+12
1e+12
1e+12
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
Figure 2: Simulated central electron-cloud density in units of electrons per cubic meter meter
as a function of time in seconds along a 40-bunch train with 4 bucket bunch spacing and a
subsequent long gap, comparing various cases: (first) photoelectron distribution confined to an
rms opening angle of 11.25◦ with and without solenoid field; (second) uniform photoelectron
distribution with and without solenoid field.
0.04
0.045
energy spectrum, solenoid, R=0
energy spectrum, no field, R=0
energy spectrum, solenoid, R=1
energy spectrum, no field, R=1
0.04
0.035
0.035
0.03
0.03
0.025
0.025
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
0
0
0
50
100
150
200
250
0
50
100
150
200
250
Figure 3: Simulated energy spectra in eV of electrons hitting the wall during and after the
passage of a 40-bunch train with 4 bucket bunch spacing, comparing various cases: (first) photoelectron distribution confined to an rms opening angle of 11.25◦ with and without solenoid
field; (second) uniform photoelectron distribution with and without solenoid field.
4
yield for the elastically scattered/rediffused part can be parametrized as [14]
2
Epe
δel = δel,0 + δel,E exp − 2 .
2σel
(1)
Measurements on LHC prototype vacuum chambers indicate that [12] δel,0 ≈ 0.0, δel,E = 0.56,
and σel = 52 eV. These values of δel,E and σel are each about 10 times higher than those
considered in previous studies [14].
The elastically scattered electrons constitute an additional source of secondary electrons,
which can lead to an increase in the saturation density by 50%–100%. This is demonstrated in
Figs. 4 and 5 depicting results for a field-free region and for a solenoid, respectively. We may
also suspect that the elastically scattered electrons will slow down the decay of the electron
cloud during a bunch-train gap. Figure 4 illustrates that this is indeed the case for a field-free
region. However, Fig. 5, shows that in a solenoid field the additional elastic component does
not appreciably affect the electron decay time. The latter thus seems to be determined by
the solenoid field alone. In the following simulations, we do no longer include elastic electron
scattering.
3.5e+10
9e+12
no field, R=1, only pure secondaries
no field, R=1, reflected electrons
no field, R=1, pure secondaries only
no field, R=1, reflected e8e+12
3e+10
7e+12
2.5e+10
6e+12
2e+10
5e+12
4e+12
1.5e+10
3e+12
1e+10
2e+12
5e+09
1e+12
0
0
0
2e-07
4e-07
6e-07
8e-07
1e-06
1.2e-06
0
2e-07
4e-07
6e-07
8e-07
1e-06
1.2e-06
Figure 4: Simulated electron line density (left) in electrons per meter and central electron
density (right) in units of m−3 as a function of time in seconds during and after the passage
of a 100-bunch train with 4 bucket bunch spacing without solenoid field, comparing cases with
and without a large component of elastically reflected electrons.
So far we have considered a sinusoidal solenoid field
Bz (x, y, z) = B0 sin kz
(2)
with transverse field components
1
Bx (x, y, z) = − B0 kx cos kz
2
1
By (x, y, z) = − B0 ky cos kz
2
(3)
(4)
This roughly represents a series of solenoids with alternating polarity. The parameter values
assumed, B0 = 50 G and λ = 2π/k = 1 m, resemble those in the real machine. Earlier
simulations [2] always referred either to this configuration or to a perfectly uniform logitudinal
solenoid field.
5
4e+10
1.2e+12
solenoid, R=1, pure secondaries
solenoid, R=1, reflected e-
solenoid, R=1, pure secondaries
solenoid, R=1, reflected e-
3.5e+10
1e+12
3e+10
8e+11
2.5e+10
2e+10
6e+11
1.5e+10
4e+11
1e+10
2e+11
5e+09
0
0
0
2e-07
4e-07
6e-07
8e-07
1e-06
1.2e-06
0
2e-07
4e-07
6e-07
8e-07
1e-06
1.2e-06
Figure 5: Simulated electron line density (left) in electrons per meter and central electron
density (right) in units of m−3 as a function of time in seconds during and after the passage of
a 100-bunch train with 4 bucket bunch spacing with solenoid field, comparing cases with and
without a large component of elastically reflected electrons.
However, in the KEKB LER arcs, the solenoids are actually arranged in a slightly different pattern. Three solenoids with equal polarity are followed by three others with opposite
polarity. While the transition between the two sets of three is represented by the above approximation, the transition between two solenoids of equal polarity is not. Adjacent solenoids
of equal polarity might give rise to magnetic bottles and show a different behavior, possibly
allowing for a long-term accumulation of electrons. To study this possibility, simulations were
performed for field components chosen as
1
Bx (x, y, z) = − B0 kx cos kz
2
1
By (x, y, z) = − B0 ky cos kz
2
Bz (x, y, z) = Bz0 + B0 sin kz.
(5)
With a constant term Bz0 = 30 G, a harmonic component B0 = 20 G, and a period length
λ = 1 m, the fields in Eqs. (5) represent two nearby solenoids of equal polarity, separated by
0.5 m, with a peak field of 50 G, and a minimum field of 10 G between them. We will call this
model the parallel or equal field configuration, and the former model, Eqs. (2) and (4) with
B0 = 50 G, the anti-parallel or opposite field configuration.
Figure 6 shows a few electron trajectories projected onto the x − y plane, illustrating how
both parallel and anti-parallel solenoid fields confine the electron motion to the vicinity of the
vacuum-chamber wall. The trajectories do not reveal any qualitative difference between the
two solenoid configurations. Figure 7 presents examples of longitudinal motion as a function
of time. It appears that for the opposite-polarity configuration, the electron oscillates around
z = 0.25 m. Indeed, Fig. 9 (right picture) below will show that z = 0.25 and z = ±0.25
correspond to maxima in the projected longitudinal electron density for the equal and oppositepolarity configurations, respectively.
Figure 8 compares the total number of electrons and the central density accumulating
during the passage of a long bunch train for the two field configurations. In the configuration
with opposite polarity, a few electrons reach the vicinity of the beam, and both build-up and
decay time of the cloud are shorter than for the equal-polarity case. For the latter, no electrons
are found near the beam.
6
0.05
0.05
equal polarity
opposite polarity
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
-0.01
-0.02
-0.02
-0.03
-0.03
-0.04
-0.04
-0.05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
-0.05
-0.05
0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Figure 6: Sample electron trajectories in x − y plane simulated for the parallel (left) and
anti-parallel (right) solenoid configurations. Again, we consider Nb = 3.3 × 1010 , σz = 4 mm,
σx = 600 µm, σy = 60 µm.
0.4
opposite polarity
equal polarity
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0
500
1000
1500
2000
2500
Figure 7: Sample electron longitudinal position in units of meter vs. time in nanoseconds,
for the parallel and anti-parallel solenoid configurations. Again, we consider Nb = 3.3 × 1010 ,
σz = 4 mm, σx = 600 µm, σy = 60 µm.
2.2e+10
2.5e+12
equal polarity
opposite polarity
equal polarity
opposite polarity
2e+10
1.8e+10
2e+12
1.6e+10
1.4e+10
1.5e+12
1.2e+10
1e+10
1e+12
8e+09
6e+09
5e+11
4e+09
2e+09
0
0
0
5e-07
1e-06
1.5e-06
2e-06
2.5e-06
0
5e-07
1e-06
1.5e-06
2e-06
2.5e-06
Figure 8: Simulated line density in units of m−1 and central cloud density in m−3 for a train
of 260 bunches spaced by 8 ns and followed by a gap of 40 missing bunches, as a function
of time in seconds, for parallel and anti-parallel solenoid configurations. Again, we consider
Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
7
Figure 9 depicts the longitudinal and radial electron distributions after the passage of the
last bunch for the same two magnet configurations. The number of electrons is enhanced in
regions where the longitudinal magnetic field is higher.
6e+09
8e+08
equal polarity
opposite polarity
equal polarity
opposite polarity
7e+08
5e+09
6e+08
4e+09
5e+08
3e+09
4e+08
2e+09
3e+08
1e+09
2e+08
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
1e+08
-0.5
0.05
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 9: Simulated radial density ∆N/∆r rp /r (left) and longitudinal density ∆N/∆z (right)
with 100 bins after the passage of 260 bunches, for the two solenoid configurations. Again, we
consider Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
Figure 10 shows the build up of the electron-cloud line density for the two solenoid configurations, comparing solenoid periods of 0.5 m and 2 m. The decay time varies strongly
with the length of the solenoid, shorter solenoids corresponding to shorter decay times. For
the equal-polarity configuration, still no electrons are found at the center of the chamber. On
the other hand, as already seen with λ = 1 m in Fig. 5, for the case of alternating polarity
there may be a non-negligible density of electrons near the beam. This is further illustrated
in Fig. 11, which also suggests that the central electron density only weakly depends on the
solenoid length. However, the statistics of the simulation is poor.
2.2e+10
2.2e+10
equal, period 0.5 m
equal, period 2 m
opposite, period 0.5 m
opposite, period 2 m
2e+10
2e+10
1.8e+10
1.8e+10
1.6e+10
1.6e+10
1.4e+10
1.4e+10
1.2e+10
1.2e+10
1e+10
1e+10
8e+09
8e+09
6e+09
6e+09
4e+09
4e+09
2e+09
2e+09
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
Figure 10: Simulated line density for a train of 80 bunches spaced by 8 ns and followed by a gap
of 40 missing bunches, as a function of time in seconds, for the parallel (left) and antiparallel
(right) solenoid configurations with period lengths λ equal to 0.5 m or 2 m. Again, we consider
Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
Figure 12 shows the electron-cloud build up for solenoids with a reduced peak field of 30
G. This should be compared with the results for 50 G in Fig. 8. Again differences appear to
be minor.
8
2.5e+12
opposite, period 0.5 m
opposite, period 2 m
2e+12
1.5e+12
1e+12
5e+11
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
Figure 11: Simulated central electron density for a train of 80 bunches spaced by 8 ns and
followed by a gap of 40 missing bunches, as a function of time in seconds, for the antiparallel
solenoid configuration, comparing period lengths equal to 0.5 m or 2 m. Again, we consider
Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
2.5e+10
3e+12
30 G, equal polarity
30 G, opposite polarity
30 G, equal polarity
30 G, opposite polarity
2.5e+12
2e+10
2e+12
1.5e+10
1.5e+12
1e+10
1e+12
5e+09
5e+11
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
Figure 12: Simulated electron line density in units m−1 (left) and central density in m−3 (right)
for a train of 80 bunches spaced by 8 ns and followed by a gap of 40 missing bunches, as a
function of time in seconds, for the parallel and antiparallel solenoid configurations, with a
peak field of 30 G. Again, we consider Nb = 3.3 × 1010, σz = 4 mm, σx = 600 µm, σy = 60 µm.
9
None of the simulations reported so far included the effect of electron image charges inside
the round chamber. Figure 13 compares the electron cloud build up computed with and
without the effect of these image charges. It suggests that the image-charge force reduces the
number of electrons at saturation by 10–20%. Figure 14 demonstrates that the image charges
tend to even further confine the electrons to the vicinity of the chamber wall.
2.4e+10
2.6e+10
opposite polarity, w/o image charges
opposite polarity, with image charges
2.2e+10
equal polarity, w/o image charges
equal polarity, with image charges
2.4e+10
2.2e+10
2e+10
2e+10
1.8e+10
1.8e+10
1.6e+10
1.6e+10
1.4e+10
1.4e+10
1.2e+10
1.2e+10
1e+10
1e+10
8e+09
8e+09
6e+09
6e+09
4e+09
4e+09
2e+09
2e+09
0
0
0
5e-07
1e-06
1.5e-06
2e-06
2.5e-06
0
5e-07
1e-06
1.5e-06
2e-06
2.5e-06
Figure 13: Simulated electron line density in units m−1 for a train of 80 bunches spaced by
8 ns and followed by a gap of 40 missing bunches, as a function of time in seconds, for the
antiparallel (left) and parallel (right) solenoid configurations in paraxial approximation, with
a peak field of 50 G, with and without including the effect of electron image charges. Again,
we consider Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
0.05
0.05
equal polarity, w/o image charges
equal polarity, with image charges
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
-0.01
-0.02
-0.02
-0.03
-0.03
-0.04
-0.04
-0.05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-0.05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Figure 14: Sample electron trajectories simulated for the parallel solenoid configurations
without (left) and with (right) the effect of electron image charges. Again, we consider
Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
E. Perevedentsev drew attention to the fact that Eqs. (2), (4) and (5) represent a paraxial
approximation to the real solenoid field. If the latter is still taken to be sinusoidal on axis,
Maxwell’s equations imply the more exact form
Bz (r, z) = B0 I0 (kr) cos kz
Br (r, z) = B0 I1 (kr) sin kz
(6)
Expanding the Bessel functions to first order in radius r this reduces to the previous formulae.
10
Further extensions are possible and more than one Fourier component can be kept in
the longitudinal field expansion. A better parametrisation of the solenoid field, derived by
E. Perevedentsev, reads [6]
Br
∞
2ka = B0
sin nkh K1 (nka) I1 (nkr) sin nkz
π n=1
Bz = B0
∞
2h 2ka +
sin nkh K1 (nka) I0 (nkr) cos nkz
L
π n=1
(7)
(8)
where the I and K are modified Bessel functions of the first order, a is the solenoid radius,
h the solenoid length, L the distance between adjacent solenoids with equal polarity, and
B0 a normalization constant, roughly equal to the field on axis inside the solenoid. in the
simulation, the infinite series is truncated at some order, e.g., n = 5. A similar formula, with
odd harmonics doubled and even harmonics set to zero, describes the case of solenoids with
alternating polarity, separated by L/2. These formulae have been programmed for another set
of simulations, where we have chosen a solenoid length of h = 0.4 mm, solenoid radius a = 70
mm, period length L = 1 m, and nominal field B0 = 50 G. The Fourier sums in Eqs. (7) were
calculated through order n = 5.
Figure 6 compares the electron-cloud densities obtained using Eqs. (7) and (8) with the
above parameters and those found in the paraxial approximation, Eqs. (2) and (4). While
the total number of electrons are quite similar, in the case of the 5th order Fourier expansion
and opposite-polarity solenoids the central electron density is a factor 2–3 higher than for
the paraxial approximation. Using the Fourier sum, for equal-polarity solenoids the central
electron density is about half that obtained for opposite-polarity solenoids, whereas using the
paraxial approximation the central density is zero. The increase in the central-cloud density
for the Fourier expansion is consistent with the evolution of individual electron trajectories
depicted in Fig. 16. With Eqs. (7) and (8), the electron trajectories are much less confined to
the chamber wall.
In order to shed more light on the difference between the Fourier expansion and the paraxial approximation, Fig. 17 compares trajectories obtained by expanding the longitudinal field
either through 5th order (as above) or only through order n = 1. The results are qualitatively
similar. Independently of the truncation order n, the central cloud density is increased compared with the paraxial approximation. In particular, for equal polarity solenoids and either
truncation order (n = 5 or n = 1) some electrons are found at the center of the chamber,
whereas in the paraxial approximation there are none. The density evolution at the center of
the chamber is illustrated in Fig. 18 for n = 1. This is the only case, where the central density
for the equal-polarity arrangement is higher than that for the opposite polarity.
Note that for an arbitray on-axis (r = 0) field profile Bz (z), the paraxial field expansion
to second order in r reads [6]
r2
Bz = B0 (z) − B0 (z) + . . .
4
r Br = − B0 (z) + . . . .
2
(9)
(10)
This could be used for studying yet other field patterns in future simulations.
We next repeated simulations for symmetric and asymmetric quadrupole C-yoke configurations. These represent an alternative to the solenoid, and it is not yet completely understood
11
2.4e+10
2.2e+10
equal, Bessel functions
equal, paraxial approximation
opposite, Bessel functions
opposite, paraxial approximation
2.2e+10
2e+10
2e+10
1.8e+10
1.8e+10
1.6e+10
1.6e+10
1.4e+10
1.4e+10
1.2e+10
1.2e+10
1e+10
1e+10
8e+09
8e+09
6e+09
6e+09
4e+09
4e+09
2e+09
2e+09
0
1e-07
2e-07
3e-07
4e-07
5e-07
3e+12
0
1e-07
2e-07
3e-07
4e-07
5e-07
5e+12
equal, Bessel functions
equal, paraxial approximation
2.8e+12
opposite, Bessel functions
opposite, paraxial approximation
4.5e+12
2.6e+12
2.4e+12
4e+12
2.2e+12
3.5e+12
2e+12
1.8e+12
3e+12
1.6e+12
2.5e+12
1.4e+12
1.2e+12
2e+12
1e+12
1.5e+12
8e+11
6e+11
1e+12
4e+11
5e+11
2e+11
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
0
1e-07
2e-07
3e-07
4e-07
5e-07
Figure 15: Simulated electron line density in units m−1 (top) and central density in m−3
(bottom) for a train of bunches spaced by 8 ns as a function of time in seconds, for the parallel
(left) and antiparallel (right) solenoid configuration, modelled by Bessel functions and Fourier
expansion through order n=5 according to Eqs. (7)–(8) and by the paraxial approximation,
Eqs. (2) and (4). Again, we consider Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
Electron image charges are included.
0.05
0.05
equal, Bessel function
equal, paraxial approximation
opposite, Bessel function
opposite, paraxial approximation
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
-0.01
-0.02
-0.02
-0.03
-0.03
-0.04
-0.04
-0.05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
-0.05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Figure 16: Sample electron trajectories simulated for the parallel (left) and anti-parallel (right)
solenoid configurations, modelled by Bessel functions and Fourier expansion through order n=5
according to Eqs. (7)–(8) — the plus symbols —, and by the paraxial approximation, Eqs. (2)
and (4) — the crosses. Again, we consider Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm,
σy = 60 µm. Electron image charges are included.
12
0.05
equal, Bessel f. 1. order
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
why their actual effect at KEKB was so small. In this renewed study, we extend the magnetic
field beyond the paraxial approximation, using the fields
1
1
B̄0
B̂0
y+
sin kz y + k 2 yx2 + k 2 y 3
r0
r0
4
12
B̄0
B̂0
1 2 2
1 2 3
By (x, y, z) =
y+
sin kz x + k xy + k x
r0
r0
4
12
B̂0
Bz (x, y, z) =
sin kz kxy
r0
Bx (x, y, z) =
(11)
As in previous studies [2] we consider two cases: (1) a symmetric configuration, which we
represent by B̄0 /r0 = 0.3 T/m and B̂0 /r0 = 0.2 T/m; (2) an asymmetric configuration, where
B̄0 /r0 = 0.0 T/m and B̂0 /r0 = 0.25 T/m. Simulation results are shown in Fig. 19. They
are consistent with previous studies. The dependence of the electron density on the distance
between adjacent magnets is rather weak.
2.2e+10
2.2e+10
symmetric, 10 cm distance
symmetric, 5 cm distance
antisymmetric, 10 cm distance
antisymmetric, 5 cm distance
2e+10
2e+10
1.8e+10
1.8e+10
1.6e+10
1.6e+10
1.4e+10
1.4e+10
1.2e+10
1.2e+10
1e+10
1e+10
8e+09
8e+09
6e+09
6e+09
4e+09
4e+09
2e+09
2e+09
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
9e+12
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
1e+13
symmetric, 10 cm distance
symmetric, 5 cm distance
antisymmetric, 10 cm distance
antisymmetric, 5 cm distance
9e+12
8e+12
8e+12
7e+12
7e+12
6e+12
6e+12
5e+12
5e+12
4e+12
4e+12
3e+12
3e+12
2e+12
2e+12
1e+12
1e+12
0
0
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
0
1e-07
2e-07
3e-07
4e-07
5e-07
6e-07
7e-07
8e-07
9e-07
1e-06
Figure 19: Simulated electron line density in units m−1 (top) and center density in units of
m−3 (bottom) for a train of 80 bunches spaced by 8 ns and followed by a gap of 40 missing
bunches, as a function of time in seconds, for the symmetric (left) and antisymmetric (right)
C yoke configurations, with a peak gradient of 0.5 T/m, and distances between C yokes equal
to 5 and 10 cm (the two curves in each picture). Again, we consider Nb = 3.3 × 1010 , σz = 4
mm, σx = 600 µm, σy = 60 µm.
14
symmetric C yoke
antisymmetric C yoke
0.04
0.02
0
-0.02
-0.04
-0.04
-0.02
0
0.02
0.04
Figure 20: Sample electron trajectories in x − y plane simulated for symmetric and antisymmetric C yoke configurations with 10 cm distance between magnets. Again, we consider
Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
3
Electron Distribution Inside the Bunch
Once established, the electron cloud may blow up the vertical beam size, either by driving a
single-bunch instability [15] or via the induced incoherent tune spread [3]. We here address
the latter possibility.
During the bunch passage the electron density increases near the center of the bunch,
thereby enhancing the incoherent tune spread at the bunch tail [3]. To investigate the density
distribution inside the bunch, in a separate simulation program a single electron bunch is sent
through an initially uniform electron distribution. The bunch is divided into many slices, and
the electron distribution is computed after each passage of a slice. In particular, we calculate
the number of electrons inside a 1 or 2 σ ellipse in the x − y plane, and determine the increase
of this number along the bunch. Normalizing this to the number in the first slice, the density
enhancement is obtained. Figure 21 compares results for three initial electron temperatures.
The density enhancement within 1σ is about a factor of 6 at the end of the bunch. It is little
sensitive to the initial temperature of the electron cloud.
Figure 22 depicts similar simulations for other parameters. The first picture illustrates that
the magnetic field of the beam does not affect the electron accumulation inside the bunch.
The second and third show that the electron density at the bunch tail increases for longer
bunches or higher charges, roughly with equal sensitivity. Experimental verification of this
scaling behavior, ρ ∝ Nb /σz , would provide evidence that the electron build up inside the
bunch might be responsible for the observed blow up, whereas, for constant synchrotron tune,
the effect of the single-bunch instability is expected to depend on the product Nb σz .
Figure 23 shows projected horizontal and vertical density distributions. A sharply peaked
distribution is confined well within 1σ. Only considering the linear part of the attractive beam
field, this distribution might be described by a K0 Bessel function [16]. Note that, because
of the projection over 10 mm in the orthogonal direction, the actual density enhancement is
underemphasized in this type of plot. Figure 24 depicts the horizontal and vertical electron
phase space after the bunch passage.
We can compute the electric field generated by the pinched electrons by distributing their
charge on a grid with step size 20µm (σy /3) and subsequent interpolation. Figure 25 displays
the vertical and horizontal electric field at the end of the bunch as a function of vertical and
horizontal distance from the bunch center, respectively, assuming an initial electron density
15
12
12
1 sigma, 0 eV
2 sigma, 0 eV
1 sigma, 100 eV
2 sigma, 100 eV
10
10
8
8
6
6
4
4
2
2
0
0
-2
-1
0
1
2
-2
-1
0
1
2
12
1 sigma, 400 eV
2 sigma, 400 eV
10
8
6
4
2
0
-2
-1
0
1
2
Figure 21: Simulated density enhancement factor ρ(z)/ρ(−σz ) as a function of position in
units of σz along the bunch, computed from the number of particles within 1 or 2σ in the
x − y plane. The bunch head is on the left. The three pictures refer to initial thermal electron
energies equal to 0.01 eV, 100 eV, and 400 eV (average values).
16
12
14
1 sigma, 0 eV, w/o magn. field
1 sigma, 0 eV, w magn. field
SigmaZ = 4 mm
SigmaZ = 6 mm
12
10
10
8
8
6
6
4
4
2
2
0
0
-2
-1
0
1
2
-2
-1
0
1
2
14
Nb=3.3e10
Nb=5e10
12
10
8
6
4
2
0
-2
-1
0
1
2
Figure 22: Simulated density enhancement factor ρ(z)/ρ(−σz ) as a function of position in
units of σz along the bunch, computed from the number of particles within 1 or 2σ in the
x − y plane. The bunch head is on the left. The first picture compares calculations with and
without the magnetic field of the beam; the second picture is for a 50% increased bunch length
(6 mm rms instead of 4 mm); and the last picture is for a 50% increase in bunch population
(Nb = 5 × 1010 instead of 3.3 × 1010 ).
2.4
2.8
horizontal distribution
vertical distribution
2.6
2.2
2.4
2
2.2
1.8
2
1.6
1.8
1.4
1.6
1.2
1.4
1
0.8
-0.003
1.2
-0.002
-0.001
0
0.001
0.002
0.003
1
-0.0006
-0.0004
-0.0002
0
0.0002
0.0004
0.0006
Figure 23: Simulated horizontal (left) and vertical (right) normalized projected density distribution as a function of the horizontal and vertical position in units of meter, for a region
of width and height 10 mm, and considering Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm,
σy = 60 µm. The horizontal distribution is shown over a range ±5σx , the vertical over ±10σy .
Initial electron energies are set to zero.
17
6e+07
6e+07
horizontal phase space
vertical phase space
4e+07
4e+07
2e+07
2e+07
0
0
-2e+07
-2e+07
-4e+07
-4e+07
-6e+07
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
-6e+07
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
Figure 24: Simulated horizontal (left) and vertical (right) phase space distributions at the end
of the bunch passage for a region of width and height 10 mm. Ordinate is in units of m/s,
abscissa in units of meter. Again, we consider Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm,
σy = 60 µm. Initial electron energies are set to zero.
of 1012 m−3 . Slope changes in the field-vs.-position curve mirror the shape of the phase-space
distribution in Fig. 24.
The electric field between ±1σy is roughly 60 V/m2 in both planes. This constitutes an
increase by about a factor of three compared with that for the initial uniform density. The
field at the bunch tail then gives rise to an incoherent tune shift
∆Qy =
1
1
∂Ey
< βy > C
≈ 0.06,
4π
∂y c(Bρ)
(12)
about three times larger than the tune shift at the head of the bunch,
∆Q =
re < βy > ρC
≈ 0.018.
γ
(13)
In the above equations C denotes the circumference, < βy > the average vertical beta function,
(Bρ) the magnetic rigidity, c the speed of light, and re the classical electron radius.
4
Post Head-Tail Threshold or Fast Blow Up
If the instability growth time is small compared with the synchrotron period, we may estimate
the instability threshold, and its dependence on chromaticity, using the theory of vertical fast
blow up developed by Ruth and Wang [4] and later by Kernel, Nagaoka, and Revol [5].
The growth time follows from the linearized Vlasov equation, if one takes into account the
longitudinal particle dispersion due to energy spread and momentum compaction, without
considering full synchrotron oscillations. Without expanding into longitudinal modes, the
threshold bunch current follows from an integral equation. It reads [4, 5]:
Ithr =
4αC (∆p/p)rms |ωC + ωξ |Eσz
eβ⊥ |Z⊥ (ωc )|c
(14)
Note that there are two minor differences between
the formulae of [4] and [5]. Firstly, the
threshold estimates differ by a numerical factor 3/2, and, secondly, one reference uses the
effective impedance, the other the peak impedance.
18
40
Ex vs x
Ey vs y
30
20
10
0
-10
-20
-30
-40
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
Figure 25: Simulated vertical and horizontal electric field in units of V/m at the end of the
bunch passage as a function of vertical and horizontal distance from the bunch center in units
of meter. Again, we consider Nb = 3.3 × 1010 , σz = 4 mm, σx = 600 µm, σy = 60 µm.
The frequency ωc is the angular frequency at the peak value of the impedance. We set
this equal to the electron oscillation frequency ωc (ωc ≈ 1.8 × 1011 s−1 ). For the value of
|Z⊥ (ωc )|, we take the effective impedance computed by Ohmi [17] for an electron-cloud density
of ρe = 1012 m−3 . The impedance varies in proportion to the electron density. Inserting further
the parameter values αC = 10−4 , (∆p/p)rms = 7.5 × 10−4 , σz = 6 mm, β⊥ = 15 m, E = 3.5
GeV, Eq. (14) predicts the current threshold
Ithr = 0.31 mA
[1.8 + 0.06Qy ]
,
ρe /(1012 m−3 )
(15)
which evaluates to 0.56 mA for Qy = 0, and to 0.84 mA for Qy = 15, assuming ρe = 1012 m−3 .
These numbers are close to the observed threshold currents for the beam-size blow up.
References [18, 19] proposed an alternative treatment of fast vertical instabilities, which,
in the future, may also be applied to the electron-cloud problem.
5
Summary
In this note, we have reported recent simulations of electron-cloud build up in the KEKB LER,
considering various solenoid and C yoke configurations. Our simulations have not shown any
evidence for a long-term accumulation of electrons suggested by beam observations. Typically
simulated 1/e electron-cloud decay times behind a bunch train are of the order of 100 ns in
a solenoid field. This may or may not be consistent with a measured total decay length of
about 600 ns. The decay times are shorter for adjacent solenoids with alternating polarity.
We have refined earlier calculations by introducing more accurate models of the magnetic
fields, e.g., including further longitudinal Fourier components in the field expansion [6]. This
increases the computed central electron densities by at least a factor 2–3 (more in the case of
equal-polarity solenoids, where the central density is zero for the paraxial approximation), and
19
it alters the characteristics of the electron trajectories. The value of the central density depends
rather strongly on the order n of truncation in the Fourier expansion of the longitudinal
magnetic field. For a truncation at n = 1, the central density is higher for equal polarity
solenoids than for alternating polarity. This is opposite to what is found for the paraxial
approximation and for n = 5 truncation. Electron image charges for the round chamber,
included in most of the simulations, can significantly change the electron trajectories and tend
to counteract the electron-cloud build up. Even with all these modifications, however, in none
of the cases do we observe any long-term accumulation of electrons along the bunch train.
Improved simulations for various C-yoke arrangements are consistent with previous results.
The electron density increases inside a bunch, during its passage through the electron cloud.
For KEKB, this density enhancement amounts to a factor of 6, i.e., it is of similar magnitude
as for PEP-II [3]. The incoherent tune spread thus enhanced may provide a second mechanism
blowing up the beam size, in addition to single-bunch higher-order head-tail instabilities driven
by the electron-cloud wake field [15].
Finally, we have applied the theory of fast vertical blow up [4], also called the post head
tail instability [5], to the electron-cloud wake field. The instability threshold estimated for
KEKB LER, assuming the simulated electron-cloud impedance [15], is remarkably close to
the observed threshold current.
6
Acknowledgements
We thank Y. Funakoshi, K. Oide, E. Perevedentsev, and Y. Suetsugu for helpful and inspiring
discussions, and for contributing many of the ideas presented.
References
[1] F. Zimmermann, “Electron-Cloud Studies for the Low-Energy Ring of KEKB”, CERNSL-Note-2000-004 AP.
[2] F. Zimmermann, “Electron Cloud at the KEKB Low-Energy Ring: Simulations of Central
Cloud Density, Bunch Filling Patterns, Magnetic Fields, and Lost Electrons”, CERN-SL2000-017 (AP).
[3] M. Furman and A. Zholents, “Incoherent Effects Driven by the Electron Cloud”, PAC99,
New York (1999).
[4] R.D. Ruth and J. Wang, “Vertical Fast Blow-Up in a Single Bunch”, IEEE Tr. NS-28,
no. 3 (1981).
[5] P. Kernel, R. Nagaoka, J.-L. Revol, G. Besnier, “High Current Single Bunch Transverse
Instabilities at the ESRF: A New Approach”, EPAC 2000 (2000).
[6] E. Perevedentsev, “Periodic Solenoid Field”, unpublished note, November 2000.
[7] F. Zimmermann, “A Simulation Study of Electron-Cloud Instability and Beam-Induced
Multipacting in the LHC”, CERN-LHC-Project-Report 95 (1997).
[8] O. Brüning, “Simulations for the Beam Induced Electron Cloud in the LHC Beam Screen
with Magnetic Field and Image Charges”, CERN-LHC-Project-Report-158 (1997).
[9] F. Zimmermann, “Electron-Cloud Simulations for SPS and LHC”, Proc. Chamonix 10,
CERN-SL-2000-007-DI.
20
[10] G. Rumolo, F. Ruggiero, F. Zimmermann, “Simulation of the Electron-Cloud Build Up
and Its Consequences on Heat Load, Beam Stability and Diagnostics”, presented at ICAP
2000, Darmstadt (2000).
[11] H. Seiler, “Secondary electron emission in the scanning electron microscope”,
J. Appl. Phys. 54 (11) (1983).
[12] I. Collins, private communication, February 2000.
[13] R. Kirby and F. King, “Secondary Electron Emission Yields from PEP-II Accelerator
Materials”, SLAC-PUB-8212 (2000).
[14] M.A. Furman and G.R. Lambertson, “The Electron Cloud Effect in the arcs of the PEP-II
Positron Ring,” KEK Proceedings 97-17, p. 170, December 1997 (Proc. MBI97 workshop,
KEK, Y.H. Chin, ed.) (1997).
[15] K. Ohmi and F. Zimmermann, “Head-Tail Instability Caused by Electron Cloud in
Positron Storage Rings”, published in PRL, October 2000.
[16] T. O. Raubenheimer, F. J. Decker and J. T. Seeman, “Beam distribution function after
filamentation,” SLAC-PUB-6850, and IEEE PAC 95, Dallas (1995).
[17] K. Ohmi and F. Zimmermann, “Wake Force and Single-Bunch Instability Induced by
Electron Cloud”, report in preparation.
[18] D. Pestrikov, “On the Beam Break-Up Instability in Storage Ring” Proc. 4th Advanced
ICFA Beam Dynamics Workshop on Collective Effects in Short bunches, KEK Report
90-21, p. 118 (1991).
[19] N.S. Dikansky, D.V. Pestrikov, “Physics of Intense Beams in Storage Rings,” S.O. Nauka
(1989).
21