Nuclear Instruments and Methods in Physics Research A 474 (2001) 197–208
Suppressing electron multipacting in ceramic windows
by DC bias
Pasi Yla. -Oijala*, Marko Ukkola
Rolf Nevanlinna Institute, University of Helsinki, P.O. Box 4, (Yliopistonkatu 5), FIN-00014 Helsinki, Finland
Received 3 January 2001; accepted 26 February 2001
Abstract
Electron multipacting can cause breakdown in high RF power components like couplers and windows. The
phenomenon starts if certain resonant conditions for electron trajectories are fulfilled and if the impacted surface has a
secondary electron yield larger than one. A general cure against multipacting is to avoid the resonant conditions by a
proper choice of the geometry. In many cases, however, it might not be possible to change the RF geometry sufficiently.
Therefore, other methods are required to suppress multipacting, like grooving the surfaces or static electric and
magnetic perturbations. In this paper, we investigate by numerical simulations the effect of a biasing DC voltage to
multipacting in two special ceramic window geometries. Those are the DESY and FNAL cylindrical symmetric cold
window designs of the TESLA input power coupler. We show that in both window designs multipacting can be avoided
by choosing an appropriate biasing voltage. # 2001 Elsevier Science B.V. All rights reserved.
PACS: 52.80; 07.05.T
Keywords: Multipacting; Input coupler; Ceramic window; DC bias
1. Introduction
Multipacting is a phenomenon of resonant
electron multiplication, in which a large number
of electrons build up an electron avalanche,
leading to remarkable power losses and heating
of the walls, so that it becomes impossible to
increase the cavity fields by rising the incident
power [1]. In superconducting structures a large
rise of temperature can lead to a thermal breakdown. Furthermore, a heavy bombing of multi*Corresponding author. Fax: +358-9191-22779.
E-mail address: pasi.yla-oijala@rni.helsinki.fi
(P. Ylä-Oijala).
pacting electrons may break the ceramic windows
leading to disastrous consequences.
A general cure against multipacting is to avoid
the resonant conditions by either a proper choice
of the geometry [2] or by coating the critical areas
with a material with a lower secondary yield [3,4].
In many cases, however, it might not be possible to
change the RF geometry sufficiently and the
coating, which is typically used to reduce the
secondary yield of a ceramic window, does not
suppress completely multipacting and the success
rate of the reduced secondary yield is often
unsatisfactory. In those cases, other suppressing
methods must be applied, like static electric [5] or
magnetic perturbations, or grooving [6] the surfaces.
0168-9002/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 8 8 2 - 8
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In Ref. [7], we analyzed multipacting (without
the bias) in the cylindrically symmetric cold
windows of the TESLA [8] input power coupler.
Both the cylindrical DESY (Deutsches Elektronen-Synchrotron) and conical FNAL (Fermi National Accelerator Laboratory) window designs
were considered. It was found that in the DESY
design multipacting can be avoided by placing the
window at the right distance from the coupler end.
In the FNAL window design this was not possible
because of a very wide existence of multipacting.
Therefore, in the FNAL design other methods are
required to avoid multipacting. A static electric
perturbation, i.e., a biasing DC voltage, is an
attractive choice in the coaxial structures [5]. Since
both TESLA input power coupler designs include
long straight coaxial sections and a DC voltage is
typically applied to avoid multipacting in coaxial
lines, one should know also in the DESY design
whether a new type of multipacting is possible with
the DC bias.
Our earlier computations in straight coaxial
lines with the DC bias [9] showed that the voltage
has to be chosen properly and a wrong DC voltage
may cause new low order multipacting with a high
secondary yield. Furthermore, we were able to find
simple rules for multipacting resonances with
respect to the dimensions of the line, frequency
of the field and the biasing voltage. In the ceramic
windows the multipacting analysis with a DC bias
becomes much more complicated, because of the
varying field conditions during the operation of
the system and of the complexity of the window
designs. In this paper, we investigate the effect of a
biasing DC voltage to multipacting in both
TESLA cold window designs and, in particular,
show that with the developed computational
methods [9] we are able to find the optimal DC
voltages in the DESY and FNAL window designs
by which multipacting can be avoided.
2. DC bias in coaxial lines
We begin by shortly summarizing our previous
results on multipacting simulations in straight
coaxial lines with a biasing DC voltage. In a
straight coaxial line, we define the DC field
corresponding to a voltage U as follows [9]
EDC ðr; zÞ ¼
U 1
er :
ln b=a r
Here a and b stand for the inner and outer radii of
the line and ðr; zÞ is a field point in a cylindrically
symmetric structure. In other words, a positive
voltage, U > 0, corresponds to an electric field
pointing from the inner conductor to the outer
conductor and a negative voltage corresponds to
an opposite field. In the ceramic windows, the DC
field is defined accordingly. Obviously, in the
ceramic windows the DC field does no longer
have an analytical solution and numerical methods
must be applied for the field calculations.
In Ref. [10], we studied DC voltage in coaxial
lines by varying the dimensions, frequencies and
wave forms. The results were summarized in Ref.
[9]. Two main points were recognized. Firstly, the
DC voltage which is required to suppress multipacting in the standing wave (SW) operation
suppresses multipacting also for all other wave
configurations (traveling waves (TW) and mixed
waves). Secondly, the suppressing DC voltage
satisfies the following scaling law
U dZf
ð1Þ
where d is the diameter of the line, Z is the
impedance of the line and f is the frequency of the
field. In other words, the DC voltage which is
required to suppress multipacting depends linearly
on the dimension, impedance and frequency.
Table 1 lists the DC voltages, both negative and
positive, by which multipacting can be suppressed
up to 1 MW in the straight coaxial sections of the
DESY and FNAL input coupler designs [10]. Note
that the coaxial lines have different dimensions on
the cold (the section from the cold window to the
cavity) and warm (the section from the cold
window towards the generator) sides of the
couplers. We may conclude that in the DESY
coupler either 3:5 or þ2:7 kV is needed to
suppress multipacting. In the FNAL coupler, the
corresponding numbers are 3:5 and þ2:5 kV.
Naturally, these results do not hold in the ceramic
windows and the suppressing voltages have to be
redefined. However, we may conclude that the
analysis in the ceramic windows can be restricted
P. Ylä-Oijala, M. Ukkola / Nuclear Instruments and Methods in Physics Research A 474 (2001) 197–208
Table 1
The suppressing DC voltages and dimensions of the coaxial
lines of the TESLA input coupler designs. Here Z is the line
impedance, d is the outer diameter and U is the voltage
DESY
Z ðOÞ
d (mm)
U (kV)
U (kV)
Cold
Warm
70
50
40
62
2:9
3:5
þ2:7
þ2:5
FNAL
Cold
Warm
50
50
40
62
2:1
3:5
þ2:3
þ2:5
to the voltages equal or higher than 3:5 kV and
þ2:7 or þ2:5 kV.
3. DC bias in ceramic windows
Before we can start the multipacting analysis the
distribution of the electromagnetic fields must be
computed. The DC field in the ceramic windows is
computed by a finite element method with a third
order approximation [11]. For the trajectory
calculations, both the DC and RF fields are given
in a grid and interpolation is used elsewhere.
During the operation of the RF pulse, the
reflection conditions on the coupler vary. Therefore, we have to consider a large number of
simulations with various RF field conditions. In a
similar fashion as in Ref. [7] (without the bias), we
define the field conditions by a complex reflection
coefficient R with jRj41. The magnitude of R
describes the amplitude ratio of the reflected and
forward waves, and the phase of R gives the phase
difference of the reflected and forward waves. In
reality only part of the reflection conditions occur
and the analysis may be restricted to these areas.
Here, however, the analysis is carried out for all
possible reflection conditions and the real operation conditions, given in Ref. [7], and indicated in
the figures (in the DESY case). The coordinate
systems are fixed similarly as in Ref. [7]. That is,
R ¼ 1 gives a SW with a voltage maximum at the
one quarter wave length from the center of the
window, R ¼ 0 is a TW and R ¼ 1 is a SW with
a voltage minimum at the distance of one quarter
wave length from the center of the window.
199
The multipacting analysis is carried out as
follows. First, we fix the DC voltage. Then we
scan through a representative sample of reflection
conditions with jRj41 and power levels up to
1 MW as described in Ref. [7]. For each fixed
reflection coefficient and RF power combination,
we send a large number of electrons from different
points on the surface of the structure, compute the
electron trajectories and count the total number of
secondary electrons after 20 impacts. This count is
called an enhanced electron counter function. The
results of the simulations are illustrated by plotting
relative enhanced counter function, which is the
number of secondary electrons after 20 impacts
divided by the number of initial electrons, as a
function of the reflection coefficient.
For each complex R, we plot the maximum of
the relative enhanced counter with respect to the
RF power, when the RF power is scanned from 0
to 1 MW. By this analysis, we are able to locate
those operation conditions at which multipacting
may occur. In order to find the multipacting power
levels we repeat the analysis by fixing the reflection
coefficient and by varying the DC voltage and RF
power. The latter analysis is, however, carried out
only with a couple of reflection coefficients.
In the figures below (Figs. 2–7 and 9–12) the
shade indicates the change of the total number of
electrons after 20 impacts. In bright areas the
Fig. 1. The secondary electron yield at different impact energies
(in eV) for a niobium surface baked at 3008C.
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number of electrons is increased and multipacting
occurs, whereas in dark areas the number of
electrons is decreased. (See the colorbar on the
right hand side of the figures.) The multipacting
level, i.e., when the relative enhanced counter is
Fig. 2. The relative enhanced electron counter for the cold side
of the DESY coupler in 10 base logarithmic scale as a function
of the complex reflection coefficient after 20 impacts when the
DC voltage is 3:5 kV and for the RF power range 0–1 MW.
The real operation conditions are denoted on the plot by
dashed lines (see Fig. 9 in Ref. ½7).
Fig. 3. The relative enhanced electron counter for the warm
side of the DESY coupler for the RF power range 0–1 MW
when the DC voltage is 3:5 kV.
one (zero in the logarithmic scale) is indicated in
the figures by a solid line.
Here, we assume that the secondary yield of the
ceramic window is the same as in the metallic
surfaces. This corresponds to a coated ceramic,
coated e.g. by TiN. The secondary yield curve
applied in the calculations is displayed in Fig. 1
[12]. This curve corresponds to a niobium surface.
However, the couplers are made of copper. The
Fig. 4. The relative enhanced electron counter for the warm
side of the DESY coupler for the RF power range 0–1 MW
when the DC voltage is þ3:0 kV.
Fig. 5. The relative enhanced electron counter for the warm
side of the DESY coupler for the RF power range 0–1 MW
when the DC voltage is þ4:0 kV.
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201
Fig. 6. The relative enhanced electron counter for the warm side (top) and cold side (bottom) of the DESY window in 10 base
logarithmic scale as a function of the RF power and DC voltage, when R ¼ 1, i.e., a SW with a voltage maximum at the window
region.
secondary yield curve for copper is rather similar,
the maximum yield 1.3 is reached at Ekin ¼ 600 eV,
and the secondary yield curve exceeds unity
between 200 and 1500 eV [1,13]. Thus, the major
difference between the secondary yield curve for
the copper and niobium is that the copper has a
lower maximum yield. This effects only to the
strength of multipacting, not to the existence.
More details about the multipacting analysis in
ceramic window are presented in Ref. [7].
3.1. DESY cylindrical window design
First we consider the DESY window design.
The design is displayed in Ref. [7] and more
details about the geometry are given, e.g., in
Ref. [14].
As the computations in the straight coaxial lines
suggest, first we consider the biasing DC voltages
of 3:5 and þ3:0 kV. Figs. 2–4 display the results
of the analysis, separately for both cold and warm
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Fig. 7. The relative enhanced electron counter for the warm side (top) and cold side (bottom) of the DESY window in 10 base
logarithmic scale as a function of the RF power and DC voltage, when R ¼ 0, i.e., a TW.
sides. Figs. 2 and 3 show that when the DC voltage
is 3:5 kV, multipacting takes place on a very
wide area of the reflection chart (bright area). In
fact, DC voltage has created a lot of new multipacting, especially on the warm side, which does
not occur without the bias. Fig. 4 shows the
relative counter function on the warm side of the
DESY window when the DC voltage is þ3:0 kV.
We can see that also in this case multipacting
occurs on a rather wide area of the reflection chart.
On the cold side, on the other hand, multipacting
can be avoided with þ3:0 kV (the results are not
presented here). These results suggest that a
positive DC voltage higher than þ3:0 kV is
required to suppress multipacting. Therefore, we
repeated the analysis with þ3:5 and þ4:0 kV.
When the bias is þ3:5 kV, the cold side is still free
of multipacting and on the warm side the multipacting region (bright area in Fig. 4) becomes
smaller, but the area of the real operation
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203
Fig. 8. Electron trajectories in the DESY window design when R ¼ 0: (a) U ¼ 3:5 kV and P ¼ 690 kW, (b) U ¼ þ3:0 kV and
P ¼ 800 kW, (c) U ¼ 2:5 kV and P ¼ 606 kW, (d) U ¼ þ1:5 kV and P ¼ 138 kW. The dimensions are in meters and the trajectories
are not to scale. All trajectories are one-point processes of order one and the corresponding impact energies are 270, 580, 785 and
460 eV. The window is denoted by shading. For the geometry, see Fig. 1 in Ref. [7].
conditions still hits the dangerous multipacting
region. This takes place near the TW operation
and therefore, it is not possible to avoid multipacting by a proper placing of the window with
þ3:5 kV and a higher voltage is required. Thereafter, Fig. 5 displays the results of the analysis on
the warm side when the voltage is þ4:0 kV. Once
again the real operation conditions hit the multipacting region. Now the multipacting region is
smaller than in the case of þ3:5 kV. A closer
analysis of the counter functions shows that for
the real operation conditions multipacting can be
avoided up to 900 kW. A slightly higher voltage
may suppress multipacting up to 1 MW. More-
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Fig. 9. The relative enhanced electron counter for the warm
side of the FNAL coupler in 10 base logarithmic scale as a
function of the complex reflection coefficient after 20 impacts
for the RF power range 0–1 MW when the DC voltage is
þ2:5 kV.
over, the analysis on the cold side with þ4:0 kV
shows no multipacting.
We repeated the analysis with higher negative
voltages, too, and found that when U ¼ 4:0 kV
the multipacting region becomes smaller, but the
region is still very large (the results are not
presented here). Thus, a positive voltage is clearly
a better choice for the DESY window design.
Next, we change the analysis so that we fix the
reflection coefficient and let the biasing voltage
vary from 4:0 to þ4:0 kV. Figs. 6 and 7 show the
enhanced counter function as a function of the RF
power and DC voltage when R ¼ 1 and R ¼ 0.
In the figures all positive values correspond to
multipacting (bright areas), while the dark areas
show the multipacting free regions. By these
figures one can determine the DC voltage so that
the RF power can be raised from zero to the
desired value without crossing any of the multipacting bands. For example, Fig. 6 suggests that
on the warm side þ1:5 kV is required to suppress
multipacting up to 1 MW, whereas on the cold
side the needed voltage is twice as much, i.e.,
þ3:0 kV. Furthermore, Fig. 7 shows that in the
TW case the corresponding voltage is þ4:0 kV. A
closer analysis of Figs. 6 and 7 shows, analogously
to the coaxial case, that when the RF power is
Fig. 10. The relative enhanced electron counter for the warm
side of the FNAL coupler for the RF power range 0–1 MW
when the DC voltage is þ3:0 kV.
increased from zero with any fixed DC voltage we
always first ‘‘hit’’ the first order process and
thereafter the second order process follows, etc.
In order to get a complete picture of the
phenomenon we would have to repeat the analysis
for all reflection coefficients.
Fig. 8 shows typical multipacting electron trajectories for four DC voltage and RF power
combinations when R ¼ 0, i.e., for a TW.
Figs. 8(a) and (b) shows the trajectories on the
warm side and Figs. 8(c) and (d) shows the
trajectories on the cold side. In the straight coaxial
lines, we found that negative voltages generate
multipacting to the outer conductor of the line and
positive voltages generate multipacting to the
inner conductor of the line [9]. The trajectories in
Fig. 8 show that this is true in the DESY window
design, too, in the TW case. For other wave forms,
the situation is more complicated.
3.2. FNAL conical window design
Next we consider the results of the multipacting
simulations in the conical FNAL window. The
design is shown in Ref. [7] and more details are
presented, e.g., in Ref. [14].
The simulations in the straight coaxial lines
suggest that in order to avoid multipacting the DC
P. Ylä-Oijala, M. Ukkola / Nuclear Instruments and Methods in Physics Research A 474 (2001) 197–208
205
Fig. 11. The relative enhanced electron counter for the warm side (top) and cold side (bottom) of the FNAL coupler in 10 base
logarithmic scale as a function of the RF power and DC voltage, when R ¼ 1, i.e., a SW with a voltage maximum at the window
region.
voltage has to be at least either 3:5 or þ2:5 kV.
The analysis shows that multipacting does not
occur in the window region when the voltage is
3:5 kV (the counter functions are not presented
here). Thus, the same negative voltage which
suppresses multipacting in the straight coaxial
sections of the FNAL coupler design will also
suppress multipacting in the ceramic window.
Next, we study whether multipacting can be
avoided with a lower positive voltage. Fig. 9 shows
the enhanced counter function as a function of the
reflection coefficient on the warm side of the
FNAL design when the DC voltage is þ2:5 kV.
The figure shows that multipacting occurs on a
wide region on the reflection chart. Therefore, we
repeated the analysis with þ3:0 and þ3:5 kV.
Fig. 10 shows that the multipacting region of
Fig. 9 becomes smaller and the intensity of multi-
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Fig. 12. The relative enhanced electron counter for the warm side (top) and cold side (bottom) of the FNAL coupler in 10 base
logarithmic scale as a function of the RF power and DC voltage, when R ¼ 0 ,i.e., a TW.
pacting decreases when the voltage is increased
from þ2:5 to þ3:0 kV. However, multipacting still
takes place on a rather wide region on the
reflection chart. This suggests that a little higher
voltage is needed. An analysis with þ3:5 kV
confirms this observation. Now the counter function stays clearly below the multipacting level for
all analyzed reflection coefficients and power levels
up to 1 MW and multipacting can be avoided.
Furthermore, the simulations show that on the
cold side all positive voltages equal or higher than
þ2:5 kV suppress multipacting. Thus, we may
conclude that either 3:5 or þ3:5 kV is required to
suppress multipacting in the FNAL design for all
reflection coefficients up to 1 MW.
Next, we repeated the analysis in the FNAL
window design by fixing the reflection coefficient
and by letting the biasing voltage vary from 4:0
to þ4:0 kV. Figs. 11 and 12 illustrate the behavior
of the multipacting power bands when R ¼ 1
P. Ylä-Oijala, M. Ukkola / Nuclear Instruments and Methods in Physics Research A 474 (2001) 197–208
207
Fig. 13. Electron trajectories in the FNAL window design when R ¼ 0: (a) U ¼ 2:0 kV and P ¼ 613 kW, (b) U ¼ þ2:0 kV and
P ¼ 389 kW, (c) U ¼ 2:0 kV and P ¼ 536 kW, (d) U ¼ þ1:5 kV and P ¼ 219 kW. The dimensions are in meters and the trajectories
are not into scale. All trajectories are one-point processes of order one and the corresponding impact energies are 460, 435, 745 and
750 eV. The window is denoted by shading. For the geometry, see Fig. 1 in Ref. ½7.
and R ¼ 0 by giving the enhanced counter function as a function of the RF power and DC
voltage. We may conclude that when R ¼ 1, i.e.,
the SW case, the voltage which is required to
suppress multipacting is roughly of the same size
as in the straight coaxial lines, i.e., 3:0 or
þ3:0 kV on the warm side and 2:5 or 2:0 kV
on the cold side (see Table 1 for the voltages in the
coaxial case). Furthermore, Fig. 12 shows that
when the wave is switched from a SW to a TW, the
SW multipacting power levels scale upwards
roughly by factor 4, similarly as in the coaxial
case [10]. The trajectory calculations confirm that
the corresponding SW and TW processes are very
similar. Fig. 13 shows samples of multipacting
electron trajectories in the TW case. Again the
same phenomenon as in the straight coaxial lines
and in the DESY design can be recognized, namely
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P. Ylä-Oijala, M. Ukkola / Nuclear Instruments and Methods in Physics Research A 474 (2001) 197–208
a negative voltage generates multipacting to the
outer surface and a positive voltage generates
multipacting to the inner surface.
As has been already mentioned, we have found
that in the straight coaxial lines it suffices to
consider the SW case only in order to find the
optimal biasing voltage for all possible wave
configurations. The present analysis shows that
also in the FNAL window design the SW case
gives a rather good estimate for the optimal DC
voltage, when several SW cases with the voltage
maximum at the window region are studied.
Again, it has to be pointed out that in order to
get a complete picture of the phenomenon we
would have to repeat the analysis for all reflection
coefficients.
4. Conclusions
In this paper, we apply the numerical methods
developed in Ref. [9] to study the effect of a DC
bias to multipacting in two special ceramic window
designs. We consider both the FNAL and DESY
cylindrically symmetric cold window designs of the
TESLA input power coupler. It is found that in
the DESY design þ4:0 kV is required to suppress
multipacting on the real operation conditions up
to 900 kW. A little higher voltage will suppress
multipacting up to 1 MW. In addition, in the
DESY window design the sign of the DC field is
important, since a negative voltage 4:0 kV
creates multipacting. In the FNAL window design
both 3:5 and þ3:5 kV suppress multipacting on
the whole reflection chart. Thus, we may conclude
that although our previous analysis without the
DC bias showed that the FNAL design is clearly
more sensitive to multipacting than the DESY
design, in the case when a DC voltage is required,
e.g., to suppress multipacting in the straight
coaxial sections, the DC bias is more effective in
the FNAL design and 500 V higher voltage is
needed in the DESY design.
In the FNAL design, multipacting phenomenon
has some similarities with the straight coaxial case.
For example, a rather good estimate for the
optimal DC voltage for all reflection conditions
can be found by studying the SW case with a
voltage maximum at the window region. In the
DESY window the phenomenon is clearly more
complicated, because of the more complex design.
Acknowledgements
The authors wish to thank S. J.arvenp.aa. from
Rolf Nevanlinna Institute for computing the DC
fields. This work has been financially supported by
the Helsinki Institute of Physics.
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