Plasma Sources Sci. Technol. 8 (1999) R21–R44. Printed in the UK
PII: S0963-0252(99)02255-0
REVIEW ARTICLE
Cold-cathode discharges and
breakdown in argon: surface and gas
phase production of secondary
electrons
A V Phelps† and Z Lj Petrovi憇
† JILA, University of Colorado and National Institute of Standards and Technology, Boulder,
CO 80309-0440, USA
‡ Institute of Physics, PO Box 57, Belgrade, Yugoslavia
Received 16 October 1998, in final form 24 February 1999
Abstract. We review the data and models describing the production of the electrons, termed
secondary electrons, that initiate the secondary and subsequent feedback avalanches required
for the growth of current during breakdown and for the maintenance of low-current,
cold-cathode discharges in argon. First we correlate measurements of the production of
secondary electrons at metallic cathodes, i.e. the yields of electrons induced by Ar+ ions, fast
Ar atoms, metastable atoms and vuv photons. The yields of electrons per ion, fast atom and
photon vary greatly with particle energy and surface condition. Then models of electron, ion,
fast atom, excited atom and photon transport and kinetics are fitted to electrical-breakdown
and low-current, discharge-maintenance data to determine the contributions of various
cathode-directed species to the secondary electron production. Our model explains measured
breakdown and low-current discharge voltages for Ar over a very wide range of electric field
to gas density ratios E/n, i.e. 15 Td to 100 kTd. We review corrections for nonequilibrium
electron motion near the cathode that apply to our local-field model of these discharges.
Analytic expressions for the cross sections and reaction coefficients used by this and related
models are summarized.
1. Introduction
This paper is concerned with the processes responsible for
the production of the initial secondary or feedback electrons
required for the growth of current at electrical breakdown
and for the maintenance of cold-cathode discharges in Ar.
These electrons are produced in collisions of Ar ions, fast
Ar atoms, metastable atoms or photons with the cathode
or in ionizing collisions of fast atoms or ions with the
neutral Ar atoms in the gas phase. We review the published
data for these processes and select the data that lead to
consistency between our model and measured breakdown
voltages and low-current, discharge-maintenance voltages
for a wide range of operating conditions. Although our
analyses make use of data obtained at low current densities,
the assembled data are also appropriate for use in models of
cold-cathode discharges at the higher current densities found
in the normal and abnormal glow modes.
Computer models of low-pressure electrical discharges
in Ar have become a very useful tool for understanding
and predicting the properties of discharges used for plasma
processing [1–4], surface sputtering [5, 6], plasma displays
[7, 8], lighting [9, 10], switching [11, 12], scintillation
0963-0252/99/030021+24$19.50
© 1999 IOP Publishing Ltd
detectors [13] and lasers [14, 15]. In order for these models
to be quantitatively useful it is necessary for the modeller to
use realistic cross sections for electron, ion, atom and photon
collisions with the gas and to use realistic probabilities for
electron, ion, atom and photon interactions with electrode
surfaces. The cross sections and rate coefficients required
for the modelling of the gas-phase portion of discharges
in Ar are relatively well known. In particular, many
people have addressed the question of electron–Ar cross
sections [3, 16–24]. Some of the cross section sets in
use are consistent with and some badly inconsistent with
simple swarm experiments [25]. The published cross section
data for Ar+ –Ar and Ar–Ar elastic and inelastic collisions
have been reviewed [26]. The accuracy of ion–atom cross
sections by modellers has been considered recently [27],
and recommendations made for avoiding common errors in
their application. Photon absorption data and experimentally
verified resonance-photon scattering treatments appropriate
to models of discharges in Ar are available in the literature
[28, 29].
The role of secondary electron avalanches in the
electrical breakdown of gases and in low-current stationary
discharges has been discussed in detail in numerous texts
R21
A V Phelps and Z Lj Petrović
and reviews [30–36]. We will include all generations
of avalanches beyond the first under the terminology of
secondary or feedback avalanches. The processes that
have been considered in the literature for the production of
the initiating secondary electrons include electron emission
from the cathode induced by positive ions, by fast atoms,
by photons and by metastable atoms and molecules;
collisional ionization by fast ions and atoms produced
by earlier avalanches; collisional ionization by electrons
backscattered from the anode and photoionization of the gas
by photons from earlier avalanches. We will not consider
photoionization of the Ar and will not review the data on the
backscattering of electrons at the anode. For convenience,
we will usually drop the qualifier ‘avalanche initiating’ and
use the terminology ‘secondary electrons’.
The emission of secondary electrons as the result of
collisions of ion beams with metallic surfaces has been
reviewed by a number of authors [37–44]. These references
contain little recent effort devoted to gas-covered, practical
discharge surfaces. Almost all of the recent beam work
[42, 43] has been concerned with what we will call clean
surfaces, with electron emission induced by multiply charged
ions and with multiple electron emission induced by a
single ion. Some measurements of electrons released as
the result of incident neutral atom beams have been reported
[45–51]. In view of the evidence for large fluxes of energetic
atoms incident on the cathode [5, 52–55] of discharges it is
important to model their contributions to electron production
at the cathode and in the gas phase of Ar discharges.
Because of our interest in cathode surfaces found in
practical devices, we consider the yields of electrons at metal
surfaces that are exposed to pure gases or gases that may
result in oxidation or other contaminating reactions. We will
find that at Ar+ energies above about 0.5 eV (E/n > 250 Td)
the electron yields for cleaned metals in very pure Ar are
close to the yields obtained for cleaned metals in ultra-high
vacuum. Here E/n is the electric field to gas density ratio
and 1 Td = 10−21 V m2 . We review data showing that the
electron yields for metal surfaces exposed to oxidizing and/or
other contaminating gases, i.e. for ‘dirty surfaces’, are very
different than for clean metals and very pure Ar. Because of
the sputtering that necessarily accompanies a cold-cathode
discharge, we will attempt to correlate the very limited data
on the effects of sputtering on the production of secondary
electrons at metal surfaces.
Opinions regarding the role of heavy-particle collisions
in the production of electrons in the electrical breakdown of
gases have varied from Townsend’s original proposal [30]
that ion–atom collisions are the only source of negatively
charged carriers (electrons) to the almost complete rejection
of this idea in favour of ion-, metastable- and photon-induced
electron ejection from the cathode as the source of secondary
electrons [32, 33, 35]. The role of heavy-particle-induced
ionization has been most extensively investigated in the case
of breakdown [56–60] of hydrogen and the cathode fall
region [61] in low-pressure hydrogen discharges. A few
investigators have indicated the importance of such a process
in the cathode fall [5, 61–63]. and in the breakdown [53]
of low-pressure Ar. In particular, Neu [61] has performed
an analysis similar to that of section 5, but for conditions
appropriate to a discharge with a well developed cathode fall.
R22
Section 2 of this paper reviews and correlates
measurements of electron emission by ion and atom
bombardment of clean and dirty metal surfaces using
beam techniques. Measurements of effective electron
yields per ion for clean and dirty surfaces obtained using
swarm techniques, including low-current discharges, are
summarized in section 3. Relevant published photoelectric
yield data are correlated and fitted in section 4. In section 5
models of breakdown in the rare gases are extended and
applied to experiment so as to indicate the roles of ions, fast
atoms, photons and metastables in electron production at the
cathode and in the gas phase. In appendix A we review two
corrections for nonequilibrium effects near the cathode that
are applied to our fluid models of electrons in discharges. The
specific parameters used in our model of breakdown data for
Ar are given in appendix B. In this paper we will sometimes
use the terminology of breakdown to refer to the conditions
for both quasi-static gas breakdown and steady-state, lowcurrent gas discharges [64].
2. Argon ion and atom beam results
In this section we consider measurements of the yield of
electrons γ as the result of the bombardment of metallic
surfaces with monoenergetic Ar+ ions γi , Ar atoms γa and
Ar metastables γm at the low and moderate particle energies
found in most discharges, i.e. below about 1000 eV.
2.1. Beam results for clean metals
In this section we review the experimental results for what
are called clean metal surfaces in the gas discharge field.
For high-boiling-point metals this means that the surface
has been heated to ≈2000 K in a very good vacuum and
that the measurements are made with the surface at room
temperature and under high vacuum. Such a heat treatment
is often called ‘flashing.’ We are not concerned with what
might be done by more thorough cleaning [65], because there
appear to be no gas discharge experiments with such surfaces.
Figure 1 shows experimental electron yields for Ar+ ions
[46, 47, 66–71] and Ar atoms [46, 47] incident on various
clean metal surfaces as a function of particle energy. We
have shown essentially all of the yield data for clean metallic
surfaces that we have found for energies below about 1 keV.
At higher ion energies, these data are only a sample that
available [37–40, 42–44]. A short summary of the cleaning
procedure is given with the reference. Note that the two sets
of values shown for Mo from the experiments of Vance [70]
were obtained with two different high-temperature cleaning
processes.
For most surfaces the electron yield per ion shown
in figure 1 is nearly independent of ion energy below
about 500 eV. This constant yield is attributed to an Auger
process called potential ejection [37, 42–44, 68]. The energy
dependent portion of the yields at above several keV is called
kinetic ejection and is less clearly understood [37–40, 42, 43].
Thresholds for kinetic ejection are associated with the onset
of multiple electron ejection [43, 72]. The electron yields
per ion measured [73–75] for most metals are about 0.1 at
1 keV. At energies near 1 keV Oechsner [73] finds a periodic
Cold-cathode discharges and breakdown in argon
10
10
Electron yield per ion or atom
Electron yield per ion or atom
dirty metals
1
Ar+
10-1
Ar
10-2
1
Ar+
10-1
clean
metals
10-2
10-3
Ar
10-3
10
102
103
10-4
10
104
Ion or atom energy (eV)
+
Figure 1. Electron yields for Ar and Ar beams incident on
various clean metal surfaces versus particle energy. The solid
symbols are for Ar+ and the open symbols are for Ar. The
symbols, metals and references are: !, W, [68]; +,#, Mo, [46];
% , , Mo, [47]; ", Mo, [70]; , Mo, [66]; #, Au, [71]; ×, Cu,
$
[67]; $, Pt, [69] and %, Ta, [69]. The curves drawn through
representative values will be used in our model.
•
correlation of yield with atomic number [73]. The low,
energy dependent yields at low energies for Ta may result
from problems in cleaning [65, 69, 76].
In contrast to the near constant yields obtained for Ar+
ions at below 1 keV, the electron yield per fast Ar atom varies
rapidly with energy with an effective threshold of roughly 500
eV. At high energies the yield per fast Ar atom approaches
that for Ar+ ions. The overall energy dependence is that
expected for kinetic ejection by a projectile with no available
internal energy [37]. This behaviour is similar to that for ions
with low ionization potentials, but comparable mass, which
show a strong energy dependence and high threshold energies
[37, 42, 43, 72].
Beam experiments in which the target is cleaned by ion
sputtering have been reported [39, 44, 71, 73, 74, 77]. Experiments using sputtering followed by yield measurements at
ion energies below 4 keV are shown for Au targets [71] in
figure 1. Small but repeatable changes in electron yield are
found [77] for 100 keV H+ ions when sputtered copper is
annealed at near 600 K. This is approximately the temperature at which some forms of damage produced by sputtering
are observed to annihilate [39]. How these results carry over
to the Ar+ ions and the much lower energies of interest for
discharges is not clear. For further discussion of sputtering
effects see section 3.3.
Measurements of γi using moderate current discharges
are sometimes interpreted as beam experiments with clean
surfaces. The crosses in figure 1 show very early results by
Güntherschulze [67] in which the copper cathode was first
cleaned by electron bombardment heating and the Ar+ ions
were obtained by operating in very low-pressure discharge.
Here the uncertainties include [37] the energy of the ions
reaching the cathode from the discharge. In spite of the
uncertainties, these results are consistent with more recent
data. Recently, Böhm and Perrin [78] have measured γi for
102
103
104
Ion or atom energy (eV)
Figure 2. Electron yields for Ar+ and Ar beams incident on
various dirty metal surfaces versus particle energy. The open
symbols are for Ar+ and the solid symbols are for Ar. The symbols,
metals and references are: $
% , Pt, [69]; $, Ta, [69]; #, !, Au, [49];
, Cu, [75]; ♦, #, Cu, [45]; (, ", Ta, [80]; ×, W, [79]; %, brass,
[81]; , unknown, [50] and , CuBe, [51]. The solid curves are
plots of the analytical yield expressions for dirty surfaces for Ar+
and Ar, while the dashed curves are the representative yield curves
for clean surfaces for Ar+ ions and Ar atoms from figure 1.
◦
•
ions from an Ar plasma-processing discharge incident on Cu
that had been cleaned by exposure to a H2 discharge. The
ions are not mass identified, but are expected to be Ar+ with
mean energies less than 60 eV. Their γi values of about 0.2
are about twice the beam results shown in figure 1.
2.2. Beam results for dirty surfaces
Next we show in figure 2 the experimental values of the
electron yield γi for Ar+ ions [45, 49, 69, 75, 79, 80] and γa
for Ar atoms [45, 49–51, 80] incident on metal surfaces with
varying degrees of surface exposure to oxygen [69], to water
[50], to ambient gas [75], or to unspecified contamination
[51]. We will refer to these surfaces as ‘dirty’, although the
terms ‘practical’ or ‘laboratory’ surfaces are sometimes used.
We have shown all of the absolute yield data that we have
found for energies below about 1 keV. At higher ion energies,
these data are only a sample of that available [37–40, 42, 43].
At energies above about 500 eV the differences in yields
among metals are small compared to the differences from
the clean metal values. The solid curves of figure 2 show fits
to the experimental beam data that will be used in section 5.3
for comparison with swarm data. The dashed curves are
averages through the experimental data of figure 1 and show
the large changes in yield that typically occur when clean
surfaces become oxidized or otherwise contaminated.
The measurements for Ar+ (open points) in figure 2
show that for low energies (<150 eV) there is more than
a two order of magnitude spread in the yields. Some of the
low-energy data [45, 81] show relatively large yields, such as
those observed on exposure of the metal [68, 69] to gases like
H2 , N2 and CO, while data [50, 69, 79] obtained following
exposure to O2 or H2 O show very small yields. Much of the
R23
A V Phelps and Z Lj Petrović
Ar+ data at energies below 100 eV suggest the relatively weak
dependence on ion energy characteristic of potential (Auger)
[37, 42, 43, 68] ejection.
Parker [69] (Pt and Ta) and Hofer [82] (W) observed the
change in γi during exposure of the surface to oxygen. We
have not found similar direct demonstrations of the increase
in γi with increasing contamination expected at Ar+ energies
above 300 eV. One does not know whether to attribute the
energy independent or weakly energy-dependent portions of
the yield to patches of clean surfaces, to reduced penetration
of the Auger electrons through the contaminants, to energydependent collisional ionization of material on the surface or
to other mechanisms.
The measured electron yields per incident fast Ar atom
γa shown by the solid points of figure 2 have much the same
energy dependence as for fast Ar incident on clean metals,
but are shifted downward in energy by about a factor of ten
so that the yields at a given energy are much larger. Haugsjaa
et al [49] found the yields at high energies to be somewhat
larger for fast Ar than for Ar+ . At very low atom energies
they found structure in the yield curves for untreated surfaces
that was interpreted as the result of the collisional ionization
of two different adsorbed atoms. Thus, the electron yields
(∼0.01 at 100 eV) are roughly consistent with the product of
the surface density for a few monolayers of contaminants
(∼1018 m−2 ) and an ionization cross section for fast Ar
colliding with atoms (10−20 m−2 for Ar target). On the other
hand, Kadota and Kaneko [51] did not find structure in the
energy dependence of yield data at low Ar atom energies for
contaminated Cu–Be. They found that for their fast atom
source and incident Ar atom energies below 95 eV most of
the electron yield appears to be the result of Ar metastable
atoms produced in collisions of metastable Ar+ ions with Ar
atoms.
of the gas-phase process. The only potentially relevant data
sets we have found are that for Ar metastables [86] colliding
with O2 , where the cross section increases by an order of
magnitude between 10 and 500 eV and that for Ar metastables
[87] colliding with Na, where the cross section decreases by
an order of magnitude between 0.01 and 10 eV. As for fast Ar
atoms, the electron yields for Ar metastables are very roughly
consistent with the probability of an ionizing collision of an
Ar metastable with a column density of contaminant equal to
that of a monolayer of gas.
3. Swarm experiment results
In this section we review and analyse the available data
for the effective electron yield per Ar ion as determined
from pre-breakdown, breakdown and low-current discharge
measurements. These will be collectively referred to as
swarm measurements. For most of the data discussed in
this section we adopt the long-standing convention [32, 33]
of expressing the results of breakdown experiments as an
effective yield per ion γeff . This convention can be regarded
as a convenient bookkeeping procedure because, as we will
see, the Ar+ ion is often not the principal source of electron
emission from the cathode of an Ar discharge and is never
dominant for Ar at low E/n for any surface of interest
here. The convenience of γeff results from the simplicity
of the model used to interpret breakdown and low-current
discharge data, from the relatively high accuracy attributed
to the ionization coefficient data used in the data reduction,
from the rough correlation between the γeff ion and the true
electron yields per incident particle and the relatively high
sensitivity of γeff to changes in electron yield mechanisms
compared to the sensitivity of the breakdown voltage.
3.1. Interpretation of swarm results
2.3. Metastable beam results
Beam measurements of electron emission induced by the
impact of Ar metastables on metallic surfaces following
a wide range of surface treatments show an apparently
uncorrelated large range of electron yields [83, 84].
Hagstrum [68] showed theoretically that electron yield per
Ar metastable γm for clean metals, such W and Mo, should
be the same as for the ions, i.e. the metastable is ionized
near the surface and the ion ejects an electron by the Auger
process. A review of experimental data [83] gives values
for near thermal impact energies from 0.02 to 0.4, although
measured yields for freshly flashed surfaces are close to the
values for Ar+ . The measured [85] energy dependence for
Ar metastables incident on Cu–Be exposed to O2 at incident
energies from 250 to 600 eV is the same as that shown for
ground state atoms in figure 2. Data for CuBeO contaminated
by alkali vapours [86] shows a similar yield at energies above
500 eV, but is much higher at the lower energies. The change
in electron yield with Ar metastable kinetic energy for other
clean or dirty surfaces is unknown. A significant increase in
yield with surface temperature has been observed [84].
If the production of electrons by metastable impact on
metallic surfaces is regarded as a form of Penning ionization
[83], then it may be instructive to note the energy dependence
R24
The model used to obtain γeff from measurement of the
steady-state current resulting from photoelectrons released
from the cathode requires the fitting of an expression for the
current as a function of the electrode spacing or gas density
at fixed E/n. This procedure has been investigated by many
authors [30–33, 35, 88, 89]. The model assumes that only Ar+
ions produce electrons at the cathode and that only electrons
ionize Ar to produce electrons. The current i for electrode
separations d greater than d0 is
i=
i0 fes exp[αei (d − d0 )]
.
1 − γeff {exp[αei (d − d0 )] − 1}
(1)
Here i0 is the photoelectric current in vacuum, αei is the
spatial (Townsend) ionization coefficient, exp[αei (d − d0 )]
is the electron multiplication in crossing the gap and
exp[αei (d − d0 )] − 1 is the number of ions arriving at
the cathode per electron leaving the cathode. The delay
distance d0 and the electron-escape fraction fes are empirical
corrections applied to these local-field, fluid model results
in order to account for electron nonequilibrium effects near
the cathode. See the discussion in appendix A. Note that
γeff includes fes , e.g., for metastable-dominated electron
emission γeff = fes γm , where γm is the electron yield per
metastable in vacuum.
Cold-cathode discharges and breakdown in argon
γeff = {exp[αei (d − d0 )] − 1}−1 .
(2)
When applying equation (2) for conditions such that
the ionization coefficient for ion–atom collisions αii is
comparable with αei , the interpretation of γeff is complicated
by departures from the usual exponential growth of
the current versus distance [91]. Failure to allow for
the possibility of comparable electron- and ion-induced
ionization coefficients has led to much debate and misleading
conclusions in the literature [30, 31].
The model used to obtain secondary electron yields from
transient measurements of the current resulting from a pulse
of electrons released from the cathode or in the gap at voltages
below that required for breakdown varies with the dominant
electron production process at the cathode and therefore with
time scale of the experiments. In general, the techniques
involve separating the initial electron avalanche from the
delayed electron and/or ion current. The results of these
experiments are expressed in terms of the electron yields for
individual processes, e.g., γi . Hornbeck [92] and Varney
[93] were concerned with electrons produced by Ar+ and
Ar+2 , while Engstrom and Huxford [94] and Molnar [95] were
primarily concerned with electron production by metastable
Ar atoms. See the original papers for the details. Note that
these techniques do not appear to have been applied to what
we call dirty metal cathodes. Also, transient techniques do
not appear to have been used to separate the contributions
of ion-induced electron emission from the cathode and
ion–atom collisional ionization to avalanche initiation.
A variant of the transient technique uses optical pumping
to change the relative populations of the metastable and
resonance states. The only application of this approach to
Ar appears to be Molnar’s use of discharge illumination to
determine the relative electron yields for metastables and
photons [95]. See section 4 for results. Laser-induced
pumping has been used for determining dominant cathode
processes for Ne discharges, but no electron yield data were
obtained [96].
An area of recent interest in which equation (1) is
applicable is that of scintillation detectors [97] that usually
operate at very low E/n and high pressures. Of particular
interest here are experiments [98, 99] with pure Ar at
E/n <10 Td, p ' 760 Torr and d ≈ 2 mm. For these
conditions αei d * 1, d0 * d and equation (1) can be written
as
fes
fes
i
=
.
(3)
=
i0
[1 − γeff αei d]
[1 − γph fes αph d]
Here γph and αph are the photoelectric yield and effective
excitation coefficient for the dominant photon as discussed
10
Electron yield per ion
The model used to convert electrical breakdown data
to γeff data is the steady-state electron avalanche model
known as the Townsend model in numerous texts [31–34, 89].
As discussed in some of our earlier papers [64, 90], a
second source of such data is the maintenance voltage
for a low-current discharge. Here the catastrophic current
growth characteristic of breakdown is prevented by a large
external circuit resistor. The breakdown and low-current
maintenance condition is obtained by setting the denominator
of equation (1) equal to zero. Solving for γeff gives
1
10-1
10-2
10-3
10
102
103
104
105
E/n (Td)
Figure 3. Electron yields per Ar+ ion for various clean metal
surfaces versus E/n as obtained from pre-breakdown swarm
experiments. The data indicated by the solid points were obtained
after heating (flashing) the cathode to a high temperature, while
the open points show data obtained after cleaning by sputtering.
The procedures for determining the yields from swarm
measurements are reviewed in the text. The symbols, metals and
references are: #, Mo, [95]; !, Mo, [93]; ", Mo, [100] and $
% , Cu,
[88]. The dashed curve shows the γi values for Ar+ calculated
from the solid curve of from figure 1. The solid curve shows the
γeff vales predicted by our model in section 5.2 and includes all
electron production processes.
in section 5.1, i.e. the Ar2 continuum II near 127 nm. The
second form of equation (3) shows that the high-pressure
breakdown condition, i.e. the denominator equals zero,
allows one to determine the product of γph and αph without
knowing αei . As discussed in appendix B, the total excitation
coefficient αph /n is very nearly equal to E/n divided by
the excitation energy over a wide E/n range. Therefore,
the voltage required for breakdown is inversely proportional
to γph . This behaviour is in contrast to the logarithmic
dependence of breakdown voltage on γi at the higher E/n
where the exponential terms of equation (1) are dominant.
From equation (3) and our lack of independent knowledge
of γeff , we conclude that determining αei from steady-state
swarm data at very low E/n will be very difficult.
3.2. Swarm results for clean metals
The solid points of figure 3 show electron yields per Ar+
ion γi incident on various clean metal surfaces versus E/n
from swarm experiments. These data were obtained using the
pre-breakdown transient techniques [93, 95, 100]. Most of
these experiments used cleaning of the electrodes by heating
at high temperatures. They also used getters of chemically
active metals sputtered or evaporated onto the tube walls to
keep the gases clean [101]. These authors separated the ioninduced emission from the metastable and photon effects so
that the yield values shown by the solid points in figure 3 are
γi values. In these experiments the γi values are very nearly
equal to the γeff values.
The dashed line of figure 3 shows the γi results of
beam experiments from figure 1 with the ion energy to
R25
A V Phelps and Z Lj Petrović
E/n conversion discussed in appendix B. Because of the
relatively low ion and fast atom energies at the E/n of these
experiments, i.e. ion temperatures [26] of less than 2 eV, the
yield of electrons produced at these clean surfaces by fast
atoms is negligible. See figure 1. The solid curve of figure 3
shows γeff values calculated using the model to be discussed
in section 5 and the γi values from figure 1.
The electron yields per ion γi determined from prebreakdown data for clean Mo surfaces for E/n from 300
to 1500 Td shown in figure 3 are in good agreement with
the beam results shown in figure 1. Note that the three
different experiments show a significant decrease in γi for
E/n less than 250 Td, i.e. for calculated Ar+ ion energies
less than about 0.5 eV. According to the model of section 5,
this qualitative behaviour is consistent with the loss of Ar+
by conversion to Ar+2 ions and with a reduced electron yield
per ion for Ar+2 . However, the E/n of the observed transition
is higher than predicted by the model of section 5 by a factor
of about 2.5 and the pressures are too low by roughly two
orders of magnitude. An unexplored possibility is that the
Ar+ ions have difficulty penetrating an adsorbed gas layer at
low ion energies. The possibility that associative ionization
in Ar∗ +Ar → Ar+2 + e collisions [102] becomes the dominant
source of ions at these E/n seems ruled out by the estimates
of the contribution of this process by Puech and Torchin [17].
Also, the highest-pressure mass spectrometer data available
[103] appears to suggest a transition to Ar +2 at pressures
significantly higher than those corresponding to breakdown
at E/n near 250 Td, i.e. roughly 1.5 Torr.
Electron yields per Ar metastable atom for clean metals
have been determined by Molnar [95] from the transient
measurements of pre-breakdown currents. For flashed Ta the
yields per metastable vary from 0.0035 at 200 Td to 0.023 at
500 Td. For flashed Mo the yields increase from 0.02 to 0.06
for the same E/n range. At E/n ' 330 Td these values are
in good agreement with the measured electron yields for Ar+
as predicted by theory [68].
3.3. Swarm results for sputtered metal surfaces
In this section we note a potentially significant correlation
among γi values for various sputtered surfaces, i.e. for
unannealed surfaces the γi values decrease with successive
sputtering cycles. This topic takes on added importance with
the recent interest in the modelling of sputtering discharges
[5, 6] and other discharges operating at high current densities
[2, 4, 9, 10].
The extreme γeff values determined by Kruithof [88] for
Ar+ incident on Cu shown by the open points in figure 3 were
obtained by analyses of pre-breakdown, steady-state current
growth data after various intense sputtering treatments of the
cathode using Ne discharges with no flashing. These data are
shown as γi values because ion-induced electron emission is
expected to be dominant for clean metals at these E/n. Of
particular interest is the decrease in averaged γi values with
successive sputtering periods.
These results are to be compared with those of Varney
[93] in which repeated cleaning of Mo by sputtering and
flashing consistently gave the results shown by the solid
inverted triangles in figure 3. These values equal those
R26
obtained with flashing alone. Similarly, reproducible cathode
fall voltages in Ne were obtained following discharge
sputtering of Mo at high cathode temperatures [101].
The annealing temperature and duration required for a
significant change in γi for heavily sputtered surfaces and
Ar+ energies of interest for discharges is unknown. A small
decrease in γi is observed [77] for 10 keV Ar+ incident on
sputtered Cu when the surface is annealed at 200–300 ◦ C.
The changes in γi with intense sputtering are qualitatively
consistent with the changes observed [39, 104, 105] in surface
morphology.
3.4. Swarm results for dirty metals
The swarm results for dirty metal surfaces are presented
in two different formats in order to make better
contact with previous work.
Firstly, we show in
figure 4 measured breakdown [88, 106–112] and low-current
discharge-maintenance voltages [90, 113, 114] as a function
of the product of pressure and electrode spacing pd, in what
is known as the Paschen curve [32, 33, 35, 88, 89]. Later,
we reformulate these data so as to emphasize the production
of secondary electrons. Generally speaking, the rise in
breakdown voltage at high pd in figure 4 is attributed to a
decrease in the ionization coefficient with decreasing E/n
and decreasing electron energy. The rise at low pd is
attributed to the difficulty in building up an electron avalanche
when the number of collisions between electrons and the
gas in crossing the gap is small. Some authors [32, 33]
have attempted to relate these breakdown data directly to
the electron yield at the cathode. The solid, dashed and chain
curves of figures 4 and 5 are the results of the model presented
in section 5 and will be discussed there. It should be kept in
mind that for many of these experiments and all of our model
results plotted, the electrode separation d is 1 cm, so that the
numerical values of pd are equal to the Ar pressure p in Torr.
In order to establish more closely the connection between
breakdown data and electron production at the cathode, we
show in figure 5 the effective yield of electrons per Ar+
ion reaching the cathode γeff . The points of figure 5
show breakdown results [88, 106–112] and low-current
discharge results [90, 109, 113, 114] for γeff determined
using equation (2). The pd values shown vary from 0.1 Torr
cm at the highest [90, 111, 112] E/n to 2000 Torr cm at
the lowest [107–109] E/n, corresponding to pressures from
0.1 Torr to about 760 Torr. As will be shown in section 5,
the very large values of γeff at E/n ( 100 Td reflect the
fact that at low E/n there are many vuv photons produced
per ion by electron collisions with Ar. The large values of
γeff at E/n ' 3000 Td result from the combination of an
increasing electron yield per ion and the ionization produced
by Ar+ ion and fast Ar atom collisions with Ar.
4. Photoelectric yields
Figure 6 shows (open points) photoelectric yield γph versus
wavelength for Au, Cu and Pt from the review by Weissler
[115] for untreated and flashed metal surfaces. Also shown
(solid and dotted points) are the results for untreated surfaces
by Cairns and Samson [116] and others [115, 117, 118].
Cold-cathode discharges and breakdown in argon
1
105
Ar resonance
lines
Electrons per photon
Breakdown voltage (V)
10-1
104
103
Ar2 band
(continuum II)
untreated
10-2
treated
10-3
10-4
102
10-2
10-1
1
102
10
103
104
10-5
10
102
Pressure × distance (Torr cm)
Wavelength (nm)
Figure 4. Breakdown and discharge maintenance voltages for Ar
and various dirty metal surfaces versus pd, i.e. the Paschen curve
for Ar. The symbols, metals and references are: $
% , Ni, [106];
, Cu, [88]; , brass, [107]; #, Cu, [108]; ♦, Ni, [109]; ∇, Au,
[90]; ×, Steel, [168]; +, Cu, [169]; !, Au, [110]; #, SS, [169] and
", SS, [112]. Low-current discharge maintenance voltages are
shown by , Au, [90]. !, Cu, [113]. (, Cu, [114]. The dashed
and solid curves show the results of application of the full model
of section 5. The chain curve shows the predictions when only Ar+
ions contribute to the production of secondary electrons.
◦
•
Effective electron yield per ion
10
1
Figure 6. Photoelectric yield versus photon wavelength. The
solid and open points are for untreated and flashed surfaces from
[115]. The points with central dots are for untreated surfaces from
[116]. The circles, diamonds and squares are for Au, Cu and Pt,
respectively. The inverted triangles are for Cu films from [119].
The crosses are for Cu surfaces treated by laser irradiation from
[120].
have been used to ‘activate’ (clean?) metallic surfaces and
to determine the photoelectric yields shown by crosses (×)
[121].
We have fitted an empirical expression to the γph data
for heat-treated Au surfaces, i.e.
γph =
10-1
10-2
10-3
10
102
103
104
105
103
106
E/n (Td)
Figure 5. Effective electron yields per Ar+ ion incident on various
dirty metal surfaces versus E/n. These γeff are calculated from
breakdown and discharge operating voltages assuming that only
Ar+ ions produce secondary electrons. The values are often very
different from the true electron yield per ion. The symbols,
cathode metals and references for the breakdown data are: $
% , Ni,
[106]; , Cu, [88]; , brass, [107]; #, Cu, [108]; ♦, Ni, [109];
×, Steel, [168]; +, Cu, [169]; !, Au, [110]; #, SS, [111]; ", SS,
[112] and (, Cu, [114]. Effective yields from low-current
discharge data are shown by , Au, [90]. The dashed and solid
curves show the results of application of the model of section 5.
◦
•
The triangles are for evaporated Cu films [119]. Data (+)
for untreated Cu and steel surfaces are available [120] for
wavelengths from 150 to 180 nm. Recently pulsed lasers
0.0015(360/λ − 1)4
[1 + (110/λ)5.5 ]
(4)
where the wavelength λ is in nm. This expression has been
chosen to decrease with decreasing wavelength as do the more
recent experiments [116].
For untreated Cu the yield falls off much more
rapidly with increasing wavelength than for treated Au,
corresponding to an apparent higher-energy photoelectric
threshold. On the other hand, the photoelectric yield is
systematically larger for untreated surfaces than for treated
surfaces at below about 130 nm. Our empirical expression
for untreated Cu is
γph =
0.13
.
[1 + (λ/110)13 ][(35/λ)3.3 + 1]
(5)
The results for other metals, e.g., W (not shown), are
generally similar in shape, although varying somewhat in
magnitude [115, 117, 118].
The wavelengths of the Ar resonance lines are indicated
by vertical lines, with the 106.7 line being the one used
in our breakdown model in section 5.
The spectral
feature (continuum I) near 110 nm occurs at essentially
the same wavelength. Also indicated by vertical lines
is the wavelength of the molecular Ar2 band designated
continuum II near 127 nm in the model of section 5. We note
that the photoelectron yield for the resonance lines and for
continuum I is about three times that for continuum II. We will
find in section 5.3 that the γph values shown in figure 6 for
R27
A V Phelps and Z Lj Petrović
untreated surfaces are larger than the values required to fit all
but the largest experimental γeff results.
The applicability of the photoelectric yield measurements made in vacuum to gas-covered metal surfaces is critical to our analysis. The only absolute yield data we have
found for metal surfaces exposed to Ar are those of Molnar
[95] for flashed Mo and Ta and incident Ar resonance radiation. He found photoelectric yields about a factor of three
lower than the solid curve of figure 6. On the other hand, several authors [100, 122] appear to find no evidence of change
in photocurrent when Ar gas is removed from and readmitted
to their apparatus.
5. Modelling the effective yield data
In this section we present a model of the Ar+ , fast Ar,
metastable and photon transport and reactions in lowcurrent, uniform-electric-field discharges in Ar. The model
will be used with the electron yield data of figures 1,
2 and 6 and, with adjustments to that yield data, to
calculate γeff for comparison with the wide range of
experimental data presented in figures 3 and 5. The
model includes the relevant features of the steady-state
limit of the model of ion, metastable atom and resonancephoton transport and of excited-state reactions developed in
[123]. Figures 4 and 5 and figures 7–12 in sections 5.2
and 5.3 show the results of application of the present
model to the calculation of electrical breakdown and lowcurrent, discharge-maintenance voltages in Ar. These results
(a) compare calculations including all processes with the
experimental breakdown data, (b) show representative spatial
distributions of particle fluxes and densities, (c) illustrate
the behaviour with E/n of the individual secondary electron
production process and (d) show the relative importance of
the various secondary electron production processes. The
effects of some of these processes on breakdown voltages
in Ar have previously been calculated using a less-complete
model and are shown figure 11 of [53].
5.1. Details of model
In this section we develop a mostly analytic, approximate
model to describe the steady-state fluxes of electrons, Ar+
ions, fast Ar atoms, metastable atoms and resonance photons
in a spatially uniform electric field. This model can be used to
predict the role of these processes in steady-state-breakdown,
optical-emission and current-growth experiments.
The assumptions of the model are that:
(a) The electron flux density leaving the cathode is
determined by
$e (0) = fes [γi $i (0) + γi2 $i2 (0)
+γa $a (0) + γm $m (0) + γph $ph (0)].
(6)
Here $e , $i , $i2 , $a and $ph are the flux densities of electrons,
Ar+ , Ar+2 , fast Ar and photons evaluated at the cathode at
z = 0; fes is the fractional escape of electrons emitted from
the cathode into the gas and γi2 is the electron yield per ion for
Ar+2 . The γph $ph term is actually a sum over the various lines
and bands making up the photon flux reaching the cathode.
R28
(b) The spatial growth of the electron flux is described
by a fluid model in which a spatially constant spatial
ionization (Townsend) coefficient is determined by the
spatially independent E/n. At moderate and low E/n
this is a good approximation for most of the gap. See
appendix A. At very high E/n this constant ionization rate
replaces our previous approximation to the nonequilibrium
behaviour of the electrons [124] in which the electron energy
and subsequent ionization rate are determined by the solution
of an electron energy-balance equation, i.e. equation (3)
of [53]. The present assumption is justified by emission
experiments [125] and Monte Carlo solutions [125, 126] that
yield an approximately exponential growth of electron flux
and electron excitation with distance even at very high E/n.
The differential equation for the electron flux is
d$e
= αei $e + αii $i + αai $a .
dz
(7)
Here αei , αii , αai are the spatial (Townsend-like) coefficients
for ionization of Ar by electrons, by Ar+ and by fast Ar;
and z is the distance measured from the cathode. The
values of the E/n-dependent αx coefficients are discussed
in appendix B. At E/n < 3000 Td only electron impact
ionization of Ar is important and equation (7) reduces to
equation (12) of [123] in the limit of α1 = αei , β = 0
and λ = 0. Electron loss processes such as electron–ion
recombination and electron production processes such as
electron–excited atom collisions can be neglected for the
low current densities of interest here. Because of the low
currents we have neglected space-charge distortion of the
electric field.
(c) The Ar+ flux decreases as one moves away from the
cathode, i.e. it increases with distance from the anode, as the
result of ionization of Ar by electrons, Ar+ and fast Ar so that
d$i
= −αei $e − αii $i − αai $a .
dz
(8)
The ion energy distribution is assumed to be a onedimensional Maxwellian with a ‘temperature’ T+ determined
by the E/n and the charge transfer cross section [27, 127].
Inelastic Ar+ –Ar collisions are assumed negligible compared
to charge transfer collisions. We neglect the deviations
from the theoretical ion energy distributions found [128]
at high ion energies and high E/n.
The ionization
coefficients αxi in equations (7) and (8) and the ion-induced
excitation coefficients are averages of the corresponding
cross section over the ion-energy distribution [53]. Formulae
for these coefficients are given in appendix B.
Very few ions are expected to be emitted from the anode
as the result of electron bombardment [129]. As in [53],
we can approximate the ionization produced near the anode
by backscattered and reflected electrons [130] by using a
finite Ar+ flux as the boundary condition at the anode, i.e.,
$i (d) = δ$e (d).
(d) In this model we neglect the production of Ar+2 in
three-body collisions and its dissociation in collisions with
Ar. This simplification is possible because γi2 $i2 /γi $i is
predicted to be significant only at E/n < 100 Td where
photons are the dominant source of secondary electrons
Cold-cathode discharges and breakdown in argon
[131]. To take this low E/n process into account one would
use relations such as equation (13) of [123].
(e) The production of fast Ar flux is by symmetric charge
transfer collisions of Ar+ with Ar. We assume that these fast
atoms have the same energy and direction as the ions. We
assume that once the fast atom has an energy-loss collision,
i.e. an elastic viscosity collision [27] or an inelastic collision,
it is effectively thermalized. Use of these approximations
has been found to give rather good agreement with observed
emission distributions [53, 54] and with Monte Carlo models
in most cases tested [126]. It would appear that the neglect of
angular scattering leading to too few collisions in the model
roughly compensates for the too-rapid loss of fast atoms at
every elastic viscosity collision. The resultant fast-atom flux
equation is
d$a
= −αct $i + (αaa + αai + αar + αam )$a .
dz
(9)
Here αct , αaa , αar and αam are the spatial reaction coefficients
for symmetric charge transfer, elastic viscosity collisions,
resonance state excitation and metastable state excitation
collisions. Analytic formulas for these coefficients are given
in appendix B.
(f) Our model for the excited states of Ar near 11.5 eV is
a simplification that is only intended to predict the fluxes
of metastables and uv radiation that reach the cathode
in our steady-state breakdown and low-current discharge
experiments. We replace the two resonance states by the
lower resonance state (4s 3 P1 ) and replace the two metastable
states by the lower metastable state (4s 3 P2 ) [132]. The total
excitation rates for the two resonance levels [17] are assigned
to the model resonance state. Similarly, for the metastable
states.
We assume that the molecular states associated with
each of the lower two atomic states [133] are formed in
three-body collisions of the excited atom with two Ar atoms
with the measured rate coefficients [134]. The vibrationally
excited molecules are assumed to relax immediately to their
ground vibrational states. We neglect collisional coupling
between the resonance and metastable states and between
the molecular states. All of the excited Ar2 molecules
relaxing to the lower vibrational states are assumed to radiate
in the 127 nm band designated continuum II, [133], i.e.
their quenching is neglected. In this model, the observed
continuum I near 110 nm is absorbed into the wings of the
resonance line.
This very much simplified four-level model is
intermediate in complexity between the afterglow models
of Millet et al [134] and of Bretagne et al [14]. A
critical advantage of this simple model is that, knowing
the rates of production by electrons, ions and fast atoms,
we can solve for the metastable atom and resonance-photon
fluxes independently. We have compared the steady-state
predictions of this four-level model with those of a model
[135] with four atomic levels and four molecular levels when
the electron excitation is nearly independent of position as at
high Ar pressures [109]. For high-purity Ar ('0.999 99) the
difference between the photoelectric flux leaving the cathode
estimated using the four-level and eight-level models peaks at
≈20% for pressures near 10 Torr, where resonance radiation
is the dominant photon flux. For low-purity Ar (0.999) the
estimated error is (10% at all pressures.
(g) Our treatment of resonance excitation is based
directly on the model of [123], as verified for Ar in the current
growth experiments by Menes [108]. The basic equation
governing the steady-state density of excited atoms in the
resonance state is [28]
! d
nr (z. )K(|z − z. |) dz.
(Ar + q)nr (z) − Ar
0
= αer $e + αar $a + αir $i
= a1 + a2 exp(s2 z) + a3 exp(s3 z).
(10)
Here nr (z) is the resonance atom density, Ar is the
radiative transition probability, q is the frequency of
molecule formation collisions and K(|z − z. |) is the radiative
transmission kernel. The right-hand side is the resonance
state collisional production rate due to electron–Ar atom
collisions αer , fast Ar–Ar atom collisions αar and ion–Ar
atom collisions αir . Here s1 = 0, s2 and s3 are the roots
of the third order determinant of the system of equations
found by taking the Laplace transforms of equations (7)–(9)
and a1 , a2 and a3 are the respective algebraic coefficients
for the components of the resonance state production. The
expressions used for the αxr /n for resonance state excitation
by electrons, ions and atoms are given in appendix B.
Absorption measurements [136] show that the theoretical dipole-dipole resonance line profile used in this model
should be valid for d ∼ 1 cm and p ( 200 Torr. At higher
pressures the calculated excess absorption in the red wing will
be partially offset by the collision-induced radiation effects
[134]. Also, at 200 Torr collisional conversion to the radiating molecules dominates resonance radiation transport, i.e.
q considerably exceeds the effective resonance state lifetime
AI from [123].
The resonance photon flux reaching the cathode $rph is
[28]
!
!
∞
d
$rph = Ar
nr (x) dx
0
K(|y|) dy.
(11)
x
Rather than solve equations (10) and (11) numerically we
have made use of the solutions obtained in [123]. When the
exponential terms increase toward the anode, i.e. negative
values of s2 and s3 , the resultant resonance atom density terms
have a spatial dependence such as shown in figure 1 of that
reference. The resonance flux transmitted to the cathode is
that described in connection with figure 2 and equation (25).
For an exponentially decreasing source term, the distance
scale in figure 1 is reversed and equation (25) is modified by
replacing the last factor by (2 − 1/[1 + 0.175(αd)2/3 ]).
(h) The effects of Ar metastables in the 3 P2 and 3 P0
states of Ar are approximated by a single metastable state
[14, 95, 123, 134] that is produced by electron, ion and
fast-atom excitation of Ar. Metastable atoms are lost by
diffusion to infinite parallel-plane electrodes, by collisioninduced emission and by three-body collisional conversion
to radiating molecules. The steady-state metastable rate
equation is
−
(nDm ) d2 nm (z)
= αem $e + αam $a + αim $i
n
dz2
−[k2m n + k3m n2 ]nm (z).
(12)
R29
A V Phelps and Z Lj Petrović
−[αa αii + αai αi ]/(s2 s3 )
−{exp(s3 z)[(αa + s3 )(αii + (δ − 1)s3 )
−1
+αai αi + (δ − 1)s3 ]}[s3 (s3 − s2 )]
$i
= {δ[exp(s2 z)(αai + αa + s2 )
$t
− exp(s3 z)(αai + αa + s3 )]}(s2 − s3 )−1
"
(αai + αa + s2 ) exp(s2 z)
+αei
s2 (s2 − s3 )
#
(αai + αa αauv ) exp(s3 z)(αai + αp + s3 )
+
+
s2 s3
s3 (s3 − s2 )
$a
= {αct [δ(exp(s2 z) − exp(s3 z))s2 s3
$t
+αei (s2 − exp(s3 z)s2 + (exp(s2 z) − 1)s3 )]}
×(s2 (s2 − s3 )s3 )−1
(13)
−4(αa αei − αa αii − αai αrc )]1/2 ).
Normalized fluxes and densities
1
Γi
10-1
Γe
nr´ /1000
-2
10
nm´ /1000
10-3
0
2
4
6
8
10
Distance from cathode (mm)
Figure 7. Calculated spatial dependence of the normalized fluxes
(solid curves) of electrons $e , Ar ions $i and fast Ar atoms $a and
of the normalized densities (dashed curves) of metastables n.m and
excited atoms in the resonance state n.r . The model does not
distinguish between Ar+ and Ar+2 . For clean metal surfaces and
calculated breakdown conditions at E/n = 50 Td, for which
pd = 63 Torr cm.
10
Γa
1
Γi
10-1
Γe
nm´ /10
10-2
nr´ /10
10-3
0
2
4
6
8
10
Distance from cathode (mm)
(15)
(16)
The source terms for the production of metastables and
resonance atoms resulting from equations (13)–(15) have the
form of a constant plus two exponential terms. The exponent
s2 obtained with the negative sign in equation (16) is negative
for all E/n and our parameters results in growth toward the
anode. The exponent s3 obtained with the positive sign in
equation (16) may be positive at high E/n corresponding
R30
Γa
(14)
where αa = αai + αap + αaq and αi = αii + αct . The roots of
the determinant for equations (7)–(9) are 0 and
s2 , s3 = 1/2(−αa − αei + αii ∓ [(αa + αei − αii )2
10
Normalized fluxes and densities
Here αem , αam and αim are the spatial coefficients for
metastable excitation by electrons, fast Ar atoms and Ar+
ions, nDm is the metastable diffusion coefficient at unit
Ar density, k2m is the rate coefficient for collision-induced
radiation, k3m is the three-body rate coefficient for excited
molecule formation and n is the Ar density. The boundary
conditions for this equation are that the metastable density
is zero at the electrodes and the desired quantity is the
metastable flux at the cathode, −(nDm /n) dm(z)/dz.
Collision-induced radiation from the metastable state
results in nonresonant emission near λ = 110 nm, i.e.
continuum I. See equation (18) of [123]. The photoelectric
yields at 110 nm from equations (4) and (5) essentially equals
that for 106.6 nm.
At the pressures for which this process is important,
i.e. (10 Torr, the Ar is optically thin [136]. At higher
pressures the excitation reaches the electrodes via by
molecule formation, vibrational relaxation and radiation of
continuum II.
(j) Equations (7)–(9) can be solved independently of
the equations governing resonance atoms and the products
of resonance atom collisions with Ar. We have done this
analytically using the Laplace transform technique [137].
The results for the electrons, ions and fast atom fluxes at
the cathode are:
$e
= −{exp(s2 z)[(αa + s2 )(αii + (δ − 1)s2 )
$t
+αai (αi + (δ − 1)s2 )]}[s2 (s2 − s3 )]−1
Figure 8. Calculated spatial dependence of the normalized fluxes
(solid curves) of electrons $e , Ar+ ions $i and fast Ar atoms $a
and of the normalized densities (dashed curves) of metastables n.m
and excited atoms in the resonance state n.r . For clean metal
surfaces and calculated breakdown conditions at E/n = 20 kTd,
for which pd = 0.17 Torr cm.
to growth toward the cathode resulting from heavy-particle
excitation and ionization.
(k) The resultant flux equations for metastables and
for the resonance photons are much too complicated to
publish, but are available on request [138]. Instead we show
representative solutions for the metastable and resonance
state densities in figures 7 and 8. These figures will be
discussed in section 5.2.
Cold-cathode discharges and breakdown in argon
(l) The radial losses of all species are neglected in our
one-dimensional model. The lateral losses of charge particles
and of fast atoms will be small because of their high axial
energies. For nonresonant photons and the experiments cited
here the solid angle of the cathode from the centre of the anode
is from 60 to 70% of that for infinite parallel planes. The
effects of the decrease of the solid angle for photons emitted
off axis is expected to be reduced because of the Bessel
function distribution of current density in the fundamental
diffusion mode [139]. We will use 0.3 for the fraction of the
nonresonant photons reaching the cathode. The effects of
a finite tube radius on the transport of resonance excitation
appears to be small for typical discharge electrodes [140].
(m) The quasi-static breakdown and low-current
discharge maintenance condition is that the sum of the fluxes
of electrons produced at the cathode by the various incident
species is equal to the flux of electrons leaving the cathode,
i.e. that the electron flux given by equation (6) equals that
given by equation (13) evaluated at the cathode. Formulation
and the numerical solution of the very complex algebraic
equations for this model were carried out on a personal
computer [137]. The numerical solutions were obtained
by assuming a value of E/n, using analytic forms for the
collision coefficients and yields from appendix B, assuming
an electrode spacing d and solving iteratively for the value
of p, i.e. the eigenvalue, that satisfied the breakdown and
low-current discharge condition. While Monte Carlo (MC)
techniques for solution of the particle and photon fluxes
[3, 5–7, 22, 126, 141] would avoid complex algebra and allow
fewer assumptions, the number of MC calculations would be
large and time consuming [142].
Tests for the contributions of the corrections for
nonequilibrium effects at the cathode, i.e. electron
backscattered to the cathode and delay in the onset of
ionization, were made in the present analyses.
See
appendix A for the backscattering correction used. The
effects of the delays in the onset of ionization on the derived
γeff were small enough compared to the uncertainties in the
electron yields so that they were not included in the final
analyses.
5.2. Application of model to clean surfaces
The results of applying the model just described to the
calculation of conditions for breakdown and pre-breakdown
for clean metal cathodes are presented in figures 7–10.
Because of the very limited range of E/n for which
experimental γeff data for clean surfaces is available, we
have used the electron yield data estimated from the beam
experiments and listed in appendix B without adjustment.
Also, we have simplified the model by neglecting electron
reflection at the anode.
Figures 7 and 8 show the calculated spatial dependences
of the normalized fluxes (solid curves) of electrons $e ,
ions $i and fast atoms $a . The calculated resonance and
metastable state densities (dashed curves) are plotted as the
dimensionless quantities n.r (z) = [(q + AI )d/ $e (d)]nr and
n.m (z) = [νd/ $e (d)]nm , respectively. Here $e (d) is the
electron flux density at the anode and ν = [k2m n+k3m n2 ]+νD ,
where νD = (3.14/d)2 (nDm )/n is the fundamental-mode,
diffusion-loss frequency for metastables. For breakdown
or a low-current discharge at E/n = 50 Td, the model
gives pd = 63 Torr cm and the spatial distributions in
figure 7. The electron flux grows exponentially with distance
from the cathode, while the Ar+ and fast Ar atom fluxes
increase toward the cathode from zero at the anode. The
electron flux is normalized to unity at the anode. In this
low-E/n case the ion and fast atom temperatures are less
than 0.1 eV. As a result of production by electrons only and
a loss dominated by collisions, the metastable atom density
n.m (z) grows exponentially with distance except very close
to the electrodes. Because of radiation transport, there are
significant departures from an exponential growth for the
resonance atom density n.r (z), just as shown in figure 1 of
[123].
For the very high E/n of 20 kTd, the calculations
for a clean cathode give pd = 0.17 Torr cm and the
spatial distributions shown in figure 8. The calculated $e (z)
curve shows significant departures from exponential growth
because of ionization by Ar+ ions and fast Ar atoms produced
by charge transfer from Ar+ . In this high-E/n case the
ion and fast atom temperatures are about 50 eV. The heavyparticle collisions also result in a continued growth of ion and
fast atom fluxes as one approaches the cathode rather than
the saturation seen in figure 7 at low E/n. The excitation
by heavy particles near the cathode and electrons near the
anode results in a relatively flat resonance state density. The
excitation by fast atoms and electrons and the large diffusion
loss of metastables at low pressures combine to give an n.m (z)
profile that is roughly a sinusoid that is shifted toward the
cathode rather than the anode.
We next consider what we call ‘limiting cases’ of γeff
as a function of E/n resulting from models in which specific
cathode directed particles or photons are assumed to produce
all of the ‘secondary’ electrons required to cause breakdown
or to maintain the discharge. The solid curve in figure 3
and repeated in figure 9, shows the calculated γeff when all
processes are included in the model. The short-dashed curve
labelled γi in figure 9 gives the values of γeff obtained when
γi was set to a value of 0.07 and all of the other yields and the
heavy-particle collision cross sections were set to zero [143].
As expected, the short-dashed curve agrees with the assumed
γi value for E/n > 1000 Td, but drops below γi for E/n <
1000 Td because of the effects of electron backscattering to
the cathode. See appendix A. The long-dashed curve shows
the γeff values calculated when only photoelectric production
of electrons at the cathode is included in the model. Here we
use the photoelectric yield given by equation (4). Although
no experimental γeff data are available for clean metals at
E/n ( 50 Td, the model shows that photoelectric processes
determine the γeff values. At high E/n the calculated γeff
values and breakdown voltages are determined by secondary
electron production by heavy particle ionization, rather than
by secondary electron production at the cathode. The curves
labelled γm and γa in figure 9 show that the calculated γeff
values are small when only metastable atoms or fast atoms
produce secondary electrons.
Although the curves of γeff presented in figure 9 show
which secondary electron production processes are dominant
at various E/n, they do not show the contributions of
R31
A V Phelps and Z Lj Petrović
Effective electron yield per ion
10
heavy particle
ionization only
1
all processes
10-1
γph
10-2
γi
γm
10-3
10
102
γa
103
104
105
E/n (Td)
Figure 9. Calculated effective electron yields versus E/n for clean surfaces. The curves are calculated for various limiting assumptions for
the electron yields at the cathode and by heavy- particle ionization and excitation using electron yields from appendix B. The labels of the
various curves are: γi , Ar ion-induced electrons at the cathode with no heavy-particle excitation or ionization; γa , Ar fast-atom-induced
electrons at cathode with no heavy-particle excitation or ionization; heavy-particle ionization with no electron production at cathode; γph ,
vuv-photon-induced electrons at cathode with heavy-particle excitation of resonance atoms; γm , Ar metastable induced electrons at cathode
with heavy-particle excitation of metastables, and all processes, all cathode emission and heavy-particle ionization and excitation processes.
10
vuv continuum
Fractional electron emission
various electron production processes at the cathode. These
contributions are indicated by curves of figure 10 for the clean
surfaces of figures 3 and 9. We see that Ar+ ions are the
dominant source of electrons at the cathode for E/n ' 60 Td.
Resonance photons produce most of the electrons needed for
breakdown for E/n near 40 Td for a clean cathode. At lower
E/n the nonresonant vuv photons emitted by excited Ar2
formed from resonance atoms and metastables, i.e. Continua I
and II, are dominant. Because of the importance of heavyparticle ionization, the curves of figure 10 should not be
interpreted as showing that the production of avalancheinitiating electrons at high E/n is dominated by ion impact
on the cathode.
1
metastables
10-1
10-2
atoms
resonance
photons
5.3. Application of model to dirty surfaces
The results of applying the models of section 5.1 to the
calculation of conditions for breakdown and for low-current
discharge maintenance for dirty surfaces are presented in
figures 4, 5, 11 and 12. The spatial distributions of fluxes
and excited state densities for E/n of 50 Td and 20 kTd are
not shown, as they are very nearly the same as those presented
for clean cathodes in figures 7 and 8.
5.3.1. Comparison with experiments. The results of
the model are compared with experiment in figures 4 and
5. Here the solid and short-dashed curves are calculated
using all of the surface and volume secondary electron
production processes of our model. The short-dashed curves
are calculated using the γi from the solid curve of figure 2
and γph from the dashed curve for untreated surfaces from
figure 6. The solid curves are calculated using 10% of the
photoelectric yield γph shown by the dashed curve of figure 6.
R32
ions
10-3
10
102
103
104
105
E/N (Td)
Figure 10. Calculated fractional contributions of various
processes to electron production at the cathode for Ar at
breakdown for clean surfaces. The model is that of section 5 using
the parameters discussed in appendix B. The processes are
indicated by the labels attached to the curves.
The chain curves are calculated using γi = 0.07, as used in
several recent low-current models [22, 139]. In this case, all
other γ values and heavy-particle ionization rates are set to
zero.
At large pd the calculated voltages of figure 4 are nearly
coincident for all three of these models. The small spread of
voltages at fixed pd arises from the rapid decrease in αi /n
Cold-cathode discharges and breakdown in argon
Effective electron yield per ion
10
heavy particle
ionization only
all processes
exp’t. γph
1
all processes
adj. γph
10-1
γa
10-2
γph
10-3
10
γi
102
103
104
105
E/n (Td)
Figure 11. Calculated effective electron yields versus E/n for dirty surfaces. The curves are calculated for various limiting assumptions for
the electron yields at the cathode and by heavy- particle ionization and excitation using our best estimates of these electron yields from
figure 1 and the model of section 5. The labelling of the curves is the same as in figure 9 except that the two curves labelled all processes are
calculated for γi and γph values based on two different approximations to the data of figures 5 and 6. See text for details.
Fractional electron emission
vuv continuum resonance
photons
ions
atoms
1
metastables
10-1
10-2
10
102
103
104
105
E/N (Td)
Figure 12. Calculated fractional contributions of various
processes to electron production at the cathode for Ar at
breakdown for dirty surfaces. The model is that of section 5 using
the parameters discussed in appendix B. The labels attached to the
curves indicate the electron production process.
with decreasing E/n. If these data are replotted as a function
of E/n they cover a wide range of breakdown voltage or pd
values. When converted to γeff , as shown in figure 5, the data
at fixed E/n are spread over more than an order of magnitude.
The comparison of the solid and dashed curves with the points
of figure 5 leads to the conclusion that the γph values to be
used in our model range from the values shown for untreated
surfaces in figure 6 to more than an order of magnitude
smaller. Unfortunately, there is a large uncertainty at very
low E/n in the αei /n values used to calculate γeff from
experimental breakdown and discharge maintenance data.
This uncertainty is evidenced by the differences in αei /n
reported by several groups [88, 109, 144] i.e. the diverging
γeff values shown in figure 4 of Golden and Fisher [109].
The lower set of γeff values from this reference is the result
of our analyses of their breakdown voltage data and the upper
set is from their fitting of equation (1) to current growth data.
An alternate approach at very low E/n is to use
equation (3) to analyse data such as those of figure 3
of Golden and Fisher [109].
If the fraction of the
metastable and resonance state excitation converted into
continuum II photons is constant over the pressure range of
the experiment, equation (3) shows that at constant E/n the
quantity (1 − fes I0 /I ) is a linear function of pressure. The
experimental data of figure 3 of [109] fit this expression very
well and yield the product fes αph γph /p = 0.0013 Torr−1 .
Using our values of αph and fes and equation (B28) for the
fraction of photons reaching the cathode, we find γph =
0.006. This photoelectric yield is about a factor of three
smaller than we expect from figure 6. In this experiment,
the effects of known impurities [109] are estimated to reduce
the vuv emission by less than a factor of two. How much
more of the discrepancy can be attributed to impurities
is unknown. In principle, equation (3) should apply to
steady-state experiments under many scintillation detector
conditions, e.g., the normalized current data of figure 4 of
[99].
At pd ≈ 0.07 Torr cm and E/n ≈ 1000 Td, the
assumption of γi = 0.07 (chain curve) leads to much too
small a breakdown voltage and to much too large γeff values.
This comparison also shows that although our adjusted γi
values are roughly constant for ion energies between 10 and
100 eV in figure 2, the decrease in γi at lower energies
is crucial to the fit to experiment. At pd near 0.03 Torr
cm, the calculated V versus pd curve (chain curve) for
γi = 0.07 becomes nearly vertical. This behaviour is
the result of a maximum in αei /n versus E/n. We see
from this example that the assumption of constant γi or
R33
A V Phelps and Z Lj Petrović
constant γeff , common in many current Ar discharge models,
[5–8, 12, 14, 15, 22, 141], is a poor approximation.
At the high E/n where heavy-particle collisions are
important, the model results shown in figures 4 and 5 are
in satisfactory agreement with the badly scattered data. The
inclusion of ionization by electrons backscattered from the
anode would lower the breakdown voltage and raise the γeff
values. It remains to be seen whether a better model of ion
and fast atom motion, i.e. a Monte Carlo model incorporating
angular scattering and elastic energy loss by heavy particles
[5, 126, 141] and electron backscattering, will significantly
change the comparisons with experiment.
As indicated in equation (3) and in the equations in
[123], the photoelectric feedback term appears as a product
of the photoelectric yield and the effective vuv excitation
coefficient. Therefore an increase in photoelectric yield to
the expected values could be compensated for by a decrease
in effective excitation coefficient caused by errors in our
kinetics model. We have not found useful measurements
of the vuv emission efficiency for Ar that test the model.
Finally, including the delay in the onset of ionization near the
cathode, as discussed in appendix A, changes the calculated
γeff values by amounts much less than the scatter in the
experimental data.
5.3.2. Importance of various processes. The discussions
of the relative importance of the various process in this section
for dirty surfaces are similar to those for clean surfaces in
section 5.2. We will therefore emphasize the differences.
The values of the coefficients used are from the appendices
or from section 4.
In figure 11 we show γeff values for dirty surfaces when
we limit the kinds of cathode directed particle or photon that
produce secondary electrons. The dotted curve labelled γi
gives the values of γeff obtained when the γi was set equal to
the values given by the solid curve labelled Ar+ in figure 2 and
then averaged over the one-dimensional energy distribution
of Ar+ energies. Other secondary processes were omitted.
For E/n > 1000 Td this curve agrees with the assumed
average γi , but is lower for E/n < 500 Td because of electron
backscattering to the cathode. Comparison of the dotted
curve of figure 11 and the short-dashed curve of figure 9
shows that our adjustment of γi to fit the breakdown data
for dirty surfaces results in much lower values than for clean
surfaces at E/n < 10 kTd or Ar+ energies below 5 eV.
The dot–dashed curve labelled γa of figure 11 shows
calculated γeff values when the yield for fast atoms equals
the values shown by the solid curve labelled Ar in figure 2.
Again, all other secondary electron production is zero. Most
of the increase in γeff for fast atoms at high E/n compared
to that for ions is caused by the large number of fast atoms
produced in charge transfer collisions between Ar+ ions and
Ar atoms. As expected from the data of figures 1 and 2, fast
Ar atoms are much more effective for dirty surfaces than for
clean surfaces.
As pointed out previously, [53, 58, 145], the ionization of
Ar by fast Ar and by Ar+ is an important electron production
processes at very high E/n. This importance is shown in
figure 11 by the double-dot–dashed curve labelled heavyparticle ionization. This curve gives the values of γeff
R34
when ionization by Ar+ and fast Ar were included, but other
secondary electron production process were omitted. The
resultant γeff values for E/n near 10 kTd agree with the
experimental values (see below) to within the large scatter
in the data. The γeff curve for ionization by ions only (not
shown) is less than 20% of that shown for ionization by atoms
and corresponds to that originally proposed by Townsend to
explain breakdown [30]. In these heavy-particle ionization
cases, we are particularly conscious that the γeff has become
a bookkeeping factor that is a measure of the voltage across
the discharge. Because of the large contribution of heavyparticle ionization, it is difficult to determine electron yields
for ions and fast atoms striking the cathode from breakdown
data at very high E/n in Ar.
The contribution to γeff for dirty surfaces of photons
produced by excitation of the resonance state and the
molecular continua are similar to those for clean surfaces.
The main difference is that for dirty surfaces the resonancephoton contribution is large to higher E/n because of the
lower values of γi . The calculated values of γeff obtained
assuming that only metastable Ar atoms produce electrons
at the cathode with a constant yield of 0.02 electrons per
metastable are too small to show in figure 11.
The relative contributions of the various processes
at the cathode found using the breakdown and discharge
maintenance parameters from the fitted curves of figures 4, 5
and 11 are indicated in figure 12. These results show that
ions and fast atoms are the dominant source of electrons
at the cathode at E/n above about 400 Td. Resonance
photons produce most of the electrons needed for breakdown
for E/n between 40 Td and 1000 Td. At lower E/n the
nonresonant photons emitted by excited Ar2 formed from
resonance atoms, i.e. continuum II, are dominant.
It should be noted that because of the importance of
secondary electron production by resonance photons at low
E/n, with their effective lifetime [28] varying as d 1/2 , the
conventional scaling [32] of the breakdown or low-current
discharge voltage (or E/n) with pd no longer applies. The
pd (Paschen) scaling is also lost because of the n2 dependent
terms in the excited state destruction. In order to test the
scaling of γeff with electrode separation d for a fixed E/n,
we repeated the calculations for the conditions of the dashed
curves of figures 5 and 11, except that the distance was
4 cm instead of 1 cm. We find that the new γeff are barely
distinguishable from those shown. Effects of the change
are more noticeable for the fractional contributions to the
secondary electron production, where there is a shift toward
higher E/n of the transition from dominance by resonance
photons to dominance by continuum radiation. Presumably
this is caused by a shift in the competition between resonance
radiation decay and Ar2 formation, with larger distances
reducing the escape rate for resonance photons. While
the departures from pd scaling are predicted to be small
compared with the overall scatter in the experimental results
shown in this paper, the effects of such departures in the
scaling on γeff may have been observed in the experiments
of Golden and Fisher [109].
Cold-cathode discharges and breakdown in argon
6. Summary and discussion
In this paper we have compiled and analysed over 60 years
of data concerned with cathode processes of importance in
quasi-static, uniform-electric-field breakdown and in lowcurrent, steady-state discharges for Ar. We have successfully
applied an updated model that describes the important
surface and gas phase collision processes for the whole
range of experimentally available data, i.e. E/n and pd
values from 15 Td and 2000 Torr cm to 100 kTd and
0.06 Torr cm. While there are many details that require
further work, the analysis delineates quantitatively the
regions of importance of several long-debated processes
responsible for secondary electron production by cathodedirected species, e.g., electron emission at the cathode
induced by ions, fast atoms, metastables and photons
versus electron impact ionization by ions and fast
atoms.
Our review of electron yield data and our model of the
role of various secondary electron production processes in
Ar has shown that:
(1) The yields of electrons per ion, fast atom or photon at
a metallic cathode of a discharge in Ar are highly dependent
on the condition of the metal surface. Unless there are
compelling reasons to the contrary, this means that any
estimates of electron yield made by a modeller on the basis
of the cathode preparation technique should be regarded as
tentative and subject to adjustment. The data presented in
this paper should allow the modellers to make informed
initial estimates as to the electron yields at the cathode of
Ar discharges. In particular, these data should dispel the
commonly held assumption that the electron yield per Ar ion
is independent of E/n and equal to the value found for flashed
metals, e.g., that found by Hagstrum [68] for molybdenum
or by Oechsner [73] for copper.
(2) For most metal surfaces cleaned by repeated
sputtering and flashing and exposed to very pure Ar
one should expect a yield per ion γi of about 0.1 that
increases slowly with E/n only at extremely high E/n.
At E/n below about 100 Td our model indicates that
photon-induced electron production at the cathode becomes
dominant. Because the high electron yields per ion for clean
surfaces also apply to Ar metastable atoms, transient current
growth and decay experiments at intermediate E/n and
long times show time constants characteristic of metastable
diffusion.
(3) For dirty surfaces the measured yields per ion γi
at ion energies below 200 eV corresponding to E/n at the
cathode below about 70 kTd can vary by orders of magnitude
depending on the surface condition. The beam studies of γi
shown in figure 2 need to be extended to much lower ion
energies, e.g., energies well below 2 eV, to be applicable to
breakdown at moderate E/n. At ion energies above 300 eV,
corresponding to E/n > 100 kTd, the yields are remarkably
independent of the substrate and the nature of the gaseous
contaminant, but are much larger than for clean surfaces.
(4) At E/n between 10 and 100 kTd our present model
confirms the previous analysis [53] of breakdown in Ar that
showed that ionization of the Ar gas by fast Ar atoms and
by Ar+ ions becomes the dominant source of electrons for
both clean and dirty cathodes. Here the fast Ar atoms are
formed by charge transfer in Ar+ collisions with Ar. As yet
there is no direct experimental evidence of the importance
of secondary electron production by fast atoms from swarm
or discharge experiments in Ar, although the closely related
phenomena of Ar excitation by fast atoms and ions has been
demonstrated experimentally and successfully modelled in
recent years [5, 53, 63, 126, 141].
An interesting rule-of-thumb brought out by the
calculations of figure 11 is that heavy-particle ionization
becomes the dominant source of secondary electrons for
breakdown and low-current, steady-state discharges at E/n
above about 7 kTd or, from figure 4, at voltages greater than
1000 V. Because of the rapid increase in spatial ionization
coefficient for ions and fast atoms with E/n, we expect this
transition voltage to decrease when the discharge current is
increased enough to form the high-field region of the cathode
fall. See appendix B.
(5) A critical need is for better independent experimental
means of determining the electron yield from surfaces under
conditions approximating those in the discharges to be
modelled.
(6) The cross sections for collisions of electrons and ions
with the more common gases are relatively well known and
techniques for calculating the electron and ion behaviour in
the uniform electric fields of swarm experiments are well
established. In the case of electrons in Ar there is good
agreement between well tested experiments and models.
Therefore modellers should be extremely cautious about
changing these electron-gas parameters on the basis of model
results for complicated discharges, e.g., discharges involving
space charge electric fields and externally applied dc or radiofrequency fields.
(7) It is obvious from the comparison of γeff from the
model and from swarm experiments at E/n below 1000 Td
in figure 5 that one must use surprisingly low photon yields
in order to fit some of the data for dirty surfaces. Such low
values appear to be inconsistent with measurements made in
vacuum. It should be kept in mind that our model essentially
assumes that at very low E/n and Ar high pressures all of
the excitation of Ar by electrons appears as nonresonance
photons. We have not found measurements of vuv emission
efficiency for high-pressure Ar by other techniques that test
this assumption.
(8) The very limited data available [88] for low energy
Ar+ ions incident on surfaces cleaned by sputtering in the
presence of Ar indicates that high-dosage sputtering of
Cu and Au without high temperature flashing results in
significant decreases the γi values. We suggest that sputtering
may cause similar reductions in the photoelectric yield.
(9) The present modelling work should be extended to
include greater detail as to the role of excited states of Ar,
to more accurate models [5, 126, 141] of the fast ion and
fast atom fluxes in Ar, to the differential voltage–current
behaviour of low current discharges in Ar, to the time
dependent growth of current in Ar discharges and to the
analyses of discharges in other gases. An analysis of ionand photon-induced electron emission in low-current H2
discharges has evaluated the contributions of H+ , H+2 , H+3 and
vuv photons [146].
R35
A V Phelps and Z Lj Petrović
1
Fraction escaping cathode
(10) Although the model applied to the analysis of
uniform-field data in this paper is applicable only for low
enough discharge current densities so that one can neglect
space distortion of the electric field, the same secondary
electron production processes will occur at the higher
current densities of cold-cathode discharges characterized
as the normal and abnormal cathode fall [32, 61]. For
example, values of γeff for Ar discharges at moderate current
densities have been inferred by comparison of models with
voltage–current measurements in the abnormal glow mode
[10]. Most models of glow discharges [1–3, 5, 7, 8, 15, 23]
use estimates of the γ values that are significantly larger than
found in this paper.
Nagorny and Drallos
10-1
Thomson - Loeb
empirical fit
Felsch and Pech
Acknowledgments
One of the authors (AVP) would particularly like to
acknowledge numerous important suggestions and critical
reviews of the drafts by A Gallagher. He also would like to
thank B M Jelenković, L C Pitchford, J Broad and B D Paul
for helpful contributions. The other author (ZLP) would like
to thank the JILA Information Center and J. Broad for their
hospitality during the preparation of a bibliography and notes
concerned with electron collision cross sections, excited
state quenching and discharge–surface interactions relevant
to discharges in Ar. We would like to thank C A N Conde and
T H V T Dias for recent references on scintillation-detector
discharges. The present work was supported in part by the
National Institute of Standards and Technology.
10-2
1
R36
102
103
104
E/n (Td)
Figure A1. Fraction of electrons escaping the cathode versus
E/n. The symbols, electron mean energies and references are:
#, 0.2 eV, [122]; , 0.6 eV, [122]; !, 0.9 eV, [100]; ", 1.4 eV,
[147]; , 3? eV, [95]; and ×, 1 eV, [148]. The solid, long-dashed
and dot–dashed curves give the theoretical predictions of the
modes of Thomson and Loeb [31], Felsch and Pech [100] and
Nagorny and Drallos [150], respectively. Each of these models is
evaluated for an average electron injection energy of 0.6 eV. The
short-dashed curve shows our empirical fit to experiment for an
average electron injection energy of 0.6 eV.
•
The Thomson model for electron escape as modified by
Loeb [31] can be written as
Appendix A. Electron backscattering and onset of
ionization
In this appendix we review the treatment of electron
nonequilibrium in Ar near the cathode through the
use of boundary conditions in local-field fluid models
[2, 7, 8, 32, 90]. Based on experimental observations of
drift-tube currents versus E/n and pd, the corrections are
conventionally divided into two parts. One part is the
backscattering to the cathode of electrons emitted from
the surface. A second part is the delay in reaching the
steady-state rates of electron excitation and ionization as
determined from either the spatial dependence of emission
or the pd dependence of the current. While these processes
are accounted for without special effort in Monte Carlo
treatments of electrons leaving absorbing cathodes, they
must be taken into account separately in ‘local field’ or
‘equilibrium’ fluid models utilizing steady-state electron
ionization and transport coefficients.
In the case of electron backscattering to the cathode,
we summarize available experimental data and models
for Ar. Figure A1 shows the results of measurements
[95, 100, 122, 147, 148] and models [31, 100, 149, 150] for
the fraction of electrons escaping the cathode versus E/n
for Ar. The measured escape fractions fes shown decrease
monotonically with increasing electron injection energy.
Some theories [149, 151, 152] find a minimum in the escape
probability as the initial electron energy is increased.
10
fes = 1/(1 + 1v2/4We )
(A1)
where 1v2 is the mean velocity of the injected electrons and
We is the electron drift velocity [19] at the E/n near the
cathode. The results of such a calculation, shown by the
solid curve of figure A1, are well below the experimental
data. Recent theoretical results by Nagorny and Drallos [150]
for E/n from 10 to 1100 Td (dot–dash curve) are barely
distinguishable from the empirical fit given next. It should
be noted that the Thomson–Loeb model works well for many
molecular gases [31, 137].
The short-dash curve of figure A1 shows a fit to
experiment of an empirical expression for the escape fraction
given by
$
[((es /0.6)2 + (E/n)/30] 100
fes = 1 +
[1 + (E/n)/30]
(E/n)
%−1/2
(A2)
where the E/n value is in Td and (es is the energy of the
electrons ejected from the surface in eV. The exponent 2 in
equation (A2) should be reduced to about 1 for (es = 1 eV. In
our models we used (es = 0.6 eV. The long-dash curve shows
that the empirical fit by Felsch and Pech [100] varies too
rapidly with E/n at the lower E/n. Experiments [153, 154]
and Monte Carlo calculations [149, 151, 152, 154–156] show
that results such as plotted in figure A1 and represented by
equation (A2) are independent of electrode separation [122]
only for a limited range of pd. At low pd, fes decreases as
Cold-cathode discharges and breakdown in argon
backscattering sets in [95]. Secondary electron production
causes a growth of current at high pd [95, 98, 13].
The escape fraction data of figure A1 are used in the
analysis of the γi data of figure 3 and of the γeff data
of figure 5. At E/n from 100 to 1000 Td, the escape
fraction is close to unity and the energy of the electrons
emitted from the cathode is not important. At the low
E/n, where photoelectric emission dominates secondary
electron production, uncertainties in the energy of the injected
electrons and the escape fraction caused by unknown surface
conditions may reach 30%, but cannot account for the low
and variable photoelectric yields derived from breakdown
experiments.
Measurements of the delay in the onset of the steadystate ionization have been reported for Ar by Druyvesteyn
and Penning [32] and by Kruithof [88]. An empirical fit to the
former data expressed as the effective value of the electrode
potential difference V0 required before the current begins to
grow exponentially with distance is given by
V0 = 16{1 + [(E/n)/1000]2 }0.5
(A3)
where V0 is in V and E/n is in Td. Monte Carlo techniques
have been used to model the initial nonequilibrium for
electrons in Ar [157]. The application of the singlebeam, energy-balance model of the nonequilibrium motion
of electrons [53, 158] gives similar values of V0 at low E/n,
but much larger values at high E/n.
The addition of this correction to the equations of the
metastable and resonance atom models of this paper at very
high E/n makes the algebra extremely complicated. Because
the effects of this correction on breakdown pd values at
intermediate and low E/n are smaller than the spread in the
experimental data, the correction is omitted throughout the
present calculations.
Appendix B. Coefficients for modelling Ar
discharges
In this appendix we assemble what we believe are the
best available analytical expressions for cross sections, rate
coefficients and electron yields for modelling electron, ion,
fast atom and photon behaviour in Ar under conditions
appropriate to electrical breakdown and to low- and
moderate-current density discharges. The data are given
in some detail because of their use in the models of this
and a number of other papers [27, 53, 54, 90, 126] and their
potential for use in future research. Analytical formulations
of model parameters often result in a considerable reduction
of computer time. In this section the αx /n values are in m2 ,
Qx values are in 10−20 m2 , the particle energies (x and kTx
are in eV and the E/n are in Td unless otherwise noted.
B.1. Electron collisions with Ar
The empirical electron ionization coefficient expression used
in this paper has been modified from that of [90] so as to be in
better agreement with the experimental values [88, 107, 144]
at very low E/n. At high E/n the expression is adjusted to
be consistent with the results of Jelenak et al [159], Božin
et al [160], unpublished calculations of Nanbu and Konoko
[23] and unpublished experimental results from our group,
rather than the higher experimental values of Kruithof [88].
An empirical fit is
αei /n = 1.1 × 10−22 exp[−72/(E/n)]
+5.5 × 10−21 exp[−187/(E/n)]
+3.2 × 10−20 exp[−700/(E/n)]
−1.5 × 10−20 exp[−10 000/(E/n)].
(B1)
This and other fits in this paper are not linear least-squares
fits, but are visual fits to data that are generally good to better
than 10% at all E/n for which data exist. We have adjusted
equation (B1) for an even closer fit to experiment [88] at E/n
from 100 to 1000 Td, where exp(αei d) is largest and has the
greatest effect on the computed γi . Note that our expression
is consistent with that proposed many years ago by Ward
[161] for 100 < E/n < 1000 Td, but differs significantly at
higher E/n. On the basis of the calculations of Puech and
Torchin [17] we conclude that the contribution of associative
ionization to αei is less than about 10% at E/n ' 20 Td and
neglect its role in our model.
The coefficients for electron excitation of Ar to the
metastable state are the sum of the calculated excitation
coefficients to the two metastable levels from Puech and
Torchin [17]. These values are about twice those measured
by Tachibana [16]. We have adopted the theoretical values
because of the overall consistency of the calculated data with
swarm experiments [17, 19]. A fit to the spatial metastableexcitation coefficient data is
αem /n = 2 × 10−21 exp[−16/(E/n)]
+2.3 × 10−21 exp[−80/(E/n)]
−1.5 × 10−21 exp[−2000/(E/n)].
(B2)
The coefficients for electron excitation of Ar to the
resonance state are also based on calculations of the total
rate of excitation of the two lowest radiating states of Ar
by Puech and Torchin [17]. The empirical fit to the spatial
resonance-state excitation coefficient data is
αer /n = 2 × 10−21 exp[−20/(E/n)]
+4 × 10−21 exp[−100/(E/n)]
−2 × 10−21 exp[−2000/(E/n)].
(B3)
An empirical fit to the mean energy of electrons in Ar
from our solutions of the Boltzmann equation [25] is
1(2 = 2.5(E/n)0.5 {1 + [(E/n)/5]2 }−0.2
×{1 + [(E/n)/1000]2 }0.7 .
(B4)
Note that for 20 < E/n < 100 Td the sum of
the excitation coefficients for the resonance and metastable
states times their respective excitation thresholds in eV is
numerically within 20% of the value of E/n, as expected
if all of the electron energy is used for excitation of the Ar.
This agreement extends to higher E/n =10 kTd when the
ionization coefficient times the sum of the ionization potential
and the electron mean energy is added to the energy input to
excitation.
The ‘single-beam’ models of electron motion in Ar of
[53] and [54] utilize the elastic momentum transfer cross
section Qmel , the total excitation cross section Qex and the
ionization cross section Qion . Also, one uses the effective
R37
A V Phelps and Z Lj Petrović
momentum transfer cross section Qmeff , where Qmeff =
Qmel + Qex + Qion in two-term solutions to the electron
Boltzmann code for Ar [3, 18, 19, 21, 22, 53].
The cross section for elastic momentum transfer
collisions of electrons with Ar Qmel has been presented
graphically in several recent papers [3, 17–19, 21–25]. Here
we express the results tabulated in [25] as
2 3.3
Qmel (() = ABS{6/[1 + ((/0.1) + ((/0.6) ]
−1.1 × ( 1.4 /[1 + ((/15)1.2 ]/[1 + ((/5.5)2.5
+((/60)4.1 ]0.5 } + 0.05/(1 + (/10)2
+0.01 × ( 3 /[1 + ((/12)6 ].
(B5)
This cross section is intended to be treated as isotropic
scattering [25] when used to determine a differential cross
section.
The sum of the cross section for electronic excitation in
collisions of electrons with Ar Qex obtained by fitting the
results tabulated in [25] is approximated by
1.1
2.8
5.5
Qex (() = 0.034(( − 11.5) [1 + ((/15) ]/[1 + ((/23) ]
+0.023(( − 11.5)/[1 + ((/80)]1.9 .
(B6)
In the single-beam model [53, 54] this cross section is
multiplied by a representative excitation energy of 11.5 eV
and used as the excitation ‘energy-loss’ function. This cross
section is intended to be treated as forward scattering [25]
when used to determine a differential cross section.
The cross section for electron impact ionization in
collisions of electrons with Ar, Qion , obtained by fitting the
results tabulated in [25] is approximated by
Qion (() = 970(( − 15.8)/(70 + ()2
+0.06(( − 15.8)2 exp(−(/9).
(B7)
This cross section is intended to be treated as forward
scattering [25] when used to determine differential scattering
cross sections.
B.2. Ion energy distribution and drift velocity
The ion energy distribution is a one-dimensional Maxwellian
with a temperature T+ . An empirical fit to the kT+ versus E/n
values of figure 8 of [26] is
KT+ = +1.9[(E/n)/1000]1.1 + 0.026.
(B8)
Note that here kT+ is twice the ion mean energy [26].
The Ar+ drift velocity is approximated closely by [26]
W+ = 4(E/n)/{1 + [0.007(E/n)]1.5 }0.33
(B9)
electron yield per fast atom. When numerically averaged
over the one-dimensional Maxwellian energy distribution of
the Ar+ the electron yield per ion is approximated by
1γic 2 = 0.07 + 1 × 10−5 (kT+ )1.2
× exp(−500/kT+ )/[1 + (kT+ /100 000)0.7 ].
(B11)
Note that this expression can be obtained from equation (B10)
by replacing the threshold factor ((i − 500)1.2 by
(kT+ )1.2 exp(−500/kT+ ), replacing (i by kT+ elsewhere and
by modifying slightly the high-energy behaviour. In all
cases examined, we find that this procedure provides a useful
first guess as to the function fitting the average over the
Maxwellian.
The approximation to the yield of electrons per fast atom
γac for clean, annealed electrodes is shown by the dashed
curve in figure 1. For atom energies above the apparent
threshold at 500 eV this curve is described by
γac = 1 × 10−5 ((a − 500)1.2 /[1 + ((i /70 000)0.7 ]
(B12)
where the yield is zero for (a below 500. As discussed in
section 5, we assume that the energy distribution of the fast
Ar atoms is the same as that for the Ar+ ions. When this yield
is averaged over the Maxwellian energy distribution the result
is
1γac 2 = 1 × 10−5 (kT+ )1.2
× exp(−500/kT+ )/[1 + (kT+ /100 000)0.7 ].
(B13)
For clean Cu surfaces that have been heavily sputtered,
as in the experiments of Kruithof, [88], we will approximate
the electron yield per ion γic at low energies by a typical value
from the open points of figure 3 and at high energies by a term
that approaches Oechsner’s value at 1 keV [73]. Thus, we
suggest
γics = 0.02 + 5 × 10−5 ((i − 300)
(B14)
where the contribution to the second term is zero for (i below
300. An even better approximation would have the constant
term decrease to as low as 0.01 with an increase in sputtering.
Unfortunately, the dependence of this decrease on ion energy
and the initial surface condition is unknown. These results
have been obtained when the ion and electron mean free
paths are large compared to the expected surface roughness.
The effects of surface roughness on the ‘effective secondary
emission coefficient’ have been examined when the ion mean
free path is small compared to the scale of the roughness
[162]. For fast Ar atoms we suggest only the second term of
equation (B14).
where W+ is in m s−1 .
B.4. Ion- and atom-induced electron yields for dirty
metals
B.3. Ion- and atom-induced electron yields for clean
metals
For dirty surfaces our approximation to the electron yield per
fast Ar+ ion γid shown by the upper solid curve in figure 2 is
given by
For clean surfaces we will approximate the electron yield per
ion γic by the solid curve in figure 1. This curve is given by
γic = 0.07+1×10−5 ((i −500)1.2 /[1+((i /70 000)0.7 ] (B10)
where the contribution to the second term is zero for (i below
500. The second term in this relation is also equal to the
R38
γid = 0.002(i /[1 + ((i /30)1.5 ]
+1.05 × 10−4 ((i − 80)1.2 /[1 + ((i /8000)]1.5
(B15)
where the second term is zero for ion energies (i below
80 eV. As pointed out above, the continued decrease in γi
with decreasing ion energy at energies below those shown in
Cold-cathode discharges and breakdown in argon
figure 2 is important for our fit to the γeff data of figure 5.
In the energy range from 10 to 500 eV, these γi values are
similar to those adopted by Neu [61].
When averaged over the one-dimensional Maxwellian
velocity distribution of the fast Ar atoms, the yield is
approximated by
1γid 2 = 0.06[kT+ − 30 exp(30/kT+ )$(0, 30/kT+ )]/kT+
+1 × 10−4 (kT+ )1.2 exp(−80/kT+ )
(B16)
×[1 + (kT+ /6000)1.5 ]
where $[n, x] is the incomplete gamma function [163].
Note that for the first term of equation (B16) we have
used the exact form of the integral obtained by averaging
over a Maxwellian, rather than the procedure discussed in
connection with equation (B11). Also, note that this first
term is set to zero for calculating the short-dashed curves of
figures 4, 5 and 11.
For dirty surfaces the approximation to the electron yield
per fast Ar atom γad shown by the lower solid curve in figure 2
is given by
γad = 1 × 10−4 ((a − 90)1.2 /[1 + ((a /8000)1.5 ]
+7.0 × 10−5 ((a − 32)1.2 /[1 + ((a /2800)1.5 ]
(B17)
where the first and second terms are zero for (a below 90 and
32 eV, respectively. The second term in equation (B17) is
basically a shift of the first term to lower energies and lower
magnitude so as to represent the structure found by Amme
[50]. In the energy range from 10 to 500 eV, these γa values
are similar to those adopted by Neu [61].
When averaged over the one-dimensional Maxwellian
velocity distribution of the fast Ar atoms the yield is
1γad 2 = 1 × 10−4 (kT+ )1.2
× exp(−90/kT+ )/[1 + (kT+ /6000)1.5 ]
+7.0 × 10−5 (kT+ )1.2
× exp(−32/kT+ )[1 + (kT+ /2100)1.5 ].
(B18)
B.5. Fast ion–atom reaction coefficients
The empirical approximation for symmetric charge transfer
cross section as obtained by averaging the cross section from
table 7 and figure 7 of [26] over the one-dimensional ion
energy distribution and transforming from a kT+ energy scale
to an E/n scale is
αct /n = 1.3 × 10
−18
−0.117
(E/n)
.
(B19)
The following spatial ionization and excitation coefficients for Ar + + Ar collisions are obtained by averaging the
cross sections from [20] and [26] over the one-dimensional
ion energy distribution and transforming from a kT+ energy
scale to an E/n scale. Thus, the ionization coefficient for the
Ar + + Ar → 2Ar + + e reaction is approximated by
αii /n = 8.2 × 10−22 exp[−19 900/(E/n)]
(B20)
+7.1 × 10−21 exp[−46 000/(E/n)].
The empirical approximation for the spatial excitation
coefficient for Ar + + Ar → Ar(1 P1 ) + Ar + is
αir /n = 1.07 × 10−22 exp[−41 700/(E/n)].
(B21)
The empirical spatial excitation coefficient for Ar + +
Ar → Ar(3 P2 ) + Ar + is
αim /n = 3.5 × 10−21 exp[−18 000/(E/n)]
(B22)
+5.0 × 10−21 exp[−50 000/(E/n)].
B.6. Fast atom–atom reaction coefficients
Our empirical expression for the isotropic scattering
approximation to the elastic viscosity collision cross section
from [20] is
−0.23
−0.28
−2
2
Qaa = 16(rel
/[1 + (rel
] + 11.3(rel
/[1 + (rel
].
(B23)
This cross section results from a more detailed analysis of
viscosity and beam experiments than that of [27]. It is in
good agreement with that used in a recent model of sputtering
experiments [6].
The corresponding approximation for the spatial loss
coefficient for fast Ar atoms as the result of elastic viscosity
collisions is obtained by replacing (rel by kT+ /2, i.e. assuming
that the atoms have the same energy as the ions from which
they were formed. We then transform from a kT+ scale to
an E/n scale using the high-energy limit of equation (B8) to
obtain
αaa /n = 1.62 × 10−19 (E/n)−0.25 /[1 + 1.1(E/n)2.2 ]
+1.15 × 10−19 (E/n)−0.31 /[1 + 0.9(E/n)−2.2 ]. (B24)
In this equation only, E/n are in kTd. Because of
the simplified energy transformation, this approximation is
intended for use at Ar atom energies well above thermal.
The following spatial ionization and excitation coefficient for collisions of fast Ar with thermal Ar are obtained by
averaging the cross section from [26] and [20] over the onedimensional ion energy distribution and transforming from a
kT+ energy scale to an E/n scale. The ionization coefficient
for the Ar + Ar → Ar + + Ar + e reaction is
αai /n = 2.1 × 10−21 exp[−25 000/(E/n)]
+4.8 × 10−21 exp[−40 800/(E/n)].
(B25)
The empirical approximation for the spatial resonance
state excitation coefficient for the Ar + Ar → Ar(1 P1 ) + Ar
reaction is
αar /n = 2.7 × 10−21 exp[−32 800/(E/n)].
(B26)
The empirical approximation for the spatial metastable
excitation coefficient for the Ar+Ar → Ar(3 P2 )+Ar reaction
is
αam /n = 2.1 × 10−21 exp[−25 000/(E/n)]
+4.8 × 10−21 exp[−40 800/(E/n)].
(B27)
B.7. Thermal reaction and transport coefficients
The collisional rate coefficients for Ar at 300 K defined
in [123] and used here are: nDa = 1.7 × 1020 m−1 s−1 ,
nDm = 1.7 × 1020 m−1 s−1 , d = 0.01 m, Ba = 1.2 × 10−21
m3 s−1 , Ca = 1.3 × 10−44 m6 s−1 , Bm = not needed,
Cm = not needed, AI = 8 × 104 s−1 , G = 1.5 × 10−45 n2 +
1.5 × 10−20 n s−1 . Here the gas density n is in m−3 .
B.8. Photon collection at the cathode
We have calculated numerically the fraction of isotropically
emitted photons that reach the cathode assuming that the
photons are produced with a distribution that is radially
uniform but varies as exp(αei z) in the cylindrical volume
of height d and radius r between the electrodes. We assume
R39
A V Phelps and Z Lj Petrović
that the photons are not absorbed or scattered by the gas. The
results can be approximated by
fgeom = 0.5/[1 + (d/r)(1 + αei d)0.55 /0.9].
(B28)
For d/r ( 3 and αei d = 0 this expression is good to
better than 5%. For d/r ( 3 and αei d ( 3 the fit is
good to better than 10%. If the photons are produced by
an electron flux that varies radially as a Bessel function, the
calculated fgeom values increase by 10 to 15% for d/r ( 3
and αei d ( 3. The fgeom factor has not been explicitly shown
in the formulas given in this paper, but approximations to it
have been applied to the calculations of the contributions of
nonresonant photons to the breakdown condition and to the
current growth equations.
References
[1] Belenguer Ph and Boeuf J P 1990 Phys. Rev. A 41 4447
[2] Lieberman M A and Lichtenberg A J 1994 Principles of
Plasma Discharges and Materials Processing
(New York: Wiley) ch 1
[3] Surendra M, Graves D B and Jellum G M 1990 Phys. Rev.
A 41 1112. These authors do not document tests of this
set against experimental ionization and transport swarm
data. One of us (AVP) has constructed a cross section set
from the published graphs and finds that the calculated
ionization coefficients are a factor of two too low and a
factor of three too high compared to experiment at E/n
of 30 and 1000 Td, respectively. One of the authors of
this paper found records of a few calculations that are
about 60% of our values.
[4] Lister G G 1992 J. Phys. D: Appl. Phys. 25 1649
[5] Bogaerts A, van Straaten M and Gijbels R 1995
Spectrochim. Acta B 5 179. This is one of a series of
models of moderate current density discharges in Ar.
[6] Serikov V V and Nanbu K 1996 J. Vac. Sci. Technol. A 14
3108
Serikov V V and Nanbu K 1997 J. Appl. Phys. 82 5948. We
thank Professor Nanbu for providing details of their
viscosity cross sections that resulted in corrections to our
original cross sections for Ar–Ar elastic collisions at low
energies. See [27].
[7] Nagorny V P, Drallos P J and Williamson W Jr 1995
J. Appl. Phys. 77 3645
[8] Meunier J, Belenguer Ph and Boeuf J P 1995 J. Appl. Phys.
78 731
[9] Waymouth J F 1971 Electric Discharge Lamps (Cambridge,
MA: MIT Press) ch 4
[10] Lagushenko R and Maya J 1983 J. Appl. Phys. 54 2255
[11] Pak H and Kushner M J 1990 Appl. Phys. Lett. 57 1619
Sommerer T J, Pak H and Kushner M J 1992 J. Appl. Phys.
72 3374
[12] Boeuf J P and Pitchford L C 1991 IEEE Trans. Plasma Sci.
19 286
Alberta M P, Derouard J, Pitchford L C, Quadoudi N and
Boeuf J P 1994 Phys. Rev. E 50 2239
[13] Conde C A N, Santos M C M, Fátima M, Ferreira A and
Sousa C A 1975 IEEE Trans. Nucl. Sci. 22 104
Santos F P, Dias T H V T, Rachinhas P J B M, Stauffer A D
and Conde C A N 1998 IEEE Trans. Nucl. Sci. 45 176
[14] Bretagne J, Godart J and Puech V 1983 Beitr. Plasmaphys.
23 295
[15] Belasri A, Boeuf J P and Pitchford L C 1993 J. Appl. Phys.
74 1553
[16] Tachibana K 1986 Phys. Rev. A 34 1007. This paper
documents tests of this electron–Ar cross section set
against experimental metastable and ion production
coefficients and transport data. This rather detailed cross
section set (eight levels) is apparently not available on
the internet.
R40
[17] Puech V and Torchin L 1986 J. Phys. D: Appl. Phys. 19
2309. This paper documents tests of this very complete
set of electron–Ar cross sections (≈30 levels) against
experimental ionization, excitation and transport swarm
data. Unfortunately, this highly recommended cross
section set is not available on the internet.
[18] Hayashi M 1990 unpublished. This set of 25 cross sections
for electrons in Ar has been found to be consistent with
swarm experiments. Contact Z Lj Petrović for details on
these tests and regarding the associated compilation of
excited state and transport data.
[19] Pack J L, Voshall R E, Phelps A V and Kline L E 1992
J. Appl. Phys. 71 5363. A tabulation of this very much
simplified cross section set for Ar atoms is available. See
[20].
[20] Phelps A V 1999 unpublished. These data are available via
the JILA web site at
http://jilawww.colorado.edu/ on the ‘Atomic
physics’ page under the subject ‘Collision data’ or at
ftp://jila.colorado.edu/collision data.
The file electron.txt also documents tests of
calculated coefficients against experimental ionization
and transport data from swarm experiments.
Other files list recommended heavy-particle collision
data.
[21] Morgan W L 1992 Plasma Chem. Plasma Proc. 12 449
Morgan W L 1992 Plasma Chem. Plasma Proc. 12 477.
This simplified cross section set is available for Ar and
other gases from The Siglo Data Base, CPAT and
Kinema Software at http://www.sni.net/siglo.
[22] Fiala A, Pitchford L C and Boeuf J P 1994 Phys. Rev. E 49
5607. According to the web site in [21], the cross section
set for electrons in Ar used in this paper has been
revised (1998) to be consistent with electron swarm
experiments.
[23] Nanbu K and Kageyama J 1996 Vacuum 47 1031. Electron
drift velocities and ionization coefficients calculated
using this cross section set agree with experiment. See
Nanbu K and Kondo S 1997 Japan. J. Appl. Phys. 36
4808 and Nanbu K 1998 private communication. The
assumed γi values are significantly larger than the values
expected for sputtered Cu from the data shown in
figure 3.
[24] Vasenkov A V 1998 Phys. Rev. E 57 2212. This cross
section set is designed for application to models of high
energy electron beams, rather than electric discharges.
Tests of calculated quantities such as the mean energy
per ion pair and electron range are presented. No
tests of electron transport and ionization coefficients are
given.
[25] Phelps A V 1997 Bull. Am. Phys. Soc. 42 1721. Here it is
strongly advocated that each publication and internet site
that makes electron cross section sets available for
modelling gas discharges provide documentation of the
tests that have been made of the consistency of the cross
section set with simple swarm experiments, such a spatial
(Townsend) ionization coefficients, drift velocity and the
ratio of the diffusion to mobility coefficients. Such a
procedure would reduce the number of inconsistent sets
in circulation and reduce the tendency of some modellers
to attribute discrepancies between complex models and
experiment to errors in the electron cross sections.
[26] Phelps A V 1991 J. Phys. Chem. Ref. Data 20 557. It should
be noted that in the absence of more detailed data the
inelastic cross sections in this and the associated series of
papers should be used with a differential scattering cross
section sharply peaked in the forward direction, as is
typical of high energy collisions. An isotropic inelastic
scattering model for the inelastic collisions leads to
excessive angular scattering and much larger
contribution to the momentum transfer than was
intended.
Cold-cathode discharges and breakdown in argon
[27] Phelps A V 1994 J. Appl. Phys. 76 747. In addition to
citations [3]–[10] of this reference, numerous recent
papers have misused and/or misstated the concepts and
published data concerned with elastic scattering and its
symmetric-charge-transfer component in collisions of
Ar+ with Ar. For example, very recently Zhong X X, Wu
J D, Wu C Z and Li F M 1998 J. Appl. Phys. 83 5069
have overestimated the isotropic component of the ion
scattering at 100 eV by more than an order of magnitude.
[28] Holstein T 1947 Phys. Rev. 72 1212
Holstein T 1951 Phys. Rev. 83 1159
[29] Payne M G, Talmage J E, Hurst G S and Wagner E B 1974
Phys. Rev. A 9 1050
[30] Townsend J S 1915 Electricity in Gases (Oxford:
Clarendon) p 313
[31] Loeb L B 1939 Fundamental Processes of Electrical
Discharge in Gases (New York: Wiley) p 313
[32] Druyvesteyn M J and Penning F M 1940 Rev. Mod. Phys.
12 87. This paper also discusses surface preparation
procedures used in gas discharge experiments.
[33] Little P F 1956 Handbuch der Physik vol 21, ed S Flügge
(Berlin: Springer) p 574
[34] Raether H 1964 Electron Avalanches and Breakdown in
Gases (London: Butterworth)
[35] Dutton J 1978 Electrical Breakdown of Gases ed J M Meek
and J D Graggs (Chinchester: Wiley) ch 3
[36] Raizer Yu P 1986 Tepolfiz. Vys. Temp. 24 984 (Engl. Transl.
1986 High Temp. (USSR) 24 744)
[37] Kaminsky M 1965 Atomic and Ionic Impact Phenomena on
Metal Surfaces (New York: Academic) ch 12 and 14
[38] Abroyan I A, Eremeev M A and Petrov N N 1967 Usp. Fiz.
Nauk 92 105 (Engl. Transl. 1967 Sov. Phys. Usp. 10 332)
[39] Carter G and Colligon J S 1968 Ion Bombardment of Solids
(New York: Elsevier) ch 3
[40] Krebs K H 1968 Fortschr. Phys. 16 419
Krebs K H 1983 Vacuum 33 555
[41] Weston G F 1968 Cold Cathode Discharge Tubes (London:
ILIFFE) ch 3
[42] Hasselkamp H D 1992 Particle Induced Electron Emission
II ed G Höhler (Berlin: Springer) p 1
[43] Varga P and Winter H 1992 Particle Induced Electron
Emission II ed G Höhler (Berlin: Springer) p 149
[44] Baragiola R A 1994 Low Energy Ion-Surface Interactions
ed J W Rabalais (New York: Wiley) ch 4
[45] Rostagni A 1934 Z. Phys. 88 55. Surface treatment of Cu
not stated. Background pressure 10−5 Torr.
[46] Arifov U A, Rakhimov R R and Dzhurakulov Kh 1962
Dokl. Akad. Nauk SSSR 143 309 (Engl. Transl. 1962 Sov.
Phys.–Dokl. 7 209). Prolonged heating of Mo at
2100–2200 K. Measurements at 1100 K following brief
heating to 2000 K at 10−7 Torr.
[47] Medved D B, Mahadevan P and Layton J K 1963 Phys. Rev.
129 2086
Mahadevan P, Layton J K and Medved D B 1963 Phys. Rev.
129 79. Yield extrapolated to immediately after flashing
Mo to 2000 K at 10−8 Torr.
[48] Hayden H C and Utterbach N G 1964 Phys. Rev. 135
A1575. Surface (Au) washed with detergent.
Background pressure 10−6 to 10−5 Torr.
[49] Haugsjaa P O, McIlwain J F and Amme R C 1968 J. Chem.
Phys. 48 527. Surface (Au) washed with detergent.
Background pressure ∼ 10−6 Torr.
[50] Amme R C 1969 J. Chem. Phys. 50 1891. Unspecified
surface and treatment. Background pressure 10−6 to
10−5 Torr.
[51] Kadota K and Kaneko Y 1974 Japan. J. Appl. Phys. 13
1554. Polished, washed and outgassed at low
temperatures. The surface was the first dynode of an
electron multiplier.
[52] See, for example, Armour D G, Valisadeh H,
Soliman F A H and Carter G 1984 Vacuum 34 295
Rickards J 1984 Vacuum 34 559
[53] Phelps A V and Jelenković B M 1988 Phys. Rev. A 38
2975. Cathode (Cu) electropolished and chemically
cleaned. Mild bakeout of system gave rate of pressure
rise of 10−4 Torr min−1 .
[54] Scott D A and Phelps A V 1991 Phys. Rev. A 43 3043
Scott D A and Phelps A V 1992 Phys. Rev. A 45 4198
[55] Mason R S and Alott R M 1994 J. Phys. D: Appl. Phys. 27
2372
[56] Townsend J S and Llewellyn Jones F 1933 Phil. Mag. 15
282. See Townsend J S and Yarnold G D 1934 Phil. Mag.
17 594 for measurements in He.
[57] McClure G W 1959 J. Electron. Control 7 439
[58] Ul’yanov K N 1970 Zh. Tech. Fiz. 40 2138 (Engl. Transl.
1971 Sov. Phys.–Tech. Phys. 15 1667)
Ul’yanov K N and Tskhai A B 1981 Tepolfiz. Vys. Temp. 19
41 (Engl. Transl. 1981 High Temp. (USSR) 19 32)
[59] Blasberg H A M and de Hoog F J 1971 Physica 54 468
[60] Lauer E J, Yu S S and Cox D M 1981 Phys. Rev. A 23
2250
[61] Neu H 1959 Z. Phys. 154 423
Neu H 1959 Z. Phys. 155 77
[62] Warren R 1955 Phys. Rev. 98 1650
Warren R 1955 Phys. Rev. 98 1658
[63] Rózsa K, Gallagher A and Donkó Z 1995 Phys. Rev. E 52
913
[64] Phelps A V, Petrović Z Lj and Jelenković B M 1993 Phys.
Rev. E 47 2825
[65] Cleaning techniques for various elements have been
summarized by Musket R G, McLean W, Colmenares C
A, Makowiecki D M and Siekhaus W J 1982 Appl. Surf.
Sci. 10 143; Grunze M, Ruppender H and Elshazly O
1988 J. Vac. Sci. Technol. A 6 1266. We thank a referee
for pointing out these references.
[66] Ferrón J, Alonso E V, Baragiola R A and Oliva-Florio A
1981 J. Phys. D: Appl. Phys. 14 1707. Cleaned Mo target
by ion bombardment until yield was independent of time.
Apparently no annealing was used. Measurements at
10−10 Torr.
[67] Güntherschulze A 1930 Z. Phys. 62 600. Probably these
data should be characterized as obtained using a
discharge technique, rather than a beam technique.
However, we have not been able to determine the
appropriate E/n values and so have adopted the author’s
assignment of the incident ion energy. Copper targets
cleaned by electron and ion bombardment.
[68] Hagstrum H D 1954 Phys. Rev. 96 325
Hagstrum H D 1954 Phys. Rev. 96 336. The tungsten was
flashed to 2200 K at 10−11 Torr.
[69] Parker J H Jr 1954 Phys. Rev. 93 1148. The system was
baked at 670 K and the targets were flashed to 1670 K
for Pt and 1670–2270 K for Ta. The measurements were
made at 2 × 10−5 Torr.
[70] Vance D W 1967 Phys. Rev. 164 372. The Mo target was
flashed to 2000 K for 15 s and measurements were made
at a pressure of 4 × 10−9 Torr. The higher results shown
in figure 1 are for Mo treated with O2 to remove carbon.
[71] Winter H, Aumayr F and Lakits G 1991 Nucl. Instrum.
Methods B 58 301
Töglhofer K, Aumayr F and Winter H P 1993 Surf. Sci. 281
143. Gold target cleaned by ion sputtering in vacuum.
Measurements made at 10−3 Torr. Lakits G, Aumayr F
and Winter H 1989 Rev. Sci. Instrum. 60 3151
cite evidence that these Au targets are ‘atomically
clean’.
[72] Alonso E V, Alurralde M A and Baragiola R A 1986 Surf.
Sci. 166 L155. For Xe+ incident on sputtered Au these
authors find an exponential decrease in electron yield
with decreasing energy at 0.5 to 3 keV. These yields are
much smaller than those of [76].
[73] Oechsner H 1978 Phys. Rev. B 17 1052. Copper target
cleaned by sputtering at 5 mA cm−2 and measurements
made at 10−5 Torr.
R41
A V Phelps and Z Lj Petrović
[74] Magnuson G D and Carlston C E 1963 Phys. Rev. 129 2409.
Targets cleaned by ion bombardment and measurements
made quickly at a background pressure of 5 × 10−8 .
[75] Szapiro B and Rocca J J 1989 J. Appl. Phys. 65 3713.
Commercial purity samples were polished and then
operated as the cathode of a glow discharge. The system
was then pumped to 10−6 Torr for yield measurements.
[76] Hagstrum H 1953 Phys. Rev. 89 244. The Mo target was
flashed periodically to 1750 K for minutes and
measurements were made at a He pressure of 10−7 Torr.
[77] Kirchhoff J F, Gay T J and Hale E B 1993 Ionization of
Solids by Heavy Particles ed R A Baragiola (New York:
Plenum) p 283
[78] Böhm C and Perrin J 1993 Rev. Sci. Instrum. 64 31. Targets
cleaned by operating discharge in H2 –Ar mixture.
[79] Hagstrum H D 1960 J. Appl. Phys. 31 897. The treatment of
the contaminated tungsten was not given. The
background pressure was at 10−9 Torr.
[80] Berry H W 1948 Phys. Rev. 74 848. Surface (Ta)
preparation and pressures not given.
[81] Ghosh S N and Sheridan W F 1957 J. Chem. Phys. 26 480
[82] Hofer W 1983 unpublished. Cited in figure 3.3 of [43].
Cleaning procedure not given, but presumably similar to
[71].
[83] Rundel R D and Stebbings R F 1972 Case Studies in Atomic
Collision Physics II ed E W McDaniel and M R C
McDowell (Amsterdam: North-Holland) ch 8. This
paper reviews surface preparation procedures.
[84] Schohl S, Klar D, Kraft T, Meijer H A J, Ruf M-W,
Schmitz U, Smith S J and Hotop H 1991 Z. Phys. D 21
25. Au and Mo surfaces prepared by evaporation on to
polished stainless steel and then chemically cleaned.
Measurements were made at 300 to 360 K at a
background pressure of 10−7 Torr.
[85] Schall H, Beckert Th, Alvariño J M, Vecchiocattivi F and
Kempter V 1981 Nuovo. Cimento B 63 378
[86] Alvariño J M, Hepp C, Kreiensen M, Staudenmayer B,
Vecchiocattivi F and Kempter V 1984 J. Chem. Phys. 80
765
[87] Neynaber R H and Magnuson G D 1977 J. Chem. Phys. 67
430
[88] Kruithof A A 1940 Physica 7 519. Tubes baked with H2 at
670 K and then under a vacuum at 740 K. These
thermally grounded, large area Cu cathodes were heavily
sputtered (∼1019 ions per cycle) with an Ne discharge at
2 mA cm−2 , but at unspecified pressure and voltage.
Earlier results with Ar in which the Cu cathode of the
same experimental tube was cleaned by sputtering and
was not heated to high temperatures also showed low γi
values (0.009 to 0.04). The breakdown data were
obtained with d = 1 cm. See Kruithof A A and
Penning F M 1936 Physica 3 515.
[89] Llewellyn-Jones F 1953 Rep. Prog. Phys. 16 216
[90] Petrović Z Lj and Phelps A V 1997 Phys. Rev. E 56 5920.
Cathode (Cu) chemically cleaned, electropolished and
gold-plated. Mild bakeout of system gave rate of
pressure rise of better than 10−4 Torr h−1 . The voltages
cited in the present paper are the discharge maintenance
voltages extrapolated to zero current. In our present
notation the γi values of this reference should be
designated as γeff values.
[91] Phelps A V 1998 unpublished. Solution of the appropriate
continuity equations shows that when a change in E/n
causes αei − αii to increase through zero the curvature of
a plot of i(nd)/i(0) versus electrode separation changes
from positive to negative. On the other hand the
curvature of equation (1) is always positive. The change
in curvature is calculated to occur for low-current
discharges in Ar and He at E/n of the order of 50 kTd.
Here we have neglected the effects of changes in γi ,
ionization by fast atoms and electrons backscattered
from the anode.
R42
[92] Hornbeck J A 1951 Phys. Rev. 83 374. Here we are only
interested in the technique, because the cathodes were
coated with metal oxides.
[93] Varney R N 1954 Phys. Rev. 93 1156. The tube baked at
670 K and the Mo electrodes repeatedly ‘glowed by
induction heating’ and sputtered at high currents. The
residual pressure was 4 × 10−9 Torr.
[94] Engstrom R W and Huxford W S 1940 Phys. Rev. 58
67
[95] Molnar J P 1951 Phys. Rev. 83 940. Repeated baking and
outgassing of the cathode with a glow discharge. The
tubes were filled and sealed off after flashing a barium
getter.
[96] Brunker S A and Haydon S C 1983 J. Physique Coll. C7
55
Ernest A D, Haydon S C and Fewell M P 1994 J. Phys. D:
Appl. Phys. 27 2531
[97] Policarpo A J P L 1981 Phys. Scr. 23 539
Policarpo A J P L 1982 Nucl. Instrum. Methods A 196
53
[98] Breskin A, Buzulutskov A, Chechik R, Di Mauro A,
Nappi E, Paic G and Piuz F 1995 Nucl. Instrum. Methods
A 367 342
[99] Di Mauro A, Nappi E, Posa F, Breskin A, Buzulutskov A,
Chechik R, Biagi S F, Paic G and Piuz F 1996 Nucl.
Instrum. Methods A 371 137. These authors say their
Monte Carlo calculations do not include photoelectron
feedback. We therefore note that the MC results shown in
their figure 4 are inconsistent with the experimental
results of [122] for E/p between 2 and 5 V cm−1 Torr,
where photoelectron feedback is expected to be small.
[100] Felsch D and Pech P 1973 Beitr. Plasmaphys. 13 197
Felsch D and Pech P 1973 Beitr. Plasmaphys. 13 253.
Electrodes were inductively heated to 1670 K for Mo and
1270 K for Ni and Fe. The cathode was then bombarded
with ions from a glow discharge.
[101] Jurriaanse T, Penning F M and Moubis J H A 1946 Philips
Res. Rep. 1 225. These small, thermally isolated Mo
cathodes were sputtered at high enough currents to raise
the cathode temperature to above 1500 K.
[102] Hornbeck J A and Molnar J P 1951 Phys. Rev. 84 621
[103] Kebarle P, Haynes R M and Searles S K 1967 J. Chem.
Phys. 47 1684
[104] Gallagher A C 1996 private communication
[105] Changes in surfaces produced by sputtering are reviewed by
Navinšek B 1976 Prog. Surf. Sci. 7 49
[106] Schade R 1938 Z. Phys. 108 353. Tube baked at 770 K and
Ni electrodes heated to bright red. Electrode spacing
from 0.3 to 5 cm.
[107] Kachickas G A and Fisher L H 1953 Phys. Rev. 91 775.
Electrode spacings were 0.3 to 3 cm.
[108] Menes M 1959 Phys. Rev. 116 481. System baked at 620 K
and residual pressure of 3 × 10−9 . Probable purity of gas
supply is 0.9999. Electrode spacing 0.3m to 1 cm.
[109] Golden D E and Fisher L H 1961 Phys. Rev. 123 1079.
Electrodes (Ni) exposed to (a) a 10−6 A discharge for
several days and (b) ‘allowing an intermittent spark
discharge to pass between the electrodes for about 50
hours’. Measured purity of gas supply was 0.999. The
electrode separation was 0.5 to 5 cm and their diameter
was 20 cm. The observed large increase in breakdown
voltage as the gas aged may have been caused by the
build-up of impurities that quenched the long-lived lower
molecular state of Ar2 [134].
[110] Heylen A E D 1968 J. Phys. D: Appl. Phys. 1 179. Baking
system at 470 K produced pressure of 10−7 Torr. Typical
electrode spacings at breakdown were 0.5 cm. Electrodes
exposed to ethane from mixture experiments.
[111] Pace J D and Parker A B 1973 J. Phys. D: Appl. Phys. 6
1525. Electropolished and chemically cleaned stainless
steel electrodes separated by 0.3 to 2 cm. System baked
to give 10−8 Torr.
Cold-cathode discharges and breakdown in argon
[112] Bhasavanich D and Parker A B 1977 Proc. R. Soc. A 358
385. Electropolished and chemically cleaned stainless
steel electrodes separated by 0.2 to 3 cm. System baked
to give 10−9 Torr. Electrodes conditioned with glow
discharge to give reproducible results.
[113] Stefanović I and Petrović Z Lj 1997 Japan. J. Appl. Phys.
36 4728
[114] Auday G, Guillot Ph, Galy J and Brunet H 1998 J. Appl.
Phys. 83 5917. Tube heated to 240 ◦ C for 24 h. Electrode
diameter 5 cm, d = 0.2 to 1 cm.
[115] Weissler G L 1956 Handbuch der Physik vol 21,
ed S Flügge (Berlin: Springer) p 304
[116] Cairns R B and Samson J A R 1966 J. Opt. Soc. Am. 56
1568
[117] Kenty C 1933 Phys. Rev. 44 891
[118] Hinterreger H E and Watanabe K 1953 J. Opt. Soc. Am. 43
604
[119] Krolikowski W F and Spicer W E 1969 Phys. Rev. 185 882.
Very similar photoelectric yields have been obtained for
Ni by Blodgett A J Jr and Spicer W E 1966 Phys. Rev.
146 390.
[120] Séguinot J, Charpak G, Giomataris Y, Peskov V,
Tischhauser J and Ypsilantis T 1990 Nucl. Instrum.
Methods A 297 133
[121] Travier C 1994 Nucl. Instrum. Methods A 340 26
[122] Theobald J K 1953 J. Appl. Phys. 24 123
[123] Phelps A V 1960 Phys. Rev. 117 619. In equation (20) of
this reference the symbol E should be Cam . In the line
following equation (22), xi = (2i − 1)d/2q and in the
line following equation (23), pi = exp[α(2i − 1)d/2q].
The lower limit to the integral for θm should be
(2m − 1)d/2q.
[124] Nonequilibrium behaviour of the electrons is expected at
breakdown and for low-current discharges for
E/n > 3000 Td or pd < 0.3 Torr cm. The condition for
equilibrium and for the applicability of the local field
model is examined theoretically in many recent papers.
See, for example, the reviews by Kolobov V I and
Godyak V A 1995 IEEE Trans. Plasma Sci. 23 503
Kortshagen U, Busch C and Tsendin L D 1996 Plasma
Sources Sci. Technol. 5 1. The experimental evidence for
nonequilibrium is summarized in appendix A.
[125] Jelenković B M and Phelps A V 1996 Phys. Rev. E 53 1852
and unpublished measurements in Ar and Monte Carlo
calculations in H2 . Deviations from the exponential
growth predicted by a constant spatial ionization
coefficient have been calculated by Hayashi in [152].
[126] Petrović Z Lj and Stojanović V D 1998 J. Vac. Sci. Technol.
A 16 329. The excitation and ionization coefficients for
Ar+ –Ar collisions calculated by these authors using the
MC technique at their lower E/n are significantly larger
than those calculated using the one-dimensional
distribution of the present paper. This difference may be
important for breakdown models at E/n near 3000 Td.
[127] Lawler J E 1985 Phys. Rev. A 32 2977
[128] Rao M V V S, Van Brunt R J and Olthoff J K 1996 Phys.
Rev. E 54 5641
[129] Madey T E and Yates J T Jr 1971 J. Vacuum Sci. Tech. 8 525
Langley R A, Bohdansky J, Eckstein W, Mioduszewski P,
Roth J, Taglauer E, Thomas E W, Verbeek H and
Wilson K L 1984 Nucl. Fusion Special Issue ch 4
[130] Klyarfel’d B N, Guseva L G and Vlasov V V 1968 Zh. Tekh.
Fiz. 38 1288 (Engl. Transl. 1969 Sov. Phys.–Tech. Phys.
13 1056)
[131] We have not found data for the yield of electrons from Ar+2
collisions with either clean or dirty metals. Because
about 14.3 eV of energy is available [164] in a vertical
transition from the lowest vibrational state of the ion to
the repulsive ground state of Ar2 we assume that the
electron yield for clean metals is similar [74] to that for
Kr+ (ionization potential = 14 eV) on clean Mo, i.e.
∼0.05 or about half that for Ar+ .
[132] Weise W L, Smith M W and Miles B M 1969 Atomic
Transition Probabilities vol II (Washington, DC: US
Government) p 192
[133] Castex M C, Morlais M, Spieglemann F and Malrieu J P
1981 J. Chem. Phys. 75 5006
[134] Millet P, Briot A, Brunet H, Dijols H, Galy J and
Salamero Y 1982 J. Phys. B: At. Mol. Phys. 15
2935
[135] The model with eight excited states is subject to
considerable uncertainty because of poorly known rate
coefficients and/or reaction products. This is particularly
true for the 4s 1 P1 resonance state and the 4s 3 P0
metastable state where the potential curves [133] suggest
no collisional coupling to lower levels at 300 K and
experiments show two and three-body collision loss
[165, 166]. Fortunately, only 25% of the electron
excitation is to these levels [17]. Most, but not all, of our
rate coefficients for the lower six levels agree with those
given by Millet et al [134]. We have not examined the
transient solutions for our eight-level or our four-level
model.
[136] Wilkinson P G 1968 Can. J. Phys. 46 315. These results can
be used to show that the vuv absorption by Ar near
106.6 nm scales as p 2 d and approximately as 1/*λ at
wavelength shifts *λ from +0.6 to +30 nm, as expected
for dipole–dipole collisional broadening. Over most of
this range the normalized absorption is about twice that
predicted by dipole–dipole broadening theory used in
our resonance-radiation transport model [28]. At
wavelengths closer to line centre the limited data
approach the theory.
[137] The calculations presented in this paper were carried out
using Mathematica 3.0 © on a 300 MHz personal
computer and required about 25 s per E/n value.
[138] Details are available on request. Send email to
[email protected].
[139] Kolobov V I and Fiala A 1994 Phys. Rev. E 50 3018
[140] Avery L W, House L L and Skumanich A 1969 J. Quant.
Spectrosc. Radiat. Transfer 9 519. From these
calculations for a Doppler broadened profile, spatially
uniform excitation, a finite cylinder with an axial optical
depth at line centre of 100 and ratios of radius to length
of 1:1 and 4:1, roughly 45% and 15% of the radiation is
lost to the side wall. A collision-broadened line profile
appropriate to breakdown in Ar, with its weaker
dependence of photon transmission on distance, will
presumably lead to somewhat larger losses to the side
wall. For comparison, the corresponding losses by
diffusion in the fundamental spatial mode are 37% and
13%.
[141] Revel I, Pitchford L C and Boeuf J P 1997 Proc. 23rd Int.
Conf. on Phenomena in Ionized Gases vol II, ed M C
Bordage and A Gliezes (Toulouse: University Paul
Sabatier) p 56
[142] For example, the determination of the steady-state
breakdown and discharge maintenance pd at a given
E/n would require interpolation between pd values for
MC solutions yielding temporally growing and decaying
discharge currents. Unfortunately, a recent application of
MC techniques for a finite growth rate during pulsed
breakdown appears to neglect the multiple avalanches
required to observe breakdown [167]. The use of Monte
Carlo techniques for the modelling of high current
cathode fall discharges [5, 6] usually does not require
multiple solutions because the nonlinear dependence of
space-charge electric fields on current density means that
one can often reach a steady-state current with an
arbitrary choice of discharge voltage and pd.
[143] In some cases it was not possible to set parameters equal to
zero because of indeterminant results. Therefore the
numerical results reported were calculated with arbitrary
values from 10−4 to 10−10 instead of zero.
R43
A V Phelps and Z Lj Petrović
[144] Specht L T, Lawton S A and DeTemple T A 1980 J. Appl.
Phys. 51 166. These authors suggest their large αei /n
values at low E/n are caused by photon feedback. If we
use equation (B28) and our αph fes values, we find
satisfactory agreement, ±50%, with their apparent αi
values for 5 < E/n < 15 Td. For higher E/n we believe
their αi values are too small because of the neglect of the
initial period of low ionization discussed in Appendix A.
[145] Pustynskii L N and Shumilin V P 1987 Zh. Tekh. Fiz. 57
1699 (Engl. Transl. 1987 Sov. Phys.–Tech. Phys. 32
1016)
[146] Šimko T, Bretagne J and Gousset G 1997 Proc. 23rd Int.
Conf. on Phenomena in Ionized Gases vol IV, ed M C
Bordage and A Gliezes (Toulouse: University Paul
Sabatier) p 184. See also Šimko T, Donkó Z and Rózsa K
1997 Proc. 23rd Int. Conf. on Phenomena in Ionized
Gases vol II, ed M C Bordage and A Gliezes (Toulouse:
University Paul Sabatier) p 64
[147] Dahlquist J A 1962 Phys. Rev. 128 1988
[148] Smejtek P, Silver M, Dy K S and Onn D G 1973 J. Chem.
Phys. 59 1374. These experiments extend the
measurements to very low E/n (0.01 Td) and up to
liquid Ar densities.
[149] Pareathumby S and Segur P 1979 Lett. Nuovo Cimento 26
243
[150] Nagorny V P and Drallos P J 1997 Plasma Sources Sci.
Technol. 6 212
[151] Thomas R W L and Thomas W R L 1969 J. Phys. B: At.
Mol. Phys. 2 562
[152] Hayashi M 1976 Proc. 4th IEE Int. Conf. on Gas Discharges
(London: Institution of Electrical Engineering) p 195
Hayashi M 1982 J. Phys. D: Appl. Phys. 15 1411
R44
[153] Orient O J 1965 Can. J. Phys. 43 422
[154] Békiarian A 1968 J. Physique 29 434
[155] Burch D S and Whealton J H 1977 J. Appl. Phys. 48 2213
Whealton J H, Burch D S and Phelps A V 1977 Phys. Rev.
A 15 1685
[156] Kuntz P J and Schmidt W F 1982 J. Chem. Phys. 76 1136
[157] Sakai Y, Tagashira H and Sakamoto S 1972 J. Phys. B: At.
Mol. Phys. 5 1010
[158] Phelps A V, Jelenković B M and Pitchford L C 1987 Phys.
Rev. A 36 5327
[159] Jelenak Z M, Velikić Z B, Božin J V, Petrović Z Lj and
Jelenković B M 1993 Phys. Rev. E 47 3566
[160] Božin J V, Jelenak Z M, Velikić Z V, Belča I D,
Petrović Z Lj and Jelenković B M 1996 Phys. Rev. E 53
4007
[161] Ward A L 1962 J. Appl. Phys. 33 2789
[162] Drallos P J, Nagorny V P, Ryutov D D and Williamson W Jr
1997 Comments Plasma Phys. Control. Fusion 18 37
[163] Davis P J 1964 Handbook of Mathematical Functions ed M
Abramowitz and I A Stegun (Washington, DC: US
Government) ch 6, equation (6.5.3).
[164] Kuo C-Y and Keto J W 1983 J. Chem. Phys. 78 1851
[165] Klots J H and Setser D W 1978 J. Chem. Phys. 68 4848
[166] Firestone R F, Oka T and Takao S 1979 J. Chem. Phys. 70
123
[167] Agache M, Fitaire M and Ludac E 1996 J. Phys. D: Appl.
Phys. 29 1217
[168] Guseva L G 1964 Investigations into Electrical Discharges
in Gases ed B N Klyarfel’d (New York: Pergamon) p 1
[169] Klyarfel’d B N, Guseva L G and
Pokrovskaya-Soboleva A S 1966 Zh. Tekh. Fiz. 36 704
(Engl. Transl. 1966 Sov. Phys.–Tech. Phys 11 520)