Full text of DOI:10.1051/epjap:2000145

Eur. Phys. J. AP 11, 59–69 (2000)
THE EUROPEAN
PHYSICAL JOURNAL
APPLIED PHYSICS
c
ꢀ EDP Sciences 2000
Transition from diffuse to constricted low current discharge
in argon
a
˘
˘
S. Zivanov, J. Zivkovi´c, I. Stefanovi´c , S. Vrhovac, and Z.Lj. Petrovi´c
Institute of Physics Belgrade, Pregrevica 118, POB 68, 11080 Zemun Belgrade, Yugoslavia
Received: 20 March 2000 / Revised and Accepted: 25 May 2000
Abstract. We have measured the basic characteristics of low current diffuse discharges in argon for low
pressures. The data for the voltage current characteristics and negative differential resistance were obtained
together with the data for the frequency and for the damping coefficient of the induced oscillations. These
data were compared with the predictions of a simple analytic model of Phelps, Petrovi´c and Jelenkovi´c [A.V.
Phelps, Z.Lj. Petrovi´c and B.M. Jelenkovi´c, Phys. Rev. E 47, 2825 (1993)] and it was found that, while
most observables may be represented correctly, there is a systematic discrepancy between the predictions
for frequency of induced oscillations based on the data for transport coefficients from the literature and
measurements. The basic idea of the present work is thus to check the fundamental assumptions of the
theory and to extend the application of the experiment to study the transition to constrictions in order to
initiate modifications of the theory to cover the transition to the constricted regime. Very good agreement
was found between the spatial profiles of the electric field calculated from the model and the data obtained
from the spatial profiles of emission and these data may be extended to follow the transition to the
constricted regime and the development of the cathode sheath.
PACS. 52.80.Dy Low-field and Townsend discharges – 52.80.Hc Glow; corona – 52.40.Hf Plasma–wall
interactions; boundary layer effects; plasma sheaths
1 Introduction
necessary to mention that the transition to the normal
glow (i.e. cathode fall formation) is not always accompa-
nied by the appearance of constrictions.
For a long time, it was considered that Townsend’s the-
ory [1] fully describes the low current non-self-sustained
and self-sustained electrical discharges, i.e. the dark
Townsend regime [2]. However, this theory cannot explain
some phenomena that were observed in experiments [3–7]
such as negative differential resistance, oscillations in cur-
rent and voltage and the well known transition to con-
striction. The region where such phenomena occur was
even labeled “subnormal”. Recently, it became possible
to develop corrections to the standard Townsend’s break-
down theory in order to explain negative differential re-
sistance and its relation to oscillations occurring for low
current diffuse regime [3–7]. Consequently, the transition
to the constricted regime became the subject of further
studies [7–9]. In addition, the kinetics of the low pressure
breakdown was found to deviate significantly, at most E/n
(where E is electric field and n is the gas number density
in units of Townsend 1Td = 10−21 V m2), from the stan-
dard assumptions of the basic Townsend’s theory [10].
Constriction is a phenomenon of transition from a
region of diffuse to a region in which the higher cur-
rent discharge operates in a narrow channel. This chan-
nel becomes wider with the increasing current. It is also
Our interest in studies of development of constrictions
at low pressures also comes from the fact that this phe-
nomenon is one of the simplest forms of self-organization
and that the reasons for the transition can be predicted
on the basis of physics of low current diffuse discharges.
Clearly, Townsend’s theory in its basic form is not suffi-
cient to achieve a totally adequate description, but some
corrected forms of this theory may prove to be sufficient.
In this paper we describe the results of measurements
in argon which enabled us to obtain well-defined experi-
mental data for quantitative comparisons with the theories
of diffuse low current discharges [3–7] and the theories of
constriction that are being developed [8,11]. The data in-
clude measurements of volt-ampere characteristics, prop-
erties of the induced oscillations such as frequency and
damping coefficient and radial and axial distributions of
emission. From emission profiles we obtained axial dis-
tributions of electric field as the transition occurs from
the uniform field distribution of the low current diffuse
discharge to the highly non-uniform distribution of the
constricted discharge.
The relevant literature, in addition to that already
mentioned, may be separated into three categories. The
a
e-mail: is@plas.ruhr-uni-bochum.de

60
The European Physical Journal Applied Physics
first consists of papers associated with charge buildup in
low current diffuse discharges [12] where it was shown that
the space charge affects the discharge and causes it to de-
part from the basic assumptions of the Townsend’s theory
leading to the development of negative differential resis-
tance and oscillations [13]. This group forms the founda-
tion of the theory of Phelps and coworkers and have been
reviewed more thoroughly in reference [3].
Oscilloscope
Probes
PC
C
A
I
V
V1
CCD
camera
Discharge
RC
(2000 V)
High voltage
supply
V2
(500 V)
MOSFET
Another set of references important for the present
work are those dealing with the development of the cath-
ode fall. Recently, well defined measurements of the tem-
poral development of the cathode fall were made in ar-
gon [14] and other gases. This paper gives an observation
of how initially, as the pulse develops, the exponential
growth starting from the cathode turns into the spatial
profile peaking closer to the cathode with a well developed
cathode fall and negative glow. However, in this paper
temporal dependence of the voltage pulse had to be mod-
ified to promote mechanisms which dominate at low over-
voltages, i.e. when the discharge is running. The present
paper, deals with the similar development but observed
in stationary, developed discharges and it is followed as a
function of the increasing current.
Finally there are papers dealing with constrictions.
Numerous references present experimental observations of
constrictions under different conditions ranging from the
results in dc discharges [15], in dc discharges in electroneg-
ative gases [16] to the constrictions in rf discharges [17]
and also under the higher pressure conditions where ther-
mal gradients may become the primary cause of constric-
tion. In this paper we deal only with the simplest possi-
ble form of constrictions occurring at lower pressures and
consequently higher E/n [7,18] where no thermal pertur-
bation occurs in the gas and on the surface of the cathode.
Theoretical descriptions of the transition from the low cur-
rent diffuse to the constricted (normal) glow include those
trying to predict the shape of the channel in imposed or
self-consistently calculated field distributions by adding
the radial fields and diffusion losses [19].
However, the onset of transition is determined by pro-
cesses at low currents. Such discharges (i.e. operating in
dark Townsend regime) may be described exactly by ex-
tending the simple Townsend theory and many aspects
of the theory may be established by comparing the theo-
retical predictions and the measured, well defined, prop-
erties such as volt-ampere characteristics, oscillation fre-
quencies and damping coefficients. Thus our point is that
low current diffuse mode theories such as that of Phelps
and coworkers [3] may be extended to predict the onset
of transition to constrictions and its principal causes even
without attempting to predict the exact properties of the
constricted discharge itself.
Pulse
generator (15V)
Fig. 1. Schematics of the experimental setup.
coefficient changes sign. Unfortunately their technique was
applied to silane discharge only, where data are somewhat
limited and there is a possibility of the existence of sev-
eral different ions including negative ions. A much clearer
picture of the correspondence between this model and our
results including the proposed physical explanations [11]
may be obtained if the model of Yamaguchi and Makabe
is applied to argon. A similar technique was used by Mar-
ode [21] who applied thermodynamic techniques to de-
scribe the transition as a condition for achieving optimal
operating conditions.
2 Experimental setup and procedure
The system that we use for our measurements in argon
consists of parallel plate electrodes inside a quartz cylin-
der that prevents the long path breakdown. The copper
cathode and the anode, which was made transparent by
depositing a thin film of gold on quartz, can be positioned
at several fixed distances, but it is necessary to open the
system to adjust the gap. The diameter of the electrodes
(2r) was 5.4 cm while the gap (z) was 1.0 cm. Recordings
of axial and radial emission were made by a CCD cam-
era. Our system is designed to make it possible to produce
a pulse of current in addition to a very small dc current
[1,4,5]. The dc current allows us to avoid long breakdown
delay times [22]. The dc current is as small as possible
in order to avoid heating and/or significant conditioning
of the cathode during the measurements. It was neces-
sary to treat the surface of the cathode from time to time
by a relatively high current (30–100 µA) in a hydrogen
discharge before the measurements in argon, in order to
achieve a stable breakdown voltage. The schematic of the
experiment is shown in Figure 1.
Gas flow is used to reduce the accumulation of impu-
rities and products of the discharge. Voltage is measured
by two probes connected as closely as possible to the an-
Yamaguchi and Makabe [20] have used the bifurca- ode and to the cathode, when a relatively high monitoring
tion theory to solve the system of fluid equations for resistor is used in the low voltage anode circuit to deter-
electrons and for the ions and obtain the transition to mine the current. After a stable, low dc current operation
the glow described as a phase transition with the condi- is achieved, pulses of higher current are triggered lasting
tion for the existence of the sudden transition (i.e. what only long enough to achieve stable final discharge and to
appears to be like subnormal glow) given by the crite- make reliable recordings of voltage and current transients.
rion that the second derivative of the effective ionization Operation in low duty cycle or single shot mode allowed

˘
S. Zivanov et al.: Transition from diffuse to constricted low current discharge in argon
61
us to make measurements without modifying the surface Here, ISS is the “steady state” value of the discharge cur-
of the cathode or heating the gas significantly [4–6].
rent (the value that discharge current reaches after a re-
laxation of the damped oscillations), C stands for stray ca-
pacitance and ∂g/∂V expresses the derivative of the round
loop electron gain g = γ(exp(αd)−1). As we can see from
equation (3b) for ISS = 0 the damping tends to a constant
value determined by the external circuit parameters.
Normalized negative differential resistance was also
used for comparisons in the form [3]:
3 Model of the low current discharge
The model that was used here is based closely on the
phenomenological model of Phelps et al. [3] but we also
use some of the results of the physical model. The ba-
sic assumption is that the negative differential resistance
and consequently all other characteristics are caused by a
small perturbation of the field due to space charge shield-
ing of the external field. Thus it is essential to calculate
the field distribution which provided motivation for us to
make measurements presented here.
Within the framework of the phenomenological model
of Phelps et al. it is possible to represent the physical pro-
cesses with effective quantities, most importantly to use
the negative effective resistance of the discharge RD. Ac-
cording to the model of Phelps et al. the magnitude of
the series resistor RS, the stray capacitance C and rel-
ative magnitude of the monitoring resistor Rm and the
discharge resistance RD determine the behavior of the dis-
charge current and the current stability regions [7]. The
negative discharge resistance and other observables may
be associated with the properties of the discharge itself
such as the ion transit time T, the ionization coefficient
α, the secondary electron yield γ and its dependencies on
voltage kV = ∂γ/∂V and on current kI = ∂γ/∂I. Thus
the negative differential resistance of the discharge may
be described by [3]:
A
d2
γˆf(αd)
ε0W+gˆ
RN := RD = −
,
(4)
where A is the electrode area, W+ is the ion drift velocity,
ε0 is dielectric constant of the vacuum, γˆ and gˆ are the
logarithmic derivatives of γ and g with the respect to E/n
evaluated at E0/n and f(αd) is a function of α and d in
a form given in reference [3] (here E/n is the ratio of the
electric field E to the gas density n, and E0 is the value
of the unperturbed electric field). Finally we calculate the
field distribution from the equation for spatial distribution
of ion density and Poisson equation (see Eqs. (37–44) in
Ref. [3]).
The basis procedure was to apply the data for T and
for α from the literature [3,7]. Ionization coefficient was
parameterized by [7]:
α/n = 5 × 10−21 exp[−170/(E/n)]
+ 3 × 10−20 exp[−700/(E/n)]
+ 1.5 × 10−20 exp[−104/(E/n)]
(5)
(6)
and used in the breakdown condition
ꢀ
ISSkI
ꢁ
kV
γSS[exp(αd) − 1] = 1
−kI
ISSkI
RD
=
1 +
γSS (∂g/∂V )
γSS(1 + γSS
)
to determine the γSS coefficient by fitting it with:
ꢀ
ꢀ
ꢁꢁ−1
ꢂ
ꢃ
R
E/n
Q
× 1 +
1 −
,
(1)
(2)
P
γSS(1 + γSS
)
γSS (∂g/∂V )
γSS = γP +
·
(7)
E/n
Q
while the frequency of oscillations ω is given by
1 +
ꢀ
ꢁ
ISS
∂g
kI
ω2 =
(RS + Rm)
−
Both our data and the data compiled by Phelps and
Petrovi´c [10] could be used in the analysis. In the analy-
sis we used equation (7) with the best choice of param-
eters: P = 3.4 × 10−2, Q = 2070 × 10−21, R = 1.1 and
γP = 0.001. The value for kV is obtained directly by dif-
ferentiation (∂γ/∂V ). The value for ∂g/∂V is obtained by
differentiating g:
RSCT
ꢀ
∂V
γSS
ꢁ−1
ISS(kI − RmkV )
× 1 +
− k2,
γSS(1 + γSS
)
and the damping coefficient k is:
ꢀ
1
ISS(kI − (RS + Rm)kV )
∂g
∂α
∂V
k =
1 +
= kV (exp(αd) − 1) + γ
d exp(αd).
(8)
2RSC
γSS(1 + γSS
)
∂V
ꢂ
ꢃꢁ
RSCISS kI
∂g
On the other hand kI is obtained from:
−
− Rm
T
γSS
∂V
kI
ꢀ
ꢁ−1
RD = −
·
(9)
ISS(kI − RmkV )
γSS∂g/∂V
× 1 +
,
(3a)
(3b)
γSS(1 + γSS
)
Drift velocities of ions in a wide range of conditions (from
5×10−19 V m2 to 1×10−16 V m2) may be represented by:
E/n
or in a more simplified form:
1
I
∂g/∂V
T
SS (Rm + RD)
·
W+ = A
,
(10)
k =
+
[1 + B(E/n)C]D
2RSC
2

62
The European Physical Journal Applied Physics
where optimal values of parameters may be established
by fitting the data from Table 8 in reference [23]: A = 47,
B = 2.1, C = 1.2 and D = 0.37. These data were used to
d
W+
obtain the time of flight T =
required in the formulae.
The calculated values of discharge parameters are
shown in Table 1.
4 Results and discussion
Two sets of measurements were made. The first consisted
of measurements of the properties of low current dis-
charges in argon at different gaps d and fixed pd values.
The measurement were performed in argon at low pres-
sures (pd = 40 Pa cm, 66 Pacm and 133 Pa cm [5,6] for dif-
ferent values of d = 1−3 cm). Low pressures were selected
because, under those conditions, volt-ampere character-
istic of the discharge allows us to probe the low current
diffuse glow since negative differential resistance (NDR) is
relatively low. Unfortunately, under those circumstances
the constriction is not as sharp as that at higher pressures.
The second set of measurements was performed with a
transparent electrode in order to observe constrictions and
consequently the gap was somewhat different and fixed
(d = 1.1 cm). It was possible to obtain constrictions at
pressures of 133 Pa and 266 Pa.
(a)
At higher pressures it was not possible to record the
properties of the discharge below the onset of constric-
tion which required to correlate the measured conditions
in the low current regime with the model in order to use
the simple theory for that regime [3] to try to explain
the non-linear effects leading to self-organization (i.e. con-
striction).
4.1 Volt-ampere characteristics and induced
oscillations
(b)
It is important to measure the volt-ampere character-
istics and properties of induced oscillations in order to
establish the dominant feedback mechanism leading to
the breakdown [3,5,6]. Our measurements of volt-ampere
characteristics were presented earlier in reference [5] the
results are in general agreement with those of Petrovi´c
and Phelps [7]. In the analysis of those characteristics it
was found that the breakdown mechanism in argon might
not be described purely by argon ion feedback mechanism
(standard Townsend’s theory) since the transit time of
the ions does not correspond very well to the frequency
of induced oscillations. In case of hydrogen the correspon-
dence was excellent [3,5,6] but in the case of argon one
may have to recognize the importance of additional mech-
anisms such as metastable and resonant photon induced
breakdown [7,10].
Fig. 2. Angular frequency of induced oscillations: a) pd =
66 Pa cm, d = 2 cm; b) pd = 133 Pa cm, d = 1 cm. Points are
the experimental data obtained with different external circuit
elements. Lines are predictions of the theory as explained.
to compare our results with the predictions of the model
we had to use the discharge parameters obtained by fit-
ting the model to our breakdown voltages and RD values,
in the manner proposed in reference [3].
In reference [5] we have shown several examples of the
volt-ampere characteristics that were obtained in the first
set of measurements. The characteristics obtained in the
second set of measurements will be presented later asso-
In the following section we present a comparison of the ciated with emission profiles. At this point we show in
measured current dependence of the volt-ampere charac- Figures 2 and 3 a few examples of the current dependen-
teristics, negative differential resistance, and also of the cies of the frequency (ω) and of the damping constants
frequency and damping coefficient to the theoretical val- (k), respectively. In these figures, the different data point-
ues calculated using formulae of the model [3,7]. In order types represent the results of frequency and damping

˘
S. Zivanov et al.: Transition from diffuse to constricted low current discharge in argon
63
Table 1. (a) Discharge parameters for argon, pd = 40 Pa cm, (b) discharge parameters for argon, pd = 66 Pa cm, (c) discharge
parameters for argon, pd = 133 Pa cm, (d) discharge parameters for argon, pd = 290 Pa cm.
(a)
d
p
VB
E/n
γSS
RD
T
kI
(cm) (Pa)
(V)
(×10−21 V m2) (×10−3
)
)
)
(kΩ)
(×10−6 s) (A−1
)
)
)
1
2
3
40
20
13
690 ꢁ 20
708 ꢁ 40
714 ꢁ 55
6990
7180
7240
31
30
30
−150
−440
−1100
2.1
4.1
6.1
5.3
15
35
(b)
d
p
VB
E/n
γSS
RD
T
kI
(cm) (Pa)
(V)
(×10−21 V m2) (×10−3
(kΩ)
(×10−6 s) (A−1
1
2
3
66
33
22
284 ꢁ 16
309 ꢁ 15
280 ꢁ 1
1730
1880
1700
18
16
18
−150
−490
−1100
4.5
8.6
14
14
34
99
(c)
d
p
VB
E/n
γSS
RD
T
kI
(cm) (Pa)
(V)
(×10−21 V m2) (×10−3
(kΩ)
(×10−6 s) (A−1
1
2
133
66
228 ꢁ 1
242 ꢁ 2
223 ꢁ 7
215 ꢁ 1
693
736
678
580
7.6
−180
−520
−1600
−201
7.5
22
6
14
51
3
44
8.4
7.55
23
256
24
1.1
133
8.96
(d)
d
p
VB
(V)
E/n
γSS
RD
T
kI
(cm) (Pa)
1.1 266
(×10−21 V m2) (×10−3
)
(kΩ)
−370
(×10−6 s) (A−1
)
212 ꢁ 1
343
6.5
12
25
measurements for different values of external circuit el- 4.2 Normalized negative differential resistance
ements RS and Rm.
and discussion
We could not avoid the free running oscillations in ar-
gon for all discharge conditions under the conditions of We have also determined the values of negative differen-
the first set of experiments. The reason for this is that tial resistance [3–5] by fitting the low current portion of
the value of the negative differential resistance is larger the measured volt-ampere characteristics. To compare the
than the value of Rm (see Tab. 1). The measured angu- measured values with the physical model (as described
lar frequencies and damping constants are compared to in the second Chapter of Phelps et al. [3]) we have used
the model calculations in both figures. For the sake of equation (4) and the results are shown in Figure 4. The
simplicity we have omitted the dependence of measured breakdown voltage in argon discharge was stable so the
frequencies on discharge circuit parameters that are much estimated discharge parameter variation is of the order of
less important as compared to the case of damping coef- the magnitude of the experimental points. For the model
ficient. We have also omitted presenting the values of the calculations there are two possibilities to obtain the data:
oscillation frequencies in the current region of free running one to fit the breakdown voltage and the other to use
oscillations, because the model ceases to be valid for these the values for γˆ and gˆ derived from figures in the paper
conditions. The different types of data points are for dif- of Phelps and Petrovi´c [7]. We have used the former ap-
ferent electrode spacing values and different line types are proach, but both gave very similar results. The calculated
for the values predicted by the model of Phelps et al. [3]. value for RN is practically independent of E/n, the same

64
The European Physical Journal Applied Physics
Fig. 4. Normalized negative discharge resistance RN. Solid
points are effective experimental data obtained for different
geometries, open points are predictions of the model, crosses
are the data of Petrovi´c and Phelps [7].
(a)
If, however, we choose to increase the drift velocity of
ions (i.e. the speed of feedback mechanism) by a factor of
2.5 a very good fit to all the experimental data is obtained.
Even the fit for the damping constants is improved. The
examples are shown in Figure 5 for properties of oscil-
lations and Figure 6 for normalized negative differential
resistance. This is consistent with the findings of Phelps
and Petrovi´c [10] that at moderate E/n, such as those cov-
ered here, ion feedback is not the dominant mechanism in
breakdown and that photons from resonant states may
make a significant, even determining contribution.
4.3 Radial and axial emission profiles
We have studied the initial stages of the transition from
diffuse to constricted discharge by observing the profiles
of emission. At low values of pd, covered below, constric-
tion was not well established so we had to increase the
pd value for these studies. At somewhat higher pd, how-
ever, it was not possible to perform measurements at dif-
ferent values of d due to a very high value of RD. At
pd = 133−146 Pa cm it was possible to do both measure-
ments and the basic characteristics of the discharge used
for constriction measurements (with somewhat larger dis-
tance between the electrodes) agree very well with the
previously measured data (as shown in the separate row
of Tab. 1c). The basic characteristics of the discharges
used for field measurements at pd = 290 Pacm are shown
in Table 1d.
(b)
Fig. 3. Damping coefficient of induced oscillations: a) pd =
66 Pa cm, d = 2 cm; b) pd = 133 Pa cm, d = 1 cm. Points are
the experimental data obtained with different external circuit
elements. Lines are predictions of the theory as explained.
as in the experiment. On the other hand the calculated
values of RN are approximately three times larger than
the experimental data. The results represented by crosses
are the experimental values from [7] and are in good agree-
ment with our experimental results.
Measurements were made of both axial and radial pro-
We note that the frequencies predicted by the model files of emission. It is in principle possible to normalize
are also systematically lower than the measured data. The the profiles to absolute values by using separately estab-
discrepancy between the model and the experimental fre- lished profiles for specific lines and the data for excitation
quencies increases with increasing electrode distance, i.e. coefficients measured in a separate, absolutely calibrated
with lowering of the gas pressure for the same value of the system [25] but we have not done that since our data were
normalized electrical field strength E/n. The discrepan- not spectrally resolved. In separate experiments [25,26] it
cies also increase with decreasing E/n.
was shown that spatial profiles of the lines contributing

˘
S. Zivanov et al.: Transition from diffuse to constricted low current discharge in argon
65
Fig. 6. Normalized negative discharge resistance RN calcu-
lated with arbitrarily increased drift velocity of ions by a factor
of 2.5. Solid points are effective experimental data obtained for
different geometries, open points are predictions of the model,
crosses are the data of Petrovi´c and Phelps [7].
(a)
(b)
Fig. 5. Properties of induced oscillations calculated with ar-
bitrarily increased drift velocity of ions by a factor of 2.5: fre-
quency of oscillations: a) pd = 66 Pa cm, d = 2 cm; damping
constant: b) pd = 66 Pa cm, d = 2 cm.
Fig. 7. Radial (observed through the semitransparent elec-
trode) profiles of emission at pd = 290 Pa cm. The profiles were
obtained at different currents where transition from a symmet-
ric radial profile corresponding to the exponential axial profile
to the asymmetric radial profile is observed.
most to the emission in visible and near infra red (where
our CCD was sensitive) have the same spatial profiles with
the same values of the corresponding ionization coefficient.
The differences occur in the contribution of the heavy par-
ticle excitation which is developed as a peak close to the
cathode [27–29] but we have not covered E/n values high
enough to allow significant heavy particle excitation in
determination of the profiles and field distributions.
emission by a zero order Bessel function [8]. As the cur-
rent is increased, the profile is not symmetric anymore and
the maximum of emission is moved from the center of the
electrode. It means that the discharge changed its mode
We show radial (in Fig. 7) and axial (in Fig. 8) pro- from a diffuse to the constricted. However, E/n is still
files in argon for pd = 290 Pacm for different discharge high enough so that a very narrow constriction does not
currents. We have chosen this pd value because the con- form. As mentioned above we were not able to make mea-
striction is more pronounced at higher pressures. At low surements below the onset of constriction under conditions
currents, it is possible to describe the radial profile of when narrow channel forms (i.e. at higher pressures). At

66
The European Physical Journal Applied Physics
Fig. 8. Axial (along the axis of electric field) profiles of emis-
sion at pd = 290 Pa cm. The profiles were obtained at different
currents where transition from a symmetric radial profile cor-
responding to the exponential axial profile to the asymmetric
radial profile is observed.
Fig. 9. Volt-ampere characteristic of the discharge in argon
operating at pd = 146 Pa cm indicating the positions corre-
sponding to different axial and radial profiles.
that at low currents the distribution is centered indicates
that the electrodes are very parallel. We use the radial
profiles as a test of how parallel the electrodes are. In case
that they are not, to the left of Paschen minimum the
low current discharge will be confined to one end (longer
distance) while to the right of Paschen Minimum the dis-
charge would be confined to the opposite end (shorter
distance). When discharge is centered in both cases the
electrodes are sufficiently parallel.
even higher currents the discharge extends to fill the whole
radius again.
The exponential emission profiles along the field axis
which are observed at low currents (Fig. 8) change signif-
icantly as current increases. Axial profiles at low current
could only be measured down to ISS = 20 µA as the sensi-
tivity of the camera did not allow recording at lower cur-
rents without significant noise. At the lowest current cov-
ered here the profile consists of a broad non-equilibrium
region next to the cathode and the growth of emission
which for the lowest currents appears to be exponential.
We have performed similar measurements in a different
apparatus [26] where very low currents and the emission
could be resolved spectrally. Under those conditions it was
clearly established that at very low current a single expo-
nential profile is observed changing slightly close to the
anode as the current increases [30].
Close to the anode leveling off and decreasing of emis-
sion occurs due to poor resolution of the CCD and in that
region one cannot establish the emission profile very accu-
rately. In the region of emission growth there are at least
two different exponentials that describe the growth at the
lowest current covered here (see Fig. 8). At higher cur-
rents the profile of radiation has a rapidly changing slope
and it levels much further from the anode indicating low
electric field. Thus the profile is consistent with the devel-
opment of the cathode fall and low field beyond it. At the
highest currents there is very little change of the profile as
the current increases and we did not carry the measure-
ments further to reduce the need to clean the glass from
the products of sputtering.
The development of constriction in our experiment is
followed to very high currents when the discharge fills the
whole volume again. The correlation between the emission
profiles and the volt-ampere characteristic is shown in Fig-
ure 9. We have not observed the rotating constrictions re-
ported by Petrovic and Phelps [7] but sometimes the con-
strictions moved suddenly. One may observe that skewing
of the axial field occurs first without constriction and that
that only at higher currents when skewing of the field be-
comes significant, the transition to asymmetric radial pro-
file occurs.
It is important to note that under some conditions a
slow, gradual, transition to the final mode was achieved.
In other words we were able to follow the transition to
constricted mode without a sudden transition or even
development of oscillations. However, one should note that
in our procedure we make discrete changes to the current
pulses and there is a possibility that such transition may
occur though it may be quite small. At such values of pd
as those covered here the voltage drop in transition to
the constricted mode would be very small or negligible
and negative differential resistance very small and thus
At the same time as the axial profiles make the change the electric circuit as used here may provide a continu-
from exponential growth to a sudden growth and a broad ous transition. When we changed the dc voltage manually
peak, the radial profiles make transition from the centered no significant transition was observed either but we could
Bessel function-like distribution to the narrower off-center not make recordings of the voltage and current under such
distribution corresponding to the constriction. The fact mode.

˘
S. Zivanov et al.: Transition from diffuse to constricted low current discharge in argon
67
Fig. 11. Comparison between the theory and the model.
Points are the measured field distributions obtained from the
axial profiles for pd = 146 Pa cm and for two different currents:
ISS = 23.7 µA and ISS = 40.4 µA. Lines are the calculated field
distributions for the same currents and the homogenous elec-
tric field.
Fig. 10. Contour plots of radial emission profiles for conditions
in Figure 7 (left: 36 µA and right: 65 µA at 290 Pa cm).
and sharper with the increase of the current yet the over-
all voltage becomes smaller. The agreement between ex-
perimentally established field and the calculated values is
very good except in the region of the plateau. Just where
one should draw the limit of the region where leveling off
does not affect the field determination is not clear so we
have presented a broader range of the results rather than
make an arbitrary division. It is however clear that level-
ing off at low current has one cause: the shadowing by the
electrode while that at higher currents the low field re-
gion is clearly established as the plateau occurs quite far
from the electrode. As the sheath is formed the accuracy
of the field established from the emission profile may be
questionable since the transport of electrons will be non-
hydrodynamic (non-local). However, we have covered here
the range which clearly includes conditions where such
determination may be made. A gradual skewing of the
field profile towards the well developed profile of the con-
stricted (glow) discharge is observed sometimes without a
sharp transition. The values of the field agree well with the
level expected for the uniform field and with the degree of
skewing of the field predicted by the model [3].
Finally we show the contour plot of the discharge for
two similar currents below and above the onset of con-
striction in Figure 10.
4.4 Calculation of the field distribution
We were able to obtain the local field distribution from the
experimental data by determining the slope of the emis-
sion profile and by associating it with the ionization coef-
ficient from the literature [7,25,31] and also we have taken
the local ionization coefficients to represent the fields. The
choice of the regions which are fitted by a single exponen-
tial may be somewhat arbitrary. Nevertheless it is clear
(see Fig. 8) that the slope changes sign and goes down
towards the anode.
A comparison between calculated field distribution and
that obtained from the axial emission profile is shown in
Figure 11. The assumptions used in the analysis are valid
in the region of swarm like transport, when the sheath is
not formed, so the results obtained with the well devel-
oped sheath should be taken just as an indication. How-
ever, the data may be fitted by Monte Carlo simulations
that would represent the non-hydrodynamic kinetics in
the sheath [32].
The independently made measurements allowing de-
termination of the axial profiles at even lower currents of
1–20 µA (but without the radial profiles) made it possible
In Figure 11 we can see the excellent agreement be- to establish the low current limit where a single expo-
tween the measured and calculated field distributions. nential is an excellent approximation and thus one could
Experimental data follow the basic characteristics of the determine the profile of the efficiency of detection all the
profile predicted by the theory. At the highest currents way to the anode. The lowest currents covered here, with
covered in the experiment the calculation is not performed the limit set by the CCD detection technique correspond
as the sheath is clearly developed which makes the model to the situation where field still changes significantly and
not valid as it is based on a small perturbation to the thus the spatial profile cannot be determined by a single
homogeneous field approximation. Yet we have observed exponential.
a continuous transition and the charge induced field de-
The field profiles determined here show symmetry, i.e.
viation becomes more and more pronounced leading to the point where field crosses the expected (uniform) value
the development of the cathode fall. At lower currents we is quite close to the middle of the gap. The reason is
see the transition from approximately two to three dis- that we operate in the region where ionization coeffi-
tinct exponentials to several, the field becomes sharper cient increases slowly, close to the point where the second

68
The European Physical Journal Applied Physics
derivative of the ionization coefficient changes sign. That leads to the negative differential resistance and eventually
is the point of weak constriction and there, sudden tran- to the development of the cathode sheath.
sition to the constricted mode according to the theory of
Yamaguchi and Makabe [20] is not expected.
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5 Conclusion
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