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A semi-empirical expression for the first Townsend coefficient in strong electric fields

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A semi-empirical expression for the ﬁrstTownsend coeﬃcient in strong electric ﬁelds
M. Radmilovi´c-Radjenovi´c, B. Radjenovi´c, M. Klas
and
ˇS. Matejˇcik
EPL,
108
(2014) 65001Please visit the website
www.epljournal.org
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EPL,
108
(2014) 65001
www.epljournal.org
doi:
10.1209/0295-5075/108/65001
A
semi-empirical expression for the ﬁrst Townsend coeﬃcientin strong electric ﬁelds
M. Radmilovi´c-Radjenovi´c
1
, B. Radjenovi´c
1
, M. Klas
2
and
ˇS. Matejˇcik
21
Institute of Physics, University of Belgrade - P.O. Box 57, 11080 Belgrade, Serbia
2
Department of Experimental Physics, Comenius University - Mlynski dolina F2, 84248 Bratislava, Slovakia
received 24 October 2014; accepted in ﬁnal form 28 November 2014published online 17 December 2014
PACS
52.90.+z
– Other topics in physics of plasmas and electric discharges
PACS
52.80.Tn
– Other gas discharges
PACS
52.80.-s
– Electric discharges
Abstract
– The ionization coeﬃcients for argon, nitrogen, hydrogen, oxygen and air are deter-mined from the measured breakdown voltage curves in microgaps. The obtained values agree verywell with the data for the standard size discharges indicating that the Townsend phenomenologydoes not need extension to describe microdischarges and that the ionization coeﬃcients for suchdischarges obey the similarity law
α/p
=
f
(
E/p
). In addition, an analytical expression for the ion-ization coeﬃcients has been derived based on the generalized theory. The appropriate constantsin derived expression have been modiﬁed by ﬁtting the ionization coeﬃcients obtained from themeasured breakdown voltage curves.
Copyright c
EPLA, 2014
Introduction. –
Developing techniques for reliablepredictions of the electron ionization coeﬃcients areof interest both for industrial applications [1–3] andfor a deeper understanding of fundamental plasmabehavior [4–7]. It is well known that the Townsend ioniza-tion coeﬃcient,
α
coeﬃcient by other name, is critical pa-rameter required to model plasma maintenance since theoperation of gas discharges is usually established at theoperating point where ionization can compensate losses.The impact ionization is deﬁned as the inverse of the av-erage distance traveled by a carrier prior to the ioniza-tion event and obeys the similarity law
α/p
=
f
(
E/p
)which dictates the relation between the ionization coeﬃ-cient and the reduced electric ﬁeld
E/p
(electric ﬁeld overthe gas pressure) [8]. This is one of the most widespreadexpressions used to represent analytically the experimen-tal results for the ionization coeﬃcient which may be use-ful in modelling the volt-ampere characteristics and thebreakdown voltage curves for low current discharges [9,10].These applications were often based on analytic approx-imations of the dependence of the ionization coeﬃcienton the reduced electric ﬁeld needed for extracting infor-mation on secondary electron yields for conductingandfor dielectric electrodes, for modelling cylindrical dis-charge geometries and for verifying the applicabilityof the models in predicting higher current dischargecharacteristics [9].Recent growing interest in studies of microdischargesfor atmospheric nonequilibrium plasma processing andplasma displays requires knowledge of the plasma param-eters, in the ﬁrst line the ionization coeﬃcients. Manymethods for measuring the ionization coeﬃcient were es-tablished according to the Townsend theory. Since theTownsend discharge is usually produced at low pressure,almost all the measurements of the ionization coeﬃcientwere carried out in the gases at a pressure much lower thanthe atmospheric pressure. Thus, the values of the
α
coef-ﬁcient obtained by the measurements at low gas pressureare sometimes useless in the gas discharges at atmosphericpressure [11].Although, measurement of the ionization coeﬃcient isnot a rarity, it is certainly crucial data especially for micro-discharges. In this paper we are primarily interested inthe ionization coeﬃcient in strong electric ﬁelds whereas,until recently, very little information existed and the exis-tence of well-deﬁned values of the rate coeﬃcients had notbeen deﬁnitely established [12,13]. The ﬁrst Townsendionization coeﬃcient is found analytically based on thegeneralized theory and following the derivation describedin [14]. A noticeable disagreement between the
α
valuesobtained by using the derived expression with the con-stants taken from [14] and values determined from themeasured breakdown voltage curves [15–18] indicate thatthe modiﬁcation of the constants are needed. For that65001-p1
M. Radmilovi´c-Radjenovi´c
et al.
purpose,
α
coeﬃcients extracted from the measurementsare ﬁtted in accordance with the derived expression andthe appropriate constants are modiﬁed. By comparingtheoretical and experimental results, the validity region of the expression has been determined.
Theoretical background. –
It was shown that thePaschen curve is very sensitive on rates of the ionizationat strong reduced electric ﬁelds where the total ionizationrate would no longer decline at high
E/p
[19]. When astrong electric ﬁeld is applied between two plane-parallelelectrodes, the current density
(
x
) at a given point
x >
0can be expressed as:
(
x
) =
0
Φ(
x
)
,
(1)where
0
is the current density at the cathode (
x
= 0).The unknown function Φ(
x
) could be determined underassumptions that all new electrons created in ionizing col-lisions initially have zero energy and that the energy lossesby the ionizing electron can be neglected. The total num-ber of ionizations in the interval d
x
can be obtained fromthe contributions of all the electrons created in the interval(0
,x
):
n
(
x
)d
x
= d
x
x
0
n
(
t
)
Nσ
(
ε
(
x
−
t
))d
t
+
j
0
Nσ
(
ε
(
x
))d
x,
(2)where
n
(
t
)
Nσ
(
ε
(
x
−
t
))d
t
d
x
is the number of electronscreated in an interval d
t
,
N
is the concentration of thegas atoms and
σ
(
ε
(
x
−
t
)) is the ionization cross-sectionevaluated at the energy
eE
(
x
−
t
). Combining eq. (1) withthe continuity equation
n
(
x
) = d
(
x
)
/
d
x
, eq. (2) can bere-written asΦ(
x
)d
x
=
x
0
dΦ(
t
)d
t ς
(
ε
(
x
−
t
))d
t
+
ς
(
ε
(
x
))
,
(3)with
ς
(
ε
(
x
)) =
Nσ
(
ε
(
x
)). The equation for the functionΦ(
x
) could be obtained by integrating eq. (3) with initialconditions Φ(0) = 1 and
ς
(
ε
(
x
−
x
)) = 0:Φ(
x
)d
x
=
−
x
0
Φ(
t
)
∂ς
(
ε
(
x
−
t
))
∂t
d
t.
(4)Putting the asymptotic solution in the form Φ(
x
) =
(
x
)
/
0
=
e
αx
into eq. (4), we obtain the equation forthe ionization coeﬃcient
α
:
α
= lim
x
→∞
x
0
exp[
−
α
(
x
−
t
)]
∂ς ∂t
d
t.
(5)If we approximate the ionization cross-section with
ς
(
ε
) =
A
1
ε
n
e
−
A
2
ε
,
(6)the partial diﬀerentiation provides
∂ς
(
ε
(
x
−
t
))
∂t
=
A
1
n
(
x
−
t
)
(
n
−
1)
·
e
−
A
2
(
x
−
t
)
−
1+
A
2
(
x
−
t
)
,
(7)where
A
1
,
A
2
and
n
are constants. Substituting (7)into (5) and introducing
y
=
α
+
A
2
and
z
=
y
(
x
−
t
)we obtain
y
−
A
2
=
A
1
y
−
n
n
x
0
z
n
−
1
e
−
z
d
z
−
A
2
y
−
1
∞
0
z
n
e
−
z
d
z
,
(8)that can be rewritten in terms of the gamma functionΓ(
n
):
A
1
Γ(
n
+ 1)
y
+
A
2
y
n
+1
−
y
n
+2
=
A
1
A
2
Γ(
n
+ 1)
.
(9)Since eq. (9) has two solutions,
y
1
=
A
2
and
y
n
−
22
=
A
1
Γ(
n
+ 1), there are two possible values for
α
:
α
= 0(with no physical meaning) and
α
= (
A
1
Γ(
n
+1))
1
/
(
n
+1)
−
A
2
[20]. After simple mathematical operations, we obtaina generalized expression for the ionization coeﬃcient ina strong electric ﬁeld, which is in accordance with thatderived in [14]:
α p
=
A
E p
n
Γ(
n
+ 1)
(1
/
(
n
+1))
−
BE p ,
(10)where
A
1
=
A
E p
n
p
n
+1
and
A
2
=
B
E p
p
. We are goingto compare the values for the ionization coeﬃcients pre-dicted by (10) with those obtained from our experimentaldata in order to determine the applicability of the derivedexpression.
Experimental set-up. –
The breakdown voltagecurves and volt-ampere characteristics of the direct cur-rent (DC) gas discharges in microgaps were measured bythe experimental set-up depicted in ﬁg. 1 and well doc-umented in previous publications [15–18]. The micro-discharge is operated in the Townsend regime betweentwo parallel-plate electrodes. The vacuum chamber con-tains three parts: in the upper part, there is a positionerfor centering the electrode position in three directions andtilting the upper electrode. There are also a glass cruxwith four fused silica windows and a positioning systemfor tilting electrode located in the middle and the bottompart, respectively. The electrode is ﬁxed in the cradle witha micrometric screw making it possible to achieve paral-lelism of the electrodes with an accuracyof 1
µ
m. The elec-trodes were mechanically polished by the ﬁnest diamondpaste (0
.
25
µ
m grain size) in order to achieve the averageroughness of the electrode better than 0
.
25
µ
m. One of the electrodes was ﬁxed while the other was moved con-tinuously with micrometer scale linear feed-through. The0
µ
m separation of the electrodes was settled by check-ing the electrical contact between the electrodes and thenthe movable electrode was pulled away by means of themicrometer screw at the upper electrode. Breakdownvoltage curves for all gases were determined on the ba-sis of the recorded current-voltage characteristics by usinga digital oscilloscope and the AD card (National Instru-ments NI USB-6211). As a ﬁrst step in measurements,a very slowly increasing potential was applied to one of 65001-p2

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