Calculating reduced electric field in diffusion regime of DC discharge positive column
The present paper outlines an analytical model of the positive column of the direct current discharge in a diffusion mode. We consider the case when charged particles are produced through direct ionization of gas molecules via electron impact and the ambipolar escape of them to discharge tube walls...
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irk-123456789-822342015-05-28T03:01:49Z Calculating reduced electric field in diffusion regime of DC discharge positive column Lisovskiy, V.A. Artushenko, E.P. Yegorenkov, V.D. Низкотемпературная плазма и плазменные технологии The present paper outlines an analytical model of the positive column of the direct current discharge in a diffusion mode. We consider the case when charged particles are produced through direct ionization of gas molecules via electron impact and the ambipolar escape of them to discharge tube walls is the sole mechanism of their loss. We solved the equation for the charged particles balance and obtained simple formulas for the reduced electric field E/p in the positive column in molecular gases. Results of our calculations for E/p values in nitrogen are in good agreement with experimental and theoretical data of other authors for low discharge current values. Представлена аналитическая модель положительного столба разряда постоянного тока в диффузионном режиме. Рассмотрен случай, в котором заряженные частицы образуются вследствие прямой ионизации молекул газа электронным ударом, а единственным механизмом потерь является их амбиполярный уход на стенки разрядной трубки. Решено уравнение баланса заряженных частиц и получены простые формулы для приведенного электрического поля E/p в положительном столбе в молекулярных газах. Результаты наших расчетов для E/p в азоте хорошо согласуются с экспериментальными и теоретическими данными других авторов для случая низких разрядных токов. Представлено аналітичну модель позитивного стовпа розряду постійного струму в дифузійному режимі. Розглянуто випадок, коли заряджені частинки утворюються внаслідок прямої іонізації молекул газу електронним ударом, а єдиним механізмом втрат є їх амбіполярний вихід на стінки розрядної трубки. Розв’язано рівняння балансу заряджених частинок і отримані прості формули для зведеного електричного поля E/p у позитивному стовпі в молекулярних газах. Результати наших розрахунків для E/p в азоті добре узгоджуються з експериментальними і теоретичними даними інших авторів для випадку низьких розрядних струмів. 2015 Article Calculating reduced electric field in diffusion regime of DC discharge positive column / V.A. Lisovskiy, E.P. Artushenko, V.D. Yegorenkov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 205-208. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 52.80.Hc http://dspace.nbuv.gov.ua/handle/123456789/82234 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Низкотемпературная плазма и плазменные технологии Низкотемпературная плазма и плазменные технологии |
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Низкотемпературная плазма и плазменные технологии Низкотемпературная плазма и плазменные технологии Lisovskiy, V.A. Artushenko, E.P. Yegorenkov, V.D. Calculating reduced electric field in diffusion regime of DC discharge positive column Вопросы атомной науки и техники |
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The present paper outlines an analytical model of the positive column of the direct current discharge in a diffusion mode. We consider the case when charged particles are produced through direct ionization of gas molecules via electron impact and the ambipolar escape of them to discharge tube walls is the sole mechanism of their loss. We solved the equation for the charged particles balance and obtained simple formulas for the reduced electric field E/p in the positive column in molecular gases. Results of our calculations for E/p values in nitrogen are in good agreement with experimental and theoretical data of other authors for low discharge current values. |
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Lisovskiy, V.A. Artushenko, E.P. Yegorenkov, V.D. |
| author_facet |
Lisovskiy, V.A. Artushenko, E.P. Yegorenkov, V.D. |
| author_sort |
Lisovskiy, V.A. |
| title |
Calculating reduced electric field in diffusion regime of DC discharge positive column |
| title_short |
Calculating reduced electric field in diffusion regime of DC discharge positive column |
| title_full |
Calculating reduced electric field in diffusion regime of DC discharge positive column |
| title_fullStr |
Calculating reduced electric field in diffusion regime of DC discharge positive column |
| title_full_unstemmed |
Calculating reduced electric field in diffusion regime of DC discharge positive column |
| title_sort |
calculating reduced electric field in diffusion regime of dc discharge positive column |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2015 |
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Низкотемпературная плазма и плазменные технологии |
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http://dspace.nbuv.gov.ua/handle/123456789/82234 |
| citation_txt |
Calculating reduced electric field in diffusion regime of DC discharge positive column / V.A. Lisovskiy, E.P. Artushenko, V.D. Yegorenkov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 205-208. — Бібліогр.: 20 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT lisovskiyva calculatingreducedelectricfieldindiffusionregimeofdcdischargepositivecolumn AT artushenkoep calculatingreducedelectricfieldindiffusionregimeofdcdischargepositivecolumn AT yegorenkovvd calculatingreducedelectricfieldindiffusionregimeofdcdischargepositivecolumn |
| first_indexed |
2025-07-06T08:39:36Z |
| last_indexed |
2025-07-06T08:39:36Z |
| _version_ |
1836886193618288640 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 205-208. 205
CALCULATING REDUCED ELECTRIC FIELD IN DIFFUSION REGIME
OF DC DISCHARGE POSITIVE COLUMN
V.A. Lisovskiy, E.P. Artushenko, V.D. Yegorenkov
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
E-mail: lisovskiy@yahoo.com
The present paper outlines an analytical model of the positive column of the direct current discharge in a
diffusion mode. We consider the case when charged particles are produced through direct ionization of gas
molecules via electron impact and the ambipolar escape of them to discharge tube walls is the sole mechanism of
their loss. We solved the equation for the charged particles balance and obtained simple formulas for the reduced
electric field E/p in the positive column in molecular gases. Results of our calculations for E/p values in nitrogen are
in good agreement with experimental and theoretical data of other authors for low discharge current values.
PACS: 52.80.Hc
INTRODUCTION
Direct current discharge in long tubes is widely
applied for pumping gas discharge lasers, separating
isotopes etc. The positive column connecting the
cathode portions and the anode and closing the
discharge circuit is the most important part for these
applications. The reduced electric field E/p (where E
is the electric field strength and p is the gas pressure)
is the important parameter describing the positive
column. Transport of charged particles (drift,
diffusion) and their collisions (elastic and inelastic)
with gas molecules depend just on E/p. Therefore it is
of interest to obtain simple analytical formulas for the
reduced electric field E/p in the positive column. The
available analytical and numerical models are usually
complicated and they are based on treating the
electron energy distribution function (EEDF).
However it is possible to construct an analytical model
employing such electron transport parameters as
mobilities and diffusion coefficients of electrons and
ions as well as the first Townsend coefficient for
ionizing gas molecules via electron impact. These
transport coefficients are integral functions of EEDF,
and their application simplifies analytical treatment
considerably.
This paper reports our analytical model of the
diffusion mode of the positive column when charged
particles are produced through direct ionization of gas
molecules via electron impact, and the rate of their
production has to compensate the rate of their escape
to the discharge tube walls due to ambipolar diffusion.
It is assumed that the concentration of charged
particles is not large therefore one may neglect the
volume recombination of electrons and positive ions.
Balancing the ionization rate and diffusion loss we get
the equation for the reduced electric field E/p. We
found the approximation formulas for different cases
that helped us to get the analytical formulas for the
reduced electric field. E/p is a decreasing function of
the product of the gas pressure and discharge tube
radius pR. These formulas contain mobilities and free
diffusion coefficients of electrons and positive ions as
well as the first Townsend coefficient. We calculated
the E/p values in the broad pR range for nitrogen
which are in good agreement with the results of
experiments [1-4] and calculations [1, 3].
1. DESCRIPTION OF THE ANALYTICAL
MODEL
In this paper we outline the analytical model of the
uniform positive column in the diffusion mode in a
discharge tube of radius R in detail. We consider a case
when direct ionization of gas molecules through electrons
moving in the uniform constant electric field E is the sole
process of charged particles production. Here we neglect
the processes in which metastable molecules participate
(stepwise and associative ionization) as well as those of
gas heating. We also neglect volume losses of charged
particles such as recombination and attachment. Let the
escape of charge particles to the discharge tube walls due
to ambipolar diffusion be the sole process of their loss.
According to papers by Schottky [5, 6], in the uniform
positive column of the dc discharge the ionization rate i
and the ambipolar diffusion coefficient Da are related as
follows
2
2
405.21
RDa
i , (1)
where is the diffusion length [7], i = Vdr, is the
first Townsend coefficient (the number of ionizing
collisions performed by an electron while moving along 1
cm path in the electric field), Vdr is the electron drift
velocity. For the description of the ionization of
molecular gases through electron impact one often
employs the following empirical formula for the
coefficient obtained already by Townsend [8, 9]
pE
B
A
p
exp
, (2)
where the A and B constants depend on the gas species.
For the electron drift velocity we write the formula
Vdr = e E, where e is the electron mobility. As
Schottky [5, 6] demonstrated, the coefficient of ambipolar
diffusion depends on the coefficients of free diffusion of
electrons De and ions Di, as well as on the mobility of
ions i and electrons e:
206 ISSN 1562-6016. ВАНТ. 2015. №1(95)
ei
eiie
a
DD
D
. (3)
Let us take into account that the electron mobility
exceeds much that of ions, e i, then expression
(3) can be cast in the form:
e
i
eia DDD
. (4)
Let us introduce the mobility and diffusion
coefficients for electrons and ions at the gas pressure
of 1 Torr (e1, i1, De1 и Di1) to find out the
dependence of ambipolar diffusion coefficient on gas
pressure, and then we get the following expression
1
1
11
1
e
i
eia DD
p
D
, (5)
whereas the relation for the electron drift velocity
assumes the form Vdr = e1 E/p. Let us insert the
expressions for the first Townsend coefficient (2), the
ambipolar diffusion coefficient (5) and the electron
drift velocity Vdr into the equation (1) and introduce an
additional variable z = B/(E/p), then we obtain the
following equation:
2
2
1111
1
405.2
exp
pR
DD
BA
zz
eiei
e
. (6)
Since the function F(z) = zexp(z) does not allow
analytical solving the equation (6) with respect to z,
we have to choose a suitable approximation formula
for F(z) to get such a solution. To this end one has to
know the range of z variation beforehand. As it
follows from the results of experiments and
calculations [1-4], the reduced electric field values in
the positive column in nitrogen lie in the range
E/p 20…80 V/(cm·Torr). The book by Raizer [10]
gives the value B = 342 V/(cm·Torr) for nitrogen (we
shall refine this value below). Therefore we obtain that
the solutions are within the limits z = B/(E/p) 4…17.
The complicated behaviour of the F(z) = zexp(z)
function makes it expedient to consider it in two
different ranges separately. In the range of high
values, z = 10…100, the F(z) function can be
described by the following approximation formula:
F(z) = zexp(z) Fa(z) = 8.84exp(z
1.006
). (7)
In the range of moderate values, z = 2…10, it is
convenient to apply the formula
F(z) = zexp(z) Fa(z) = 3exp(z
1.05
) – 9. (8)
From Fig. 1 it is clear that formulas (7) and (8) furnish
a good description of the function F(z).
Substitution of expression (7) into relation (6)
enables one after simple transformations to get the
expression for the reduced electric field E/p in the
range z = 10...100:
994.0
2
2
1111
1
405.284.8
ln
pR
DD
BA
B
p
E
eiei
e
, (9)
and in the range z = 2…10 one can find similarly from
relations (6) and (8) that
952.0
2
2
1111
1 .
405.23
3ln
pR
DD
BA
B
p
E
eiei
e
(10)
Relations (9) and (10) demonstrate, that in the positive
column existing in the diffusion mode with direct
ionization of gas molecules, the reduced electric field E/p
is the sole function of the pR product depending on the
gas species as well. Note also that to derive the relations
(9) and (10) we applied minimum requirements to the gas
species (direct ionization domination over step one,
absence of strong attachment of free electrons to gas
molecules), therefore these expressions can be used for
E/p determination in the broad range of molecular gases.
2. CALCULATION DATA FOR NITROGEN
Let us now check the applicability of expressions (9)
and (10) for describing the reduced electric field E/p in
the positive column of the dc discharge in nitrogen.
Nitrogen is an electropositive gas not producing negative
ions. As it was shown in the paper [3], for low discharge
current values, the direct ionization of molecules through
electron impact dominates over step and associative
ionization with the participation of metastable molecules,
their concentration being small under these conditions. In
this case the low concentration of charged particles
enables one to neglect the dissociative recombination of
electrons and positive molecular nitrogen ions as
compared to their ambipolar escape to the tube walls.
Relations (9) and (10) contain the mobilities and
diffusion coefficients of electrons and ions e1, i1, De1
and Di1. Electron mobility e1 = 4.210
5
cm
2
·Torr/(V s)
for nitrogen is determined from experimental values of
the electron drift velocity Vdr presented in papers [11, 12].
From papers [13-16] we determine the coefficient of free
electron diffusion De1 = 910
5
cm
2
·Torr/s.
For positive nitrogen ions we employ in our
calculations the following values of the mobility
i1 = 1.5410
3
cm
2
·Torr/(V·s) and diffusion coefficient
Di1 = 39.7 cm
2
·Torr/s, which are taken from book [17].
To calculate the reduced electric field from expressions
(9) and (10) we need A and B constants entering the
formula for /p (2). To find them we use the experimental
data for
0 20 40 60 80 100
10
0
10
5
10
10
10
15
10
20
10
25
10
30
10
35
10
40
10
45
F(z)
Fa(z)
F
(z
),
F
a
(z
)
z
Fig. 1. F(z) function (points) and its approximation
function Fa(z) (solid line) versus the ratio z = B/(E/p)
ISSN 1562-6016. ВАНТ. 2015. №1(95) 207
10
1
10
2
10
3
10
-3
10
-2
10
-1
10
0
10
1
/p
,
c
m
-1
T
o
rr
-1
E/p, V/(cm Torr)
Posin [18]
Haydon [19]
Maller [20]
Bolsig+ [9]
1
2
Fig. 2. Ratio /p versus the reduced electric field E/p.
Empty symbols correspond to experimental data from
papers [18-20]. Filled symbols correspond to our
calculation data using the Bolsig+ code [9]. Solid line
is calculated according to formula (2) with the
constants A = 21 cm
1
·Torr
1
and
B = 469 V/(cm·Torr) [9]. Broken line is calculated
according to formula (2) with the constants
A = 7 cm
1
·Torr
1
and B = 270 V/(cm ·Torr)
the first Townsend coefficient /p in nitrogen [18-20]
in the broad range of the reduced field
E/p = 30…1000 V/(cm·Torr). The book by Raizer
[10] gives the values A = 12 cm
1
·Torr
1
and
B = 342 V/(cm ·Torr), but they provide a good
description of the /p pattern only in the range of
large values E/p = 100 … 600 V/(cm·Torr). In paper
[9], it was found that in the range of high values
E/p = 200 … 1000 V/(cm·Torr) the /p values
demonstrate a good description by equation (2) with
the constants A = 21 cm
1
·Torr
1
and
B = 469 V/(cm ·Torr) (see Fig. 2). However, as it was
shown in papers [1-4], the reduced electric field values
in the positive column in nitrogen are in the
0,01 0,1 1 10
20
40
60
80
z > 10
z < 10
E
/p
,
V
/(
c
m
T
o
rr
)
pR, Torr cm
Model [1]
Probes [1]
Probes [2]
Probes [3]
Model [3]
Probes [4] 5 mA
Probes [4] 75 mA
Equation (9)
Equation (10)
z = 10
Fig. 3. Reduced electric field strength E/p versus pR
product in nitrogen. Solid line depicts the calculation
data according to formula (10). Broken line depicts
the calculation data according to formula (9). Dots
are the probe data from papers [1-4]. Dash-dot line is
for data from the model [1], and dash-double dot line
is for data from the model [3]
range E/p 20…80 V/(cm ·Torr). Therefore in this E/p
range it is expedient to employ the constants
A = 7 cm
1
·Torr
1
and B = 270 V/(cm ·Torr).
In Fig. 3 we present our calculation data obtained
from relations (9) and (10) as well as the measured data
[1-4] and modelling ones [1, 3]. With the horizontal line
we show the value E/p = 27 V/(cm ·Torr), for which we
find z = B/(E/p) = 270/27 = 10. The values E/p for z < 10
have to be calculated with expression (10) whereas for
z > 10 (lower reduced electric field values) one has to
apply the expression (9). The dependence of the reduced
electric field E/p versus the pR product we calculated is in
good agreement with the registered data of papers [1-4] as
well as with calculation data of papers [1, 3].
CONCLUSIONS
The present paper outlines the analytical model for the
diffusion mode of the positive column of the dc discharge
when the charged particles are produced under direct
impact of electrons with gas molecules and lost due to
ambipolar diffusion to discharge tube walls. The
approximated expressions chosen enable us to solve the
balance equation for charged particles and obtain simple
expressions from which it follows that the reduced
electric field depends on the product of the gas pressure
and discharge tube radius pR as well as on the gas
species. The results of our calculations for the positive
column in nitrogen are in good agreement with the probe
measurement data of papers [1-4] and calculated data of
papers [1, 3].
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Article received 14.11.2014
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ПОЛОЖИТЕЛЬНОГО СТОЛБА РАЗРЯДА ПОСТОЯННОГО ТОКА
В.А. Лисовский, Е.П. Артюшенко, В.Д. Егоренков
Представлена аналитическая модель положительного столба разряда постоянного тока в диффузионном
режиме. Рассмотрен случай, в котором заряженные частицы образуются вследствие прямой ионизации
молекул газа электронным ударом, а единственным механизмом потерь является их амбиполярный уход на
стенки разрядной трубки. Решено уравнение баланса заряженных частиц и получены простые формулы для
приведенного электрического поля E/p в положительном столбе в молекулярных газах. Результаты наших
расчетов для E/p в азоте хорошо согласуются с экспериментальными и теоретическими данными других
авторов для случая низких разрядных токов.
РОЗРАХУНОК НАВЕДЕНОГО ЕЛЕКТРИЧНОГО ПОЛЯ В ДИФУЗІЙНОМУ РЕЖИМІ
ПОЗИТИВНОГО СТОВПА РОЗРЯДУ ПОСТІЙНОГО СТРУМУ
В.О. Лісовський, К.П. Артюшенко, В.Д. Єгоренков
Представлено аналітичну модель позитивного стовпа розряду постійного струму в дифузійному режимі.
Розглянуто випадок, коли заряджені частинки утворюються внаслідок прямої іонізації молекул газу
електронним ударом, а єдиним механізмом втрат є їх амбіполярний вихід на стінки розрядної трубки.
Розв’язано рівняння балансу заряджених частинок і отримані прості формули для зведеного електричного
поля E/p у позитивному стовпі в молекулярних газах. Результати наших розрахунків для E/p в азоті добре
узгоджуються з експериментальними і теоретичними даними інших авторів для випадку низьких розрядних
струмів.
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