Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system
A new general approach, laser representations and catastrophe theory have been taken to describe plasma generation of the self- sustained gas D.C. discharge. Two regions of a self-consistent effective electrical field with qualitatively different structural properties of the positive column are foun...
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| citation_txt | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system/ P.F. Kurbatov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 157-160. — Бібліогр.: 15 назв. — англ. |
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| description | A new general approach, laser representations and catastrophe theory have been taken to describe plasma generation of the self- sustained gas D.C. discharge. Two regions of a self-consistent effective electrical field with qualitatively different structural properties of the positive column are found. All allows us to propose a logically self- contained classification and a description of the low- pressure steady gas D.C. discharge with the well- defined positive column for an atomic gas system and demonstrate the existence of the similarity laws for all regimes of such a discharge.
Новий узагальнений метод, лазерні зображення і теорія катастроф були використані для опису генерації плазми самостійного газового розряду, що збуджується постійним струмом. Виявлено два види областей самоузгодженого ефективного електричного поля з якісно різними структурними властивостями позитивного стовпа. Усе це дозволяє запропонувати логічно самоузгоджену класифікацію й опис стаціонарного газового розряду постійного струму з добре визначеним позитивним стовпом для атомарної газової системи і продемонструвати існування законів подібності всіх режимів такого розряду.
Новый обобщенный метод, лазерные представления и теория катастроф были использованы для описания генерации плазмы самостоятельного газового разряда, возбуждаемого постоянным током. Обнаружены два вида областей самосогласованного эффективного электрического поля с качественно различными структурными свойствами положительного столба. Все это позволяет предложить логически самосогласованную классификацию и описание стационарного газового разряда постоянного тока с хорошо определенным положительным столбом для атомарной газовой системы и продемонстрировать существование законов подобия всех режимов такого разряда.
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Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 157-160 157
PLASMA GENERATION IN THE LOW- PRESSURE GAS D.C. DISCHARGE
WITH A SIMPLE EXAMPLE OF THE NOBLE GAS SYSTEM
P.F. Kurbatov
Institute of Laser Physics SB RAS, 630090 Novosibirsk, Russia, e-mail: ion@laser.nsc.ru
A new general approach, laser representations and catastrophe theory have been taken to describe plasma generation
of the self- sustained gas D.C. discharge. Two regions of a self-consistent effective electrical field with qualitatively
different structural properties of the positive column are found. All allows us to propose a logically self- contained
classification and a description of the low- pressure steady gas D.C. discharge with the well- defined positive column
for an atomic gas system and demonstrate the existence of the similarity laws for all regimes of such a discharge.
PACS: 05.40.-a, 05.45.+b, 51.10.+y, 51.50+v, 52.80.-s, 52.90+z, 64.60.-I, 64.70.-p
1. INTRODUCTION
The positive column is an elementary example of a
real discharge non-isothermic plasma, but until now a
satisfactory theoretical description has not been available.
The classical description of the positive column is given
in [1-5]. Its prediction of lack of striations, jumps, shock
waves, hysteresis and other phenomena is contrary to
fact, and the more so as a unit. The catastrophe model [6]
provides a good example for plasma states, jumps, and
hysteresis phenomena of the discharge plasma to be
described by means of a potential. How does this potential
come into existence in physics and what physical
processes are associated with it in the gas discharge?
It is well known that ‘the processes of generation and
absorption of charged particles are processes that not
only determine basically the electrical properties of the
positive column, but also lead to the generation of a self-
sustained gas discharge by itself’ [7-9] and ‘this behavior
is related with an influence of the electric field on the
rate of ionized reactions and the reactions impart wave
nature to the behavior’ [10].
Fig. 1. The typical current- voltage characteristic
If one notices the presence of structurally similar
regions at the current- voltage characteristic presented in
Fig. 1 one can see that they are similar. What is mainly
physical responsible for these similar laws?
The purpose of this paper is to show that the
consideration of the physical processes that take place in
the positive column allows one to answer all the above-
mentioned questions.
We consider that it is just the positive column and that
the physical processes going on in the gas discharge are
an integral of it, and we can determine properties of the
discharge and unambiguously characterize its type. This
is illustrated by the example of an atomic gas system as
ranked among the simplest and best to study. The realistic
model of the given system as the Van der Vaals model of real
gas and the Landau- Ginzburg model of superconductivity
contains the basic properties of real gas discharge.
2. MASTER EQUATION. GENERATION
AND SIMILARITY LAWS
From here on, a self- sustained gas discharge has a
positive column at a gas pressure of 1-100 mm of
mercury. The step processes and the volume
recombination of charged particles are of primary
importance in the positive column (see, for example, [1]
(pp. 238, 278)). An effort must be made to describe these
processes and analyze their effect on the properties of
this gas discharge.
There are two principal but mainly different
generation channels of electrons through the gas
ionization, like direct and step ionizations, in the
discharge. The absorption processes of charged particles,
in fact they all, reduce to their own mutual charge
neutralization; for example, with the assistance of a
geminate dissociative or an electron- ion recombination
(and other types of dissociative recombination) or a three-
body electron- ion recombination in a discharge plasma.
The latter process is known as the volume recombination
of charged particles.
In a discharge plasma, all available experimental data
(shown schematically in Fig. 1) show that the average
energy of electrons is always much less than the potential
of ionization of atoms in regions after the dark discharge,
such as regions 3,4,5,6,7. This fact indicates that the
mechanism of step ionization always dominates over the
other. Simple estimates, for example those given in [2],
confirm this assumption.
Let us derive a balance master equation for the density
of electrons (and, naturally, due to the plasma
quasineutrality for ions as well). Although we do not take
into account the phenomena in the vicinity of electrodes,
this equation allows us to explain correctly the major part
of the available experimental data and, in particular, the
behavior of the current- voltage characteristics. In what
follows, a modification of this equation made in order to
adjust it to more complete simultaneous equations
provides an understanding of the emergence of striations,
condition of their formation and hysteresis effect, etc. [7-
9].
It should be noted that even in this case so for the arc
discharge the parameter Ne / Na <<1, which governs the
mailto:ion@laser.nsc.ru
158
degree of ionization, does not exceed 0.1. Here, Na is the
density of neutral atoms. Away from the electrodes, the
balanced master equation for the density of electrons Ne
may be written as [7, 11]:
( ) ( ) ( )
2,,
2
U N EN x t Nee eD E F
t N xe
∂∂ ∂
= − + +
∂ ∂ ∂
, (1)
where
( ) ( ) ( ), 2, 0 1
U N Ee K N E N Ne e eNe
µ α α β
∂
− = = + − + +
∂
3 3...2N Ne eβ γ+ + − (2)
is the generalized force K (Ne , E) associated with the
potential ( ),U N Ee , called below a ionization potential.
The coefficients in the ionization potential ( ),U N Ee
depend on the inherent self-consistent electric field E
and are determined by the step ionization and other step
plasma reactions. The different plasma reactions give the
contributions to the coefficients , , , , ...1 2µ α β β . These
quantities are found by averaging over a plasma
ensemble. One can see that the structure of this potential
corresponds to the phenomenological potential of type
introduced in [6] to explain of the hysteresis phenomena
in a gas discharge. The ionization potential binds the
processes of generation and absorption of electrons
together. It has been just these processes, which have
lead to the form of the potential as being used for
description of the basic properties of the gas discharge
plasma system as a unit. Here, we mean that the local
quasineutrality condition N Ne i≈ takes place in the
positive column, where the step processes are essential
and the master equations are averaged over the typical
spatial size of the order of Debye radius dr , i.e. the given
grain size is considered as a point. Here, for simplicity we
consider that ( )D E is the ambipolar diffusion
coefficient, which does not depend on the electric field.
Equation (1) may be cited as a typical representative
of diffusion – plasma chemical reaction equations and
Langevin equations also [7, 11, 12]. Similar Langevin
equations form a basis for the superconductivity theory
(The Landau-Ginzburg equation), laser, etc. [13]. A
distinctive feature of our approach is that we have
refined the mechanisms of electron ionization and
absorption and have dropped the ambipolar diffusion
term and the fluctuating force F for the present.
Extremes of the ionization potential U(Ne,E), namely,
the condition
( ) ( ),
, 0
U N Ee K N EeNe
∂
= − =
∂
, (3)
define plasma states of gas discharges. By
transformations of a shift, an expansion etc., the
condition can be put in a given initial canonical form
( ) ( )( )
( ) ( ) 0
, , 3U N p E q Ee
N p E N q Ee e
Ne
=
′∂
′ ′= + +
′∂
. (4)
Taking to zero the derivative of the ionization
potential U (Ne, p, q) (4) with respect Ne can be conceived
of as a surface, p (E) and q (E) being coordinates in a
generalized three- dimensional (Ne, p, q) space in which
the structure of the potential to be reduced to a canonical
form after Thom [14]. The surface called as the
ionization equilibrium surface is a universal form called
the ‘wrinkle’. The projection of the surface on the (p, q)
plane is shown on Fig. 2. The properties of physical
systems, whose ionization equilibrium surfaces are
described by equations (3) and (4), are similar to each
other, as these transformations do not change the
topology of surfaces. The similarity laws for types of such
a discharge are a result of this fact.
All functions parameterized by control parameters
from range I (see Fig. 2) have a unique minimum. All
functions of the considered form parameterized by points
of range III should have the two local minima and only
the one local maximum.
Fig. 2. The projection of the surface on the (p, q) plane
There are paths (p, q) corresponding to its V-A
characteristic and its local ionization potential
U (Ne, p, q) for any one of gases in the gas discharge.
Every this path has a behavior of a given physical system
The physical system deals with the terminal group of the
behavior despite the fact that it moves over the equilibrium
surface along a path of complex shapes. From this
consideration of the ‘static’ properties of the potential
U (Ne, p, q) it follows that the qualitative behavior of the
plasma system depends on the parameters
(p,q) = (p(E), q(E)). Range I and regions with even
numbering (2, 4, 6…) in the current- voltage characteristic
[7] can be said with certainty to correspond to solutions with
spatial homogeneity of a certain type of the self- sustained
discharge. Range III and regions with odd numbering (1,3,5)
correspond to the transition zones between adjacent types of
self- sustained discharge [8,9]. Over these regions of physical
parameters, the plasma system takes place in either the
discharge mode with the appearance of jumps, hysteresis and
stratified phenomena. There are sheets (floors) of states with
various densities of electrons and ions in gas discharges, and
the transition regions are interconnected ‘escalators’.
Let us analyze the stationary solutions of equation (4)
in order of increasing ionization parameter Ne / Na. The
discharge shown in Fig.1 in region 1 of the current-
voltage characteristic can be determined from the linear
approximation of equation (1):
Ne1≅ -µ /(α - α0 ). (5)
If the direct contribution of the coefficient α can be
neglected ( )0α ≈ due to the threshold dependence on the
electrical field E , equation (5) takes the following form:
159
Ne1≅ µ /α0. (6)
A breakdown occurs as the field E reaches its threshold
or critical value 0 1,2
c cE E E= = . Expression (5) becomes
infinite at this point. It shows that it is necessary to take
into account the saturation effects in electron
multiplication. The two-particle electron-ion dissociative
recombination is mainly responsible for the electron
losses at the stage of the Townsend (shaping dark)
discharge [1-5], saturation being determined by a
nonlinearity that is quadratic on Ne .
Fig. 3. The modification of the ionization potential
U (Ne,, E) in close vicinity to the point of breakdown
Fig. 2 a show modifications of the ionization
potential U (Ne,, E) versus the strength of the (self-
consistent) electrical field E. The structures of transitions
at critical points are identical.
The physically inaccessible range, negative in Ne, in
the ionization potential U (Ne,, E) is isolated. It is shown
by a dashed line. These two minima, namely the usual
and the conditional, will be realized as two stationary
stable solutions of equation (3) in this approximation.
The conventional minimum of the ionization potential in
the neighborhood of zero is actually displaced and
determined by approximate formula (6) obtained earlier.
The stable solution is associated with the usual minimum
of the ionization potential U (Ne,, E) and corresponds to
the self- sustained mode, called the dark or Townsend
discharge. The densities of electrons and the current
within region 2 are determined by the formula
2
0 0
2 2 21 1 1
Ne
α α α αµ
β β β
− −
− + − +
;
. (7)
The transition from the non-self- sustained (chaotic)
mode to the self- sustained mode of the gas discharge
takes place in the vicinity of the critical point 0 1,2
c cE E=
and is associated with the Townsend breakdown. This
transition can be qualitatively distinguished. The
transition zone under study is an isolated point and is
separated from other transition zones corresponding to
the finite interval of the electric field between other types
of the self- sustained gas discharge. In the latter case,
stratified or hysteresis phenomena are observed.
Thus, the first self-consistent state, named the dark or
Townsend discharge, has the density of electrons (and
‘ionic skeleton’) (7). The existence of falling region 3 in
Fig. 1, when passing from the dark (Townsend) discharge
(2) to the glow one (4), suggests that it is necessary to
take into account the next expansion terms in equation
(4) down to the third degree on Ne. The mechanisms
providing the effects of saturation of the following order
are directly connected with volume recombination.
Region 4 of the current- voltage characteristic is in
agreement with the self- sustained mode called a glow
discharge. Here, the coefficient 1β met with in 2
1Neβ
has the sign opposite (plus) to that in the case discussed
above (namely, for the quadratic-law mechanism of
saturation, where this coefficient has the minus sign). It
is mainly determined by step ionization. The stationary
solutions of equation (4) in this approximation are found
from the expression [15]
4
3 2 3 2
3 3
2 3 2 2 3 2e
q p q q p q
N = − + + + − − +
, (8)
where
( ) ( )3
1 012 23 3
q q E
β α αβ µ
γ γγ
−
= = − + +
( )
2
1 01
3
p p E
α αβ
γ γ
−
= = − +
. (9)
Region 6 of the current- voltage characteristic is in
agreement with the self- sustained mode called an arc
discharge. The density of electrons (ions) is given by
similar formula (8) provided that
( ) ( )
( )
( )
3
1 012 23 232 2
q Eq
β α αβ µ
γ βγ β γ β
−
= − + +
−− −
=
( )
2
1 01
3 2 2
p p E
α αβ
γ β γ β
−
= = − +
− −
. (10)
In upper hardly rising region 4, where ( ) 0p p E= > , the
possibilities for the existence of a multiplication channel
through the metastable excited atoms have reached a
certain limit, and the competition between the direct and
stepwise processes is beginning again. The mode states of
the discharge corresponding to falling regions 3 and 5 are
unstable from stratification. We do not consider transition
region 5 for the same reason, and come to region 6 of the
current- voltage characteristic, which is called the arc
discharge or the electrical arc. Here, the multiplication
process of electrons takes place through the metastable
state of singly charged ions or ionic complexes, and the
main mass of ions becomes double ionized at the end of
region 6. Experimental data [1] correspond to sharp rise
region 6. They indicated the limit of the possibilities of
saturation of electron multiplication by stepwise processes
through singly charged ions in the discharge
characteristic. In region 6 of this characteristic, the rise of
the discharge characteristic is the same as that of region
4. The falling (transition) region 7 and subsequent
regions 8, 9, 10 ... have not yet been observed
experimentally. It seems that the following stages of
stripping are possible, and there should be other types of
self- sustained discharge.
160
Thus, the physical basis of the similarity between the
properties of the regions of current- voltage characteristic
represented in Fig. 1 is a replacement of one type of
stepwise processes, 2A e A e∗ ++ → + , by another
related stepwise process, 2A e A e+∗ +++ → + . Here,
,A A∗ +∗ are the metastable excited atoms and the singly
charged atoms, respectively, and ,A A+ ++ are the
remaining states of singly and double charged ions. These
plasma reactions are analogous to some chemical ones
[12, 13]; for example, Belousov-Zhabotisky reactions (see
references in [13]) in which, under homogeneous
conditions, there were periodic spatio-temporal
structures analogous in conception to such phenomena as
striations. What this means that Pekarec’s prediction
[10] has been justified.
3. CONCLUSIONS
It is clear that analysis of the balance control equation
(1) can be complicated by the inclusion of ambipolar
diffusion, the electron affinity electron, etc. The previous
classification, however, will remain and can be applied to
gas discharges in the pressure range where stepwise
processes are essential. In our opinion, this method for
attacking the problem of plasma generation gives a clue
to the understanding of the nature of globe lighting.
The introduction of a general potential provides a
basis understanding of the general concept of the
phenomena in gas discharges. The potential gives an
insight as Townsend, glow and arc types (or modes) of
the gas discharge and relations in between are closely
associated with plasma reactions. Analysis of plasma
system dynamics with the potential offers a clearer view
of how the phenomena in the gas discharge as a unit
come into being.
I wish to thank S. Bagaev and A. Tumaikin for
valuable discussions and support of this work.
REFERENCES
1. V.L. Granovsky. Electric current in a gas: steady-state
current /Ed. L.A. Sena and V.E. Golant. M.: “Nauka”, 1971 (in
Russian).
2. B.M. Smirnov. Introduction to physics of plasma. Moscow:
“Mir“, 1977.
3. Ir.P. Raizer. Gas discharge Physics. Berlin: “Springer-
Verlag“, 1997.
4. A.M. Howatson. Introduction to gas discharges. Oxford:
“Pergamon Press”, 1976.
5. Encyclopedia of Low Temperature Plasma. 2 vols. / Ed. V.E.
Fortov. Moscow: ”Nauka”, 2000 (in Russian).
6. G. Knorr. Hysteresis phenomena in plasma and catastrophe
theory // Plasma Phys. Contr. Fusion. 1984, v.26, p.949-953.
7. P.F. Kurbatov. Modern view on physics of low- pressure gas
D.C. discharge: Preprint. Novosibirsk: Institute of Laser
Physics SB RAS, 3 – 2001 (in Russian).
8. P.F. Kurbatov. Striations as if they were lasing mode// Proc.
of the 4th International Symposium Modern Problems of Laser
Physics, Novosibirsk, Russia, August 22-27, 2004 /
Novosibirsk: Institute of Laser Physics SB RAS, 2005, p. 263-
276.
9. P.F. Kurbatov. Jumps in current, shock waves and hysteresis
phenomena in pressure gas D.C. discharge plasma // Book of
abstracts of the 13th International Congress on Plasma Physics,
Kiev, Ukraine, May 22-26, 2006, part 1, p. 26.
10. L. Pecarek. Ionization waves (striations) in discharge
plasma // Physics-Uspekhi. 1968, v. 94, p. 463-500.
11. H. Wilhelmsson and E. Lazzaro. Reaction- diffusion
problems in the physics of hot plasmas. Bristol and
Philadelphia: “IOP Publishing”, 2001.
12. G. Nicolas and I. Prigogine. Self-organization in
nonequilibrium system. From dissipative structures to order
through fluctuations. New York, London, Sydney, Toronto:
“John Wiley & Sons”, 1977.
13. H. Haken. Synergetics. Berlin: „Springer-Verlag“, New
York: „Heidelberg“, 1978.
14. R. Gilmore. Catastrophe theory for scientists and
engineers. New York: “Chichester”, Brisbane, Toronto: “John
Wiley & Sons”, 1981.
15. G.A. Korn and T.V. Korn. Mathematical handbook for
scientists and engineers. Definition, theorems and formulas for
reference and review. New York, San Francisco, London,
Sydney, Toronto: “McGraw- Hill Book Company”, 1968.
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|
| id | nasplib_isofts_kiev_ua-123456789-110507 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:39:34Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kurbatov, P.F. 2017-01-04T17:20:25Z 2017-01-04T17:20:25Z 2007 Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system/ P.F. Kurbatov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 157-160. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 05.40.-a, 05.45.+b, 51.10.+y, 51.50+v, 52.80.-s, 52.90+z, 64.60.-I, 64.70.-p https://nasplib.isofts.kiev.ua/handle/123456789/110507 A new general approach, laser representations and catastrophe theory have been taken to describe plasma generation of the self- sustained gas D.C. discharge. Two regions of a self-consistent effective electrical field with qualitatively different structural properties of the positive column are found. All allows us to propose a logically self- contained classification and a description of the low- pressure steady gas D.C. discharge with the well- defined positive column for an atomic gas system and demonstrate the existence of the similarity laws for all regimes of such a discharge. Новий узагальнений метод, лазерні зображення і теорія катастроф були використані для опису генерації плазми самостійного газового розряду, що збуджується постійним струмом. Виявлено два види областей самоузгодженого ефективного електричного поля з якісно різними структурними властивостями позитивного стовпа. Усе це дозволяє запропонувати логічно самоузгоджену класифікацію й опис стаціонарного газового розряду постійного струму з добре визначеним позитивним стовпом для атомарної газової системи і продемонструвати існування законів подібності всіх режимів такого розряду. Новый обобщенный метод, лазерные представления и теория катастроф были использованы для описания генерации плазмы самостоятельного газового разряда, возбуждаемого постоянным током. Обнаружены два вида областей самосогласованного эффективного электрического поля с качественно различными структурными свойствами положительного столба. Все это позволяет предложить логически самосогласованную классификацию и описание стационарного газового разряда постоянного тока с хорошо определенным положительным столбом для атомарной газовой системы и продемонстрировать существование законов подобия всех режимов такого разряда. I wish to thank S. Bagaev and A. Tumaikin for valuable discussions and support of this work. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Low temperature plasma and plasma technologies Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system Генерація плазми в газовому розряді, що збуджується постійним струмом, на простому прикладі системи із благородного газа Генерация плазмы в газовом разряде, возбуждаемом постоянным током, на простом примере системы из благородного газа Article published earlier |
| spellingShingle | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system Kurbatov, P.F. Low temperature plasma and plasma technologies |
| title | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system |
| title_alt | Генерація плазми в газовому розряді, що збуджується постійним струмом, на простому прикладі системи із благородного газа Генерация плазмы в газовом разряде, возбуждаемом постоянным током, на простом примере системы из благородного газа |
| title_full | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system |
| title_fullStr | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system |
| title_full_unstemmed | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system |
| title_short | Plasma generation in the low- pressure gas D.C. discharge with a simple example of the noble gas system |
| title_sort | plasma generation in the low- pressure gas d.c. discharge with a simple example of the noble gas system |
| topic | Low temperature plasma and plasma technologies |
| topic_facet | Low temperature plasma and plasma technologies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110507 |
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