Table Of ContentIonization Waves in Electrical
Breakdown of Gases
A.N. Lagarkov 10M. Rutkevich
Ionization Waves in
Electrical Breakdown
of Gases
With 50 Figures
Springer-Verlag
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Tokyo Hong Kong Barcelona Budapest
A.N. Lagarkov I.M. Rutkevich
Center for Applied Problems Department of Mechanical Engineering
of Electrodynamics Ben-Gurion University of the Negev
Russian Academy of Sciences Beer Sheva 84105
Izhorskaya 13/19 Israel
127412 Moscow
Russia
-Library of Congress Cataloging-in-Publication Data
Lagar'kov, A. N. (Andrei Nikolaevich)
Ionization waves in electrical breakdown of gases / A.N.
Lagarkov and I.M. Rutkevich.
p. cm.
Includes bibliographical references.
ISBN-13:978-1-4612-8727 ·8
1. Breakdown (Electricity). 2. Electric discharges through gases.
3. Gases, Ionized. 4. Plasma waves. I. Rutkevich, I. M. (Igor
Maksimovich) II. Title.
QC71 1.8 .B7L34 1993
537.5' 32 - dc20 93-27847
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© 1994 Springer-Verlag New York, Inc.
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987654321
ISBN-13:978-1-4612-8727 -8 e-ISBN-13:978-1-4612-4294·9
DOl: 10.1007/978-1-4612-4294·9
Preface
In the years since the book of Lozanskii and Firsov "The Theory of Spark"
[1975] was published, a number of experimental and theoretical studies
in the physics of electric breakdown in gases were conducted. As a result
of these studies, the concept of a wavelike nature of breakdown initiated
by single high-voltage electric pulses or by a constant electric field was
confirmed.
Theoretical models in which the concept of breakdown in a constant
external field was developed were first exposed in the above-named book in
the chapter "Development of a streamer regarded as an ionization wave,"
written by Rodin and Starostin.
This book treats the initial stage of electric breakdown as a wave pro
cess. The wavelike nature of the phenomena under consideration is pre
sented for streamers and sliding discharges, for electric breakdown develop
ment in long discharge tubes as well as in gas-filled gaps.
Chapter 1 gives a qualitative consideration of phenomena determin
ing the electric breakdown of gases. The experimental data and theoretical
results are exposed and discussed with application to streamers, plane ion
ization waves, breakdown waves in long tubes, and propagation of sliding
discharges. The subject of this chapter may be considered as an area of
applications of different theoretical models, formulas, and estimates that
are presented in other chapters of the book.
Chapter 2 addresses the problem of obtaining a closed system of macro
scopic equations intended for the description of the electric breakdown
phenomena. In the case of strong electric fields existing in high-voltage
discharge devices, the deviation of the electron distribution function from
the equilibrium Maxwell distribution becomes essential. In this case, the
problem of calculating correctly the effective frequency of ionization, the
electron mobility, and other quantities entering into the equations of hydro
dynamics and macroscopic electrodynamics arises. This chapter is written
as a review and contains formulas for macroscopic description of ioniza
tion waves, including the estimations of their validity. In most theoretical
models of electric breakdown waves, a local dependence of the frequency of
ionization upon the electric field is used. In this connection, great attention
vi Preface
is paid to the discussion of the conditions for applicability of such a local
approach.
In Chapter 3, the theory of plane electric breakdown waves is devel
oped for the cases of stationary and time-dependent propagation. Station
ary ionization waves belong to the class of autowaves in active media that
can propagate only by the permanent action of an external source of en
ergy. From the mathematical point of view, the theory of plane breakdown
waves is similar to the theory of biological population waves proposed by
Kolmogorov et al. The problem of determining stationary ionization wave
in a given constant applied field has a continuum of solutions corresponding
to a continuous spectrum of possible velocities of propagation. Analysis of
the time-dependent problem based on the solution of the Cauchy problem
gives a method of selection of any stationary wave with a given velocity of
propagation regarded as the asymptotics of a nonstationary solution with
proper initial data.
Chapter 3 also contains an analysis of the formation of the moving
anode-directed discontinuities of electron density. The mechanism of the
formation of these discontinuities (ionization shocks) differs from the well
known mechanism of breaking the Riemann waves, leading to the formation
of shock waves in classical gas dynamics.
Chapter 4 treats the propagation of ionizing waves of electric field in
long shielded discharge tubes filled with a preliminary ionized gas. Accord
ing to Loeb, these waves may be called "the ionizing waves of potential
gradient." Experimental studies of breakdown waves in long tubes were
carried out for many years. First observations of such waves were made as
long ago as the 19th century by Wheatstone and Thompson. For a long
period, rich experimental material has been stored, and the main observed
properties of breakdown waves in tubes have been systematized. However,
the closed theoretical description, which allows us to calculate the structure
of these waves and the velocities of their propagation, has appeared only
recently.
Unlike the case of a plane discharge gap, the electric field in a shielded
tube attenuates far ahead of the breakdown wave front. The leading edge of
such a wave is determined by a wave of surface charge propagating along the
plasmald ielectric wall boundary. The spatial scale of the ionization wave
depends upon the geometrical parameters of a discharge device as well as
on the degree of preliminary ionization. In order to describe the ionization
wave propagation, a proper quasi-one-dimensional system of equations is
formulated. Numerical and analytical solutions of this system are obtained
in the form of stationary waves. These solutions show that the electric
breakdown development in a long tube is connected with propagation of a
solitary wave of an electric field. The equations describing solitary ionizing
waves did not appear previously in the nonlinear theory of wave propagation
in active media.
Preface vii
In Chapter 4, some peculiarities of the breakdown wave structures are
considered, including the oscillating structure of the trailing edge of the
ionization front and the solitary waves in a longitudinal magnetic field.
Chapter 5 contains a generalization of the theory of ionizing solitary
waves to the case of wave propagation in an unpreionized inert gas. It is
assumed that generation of electrons ahead of the ionization front is deter
mined by the associative ionization and the resonance radiation transfer.
The results of numerical modeling of stationary waves propagating along
the plasma/dielectric boundary in shielded tubes and in sliding discharge
systems are presented.
Similar to the motion of a breakdown wave in a tube, the propaga
tion of a sliding discharge front is stipulated by a two-dimensional wave
of an electric field. The transverse component of the electric field near the
dielectric surface is shown to play an important role in the development
of impact ionization and in the formation of a plasma sheet behind the
ionization front.
The theoretical model describing the propagation of slow stationary
breakdown waves in tubes is presented. An important feature of these waves
is the influence of ionic current on the mechanism of propagation. The effect
of a longitudinal magnetic field on the structure of a slow solitary wave also
is examined.
The final section touches upon the problem of high-voltage breakdown
waves. Runaway electrons arising at the wave front may be accelerated
in the electric field of a solitary wave up to very high energies, and the
relativistic approach to the problem in question is developed to describe
this situation. The value of limiting electron energy that may be reached
under these conditions is found.
The comparison of theoretical and experimental results is given in the
text, where possible. Since the book treats special topics, it does not cover
all of the various phenomena existing in the electric breakdown of gases.
The books concerning other aspects of the electric breakdown are listed in
the references.
We are grateful to E.L Asinovskii, L.M. Biberman, G.A. Lyubimov,
A.Kh. Mnatsakanyan, A.V. Nedospasov, Yu.P. Raizer, A.A. Rukhadze,
O.A. Sinkevich, A.N. Starostin, and LT. Yakubov for valuable discussion
of a number of problems considered in this book. We also thank V.A. Vino
gradov for his assistance in preparation of the manuscript.
We wish to acknowledge E.E. Kunhardt and R. Morrow, who sent us
the reprints of their papers used in Chapter 1.
Finally, we wish to express our thanks to Springer-Verlag New York
for their encouragement in preparing this book and excellent editorial work
with our manuscript.
Contents
Preface .......................................................... v
Chapter 1. Wave Phenomena Determining Discharge Development
in Gas Gaps .................................................... " 1
1 DynaIllics of StreaIllers ......................................... " 2
1.1 Development of an Electron Avalanche ........................ 2
1.2 Propagation of Anode- and Cathode-Directed StreaIllers ....... 11
2 Ionization Waves in Discharge 'lUbes and in a Sliding
Discharge Formation System .................................... 23
2.1 Experimental Study of Ionization Waves in
Discharge 'lUbes ............................................ 24
2.2 Formation of a Sliding Discharge ............................. 30
Chapter 2. Macroscopic and Kinetic Description of a Weakly
Ionized Gas in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
1 Basic Macroscopic Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35
2 Local Approach for the Frequency of Impact Ionization . . . . . . . . . . .. 48
2.1 The Townsend Ionization Coefficient and the Frequency
of Ionization by an Electronic Impact . . . . . . . . . . . . . . . . . . . . . . . .. 48
2.2 Conditions of Applicability of the Local Approach. Equation
for the Electron Distribution Function over Energies in a
Nonuniform, Nonstationary Plasma ........................... 51
Chapter 3. Theory of Plane Ionization Waves. . . . . . . . . . . . . . . . . . . . . .. 59
1 Stationary Plane Electric Breakdown Waves ......... , ............ 59
1.1 Ionization-Drift Models of Anode- and
Cathode-Directed Waves ..................................... 60
1.2 Influence of Diffusion and Photoprocesses on the Plane
Breakdown Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75
2 General Properties of Nonstationary Ionization Fronts ............. 94
2.1 Integrals of Nonstationary Equations. Reduction of a
General Problem to the Cauchy Problem for the Electric
Field Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
x Contents
2.2 Solution of the Cauchy Problem by the Method of
Characteristics. Conditions for the Breaking of a
Continuous Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
2.3 Propagation of Strong and Weak Discontinuities
of Electron Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103
3 Dynamics of Formation of the Anode- and Cathode-Directed
Waves from Initial Nonuniformities. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 118
3.1 Asymptotic Behavior of the Solution of the Cauchy
Problem for a Finite Initial Distribution of
Electron Concentration ..................................... 118
3.2 Development of Ionization Waves from Infinitely
Extended Distribution of Electron Concentration 127
Chapter 4. Propagation of Ionizing Electric-Field
Solitary Waves in Shielded Discharge
Tubes with Preionization ........................................ 145
1 Basic Equations and Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 145
2 The Effect of the Surface Wave on the Formation
of the Ionization Wave ......................................... 147
3 Averaging Two-Dimensional Equations and
Formulation of a Quasi-One-Dimensional Model .................. 149
4 Numerical Simulation of Stationary Waves ....................... 153
5 Analytical Model of an Ionization Wave .. . . . . . . . . . . . . . . . . . . . . . .. 160
6 Specialized Problems of the Theory of Breakdown
Waves in Tubes with Preionization .............................. 164
6.1 Limiting Transition to a Nonlinear Model of the Electric
Potential Diffusion. Conditions of Nonmonotonic Increase
of Current in a Wave ....................................... 164
6.2 Emergence of the Oscillating Structure of an
Ionization Wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 169
6.3 The Effect of a Longitudinal Magnetic Field on the
Structure of a Fast Ionization Wave. . . . . . . . . . . . . . . . . . . . . . . . .. 175
Chapter 5. Propagation of Electric Breakdown Waves Along
a Gas-Dielectric Boundary With No Preionization ................. 185
1 Breakdown Waves in Shielded Tubes Without Preionization . . . . . .. 186
1.1 Taking Account of Associative Ionization and
Resonance Radiation Transfer ............................... 186
1.2 Results of Numerical Calculations of Breakdown
Stationary Waves .......................................... 188
1.3 Analytical Estimate of Breakdown Wave Velocity ............. 192
2 Propagation of a Sliding Discharge Front as an
Ionization Wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 194
2.1 Assumed Equations and Problem Statement .................. 194
2.2 Formulating a Calculated Model of a Stationary Wave ......... 196
Contents xi
2.3 Structure and Velocity of Front Propagation ................. 201
3 Slow Breakdown Waves in Shielded Thbes ........ . . . . . . . . . . . . . .. 207
3.1 Features of a Quasi-One-Dimensional Solution
Describing Slow Waves ..................................... 208
3.2 Influence of a Longitudinal Magnetic Field on the
Structure of Slow Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 212
4 Solitary Wave of an Electric Field as a Source of
Runaway Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 214
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 221
Chapter 1
Wave Phenomena Determining Discharge
Development in Gas Gaps
The phenomena of breakdown in gas under the effect of impulse voltage are
extremely variegated. As shown in the monographs by Meek and Craggs
[1953], Raether [1964], and Korolev and Mesyats [1991]' the process of
breakdown formation is governed by numerous external conditions includ
ing, to a considerable extent, the value of overvoltage and the specifics of
the emergence of initiating electrons [the overvoltage is characterized by the
coefficient K = (Uo - Ubr)/Ubn where Uo is the pulse amplitude and Ubr
the value of static breakdown voltage corresponding to Paschen's law]. Best
studied at present is the Townsend mechanism of breakdown starting with
the development of solitary avalanches. In the case of Townsend breakdown,
the appearing space charge has little effect on the value of the electric field
inside the breakdown gap. The Townsend mechanism of breakdown usually
is observed at low values of K Ph, where P is the gas pressure and h is
the size of the interelectrode gap. In the case of high values of the K Ph
parameter, a situation is possible when the appearance and propagation of
a space charge serves one of the most important features characterizing the
breakdown development. In this case, the breakdown exhibits a clearly pro
nounced wave behavior. The closing of the breakdown gap by ionized gas
occurs owing to the motion of the high ionization region from one electrode
towards another, as observed, for instance, in discharge tubes or, as often
happens in the case of the streamer mechanism of breakdown, from the gap
center in both directions to each one of the electrodes. In some cases, the
wave breakdown occurs even at low values of K Ph. For instance, Korolev
and Mesyats indicate that if numerous electrons are initiated on a cathode
area whose size is less than the avalanche diffusion radius, the breakdown
will be of the streamer type in spite of low overvoltages.
The specific features of breakdown waves that will be regarded are
determined by the type of gas, the value of pressure, the value and rate of
variation of voltage at the electrodes, and by the geometry of a discharge
cell. The geometry determines the space distribution of the applied electric
field, so structure and dynamics of ionization front are greatly dependent
on it. The ionization waves described below are always characterized by
the presence of space charge, which leads to external electric field screening
