PLASMA PHYSICS
`
`Ionization relaxation in a plasma produced by a pulsed inert-gas
`discharge
`
`A. A. Kudryavtsev and V. N. Skrebov
`A. A. Zhdanov State University, Leningrad
`(Submitted September 29, 1931; resubmitted January 29, 1982)
`lb. Tekh. Fiz. 53‘ 53—61 [January 1983)
`
`A model is developed for the initial stage of ionization relaxation in a pulsed inert—gas discharge plasma at
`moderate pressures for E/n, values corresponding to ionization levels nf/nf>10““. It is shown that the
`electron density increases explosively in time due to accumulation of atoms in the lowest excited states. An
`approximate analytic solution is found for describing the behavior of the time and spatial increase in n‘, as a
`function of the Specific conditions. The proposed model is verified experimentally.
`
`PACS numbers: 52.25.Lp, 51.50. + v, 52.80.Dy
`
`The study of ionization relaxation in a plasma when
`the external electric field suddenly increases is of great
`importance in many areas of gas discharge physics and
`its applications.
`Interest in ionization relaxation in inert
`gas discharges has been stimulated recently by the rapid
`development of excimer lasers excited by pulsed electri—
`cal discharges (see, e.g., Ref. 1).
`In the present work
`we study ionization relaxation in an inert gas plasma at
`moderate pressures when the electric field strength in—
`creases discontinuously and the ratio E/n1 and degree of
`ionization ne/n1 vary over a wide range.
`In this paper we consider the initial stage of ioniza-
`tion relaxation when the ionization ne/n, increases from
`the initial value neg/n1 = 10‘8—10-7 to 10—5—10“. We are
`mainly interested in analyzing the buildup of ne and the
`behavior of the spatial distribution of he during the relaxa—
`tion process. We report experimental data on ionization
`relaxation in the range 8 < E/ni 2 30—40 Td.
`
`THEORY
`
`In general, the increase in the electron density ne in
`a plasma when the electric field E increases abruptly is
`described by a nonlinear system of kinetic balance equa—
`tions which cannot be solved analytically because of the
`usual difficulties.
`
`In this section we derive an approximate dynamic
`model of ionization that is based on theoretical results on
`various aspects of ionization relaxation in Refs. 2—9.
`
`1. When E /n, increases discontinuously, the relaxa—
`tion time for the electron energy distribution function for
`E/n1 S 3 Td is much less than the characteristic ioniza—
`tion growth times (tr < 1 t1.S).2’3 Therefore, the kinetic
`electron coefficients do not depend on time explicitly but
`are functions of the parameters E/nl, ne/nl, and the na—
`ture of the ionized atoms.
`In what follows, we assume that
`these parameters are known either from published experi—
`mental data or from numerical solution of the Boltzmann
`kinetic equation}!6
`2'. If we use k to label the excited states (with k : 2
`corresponding to the first excited state), the populations
`
`of the excited states with k > 2 are almost always quasi—
`stationary. However, quasistationarity may be violated
`for the lower metastable or resonance levels when He is
`small and radiation capture predominates. The balance
`equation for these levels must therefore be formulated
`in differential form after first combining the states into a
`single effective level characterized by an average energy
`and total statistical weight.
`
`3. Because of the importance of the transition k ‘1 ki
`1 in the collisional transition kinetics between excited
`states, it is helpful when describing the atomic distribu-
`tion over the excited states with k > 2 to use the machinery
`of the modified diffusion approximation (MDA) theory,“t5
`which gives the quasistationary populations in terms of
`ne and n2.
`
`4. Since for E/nl in the range of interest the ioniza-
`tion ng/n? is much greater than 104, we may neglect ra—
`diative processes and three-body collisions.
`
`5. The data in Refs. 7, 8 on the rate constants for
`formation of molecular ions and excimer molecules show
`that we may assume that these processes have little in—
`fluence on the ionization growth rate when an electric field
`is suddenly applied to a weakly stabilized inert gas at mod'
`erate pressures [n1 < (5—?) ~ 10” em'”].
`It can be shown
`using the arguments in Ref. 4 that under these conditions
`the associative ionization channel is of minor importance
`compared with the stepwise ionization channel.
`
`Using the above remarks, we can deseribe the ionizav
`tion relaxation using the following system of equations:
`
`a E
`% : nineBla + ”271,925 + ”Z’HEI “' \‘Ta,
`
`
`(in
`mirnantw ngnckm
`rig/1,9,9
`”and;
`nQAMO.‘u Vi}
`
`(1)
`
`(2)
`
`Here 11,, ha, and he are the atomic densities in the ground
`and first excited states and the electron density, respec'
`tively; km, k2, are the rate constants for the collisional
`transitions 1 7— 2; file and p28 are the rate coefficients fol‘
`
`130
`
`Sov. Phys. Tech. Physi 28(1 ). January 1983
`
`0038-5662/83/01 003006 $03.40
`
`30
`
`4
`GILLETTE 1404
`
`GILLETTE 1404
`
`

`

`
`
`
`
`
`FIG. 1. Diagram showing the relative Sizes of the electron
`fluxes in terms of the atomic energy levels for the slow (a)
`and fast (17) stages. The width of the arrows indicates the
`magnitude of the electron flux. The horizontal arrows give
`the diffusion fluxes of electrons and excited atoms reaching
`the walls of the discharge tube.
`
`Estimates using the equations in Ref. 4 show that
`under typical conditions (R 3 1 cm, n‘ 3 10“5 cm‘3), the
`radiation capture factor is 62‘ 2 10’3—10‘4. Therefore, the
`effective radiative lifetime of level 2 is long and the level
`may be regarded as quasimetastable.
`Under these same conditions, the characteristic ex~
`cited atom diffusion times are TD2 2 10'3—10~2 s, so that
`we may neglect V F2 compared with nznefi26 in (2).
`
`The above arguments Show that the three—level approx—
`imation can be used to describe ionization buildup under
`our assumptions.
`In dimensionless variables. the equa—
`tions for the ionization kinetics take the form
`
`aN/BtszM—l—CNde,
`aM/ar : N — bNM,
`
`(5)
`
`4: = tnlku, M it“) :11“. N lfl, = 1,
`IV 215/1260,
`M = [IQ/72:0,
`b : ”mph/”1km C : lain/kw»
`d = “DU/”1’5?
`(6)
`
`Equations (5)-(6) easily yield the following relation
`between N and M:
`
`N z 41 + Mo -_ 111+ (1+?er 1n [*8: ml]
`
`(7)
`
`so that the solution reduces to a quadrature.
`
`Using the rate constant data in Ref. 5, we find that
`b << 1 and 0 << 1 in all cases of practical interest (E/ni <
`300 Td, ne/nl < 10‘5).
`in a steady-state plasma, we usually
`have Mo €
`1 [Ref. 9].
`
`Using the smallness of b and c, we find from (5)—(7)
`that dN/dM m c — d + bM0 << 1 in the initial stage, i.e., the
`number of atoms in the first excited states increases
`rapidly for a relatively slow change in the electron density.
`The rate of ionization then increases with time and rises
`most steeply for M > c/b, For nearly stationary n2 values
`(M = l/b), so that n2 is changing slowly, there is an ex~
`plosive increase in he. The subsequent increase in he then
`reaches its maximum value, equal to the rate of excitation
`LIN/d7 = N(i + c — d), which is several orders of magnitude
`greater than the ionization rate during the initial stage.
`
`The behavior of the increase in ne with time thus
`enables us to arbitrarily divide the ionization process into
`two stages, which we will call the slow and fast growth
`stages. Figure 1 illustrates the relationships between
`the main electron currents in terms of the atomic energy
`levels during the slow and the fast stages.
`
`Since ”6 rises at an ever increasing rate (which is
`several orders of magnitude larger than the initial rate)
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`31
`
`
`fiect ionization from the ground state to the first excited
`i 1; list is a generalized coefficient for stepwise ioniza—
`
`from the quasistationary levels (k > 2); A21 is the
`mucous transition probability 2 —> 1; 921 is the capture
`
`ctor for resonance radiation, calculated using the Biber—
`-—Holstein equations“; \7I‘e and V P2 are the diffusion
`
`axes of the electrons and excited atoms, and are given
`Troximately by VI‘k 9: nkI/Dk [for a cylindrical geometry,
`(2 (2.4)2Dk/R2l.
`
`‘ It can be shown using the MDA theory“ that
`
`2K
`1 my e—Em
`nfinulisrt = Hznfi dr—l/Eg; fi;
`1‘;
`
`6 e
`
`xp (Ola/Eta,
`
`
`
`
`
`
`
`7 ems/S.
`
`l4)
`
`To facilitate comparison, we have retained the nota—
`
`used in Ref. 4,
`
`
`:The nonlinear system (1)—(2) can be solved numerical—
`the coefficients and initial conditions are specified.
`
`g the rate constants given in Refs. 5—8 for the various
`
`sses, we can make a series of simplifications making
`
`sibio to solve (1)-—(2) analytically for E/nl and ne/nl
`
`range of interest.
`
`
`ince Bze a 104—10“? cma/s [Refs. 5, 8], we see from
`
`d (4) that the ratio fire/B st of the coefficients for
`
`and stepwise ionization from the first excited level
`
`ds on ne and on the electron temperature Te. For
`
`.5' 1013 cm"3 and Te 3 1 eV, the stepwise ionization
`
`e1 from levels with k > 2 is unimportant (3st << B26)
`miss most of the radiation escapes.
`
`
`
`
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`
`

`

`taulfl-cm-s/z
`5§79fivnfln‘"3.
`
`15
`
`:3
`[/n,. Td
`
`s.
`
`7d
`
` FIG. 3, Duration T; for a discharge in argon. p = 11.4 (a) and 3.7 in"
`
`a: 1) ago — 1.6- in” cm'a: 2) 2.7-10”; 3) 4.5-10”; 4) 6.4.1010;
`(b).
`591011; 6) 1,6 . 10“; b: 1) 9.7. 10’; 2) 3.101“; a) 6.8'1010; 4) 1.5-1011
`
`once steady conditions have been reached during the fast
`stage, ionization builds up explosively when the external
`field is constant (the kinetics of gas—phase reactions are
`classified in Ref. 10).
`
`We can solve Eqs. (El—(’7) analytically and thus analyze
`in more detail the behavior of ne(t) and n2 (t) as ionization
`develops.
`
`Since bM < 1 during the slow stage, if we keep only
`the linear terms in bM in (7) we find from (5)—(7) that
`
`N = blip/2 + (6 —~ d) M + i — hing/2 % (c i d) Mm
`
`T 2 b VET;
`
`X ami2:12:massages:—zsii
`arctg [3W] — arotg [W] ,
`A > 0,
`
`where A is given by
`
`A S (2/5) ~ ”[0 + (c ~ d)fl’l2
`
`(3)
`
`(9b)
`
`(10)
`
`and describes the relative contribution from the various
`processes in (5) for small times.
`
`The conditions for ne to increase with time are that
`ionization should always develop eventually if A > 0; if
`A s 0, the condition is that bMo + c > d.
`
`It can be shown using (6), (7) that the curve M(T) has
`an inflection point at M1 = N, - (c - d)/b «2 0.8/1), after
`which M changes slowly. Since M1 differs from the sta—
`tionary value by 20% and the various rate constants are
`only known to within a factor of two,5 may assume without
`any loss of accuracy that at subsequent times the ioniza—
`tion has become stationary:
`
`1
`1
`(1 + 6 ~ d)
`N1 '
`ln~N—
`M=—b-, 1:
`The rate constants for the elementary processes are
`known accurately enough to permit Eqs. (9) above to be
`used to describe the time changes in ne and n2 as far as
`the inflection point M = M1.
`
`(11)
`
`Equations (9), (11) derived above readily yield expres—
`sions for the characteristic times of the slow and fast
`stages. Since we have bM + c - (1 >> inf-{Kl prior to the
`start of the abrupt rice in ne, the duration T5 of the slow
`stage can be found with sufficient accuracy from (9) by set—
`ting the first quotient in the logarithm in (9a) equal to one
`
`and taking the first arctangent in (9b) equal to 7r/2. Equa‘
`tion (11) shows that the characteristic time for the last
`stage is 7 ye ln(N/N1) << TS. We see by inspecting the fem
`of the above solutions that “e builds up explosively with
`time.
`
`It should be emphasized that the reason for this be-
`havicr in ne(t) is quite universal and can be traced to the
`fact that we almost always have 516 << km-
`
`The explosive increase in ne(t) is most apparent when
`A > O, which corresponds to early times and small direct
`ionization.
`In this case, he does not increase more than
`tenfold prior to the onset of explosive growth [M > #112751},
`
`These expressions can be used to determine the limirs
`of applicability of the two simplest ionization models which
`are often used in practical calculations. These are the
`direct ionization model, in which the ionization is deter-
`mined by the appropriate rate constant or by the first
`Townsend coefficient, and the "instantaneous ionization“
`model, in which the ionization rate is taken equal to the
`rate of excitation. Equations (8)—(1l) imply that direct
`ionization predominates only for times 'r < 1', (M = V275)
`after the field is applied, and that the "instantaneous ioni-
`zation“ approximation does not become valid until times
`t e Ts after the field was first switched on.
`
`EXPERIMENTAL RESULTS
`
`We studied ionization relaxation in He, Ne, Ar, and Kr
`for initial gas densities in : (0.5~5) .1017 cm"3 by applying
`an additional electric field to the preionized gas. The
`discharge occurred inside a cylindrical tube of diameter
`2R = 2.5 cm and the distance between the electrodes was
`L = 52 cm. The gas was preionized by applying a do cur-
`rent ip = 0.5—20 mA. The parameters of the positive plas'
`ma column were calculated using the theory developed
`in Ref. 9 from the experimentally recorded current dens“)E
`and EO/ni values. The initial density “so on the axis valid
`in the range 5-109—4 .10“ cm'g.
`
`A voltage pulse with rise time (1—2) - 10-7 s of positii’i,
`polarity with respect to the cathode was applied to the
`tube using a specially designed electrical circuit. The
`electric field was measured using several detectors
`soldered into the tube along its axis. The total voltage
`across the tube and the voltage between the probes were
`recorded bya capacitative divider (Ci: 2 pF and (32 = 100
`pl”) capable of transmitting rectangular pulses with rise
`time ~ 10"7 5 without appreciable distortion. The dis—
`charge current was recorded using a zero—inductance
`
`32
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov.
`
`32
`
`

`

`75
`
`._.|
`
`E/n,, Td
`
`Tiesistor connected in series with the oathode—to—ground
`otion of the tube. The local dependences ne(t) were re—
`
`rded by observing the plasma emission intensity, which
`proportional to 11% (p 2 1). The density of the neutral
`
`gas was monitored using the interferometric technique
`'fiiescribed in Ref. 11 and adual—trace oscillOSCOpe was used
`£614 all the measurements. Special experiments were con—
`ducted to verify the unimportance of such factors as the
`proximity of the shields and grounded objects or the shape
`and composition of the electrodes (we had pL > 200 cm-
`‘torr (Ref. 12]), which do cause appreciable effects during
`breakdown of a cold gas .12,” We were also able to repro—
`iduce the experimental conditions with high accuracy. We
`t'nieasured E and the plasma emission intensity at different
`
`distances along the tube axis in order to find how the pa—
`
`rameters of the gas discharge plasma vary along the tube
`{and ascertain the importance of various mechanisms in;
`lved in discharge formation after an abrupt increase in
`
`the field strength.
`
`Figure 2 shows some typical measured curves. The
`_'oscilloscope traces of the current (a), tube voltage (b),
`ltage differences between the probes (c, d), and the emis—
`
`ant. This is followed by a sudden rise in the current, ac~
`
`mpanied by a voltage drOp across the tube. We also see
`
`at the field increases almost simultaneously (to within
`
`'7 s) over the entire length of the positive column and
`en remains constant for t < "rs. The plasma emission
`
`tensity from different regions along the length of the tube
`so starts to increase almost simultaneously and repeats
`
`e current trace.
`
`
`
`
`Our measurements revealed that for t < TS, ne/n1
`creases by less than a factor of 102, i,e., we have ne/in <
`at the end of the slow stage.
`
`
`A similar delay in the current increase has been noted
`many other workers when an electric field is suddenly
`
`ied (see e.g., Ref. 12). This lag might be caused by
`
`6 small velocity of the ionization wave down the tube from
`
`é hifill—voltage electrode to the grounded electrode,”"4
`y onset of instability in the uniform quasistationary
`charge when the field is applied [possibly caused by
`
`ooesses near the electrodes; of. Ref. 15]. Taken to—
`‘ther the above findings Show that in any case, neither
`
`pagation of an ionization wave nor processes at the
`
`eotrodes determine the duration 'rs of the slow stage.
`
`The lag in rapid current buildup has often”!” been
`
`:::;:\ss\\\\
`
`
`
`
`WI i v I l | J,,,,,1
`
`
`74
`48
`7B
`20
`22
`2”
`26
`38
`30
`
`FIG. 4. Curves for T3 vs. E/nl for a discharge in krypton; p :
`10.5 (a) and 6 tan (b). 3: 1151160: 4.1010 0111-3; 2) 1011. 3)
`1.5-10“; 4) 3.3-10“; b: 1) 3.5-10”; 2) 9.5.10“; 3) 1,6.
`10‘“: 4) 3.2-10“.
`
`L
`
`FIG. 5. Radial distribution ne(r) as a function of time for A > 0 in an argon
`discharge. p = 11.4- torr, n80 = 101" cm”, 15 = 39 us.
`
`attributed to the finite time required for growth of ther—
`mal ~ionization instability. Interferometric measurements
`show that for t < 13 the gas density does not drop by more
`than 1%. Estimates using the equations in Ref, 16 reveal
`that for such small changes in iii, the growth time for
`thermal-ionization instability is much larger than Ts.
`
`Thus, under our conditions the slow increase in ne
`during the initial stage is due to the combined effects of
`kinetic processes occurring in the bulk of the discharge
`for constant E/nl, quasistationary electron energy dis-
`tribution functions, and ne/n1 values < 10—5.
`We now compare the experimental results for Ar and
`Kr with the theory developed above.
`
`The points in Figs. 3, 4 give the experimental values
`as a function of the specific conditions (n1, n90, E/nx).
`Since in our experiments we had A > 0 [cf. (10)], Eq. (9b)
`implies that the duration of the slow stage is given by
`
`
`
`T 4/2 ”—1,.
`bVwiW+fit
`
`
`m‘ccthW“
`
`[V1— % (Mo+ 67’):
`
`.
`
`(12)
`
` Sov. Phys. Tech. Phys. 281 l ), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`33
`
`

`

`cm‘2
`
`nyR,
`
`3.9
`
`#5
`
`.775 Z7
`
`Z7
`
`.73
`
`The form of the radial distribution ne(r) will then be simi.
`lar to the initial distribution and no well —defined plasma
`column is produced. Our model thus predicts slight de-
`formation of the initial distribution nen(r) for A < 0 but
`substantial deformation when A > 0.
`
`Experiments (of. Refs. 18, 19) have shown that ioniza.
`tion occurs uniformly over a cross section of the discharge
`tube when a field is applied to a preionized gas; however,
`if p > p0 the ionization is highly nonuniform and a narrow
`plasma column forms on the axis.
`In particular, for argon
`p0 = 1 torr for R = 2—4 cm (R is the tube radius) [Ref. 18]
`and p0 = 2—3 torr for R : 1 cm [Ref. 19]. For discharge
`in helium, the experiments indicate that ionization is uni-
`form for pressures € 10 torr when R : 1 cm.
`
`Our model accounts well for these experimental find.
`ings. Figure 6 shows calculated curves for a discharge in
`argon. We see that for pH < 1 cm -torr, ne cannot in-
`crease unless A < 0.
`In this case ionization develops more
`uniformly in the bulk, in good agreement with experiment,
`Since for the heavy inert gases (Ne, Ar, Kr, Xe), km, me.
`[323, VDa all have the same order of magnitude, the cor—
`responding curves behave similarly for similar values of
`pH. Helium is an exception, since under our conditions
`VD“ is much larger and [316 much smaller than for the
`heavy inert gases. The region A < 0 will therefore cor‘
`respond to larger products pH, and this is also in agree—
`ment with the experimental findings.
`
`We have thus develOped a model for the initial stage
`of ionization relaxation in a pulsed inert—gas discharge
`plasma at moderate pressures for E/n1 values correspond-
`ing to equilibrium ionizations rig/n? >> 10'4. We conclude
`from a comparison of the experimental Spatial and time
`dependences of he that‘the model is quite accurate. We
`have shown that the increase of he with time is explosive
`because atoms accumulate in the lowest excited states.
`Our results are important for analyzing the role of step—
`wise ionization processes in the buildup of instabilities
`in self—sustained and externally maintained discharges.
`Under our conditions, the above equations have the ad«
`Vantage that they clearly exhibit the various ionization
`mechanisms, so that their specific effects can be studied
`as a function of the experimental conditions. Since the
`effects studied in this work are characteristic of ioniza-
`tion whenever a field is suddenly applied to a weakly
`ionized gas, they must be allowed for when studying emis-
`sion mechanisms in pulsed gas lasers, gas breakdown,
`laser sparks, etc,
`
`17,20
`
`1A. V. Eletskii, Usp. Fiz. Nauk 3.2g, 279 (1978) [Sov, Phys. Usp. a 502
`(1978)].
`2c. v, Naidis, Zh. Tekh. Fiagl. 941 (1977) [Sov. Phys. Tech. PhyS. 23.
`562 (1977)].
`3A, A. Belevtsev, Teplofiz. Vys. Temp. 31, 1138 (1979).
`4L. M. Bihei'rnan, V. S. Vorob‘cv, and I. T. Yakubov, Usp. Fiz. Nauk 191.
`353 (1972): 1233,. 233 (1979) [Sov. Phys. Usp. _1_§_. 375 (1973); 2?: 411 (1979»
`5N. L. Aleksandrov, A.,M. Konchakov, and if. E, Son, Zh, Tekh. Fiz. if].
`481 (1980) [Sov. Phys. Tech. Phys. E, 291 (1980)].
`‘w. L. Nighan, Appl. Phys. Lett. 3_2,, 424 (1978).
`7B. M. Smirnov, ions and Excited Atoms in Plasmas [in Russian), Atomizdfll:
`Moxow (1974).
`3M. G. Voitik. A. G. MolChanov, and Yu. G. Popov, Kvantovaya Elektron-
`(Moscow) 5 1722 (1977) [Sov.1. Quantum Electron. 3 976 (1977)].
`
`E/nu'l'd
`FIG. 6. The behavior of “e in the bulk of an argon discharge. 1) neg/n; =
`10‘s; 2) 10‘7. Stepwise ionization predominates in region 1, direct ioniza-
`tion processes predominate in region 11, and “a does not increase in region
`111.
`
`The solid curves in Figs. 3, 4 give 75 calculated from
`(12) using values for km B1ev and .323 from numerical cal-—
`culations in Ref. 5.
`
`Equation (12) shows that for small E/ni, when A a
`2/b, we have “"5 = w/JT‘ZF.’ It follows that (ninety/275 =
`(klzfizeri/Z : const for a fixed value of E/ni. For large
`E/nl we have A “r 0, and 7s —> 2/(neofize + nlfile -— VDa)
`is only weakly dependent on neo-
`
`Figure 3a also gives experimental values of the pa—
`rameter (n1n60)1/2'rs, which for E/n1 = const remains con—
`stant to within the experimental error for n60, hi, and 7s
`varying over wide limits. For large E/n1 [Fig. 3b, Fig.
`4b] Ts becomes almost independent of he“, as predicted
`by our theory.
`
`We also note that thefunction ne(t) calculated using
`(8)—(9) accurately describes the experimentally observed
`increase in the current and plasma emission. We thus
`conclude that theory and experiment are in both qualita—
`tive and quantitative agreement.
`
`The above equations can be used to analyze the time
`change of the radial distribution ne(r, t) when an electric
`field is suddenly applied to a gas for which the initial dis—
`tributions ne0(r) and n20(r) are known. This problem is of
`interest in terms of understanding the mechanism re—
`sponsible for formation and constriction of the current
`channel in a pulsed discharge}7
`
`The solutions of system (5), (6) show how the initial
`distribution neo(r) is deformed when a field is switched on.
`In the situation discussed above (corresponding to A > 0)
`Eqs. (8) -(9) imply that the growth rate of ne(t) depends
`strongly on neg, particularly for large values of A (A m
`2/b). This results in a sudden increase in the magnitude
`of the initial irregularities in ne(r) for t < TS. As an il~
`lustration, Fig. 5 shows curves giving the time dependence
`of ne(r) calculated from (9b) for E/ni = 12 Td and neo :
`10” cm"3 for an argon discharge. According to Ref. 9,
`the initial “co and n2.) distributions in a glow discharge
`at moderate pressures can be approximated by Bessel
`functions, and we took M0 = 0.1. Under these assumptions,
`we see that the theory predicts that the distributions will
`become highly nonuniform at times t 3 TS after the field is
`turned on.
`
`For large E/nt, A < 0 and the growth of ne during the
`slow stage lot. (9a)] is determined by direct ionization.
`
`34
`
`Sov. Phys. Tech. Phys. 28H ), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`34
`
`

`

`L. Gmnovskil, Electrical Currents in Gases. Steady Currents [in Russian],
`uka. Moscow (1971).
`
`N, Kondrat‘ev and E. E. Nikitin, Kinetics and Mechanisms in Gas-
`356 Reactions [in Russian], Nauka, Moscow (1974)‘
`N, Skrebov and A. 1. Skripchenko, Teplofiz, Vys, Temp, _]__9_, No, 3
`
`1981).
`
`5.1), Lozanskii and O. B. Firsov. Spark Theory [in Russian], Atnmizdat,
`Moscow (1975).
`' p_ Abramnv. P. I. Ishchenko, and I. G. Mazan'ko, Zh. Tekh. Fiz. fig.
`5 (1930) [Sov. Phys, Tech. Phys. gg. 449 (1980)).
`
`’ AsinovSkiiy V, N, Markov, N, S. Samoflov. and A. M. Ul‘yanov.
`
`ploflz- vys. Temp. $6.: 1309 (197B).
`73, Aleksandrov. R. B. Gurevich, A. V. Kulagina, et a1., Zh. Tckh. F12.
`
`45
`05 (1975) [Sov. Phys. Tech. Phys. _2_0, 62 (1975)].
`
`“E, P, Velikhov, V. D. Pis‘mennyi. and A. T. Rakhimov. Usp. Fiz. Nauk
`1‘22, 419 (1977) {Sm}. Phys. Usp. _2_0_. 586 [1977)],
`"KP. Napartovich and A. N. Starostin.
`in: Plasma Chemistry, B. M.
`Smirnov, ed., [in Russian] (1979). pp. 6, 153.
`”D, N. Novichkov, in: Topics in Low-Temperature Plasma Physics [in
`Russian], Nauka i Tekhnika, Minsk (1970), p. 459.
`19M. N. Polyanskii. V. N. Skrebov, and A. M. Shukhtin. Opt. Spektrosk.
`84, 28 (1973).
`25-1). Dautherty, J, A. Mangano, and I. H, Jakob, Appl, Phys, Lett. :2};
`581 (1976).
`
`Translated by A. Mason
`
` Sov. Phys. Tech. Phys. 28(1), January 1983
`
`0038-5662/83/01 0035-04 $03.40
`
`35
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