`
`Ionization relaxation in a plasma produced by a pulsed inert-gas
`discharge
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`A. A. Zhdanou State University, Leningrad
`(Submitted September 29, 1981; resubmitted January 29, 1982)
`Zh. 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 n.°/n{’>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 ina 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/n1 increases from
`the initial value neo/n1 =10‘8-10'7 to 1O"5-10"‘. We are
`mainly interested in analyzing the buildup of ne and the
`behavior of the spatial distribution of ne during the relaxa-
`tion process. We report experimental data on ionization
`relaxation in the range 8 < E/n1 € 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 ps).2'3 Therefore, the kinetic
`electron coefficients do not depend on time explicitly but
`are functions of the parameters E/n1, ne/n1, 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.”
`
`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 7* ks
`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,“v5
`which gives the quasistationary populations in terms of
`he and n2.
`
`4. Since for E/n1 in the range of interest the ioniza-
`tion ng/n9 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 < (5-7) - 10” cm'3].
`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 describe the ioniza-
`tion relaxation using the following system of equations:
`
`W : nlneiala + n2n:p2e + nilnapst — VP»
`
`% = nlnekfl " n2"aka1 _’ "£714.32: — ”a”.Et '“ ”2A-41°21 ‘ VP?
`
`(2)
`
`Here 11,, n2, and ne are the atomic densities in the ground
`and first excited states and the electron density, respec-
`tively; kn, km are the rate constants for the collisional
`transitions 1 —.-_ 2; file and B23 are the rate coefficients fol‘
`
`30
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`0038-5662/83/01 0030-06 $03.40
`
`INTEL 1104
`
`3°
`
`INTEL 1104
`
`
`
`the walls of the discharge tube.
`
`FIG. 1. Diagram showing the relative sizes of the electron
`fluxes in terms of the atomic energy levels for the slow (a)
`and fast (b) 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
`
`Estimates using the equations in Ref. 4 show that
`under typical conditions (R S 1 cm, n, S 1016 cm'3), the
`radiation capture factor is 02, 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 7-D2 2 10'3—10’2 s, so that
`we may neglect V P2 compared with nznefize in (Z).
`
`The above arguments show thatthe 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
`
`ON/at:bN1l1+CN—dN,
`OM/01: = N — ZDNIVI,
`
`(5)
`
`M : "2/"pow N = no/"'20:
`b = ’7~aol32a/711,512:
`
`" = tn1k12I M |:=o = M0» N |1=o :1-
`C :B1a/kl21
`d = “17n/n1k12-
`(6)
`
`Equations (5)—(6) easily yield the following relation
`between N and M:
`
`N—l
`
`; M, M ; ‘‘+°"‘‘’1n[‘‘“’’’’’°’]
`b
`(1— MI)
`
`(7)
`
`so that the solution reduces to a quadrature.
`
`Using the rate constant data in Ref. 5, we find that
`b << 1 and c << 1 in all cases of practical interest (E/n1 <
`300 Td, ne/n1 < 10'5).
`In a steady—state plasma, we usually
`have M0 €
`1 [Ref. 9].
`
`Using the smallness of b and c, we find from (5)-(7)
`that dN/dM 2 c — d + bM,, << 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 = 1/b), so that n2 is changing slowly, there is an ex-
`plosive increase in ne. The subsequent increase in ne then
`reaches its maximum value, equal to the rate of excitation
`dN/d7 = N(1 + 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 ne rises at an ever increasing rate (which is
`several orders of magnitude larger than the initial rate)
`
`direct ionization from the ground state to the first excited
`\.1‘é‘,e1; gst is a generalized coefficient for stepwise ioniza—
`figin from the quasistationary levels (k > 2); A2, is the
`if ntaneous transition probability 2 —+ 1; 021 is the capture
`1
`fiictor for resonance radiation, calculated using the Biber—
`.
`‘ n—-Holstein equations4; vre and V P2 are the diffusion
`:'f‘fi1'xes of the electrons and excited atoms, and are given
`gfiproximately by VI‘k 2 nkn/Dk [for a cylindrical geometry,
`( .~.:(2.4)2D1(/R2].
`
`if
`if It can be shown using the MDA theory4 that
`
`
`
`23
`1 Hg/3 9-WT»
`(T2)
`El:
`nflneflgt : nflnefi 3 ‘/Egg H2
`dz)
`
`’
`
`ere
`
`exp (0.2E?_;/Efi),
`
`
`
`: (n,/4.5 - 1015'/« T;'/a (n, — in cm'3. T, — .in'eV),
`,
`4
`X _ .
`go.2, Hg/—_—’ld.6 eV, x(.»c)—.=mSe ’tl=dt,
`0
`\/2
`4
`_
`(4,
`....a,.,
`E135,/2521.7 . 10 7
`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.
`H gthe rate constants given in Refs. 5-8 for the various
`ocesses, we can make a series of simplifications making
`ssible to solve (1)-(2) analytically for E/n1 and ne/n1
`e range of interest.
`
`Since [329 2 10'3—10‘7 cm3/s [Refs. 5, 8], we see from
`:3C- 3
`g
`) that the ratio fize/[3 St of the coefficients for
`ct and stepwise ionization from the first excited level
`nds on ne and on the electron temperature Te. For
`5- 1013 cm‘3 and Te 3 1 eV, the stepwise ionization
`nel from levels with k > 2 is unimportant (List << B26)
`use most of the radiation escapes.
`
`Sov. Phys. Tech. Phys. 28(1), January 1 983
`
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`31
`
`
`
`
`
`FIG. 3. Duration ‘rs for a discharge in argon. p = 11.4 (a) and 3.7 to"
`(b).
`a: 1) H60 = 1.6-101° cm-’; 2) 231-10”; 3) 4.5-101°; 4) 6.4401»,
`5) 10”; 6) 1.6-10”; b: 1) 9.7- 10*‘; 2) 3-10”; 3) 6.8-101°; 4) 1.6-1gl1
`
`
`
`22
`
`211
`
`E/nu 'Td
`
`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. (5)-(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
`
`1V=bM”/3-i-(0—d)M-i-'1*bM%/2—(C—d)Mo.
`
`(8)
`
`1‘:
`
`2
`an/W
`
`1
`_1
`
`2
`
`'1
`
`(bIlI+c—d—b\/|Tl|)(bzl[.,+c—d+b\/W)
`_
`_
`
`(bM+c—d +12)/1 A|)(bMa+c—d—-b\/|A]) A <(%’a)
`
`where A is given by
`
`A =(2/b)~ [Mo+(C— d)/bi”
`
`(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 5 0, the condition is that bMo + c > d.
`
`It can be shown using (6), (7) that the curve M('r) has
`an inflection point at M, = N, - (c - d)/b 2 0.8/b, 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:
`
`N
`1
`1
`=I~ “=<f—T>‘“m-
`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 — d >> bx/-IX‘ prior to the
`start of the abrupt rise in ne, the duration Ts 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 1r/2. Equa.
`tion (11) shows that the characteristic time for the fast
`
`stage is 1- R5 1n (N/N1) << 1-S. We see by inspecting the form
`of the above solutions that ne builds up explosively with
`time.
`
`It should be emphasized that the reason for this be.
`havior in ne(t) is quite universal and can be traced to the
`fact that we almost always have 3,8 << km.
`
`The explosive increase in ne(t) is most apparent when
`A > 0, which corresponds to early times and small direct
`ionization.
`In this case, ne does not increase more than
`tenfold prior to the onset of explosive growth [M > x/'(27i)‘j]_
`
`These expressions can be used to determine the limits
`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)—(11) imply that direct
`ionization predominates only for times 7' < T1 (M = x/E75)
`after the field is applied, and that the "instantaneous ioni-
`zation“ approximation does not become valid until times
`.3 rs after the field was first switched on.
`
`EXPERIMENTAL RESULTS
`
`We studied ionization relaxation in He, Ne, Ar, and Kr
`for initial gas densities n, = (0.5—5) - 10” cm'3 by applying
`an additional electric field to the preionized gas. The
`discharge occurred inside a cylindrical tube of diameter
`ZR = 2.5 cm and the distance between the electrodes was
`L = 52 cm. The gas was preionized by applying a dc 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 densiiigf
`and E0/n1 values. The initial density neo on the axis varia’;
`in the range 5 -109-4 -10“ cm"3.
`
`A voltage pulse with rise time (1-2) - 10‘7 s of positiii
`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 byacapacitative divider (C1: 2 pF and C2 = 100
`pF) capable of transmitting rectangular pulses with rise
`time ~ 10"7 s 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
`
`
`
` I
`
`L
`14
`1a
`E/n, , Td
`
`1!?
`
`11';
`
`10
`
`72
`
`14
`
`15
`
`FIG. 4. Curves for rs vs. E/n, for a discharge in krypton: p =
`10.5 (a) and 6 ton (b). a: 1) me, = 4-10” cm-3; 2) 10“; 3)
`1.5« 10“; 4) 3.3- 10“; b: 1) 3.5- 10”; 2) 9.5-101°; 3) 1.6-
`10”; 4) 3.2- 10“.
`
`Ijl
`20
`22
`
`I
`24
`
`:1
`26‘
`
`|_ l
`26
`J0
`
`“resistor connected in series with the cathode-to-ground
`'7secti0n of the tube. The local dependences ne(t) were re-
`garded by observing the plasma emission intensity, which
`proportional to ng (p 2 1). The density of the neutral
`was monitored using the interferometric technique
`escribed in Ref. 11 and adual—trace oscilloscope was used
`or all the measurements. Special experiments were con-
`cted to verify the unimportance of such factors as the
`
`
`
`and composition of the electrodes (we had pL > 200 cm -
`{arr [Ref. 12]), which do cause appreciable effects during
`Breakdown of a cold gas.”s13 We were also able to repro-
`iiuce the experimental conditions with high accuracy. We
`asured E and the plasma emission intensity at different
`{stances along the tube axis in order to find how the pa-
`ameters of the gas discharge plasmavary along the tube
`hd ascertain the importance of various mechanisms in—'
`olved in discharge formation after an abrupt increase in
`lfe field strength.
`
`
`
`Figure 2 shows some typical measured curves. The
`Wzcilloscope traces of the current (a), tube voltage (b),
`oltage differences between the probes (c, d), and the emis-
`n from different regions along the length of the dis-
`harge (e, i) show that after a high-voltage pulse is sud-
`penly applied, the discharge current rises very slowly
`(Sr times t < ‘rs and the tube voltage remains almost con-
`tant. 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
`9'7 s) over the entire length of the positive column and
`hen remains constant for t < T5. The plasma emission
`ntensity from different regions along the length of the tube
`0 starts to increase almost simultaneously and repeats
`he current trace.
`
`Our measurements revealed that for t < -rs, ne/n1
`increases by less than a factor of 102, i.e., we have ne/n1<
`‘5 at the end of the slow stage.
`
`_ A similar delay in the current increase has been noted
`Mmany other workers when an electric field is suddenly
`" lied (see e.g., Ref. 12). This lag might be caused by
`be small velocity of the ionization wave down the tube from
`high-voltage electrode to the grounded electrode,”-14
`by onset of instability in the uniform quasistationary
`charge when the field is applied [possibly caused by
`cesses near the electrodes; cf. Ref. 15]. Taken to-
`tether the above findings show that in any case, neither
`agation of an ionization wave nor processes at the
`rodes determine the duration 73 of the slow stage.
`
`
`
`The lag in rapid current buildup has often1°'” been
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`FIG. 5. Radial distribution new as a function of time for A > 0 in an argon
`discharge. p = 11.4 ton‘, neo = 101° cm'3, rs = 39 us.
`
`attributed to the finite time required for growth of ther-
`mal-ionization instability. Interferometric measurements
`show that for t < T3 the gas density does not drop by more
`than 1%. Estimates using the equations in Ref. 16 reveal
`that for such small changes in nl, the growth time for
`thermal-ionization instability is much larger than T5.
`
`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/n, 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 (nl, neo, E/nl).
`Since in our experiments we had A > 0 [cf. (10)], Eq, (9b)
`implies that the duration of the slow stage is given by
`
`‘ WW [Mo + (c - Ii)/bl
`“"°"‘3L“:.%#
`I/‘“f(M°+ b
`
`) (
`
`13)
`
`33
`
`_ E-j_‘_j
`
`",——]/1, l/1_%(Mo+%)2
`
`
`
`
`
`.9
`
`7.5
`
`27
`
`27
`
`3.7
`
`3.9
`
`‘/5
`
`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 neo(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 > po the ionization is highly nonuniform and a narrow
`plasma column forms on the axis.
`In particular, for argon
`po = 1 torr for R = 2-4 cm (R is the tube radius) [Ref. 13]
`and po = 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 pR < 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), kw, 318,
`file, VD‘: all have the same order of magnitude, the cor-
`responding curves behave similarly for similar values of
`pR. Helium is an exception, since under our conditions
`VD“ is much larger and ,B1e much smaller than for the
`heavy inert gases. The region A < 0 will therefore cor-
`respond to larger products pR, 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 ng/n? >> 10". We conclude
`from a comparison of the experimental spatial and time
`dependences of ne thatthe model is quite accurate. We
`have shown that the increase of ne 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.”v"'
`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.
`
`‘A. V. Eletskii, Usp. Fiz. Nauk E, 2'19 (1978) [Sov. Phys. Usp. 2_1, 502
`(19‘78)].
`2G. V. Naidis, Zh. Tekh. Fiz. _t1_'7, 941 (1977) [Sov. Phys. Tech. Phys. E.
`562 (19'7’7)].
`“A. A. Belevtsev. Teplofiz. Vys. Temp. _11, 1138 (1979).
`‘L. M. Biberman, V. S. V01-ob'ev, and I. T. Yakubov, Usp. Fiz. Nauk £11.
`353(19'72):_1§, 233 (1979) [Sov. Phys. Usp. 1_& 3'75 (1973): Q, 411 (19'19)l«
`5N. L. Aleksandrov, A.,M. Konchakov, and Pf. E. Son, Zh. Tekh. Fiz. gg,
`481 (1980) [Sov. Phys, Tech. Phys. 2_5, 291 (1980)].
`‘w. L. Nighan, App1. Phys. Lett. 3_2, 424 (1978).
`7B, M. Smirnov, Ions and Excited Atoms in Plasmas [in Russian], Atomizdat.
`Mowow (1974).
`“M. G. Voitik. A. G. Molchanov, and Yu. G. Popov, Kvantovaya Elektron.
`(Moscow) z_1, 1722 (1977) [Sov. J. Quantum Electron. 1, 976 (1977)].
`
`E/n, , Td
`FIG. 6. The behavior of ne in the bulk of an argon discharge. 1) neg/111 =
`10'”; 2) 10‘7. Stepwise ionization predominates in region 1. direct ioniza-
`tion processes predominate in region 11, and ne does not increase in region
`111.
`
`The solid curves in Figs. 3, 4 give ‘rs calculated from
`(12) using values for kn, 316, and Bze from numerical cal-
`culations in Ref. 5.
`
`Equation (12) shows that for small E/n1, when A u
`2/b, we have rs = 7r/\/Elf It follows that (n1ne0)1/275 =
`(k12B2e)'1/2 = const for a fixed value of E/n1. For large
`E/n1 we have A - 0, and TS —~ 2/(neofize + n1B1e — VD“)
`is only weakly dependent on neg.
`
`Figure 3a also gives experimental values of the pa-
`rameter (n1neo)‘/2-rs, which for E/n1 = const remains con-
`stant to within the experimental error for neo, n1, and Ts
`varying over wide limits. For large E/n1 [Fig. 3b, Fig.
`4b] 18 becomes almost independent of nee, as predicted
`by our theory.
`
`We also note that the function 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 theradial distribution ne(r, t) when an electric
`field is suddenly applied to a gas for which the initial dis-
`tributions neo(r) and n2,,(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.”
`
`The solutions of system (5), (6) show how the initial
`distribution ne0(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 neo, particularly for large values of A (A r2
`2/b). This results in a sudden increase in the magnitude
`of the initial irregularities in ne(r) for t < 75. As an il-
`lustration, Fig. 5 shows curves giving the time dependence
`of ne(r) calculated from (9b) for E/n, = 12 Td and neo =
`101° cm'3 for an argon discharge. According to Ref. 9,
`the initial neo 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/n1, A < 0 and the growth of ne during the
`slow stage [cf. (9a)] is determined by direct ionization.
`
`34
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`34
`
`
`
`“E, P, Velikhov, V. D. Pis‘mennyi, and A. T. Rakhimov, Usp. Fiz. Nauk
`122, 419 (1977) [Sov. Phys. Usp. _2_0_. 586 (19'7’7)].
`nfi. 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.
`"M, N. Polyanskii, v. N. Skrebov, and A. M. Shukhtin, Opt. Spektrosk.
`84, 28 (1973).
`”I.—D. Dautlierty. I. A. Mangano, and I. H. Jakob, Appl. Phys. Lett. E.
`581 (1976).
`
`Translated by A. Mason
`
`
`
`9
`
`, G1-anovskii, Electrical Currents in Gases. Steady Currents [in Russian],
`kal Moscow (1971).
`'N, Kondl-at'ev and E. E. Nikitin, Kinetics and Mechanisms in Gas-
`59 Reactions [in Russian], Nauka, Moscow (1974).
`N‘ Ski-ebov and A. I. Skripchenko, Teplofiz. Vys. Temp. _1_9. No. 3
`'\g.1)1:ozanskii and O. B. Firsov. Spark Theory [in Russian], Atomizdat,
`scow (1975).
`P. Abrarnov, P. I. Ishchenko, and I. G. Mazan'ko, Zh. Tekh. Fiz. a).
`[(1980) [Sov. Phys. Tech. Phys. E 449 (1980)'].
`, Astnovskii, V. N. Markov, N. S, Samoilov. and A. M. Ul'yanov,
`Plof-1z_ Vys-_ Temp. E, 1309 (1978).
`Ya. Aleksandrov, R. B. Gurevich, A. V. Kulagina, et a1.. Zh. Tekh. Fiz.
`105 (1975) [Sov. Phys. Tech. Phys. E, 62 (1975)].
`
`‘”7;"‘?s
`fig,
`
`Sov. Phys. Tech. Phys. 28(1 ), January 1983
`
`0038-5662/B3/O1 ooas-o4 $03.40
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