PLASMA PHYSICS
`
`Ionization relaxatien in a plasma produced by a pulsed inert-gas
`discharge
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`A. A. Zhdamw 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 nf/n1°>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/ni increases from
`the initial value neo/n1 =10‘3—10‘7 to 10'5-10"“. 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 é 30-40 Td.
`
`THE ORY
`
`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/n1 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 ]J.S).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 low er 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 2 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,“
`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/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-'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:
`
`dn,
`-5,‘ : nxneiale + n2nep2e _i" nznepst — VPGJ
`
`dn
`72 —— n1nekl2
`
`n2"e]"21
`
`n2"':.B2..»
`
`nazzeflst
`
`nzA21e21
`
`VP?
`
`(2)
`
`Here n1, n2, and ne are the atomic densities in the ground
`and first excited states and the electron density, respec-
`tively; k12, kn are the rate constants for the collisional
`transitions 1 —.~_ 2; me and 523 are the rate coefficients fol‘
`
`30
`
`Sov. Phys. Tech. Phys. 2811 ), January 1983
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`0038-5662/83/O1 003006 $03.40
`
`GILLETTE 1108
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`.30
`
`GILLETTE 1108
`
`

`
`direct ionization from the ground state to the first excited
`\.ié‘,e1; gst is a generalized coefficient for stepwise ioniza—
`figin from the quasistationary levels (k > 2); A2, is the
`ntaneous transition probability 2 —+ 1; 021 is the capture
`fiictor for resonance radiation, calculated using the Biber—
`n—-Holstein equations4; V Fe and V P2 are the diffusion
`ififlfixes of the electrons and excited atoms, and are given
`gzafiproximately by VI‘k 2 nkn/Dk [for a cylindrical geometry,
`( .~.:(2.4)2D1(/R2].
`T It can be shown using the MDA theory4 that
`
`if
`
`1 Hg/3 9-WT»
`23
`(T2)
`El:
`nflneflgt : nflnefi 3 ‘/Egg E
`dz)
`
`’
`
`exp (0.2E;/Eé,),
`
`: (n,/4.5 - 1015'/« T;'/a (n, — in cm'3. T, — .in'eV),
`.1’
`
`0.2, By: 13.6 eV, x(:v)~_=% S e"t’/vdt,
`0
`
`‘/211:: 1.7 . 10-7 cm3/._
`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
`p
`) 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.
`
`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
`the walls of the discharge tube.
`
`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
`
`azv/ax : bNM + czv — dN,
`
`3111/at = N — b1vM,
`
`M: ’12!’2/207
`N = I20/77.20,
`b = nc0B2a/n1k121
`
`1 = tnlku,
`C : l31a/k12y
`
`/11 |1:=D : AID)
`d = v1;,,/nlku.
`
`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)
`
`Sov. Phys. Tech. Phys. 28(1), January 1 983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`31
`
`

`
`
`
`
`
`

`
`1»
`Ax
`J
`20
`
`I
`’/£7
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`11')‘
`
`11]!
`E/n,, Td
`
`2'2
`
`2'4
`
`223
`
`22:
`
`3})
`
`FIG. 4. Curves for rs vs, E/n1 for a discharge in krypton: p =
`10.5 (a) and 6 torr (b). a: 1) me. = 4-10” cm-3; 2) 10“; 3)
`1.5-10“; 4) 3.3-10“; b: 1) 3.5- 101°; 2) 9.5-10”’; 3) 1.6-
`10”; 4) 3.2- 10“.
`
`resistor connected in series with the cathode—to—ground
`section of the tube. The local dependences ne(t) were re-
`corded by observing the plasma emission intensity, which
`E proportional to ng (p 2 1). The density of the neutral
`gas was monitored using the interferometric technique
`described in Ref. 11 and adual—trace oscilloscope was used
`for 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 -
`tom» [Ref. 12]). which do cause appreciable effects during
`breakdown of a cold gas.12s13 We were also able to repro-
`duce the experimental conditions with high accuracy. We
`measured 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 plasmavary along the tube
`and ascertain the importance of various mechanisms in-'
`volved 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),
`voltage differences between the probes (c, d), and the emis-
`sion from different regions along the length of the dis-
`charge (e, i) show that after a high—voltage pulse is sud-
`denly applied, the discharge current rises very slowly
`for times t < TS and the tube voltage remains almost con-
`‘stant. This is followed by a sudden rise in the current, ac-
`companied by a voltage drop across the tube. We also see
`[that the field increases almost simultaneously (to within
`f1.0'7 s) over the entire length of the positive column and
`‘then remains constant for t < T5. The plasma emission
`intensity from different regions along the length of the tube
`jalpso starts to increase almost simultaneously and repeats
`‘the current trace.
`
`Our measurements revealed that for t < 75, ne/n1
`increases by less than a factor of 102, i.e., we have ne/n1<
`il70"5 at the end of the slow stage.
`
`A similar delay in the current increase has been noted
`Qilysmany other workers when an electric field is suddenly
`fiétpplied (see e.g., Ref. 12). This lag might be caused by
`file small velocity of the ionization wave down the tube from
`[the high-voltage electrode to the grounded electrode,13v”
`1‘ by onset of instability in the uniform quasistationary
`lficharge when the field is applied [possibly caused by
`drocesses near the electrodes; cf. Ref. 15]. Taken to-
`gether the above findings show that in any case, neither
`Propagation of an ionization wave nor processes at the
`ctrodes determine the duration -rs of the slow stage.
`
`The lag in rapid current buildup has often16'17 been
`
`FIG. 5. Radial distribution ne(r) as a function of time for A > 0 in an argon
`discharge. p = 11.4 torr, neo = 101° cm'3_ rs = 39 pg,
`
`attributed to the finite time required for growth of ther-
`mal-ionization instability. Interferometric measurements
`show that for t < 73 the gas density does not drop by more
`than 1%. Estimates using the equations in Ref. 16 reveal
`that for such small changes in n1, the growth time for
`thermal-ionization instability is much larger than 1-5.
`
`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/nl 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
`
`V172 [Mo+(a-d)/bl ”
`-V1-%<M»+“:">”
`(1-2)
`
`Sov. Phys. Tech. Phys. 23(1), January 1933
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`33
`
`

`
`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 ne0(r) for A < 0 but
`substantial deformation when A > 0.
`
`Experiments (cf. Refs. 18, 19) have shown that ionizm
`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
`p0 = 1 torr for R = 2~4 cm (R is the tube radius) [Ref. 13]
`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 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), km, 316,
`B29, 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
`vDa 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/n9 >> 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.”v2°
`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, 279 (1978) [Sov. Phys. Usp. g_1, 502
`(1978)].
`2G, V. Naidis_ Zh. Tekh. Fiz. fl, 941 (1977) [Sov. Phys. Tech. Phys. 2.
`562 (19'7’7)].
`3A. A. Belevtsev, Teplofiz. Vys. Temp. 11. 1138 (1979).
`‘L. M. Biberman, V. S. Vorob'ev, and I. T. Yakubov, Usp. Fiz. Nauk E1.
`353 (1972); E, 233 (1979) [Sov. Phys. Usp. E, 375 (1973); E, 411 (1979)l-
`5N. L. Aleksandrov, A.,M. Konchakov, and
`E, Son, Zh, Tekh. Fiz. ég,
`481 (1980) [Sov. Phys. Tech. Phys. 2_5, 291 (1980)].
`‘w. L. Nighan, Appl. Phys. Lett. 3_2, 424 (1978).
`7B, M. Smirnov, Ions and Excited Atoms in Plasmas [in Russian], Atomizdaf.
`Mowow (1974).
`“M. G, Voitilr. A. G. Molchanov, and Yu. G, Popov, Kvantovaya Elektron.
`(Moscow) 4_1, 1722 (1977) [Sov. I. Quantum Electron. 1, 976 (1977)].
`
`II_I_|_I_r_|-_1xJ_r_I
`.9
`7.5
`Z7
`Z7
`3.7
`3.9
`#5
`
`E/17,, Td
`1) neg/n1 =
`FIG. 6. The behavior of ne in the bulk of an argon discharge.
`10'!‘-, 2) 10‘-'. Stepwise ionization predominates in region 1, direct ioniza-
`tion processes predominate in region 11, and me does not increase in region
`111.
`
`The solid curves in Figs. 3, 4 give 7-S calculated from
`(12) using values for kw, ,B1e, and ,B2e from numerical cal-
`culations in Ref. 5.
`
`Equation (12) shows that for small E/n1, when A u
`2/b, we have TS = 7r/\/Elf It follows that (n1neo)1/275 =
`(k12,B2e)'1/2 = const for a fixed value of E/n1. For large
`E/n1 we have A -> 0, and TS -> 2/(ne0,B2e + n1,B1e - vDa)
`is only weakly dependent on neg.
`
`Figure 3a also gives experimental values of the pa—
`rameter (n1ne0)1 27-5, which for E/n1 = const remains con-
`stant to within the experimental error for nee, n1, and Ts
`varying over wide limits. For large E/n1 [Fig. 3b, Fig.
`4b] 7-S becomes almost independent of neo, 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 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 n.,D(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 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/n1 = 12 Td and neo :
`101° cm'3 for an argon discharge. According to Ref. 9,
`the initial nee 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
`
`

`
`‘
`
`1,, G1-anovskii, Electrical Currents in Gases. Steady Currents [in Russian],
`auka, Moscow (1971).
`7 N, Kondl-at'ev and E. E. Nikitin, Kinetics and Mechanisms in Gas-
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`1981).
`125 D. Lozanskii and O. B. Firsov, Spark Theory [in Russian], Atomizdat,
`1 Moscow (1975).
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`9755 (1980) [Sov. Phys, Tech. Phys. E, 449 (1980)],
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`:_TeP1Oflz_ Vyg_ Temp. 1_6_, 1309 (1978).
`jsv Ya, Aleksandrov. R. B. Gurevich, A. V. Kulagina, et al., Zh. Tekh. Fiz.
`0,945, 105 (1975) [Sov. Phys. Tech. Phys. E, 62 (1975)].
`
`“E, P, Velikhov, V. D. Pis'1-nennyi, and A. T. Rakhimov, Usp. Fiz. Nau1<
`E, 419 (1977) [Sov. Phys. Usp. £51. 586 (1977)].
`"A. P. 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. Ski-ebov, and A. M. Shukhtin, Opt. Spektrosk.
`84, 28 (1973).
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`581 (1976).
`
`Translated by A. Mason
`
`Sov. Phys. Tech. Phys. 23(1 ), January 1983
`
`0038-5662/B3/O1 0035-04 $03.40