PLASMA PHYSICS
`
`
`Ionization relaxatien 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)
`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/nlo>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/n, 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 “ea/n1 210-8-10'7 to 10'5—10'4. 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/n, S 3 Td is much less than the characteristic ioniza—
`tion growth times (tr < 1 [18).2’3 Therefore, the kinetic
`electron coefficients do not depend on time explicitly but
`are functions of the parameters E/nl, ne/ni, 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 equationsr”6
`
`2'. If we use k to label the excited states (with k = 2
`corresponding to the first excited state), the populations
`
`30
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`0038-5662/83/01 0030»06 $03.40
`
`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 n6 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},5
`which gives the quasistationary populations in terms of
`he and n2.
`
`4. Since for E/n, in the range of interest the ioniza-
`tion hoe/n? is much greater than 10—4, 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) - 1017 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:
`
`drz,9
`W : nineple + ”2712922 + nznepst — VP”
`
`(1)
`
`
`(3712
`”21121021
`W__n1n,lr12
`n2n,lr.,ll
`72211.82,
`”2,19%:
`
`V1}.
`
`(2)
`
`Here 11,, n2, and ne are the atomic densities in the ground
`and first excited states and the electron density, respec-
`tively; k”, kn are the rate constants for the collisional
`transitions 1 7— 2; me and fize are the rate coefficients for
`GILLETTE 1206
`GILLETTE 1206
`
`.30
`
`

`

`
`
`
`
`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.
`
`direct ionization from the ground state to the first excited
`:level‘ fist is a generalized coefficient for stepwise ioniza—
`Yuan from the quasistationary levels (k > 2); A21 is the
`
`ntaneous transition probability 2 —+ 1; 021 is the capture
`'
`
`9actor for resonance radiation, calculated using the Biber—
`.___
`n—Holstein equations4; V I‘e and V P2 are the diffusion
`
`fluxes of the electrons and excited atoms, and are given
`-_;approximately by VI‘k ~ nkVDk [for a cylindrical geometry,
`a(z .4)2Dk/R2].
`
`
`
`It can be shown using the MDA theory4 that
`
`
`2
`1 (BUY e—E,/T,,
`2K
`nzngflgt 212212433 Vqu H—2 T ——Ek—’
`19—)2
`‘
`
`exp (0. 2E4/Eff),
`
`
`
`
`: (Ila/4.5 - 1013)”! Try—V!s (n, — in cm”. T. — .in'eV),
`.r
`0.2, Ry=13.6 eV, unzfi S e_’t’l1dt,
`
`___
`4 21.7 -10_7 ems/s.
`
`
`
`(4)
`
`The nonlinear system (1)-(2) can be solved numerical-
`the coefficients and initial conditions are specified.
`__ 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/ni
`e range of interest.
`
`Since Bze a 10—3—10‘7 cm3/s [Refs. 5, 8], we see from
`_nd (4) that the ratio fize/fi 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 (fist << B26)
`use most of the radiation escapes.
`
`
`
`
`
`
`
`
`
`
`Estimates using the equations in Ref. 4 show that
`under typical conditions (R S 1 cm, 11, S 1016 cm'3), the
`radiation capture factor is 021 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 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
`
`0N/61:bNM +cN—dN,
`OM/Ot : N — bNM,
`
`(5)
`
`M = “2/”207 N = ”0/7120:
`b : noOBfi/“lklw
`
`1 : tnlklfll M lr=o ——Mo» N |1=o 21:
`C : fan/km:
`(6)
`d—— “Du/”'1"12-
`
`Equations (5)—(6) easily yield the following relation
`between N and M:
`
`N—l
`
`{ MD M { (1+C_d)ln[(l_bM°)]
`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/ni <
`300 Td, ne/nl < 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 + bMD << 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/d’T = 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
`
`

`

`7W
`
`7.7
`
`5.us1’
`
`r...%1;)'49—”a‘m-Ws-torrl/2-crn'a/z
`
`
`E),
`
`I»g
`
`1M
`
`&
`
`1's,#5
`
`75
`
`1L7
`
`I3
`E/n,. Td
`
` FIG. 3. Duration TS for a discharge in argon. p = 11.4 (a) and 3.7 ton
`
`1B
`
`20
`
`24
`
`22
`E/nu 'Td
`
`26‘
`
`1) me. = 1.6-1010 cm'“: 2) 2.7 - 101°: 3) 4.5 - 101°; 4) 6.4- 1010;
`a:
`(b).
`5) 1o”: 6) 1.6-10“; b: 1) 9.7- 109; 2) 3-10”; 3) 6.8-10”; 4) 1.6-1011
`
`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. (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
`
`N = bMZ/Z + (c — d) M +1— mug/2 — (c — d) Mo,
`T:
`2
`MT)
`
`(8)
`
`><
`
`
`1]
`[(billuc—d—b\/m)(bzllo~—c—d+b\/m)]
`
`2 n
` (bill ~—c —d +bvm)(bMa—4c_d~ W171)
`bM—i—c—d
`_
`Milena—d
`arctg[ mm ]
`aictg[
`(NW ], A>O,
`
`,
`
`A <(%a)
`(9b)
`
`where A is given by
`
`A = (2/17) _ [2110 + (c — d)/b]2
`
`(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 bM0 + c > d.
`
`It can be shown using (6), (7) that the curve M('r) has
`an inflection point at M, = N1 - (c - d)/b m 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:
`
`I N
`1
`_
`_i
`‘b’ T”(H‘s—d) “T1“
`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/TAT 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 77/2. Equa.
`tion (11) shows that the characteristic time for the fast
`
`stage is 'r w ln(N/N1) << Ts. We see by inspecting the fem
`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 Bie << 1:12.
`
`The explosive increase in ne(t) is most apparent when
`A > O, 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 > @275].
`
`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‘1
`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 'r < T1 (M = #275)
`after the field is applied, and that the "instantaneous ioni-
`zation“ approximation does not become valid until times
`t m Ts after the field was first switched on.
`
`EXPERIMENTAL RESULTS
`
`We studied ionization relaxation in He, Ne, Ar, and Kr
`for initial gas densities n1 : (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 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 densitlgg
`and Eo/nj values. The initial density neo on the axis varieé
`in the range 5 - 109—4 . 1011 cm'3.
`
`A voltage pulse with rise time (1—2) - 10-7 s of positive:
`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 02 : 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.
`
`37
`
`

`

`
`
`FIG. 4. Curves for 1'S vs. E/nl for a discharge in krypton: p =
`10.5 (a) and 6 ton (b). a: 1) “e0 = 4.1010 crn'az 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“.
`
`n
`26
`
`l
`30
`
`|
`24/
`
`x
`26'
`
`N .
`\Z\
`l
`I
`l
`14
`16
`1t?
`
`J
`20
`
`l
`22
`
`18
`
`E/n,, Td
`
`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
`is proportional to 11$ (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 -
`torr [Ref. 12]), which do cause appreciable effects during
`breakdown of a cold gas.12’13 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 dr0p 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 < TS. The plasma emission
`___i_ntensity from different regions along the length of the tube
`£3150 starts to increase almost simultaneously and repeats
`lithe current trace.
`
`Our measurements revealed that for t < TS, ne/n1
`,
`increases by less than a factor of 102, i.e., we have ne/n1<
`il-O‘5 at the end of the slow stage.
`
`A similar delay in the current increase has been noted
`:meany other workers when an electric field is suddenly
`fépplied (see e.g., Ref. 12). This lag might be caused by
`the small velocity of the ionization wave down the tube from
`the high-voltage electrode to the grounded electrode,”'!14
`
`1‘ by onset of instability in the uniform quasistationary
`
`lSCharge when the field is applied [possibly caused by
`
`‘l‘Ocesses near the electrodes; cf. Ref. 15]. Taken to—
`gether the above findings show that in any case, neither
`
`prWagation of an ionization wave nor processes at the
`
`ctrodes determine the duration TS of the slow stage.
`
`The lag in rapid current buildup has often”,17 been
`
`
`
`FIG. 5. Radial distribution 1160‘) as a function of time for A > 0 in an argon
`discharge. p = 11.4 tort, neo = 1010 cm'a, Ts = 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 n1, the growth time for
`thermal-ionization instability is much larger than 7'5.
`
`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 (n1, 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
`
`1
`1 :l/3
`,, Vh—Qx—amw
`
`are ctg
`
`
`
`- V172 [Mo-l-(G~d)/b]
`
`V1_;(M.+c;d)2
`({2)
`
`fl
`
`
`
`33
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`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 neo(r) for A < 0 but
`substantial deformation when A > 0.
`
`Experiments (cf. Refs. 18, 19) have shown that ionizh
`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 sum,
`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, 131e,
`B29, VDa 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
`”Dr: is much larger and :31e 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 that‘the 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.”'2°
`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.
`
`1A. V. Eletskii, Usp. Fiz. Nauk E, 279 (1978) [Sov. Phys. Usp. a, 502
`(1978)].
`2G. V. Naidis, Zh. Tekh. Fiz. E, 941 (1977) [Sov. Phys. Tech. Phys. _2_2_.
`562 (1977)].
`3A. A. Belevtsev. Teplofiz. Vys. Temp. 1_7. 1138 (1979).
`4'L. M. Biberman, V. S. Vorob'ev, and I. T. Yakubov, Usp. Fiz. Nauk El.
`353 (1972); E, 233 (1979) [Sov. Phys. Usp. E, 375 (1973); _2_2, 411 (1979)]:
`5N. L. Aleksandrov, A.,M. Konchakov, and If E. Son, Zh. Tekh. Fiz. fig.
`481 (1980) [Sov. Phys. Tech. Phys. 2_5, 291 (1980)].
`6w. L. Nighan, Appl. Phys. Lett. Q, 424 (1978).
`7B, M. Smirnov, Ions and Excited Atoms in Plasmas [in Russian], Atomizdat.
`Momow (1974).
`8M. G. Voitik. A. G. Molchanov, and Yu. G. Popov, Kvantovaya Elektron.
`(Moscow) 11; 1722 (1977) [Sov. J. Quantum Electron. 1, 976 (1977)].
`
`(7
`
`N.
`
`E d
`
`.C‘.
`
`70
`
`10”
`
`
`
`
`I
`I_I_|_\_|_|
`I_i
`x
`J_.I__|
`3
`75
`27
`27
`.33
`3.9
`#5
`
`E/nflTd
`1) neg/r11 =
`FIG. 6. The behavior of ne in the bulk of an argon discharge.
`10"]; 2) 10‘7. 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 TS calculated from
`(12) using values for 1‘12, 131e, and ,82e from numerical cal-
`culations in Ref. 5.
`
`Equation (12) shows that for small E/ni, when A :1
`2/b, we have TS = 7r/s/2IT It follows that (nineo)1/ZTS =
`(kmflzeri/Z : const for a fixed value of E/n1. For large
`E/n1 we have A —> 0, and TS —> 2/(ne0,BZe + 111,31e - VDa)
`is only weakly dependent on neo-
`
`Figure 3a also gives experimental values of the pa—
`rameter (nineo)1 27's, which for E/n1 = const remains con—
`stant to within the experimental error for neo, n1, and 7's
`varying over wide limits. For large E/nl [Fig. 3b, Fig.
`4b] 7's becomes almost independent of “em 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.2,)(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.17
`
`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 neo, particularly for large values of A (A a
`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 :
`10‘” cm‘3 for an argon discharge. According to Ref. 9,
`the initial neo and n20 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 7's after the field is
`turned on.
`
`For large E/ni, 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,, Granovskii, Electrical Currents in Gases. Steady Currents [in Russian]I
`aukflu Moscow (1971).
`”V. N. Kondrat'ev and E. E. Nikitin, Kinetics and Mechanisms in Gas-
`
`Phase Reactions [in Russian], Nauka, Moscow (1974).
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`
`‘ N. Skrebov and A. I. Skripchenko, Teplofiz. Vys. Temp. _1_9, No. 3
`
`
`”1981).
`21‘2" D. Lozanskii and O. 3. Firsov. Spark Theory [in Russian], Atomizdat,
`_ ‘MoSCOW (1975)-
`(flail. p_ Abramov, P. I. Ishchenko, and I. G. Mazan'ko, Zh. Tekh. Fiz. fl.
`‘
`.155 (1930) [Sov. Phys. Tech. Phys. E, 449 (1980)].
`”15.1 Asinovskii, V. N. Markov, N. S, Samoilov. and A. M. Ul'yanov.
`______._.Tep10flZ- Vys_ Temp. E, 1309 (1978).
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`,45i 105 (1975) [Sov. Phys. Tech. Phys. _2_0, 62 (1975)].
`
`16E. P, Velikhov, V. D. Pis'mennyi, and A, T. Rakhimov. Usp. Fiz. Nauk
`122. 419 (1977) [Sov. Phys. Usp. fig, 586 (1977)].
`"n. Napartovich and A. N. Starostin,
`in: Plasma Chemistry, B. M.
`Smirnov. ed., [in Russian] (1979)' pp. 6, 153.
`18D. N. Novichkov.
`in: Topics in Low -Temperature Plasma Physics [in
`Russian]I Nauka i Tekhnika, Minsk (1970), p. 459.
`19M. N. Polyanskii, v. N. Skrebov, and A. M. Shukhtin, Opt. Spektrosk.
`84, 28 (1973).
`”TD, Dautherty, J. A. Mangano, and]. H. Jakob, Appl, Phys. Lett. g.
`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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