PLASMA PHYSICS
`
`ionization reiaxaticn in a plasma produced by a pulsed inerbgas
`discharge
`
`A. A. Kudryavtsev and V. N. Skrebov
`A. A. Zhdmzou 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/21, values corresponding to ionization levels 219°/rzf’>l0"“. 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 ng 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 relaxationin 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 13/!!! and degree of
`ionization ne/n, 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 neo/n, = 10‘3-10"7 to 10"5—10“‘. We are
`mainly interested in analyzing the buildup of he 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/n, '6 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/ny 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 ;zs).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 it 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 rcsonancelevels when ne 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 lr 2 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 machixierv
`of the modified diffusion approximation (MDA) theory,“v5
`which gives the quasistationary populations in terms of
`ne and r12.
`
`4. Since for E/n, in the range of interest the ioniza-
`tion ng/n? is much greater than 10“‘, 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 [n, < (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:
`
`%= n.n.@..+ntna..+n2n.igt — vru.
`
`<1)
`
`.7
`:d"T2“ni77*J"i2
`
`n2”a]"a1
`
`"'1-"1322
`
`s
`”2"ai3t
`
`"211-neat
`
`VFW
`
`lzi
`
`Here 11,, n2, and ne are the atomic densities in the ground
`and first excited states and the electron density, respecv
`tively; kn, R21 are the rate constants for the collisional
`transitions 1 —.-_ 2; [he and /323 are the rate coefficients fol‘
`
`230
`
`Sov. Phys. Tech. Phys. 28(1 ), January 1983
`
`0038-5662/83/Oi 0030-06 $03.40
`
`GILLETTE 1006 .30
`
`GILLETTE 1006
`
`

`
`
`
`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.
`
`friireot ionization from the ground state to the first excited
`five}; gist is a generalized coefficient for stepwise ioniza-
`
`
`"
`n from the quasistationary levels (k > 2); A2, is the
`‘ taneous transition probability 2 —~ 1; 921 is the capture
`for for resonance radiation, calculated using the Biber—
`111-Holstein equations4; vre and V T2 are the diffusion
`_ es of the electrons and excited atoms, and are given
`proximately by V1"k :2 nkvnk [for a cylindrical geometry,
`(D1,; 9-’ (2.4)2Dk/R2}.
`
`It can be shown using; the MDA theory‘ that
`
`
`
`23
`
`1
`
`3,, -1
`(Ta)
`
`9-'—°2'Te
`El:
`xtg)
`
`'
`
`exp (0.23;/E;}),
`
`(n,/4.5 -
`
`lO“)'»’« Ty’/e (I15 — in cm“. Tc — .in'eV),
`
`Estimates using the equations in Ref. 4 Show that
`under typical conditions (R 3 1 cm, n, S 10”‘ cm“3), the
`radiation capture factor is 621 2 10‘3—1O". 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—1O”2 s, so that
`we may neglect V F2 compared with n2nefi2e 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
`
`C1V—— dN,
`ON/0*» = DNM
`am/at .—_- N ~ (WM,
`
`(5)
`
`M =11,/nee, N :/in/12,9.
`(7 :1 ’hoB2a/’‘1ki2~
`
`1 = trzlkm, M [14, = 1110, N },_,o = 1,
`C 2 .3ia/ki2»
`9’ = "Da/77'iA'i2-
`(6)
`
`Equations (5)—(6) easily yield the following relation
`between N and M:
`
`1’
`.
`zv—i+n1,. 111+‘ T:
`
`——d
`
`l— I
`)ln[:l_%[‘;)]
`
`(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, he/I11 < 1O'5).
`In a steady—state plasma, we usually
`have M0 '6 1 {Ref. 9].
`
`Using the smallness of b and c, we find from (5)~(7)
`that dN/dM .~. c — d + bi\/I, << 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'r = N(1 + c r 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)
`
`
`
`~ 1.7 . 10-7 ms 5.
`
`(4,
`
`v3To 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,
`9; the rate constants given in Refs. 5-8 for the various
`vesses, we can make a series of simplifications making
`‘ssible to solve (1)-(2) analytically for E/111 and ne/n1
`‘e range of interest.
`
`‘Since 338 2 1O'8—1O‘7 cm3/s [Refs. 5, 8], we see from
`and (4) that the ratio 1629/,8 St of the coefficients for
`and stepwise ionization from the first excited level
`ds on ne and on the electron temperature Te. For
`-1013 cm"3 and Te 3 1 eV, the stepwise ionization
`nel from levels with k > 2 is unimportant {fist << fin)
`8,use most of the radiation escapes.
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`A. A. Kudryavisev and V. N. Skrebov
`
`31
`
`

`
`e
`
`s
`
`73
`E/n,, Td
`
` FIG. 3. Duration T5 for a discharge in argon. p = 11.4 (a) and 3.7 to"
`
`1) neg = 1.6-1.0” cur“; 2) 2.7-1o‘°; 3) 4.5401“; 4) 6.440”;
`a:
`(b).
`5)10”; 6)1.6-10”;b:
`1). 9.7- 109; 2) 3.10”; 3) 5.8-10”; 4) 1.s.mu_
`
`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=WWfl+®~®M+i~Mm2~@—®Mw
`
`(m
`
`T:
`
`E"
`b\/|
`1
`
`!
`
`X
`
`— °
`~— .
`(b:l[+c—~d—-b\/i_}T[)(b.-if +c_¢+t»/;T;)]
`1
`(l:1;1.v:c~(I:-i-17V! -‘1|)(3":‘/lg—{«c——d—»»»-bi/{A1).
`2 n
`_c~ __ ‘b +c-11
`\ arctg[—:—W]
`
`aictg[——~———-3)\/.7.‘
`J,
`.4>0,
`where A is given by
`
`A
`
`0,
`
`<(9a)
`(919)
`
`A = (2/17) - [zl[0+ (c — d)/17]’
`
`(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 O, the condition is that bM0 + c > d,
`
`It can be shown using (6), (7) that the curve M(T) has
`an inflection point at M1 = N1 - (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
`n:::fimm-
`M—r»
`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 n? 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/in prior to the
`start of the abrupt rise in ne, the duration 73 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 71/2. Equa.
`tion (11) shows that the characteristic time for the fast
`stage is 7 1:: 1n(N/N‘) << -rs. 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 Bic << 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/(E7‘b‘i],
`
`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)—(1l) imply that direct
`ionization predominates only for times "r < T, (M = x/275)
`after the field is applied, and that the "instantaneous ioni-
`zation" approximation does not become valid until times
`t e rs after the field was first switched on.
`
`E XPERIMENTAL RESULTS
`
`We studied ionization relaxation in He, Ne, Ar, and Kr
`for initial gas densities n, = (O.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 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 densilfy
`and ED/n, values. The initial density neg on the axis variti.
`in the range 5 -1094 - 10" cm'3.
`
`A voltage pulse with rise time (1 -2) - 10”’ 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 pl’ and C2 -~-- 100
`pi?) capable of transmitting rectangular pulses with rise
`time ~ 10"7 s without appreciable distortion. The di-s~
`charge current was recorded using a zero~inductance
`
`32
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`A. A. Kudryavtssv and V. N. Skrebov.
`
`32
`
`

`
`.Jj.
`
`
`
`75
`
`07
`
`as
`
`24
`
`25
`
`X 0
`.. _._L..,
`,.
`7‘!
`
`10
`
`20
`
`22
`
`FIG. 4. Curves for ‘rs vs. 13/11; for a discharge in krypton; p =
`10.5 (a) and 5 ton (b). a:
`1.) Flag = 4.10” cm”; 2) no"; 3)
`1.5-10": 4) 3.3-10“; b: 1) 3,5-10”; 2) 9.5-10”‘; 3) 1.8-
`10”; 4) 3.2210“.
`
`ijl...
`‘ad
`30
`
`70
`
`70?
`
`70
`
`‘is
`>x’;
`::°‘
`
` ....._...
`
`._..
`l
`.
`0 0.2 0.9 0.6‘ 0,8
`r/A7
`Radial distribution net!) as a function of time for A > 0 in an argon
`discharge.
`p = 11.4 ton, neg = 101° cm"
`3. 1'3 = 39 us.
`
`FIG. 5.
`
`attributed to the finite time required for growth of ther-
`mal -ionization instability. Interferometric measurements
`show that for t < TS 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/n1, quasistationary electron energy dis—
`tribution functions, and ne/n1 values < 10"“.
`
`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, neg, E/n1).
`Since in our experiments we had A > 0 [cf. (10)], Eq. (9b)
`implies that the duration of the slow stage is given by
`
`‘..s :2
`
`'2“,
`-5-
`
`i
`
`1/) _ 7 (M +
`
`‘___~/W lMo + (c ~ am ”
`
`
`__V" - % (“H Ti):
`(12)
`
`E/n,, Td
`
`‘resistor connected in series with the cathodc—to—gr-ound
`‘section of the tube. The local dependences ne(t) were re-
`corded by observing the plasma emission intensity, which
`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
`i/and composition of the electrodes (we had pL > 200 cm -
`)5“ [Ref. 12]), which do cause appreciable effects during
`7b‘real(dOWi‘i of a cold gasfzrm 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 piasmavary 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-
`"cliarge (e, i) show that after a high—voltage pulse is sud-
`rdenly applied, the discharge current rises very slowly
`times t < ‘Ts and the tube voltage remains almost con—
`fstant. This is followed by a sudden rise in the current, ac-
`tuompanicd by a voltage drop across the tube. We also see
`t the field increases almost simultaneously (to within
`
`_ 0“7 s) over the entire length of the positive column and
`‘then remains constant for t < Ts. The plasma emission
`{intensity from different regions along the length of the tube
`false starts to increase almost simultaneously and repeats
`current trace.
`
`Our measurements revealed that for t < rs, he/F11
`ficreases by less than a factor of 102, i.e., we have ne/I11 <
`5 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
`lied (see e.g., Ref. 12). This lag might be caused by
`small velocity of the ionization wave down the tube from
`high-voltage electrode to the grounded electrode,‘3v”
`by onset of instability in the uniform quasistationary
`7 charge when the field is applied [possibly caused by
`;-processes near the electrodes; of. Ref. 15]. Taken to-
`ll éther the above findings show that in any case, neither_
`sagation of an ionization wave nor processes at the
`lectrodes determine the duration 1-3 of the slow stage.
`
`
`
`The lag in rapid current buildup has often‘69” been
`
`
`
`
`
`Sov. Phys. Tech. Phys. 2B( l ), 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 wel1—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 (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 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 3 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 < O.
`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, 516,
`.829, 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
`VD“ is much larger and B19 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/n, values correspond-
`ing to equilibrium ionizations ng/nil >> 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-
`‘)9;
`'1
`wise ionization processes in the buildup of instabilities
`x.,..
`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.
`
`‘A. V. Eletskii, Usp. Fix. Nauk _1_’§_, 2'79 (1978) [Sov. Phys. Usp. _2_1_, 502
`(19’?8)].
`2G. V. Nziidis. Zh. Tekh. Fix. fl, 941 (1977) .FSov. Phys. Tech. Phys. 22,
`562 (19‘7’7)].
`3A. A. Belevtsev. Teplofiz. Vys. Temp. _1_'I_. 1138 (1979).
`‘L. M. Biberman, V. S. Voi-ob'ev. and l. T. Yakubov, Usp. Fiz, Nauk 3)],
`853 (1972): __1_2_8_, 233 (1979) [Sov. Phys. Usp. _1_& 375 (1973); Z; 411 (19'i9)lr
`5N, L, A1eksandrov_ A.,M. Konchakov, and if. E. Son. Zh. Tekh. Fiz. §Q.
`4-81(1980)[Sov. Phys. Tech. Phys. E, 291 (1980)].
`‘w. L. Nighan, Appl. Phys. Lett. gg, 424 (1978).
`7B, M. Smirnov, Ions and Excited Atoms in Plasmas [in Russian], Atomizdflh
`Moscow (1974).
`3M. G. Voitilt, A. G. Molchanov, and Yu, G. Popov, Kvantovaya Elektron.
`(Moscow) 5, 1722 (1977) [Sov. J. Quantum Electron. _’_7_, 976 (1977)].
`
` 7/J
`
`I
`
`I
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`
`I_I_s_.L ni L_l.:
`75
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`33
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`
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`
`o 0
`
`?.£2‘
`
`7flr7
`
`10”
`
`E/n,,'I‘d
`1) neg/n1=
`FIG. 6. The behavior of ne in the bulk of an argon discharge.
`10”; 2) 1O‘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 1:12, 313, and B29 from numerical cal-
`culations in Ref. 5.
`
`Equation (12) shows that for small E/n1, when A :=
`2/b, we have rs = 7r/x/-2-ET It follows that (n1neo)1/215 =
`(k12fi2e)‘1/2 = const for a fixed value of E/n1. For large
`E/n, we have A —v 0, and ‘rs —— 2/(neoflze + n1,81e - upa)
`is only weakly dependent on neg.
`
`Figure 3a alsogives experimental values of the pa-
`rameter (n1ne0)‘/Zrs, which for E/n, = const remains con»
`stant to within the experimental error for nee, hi, and 75
`varying over wide limits. For large E/n‘ (Fig. 3b, Fig‘.
`4b] 78 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 qua1ita-
`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.”
`
`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 2
`2/b). This results in a sudden increase in the magnitude
`of the initial irregularities in ne(r) for t < vs. 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 nee and nag 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. Skrabov
`
`34
`
`

`
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`1_2_2_, 419 (1977) [Sov. Phys. Usp. g. 586 (19'77)1.
`“A. P, Napartovfch 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], Nauka i Tekhnika, Minsk (1070), p. 459.
`“M, N. Polyanskii, v. N. Skrebov, and A. M. Shukhtin, Opt. S-pektrosk.
`84, 28 (1973).
`“E-D. Dautherty, J, A. Mangano, and I. H. Jakob. Appl. Phys. Lett. E,
`581 (1976).
`
`Translated by A. Mason
`
`‘L, Granovskii, Electrical Currents in Gases, Steady Currents [in Russian],
`“Ra, Moscow (1971).
`0 N, Kondraflev and E. E. Nikitin. Kinetics and Mzchanismsin Gas-
`:-
`‘ase Reactions [in Russian]. Nauka, Moscow (1974).
`, N‘ Skrebov and A. 1. Skripchenko, Tcplofiz. Vys. Temp. _1_‘.1, N0. 3
`981).
`’ D, Lozanskii and O. B. Firsov, Spark Theory [in Russian], Ammizdat,
`cu.r:O2 As—\(D.2 01
`).
`P, Abramov, P. I. Ishchenko, and I. G. Mazan'ko, Zh. Tekh, Fiz. §_Q,
`Q5 (1980) [Sov. Phys. Tech. Phys, 33, 449 0980)].
`I; Asgnovskii, V. N. Markov, N. S, Samoilov, and A, M. U1'yanov,
`feplofiz. Vys. Temp. ;§, 1309 (1978).
`' ya‘ Aleksandmv. R. B. Gurevich, A, V. Kulagina, et a1., Zh. Tekh. Fiz.
`105 (1975) [Sov. Phys. Tech. Phys. _2_0, 62 (1975)].
`
`
`
`o 1
`
` Sov. Phys. Tech. Phys. 28( 1 ), January 1983
`
`0038-5662/83/O1 0035-04 $03.40
`
`35