`----------------------------- ~----------------------·-
`Ionization relaxation in a plasma produced by a pulsed inert .. gas
`discharge
`A. A. Kudryavtsev and V.l\1. Skrebov
`
`A. A. Zhdanov 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 1 values corresponding to ionization levels n~ In~> 10-4
`• 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: S2.2S.Lp, 51.50. + v, S2.80.Dy
`
`The study of ionization relaxation in a plasma when
`the external electric field suddenly increases is of great
`importance in many areas of gas discharge physics and
`its applications. Interest in ionization relaxation in inert
`gas discharges has been stimulated recently by the rapid
`development of excimer lasers excited by pulsed electri(cid:173)
`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(cid:173)
`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(cid:173)
`tion relaxation when the ionization ne/n1 increases from
`the initial value ne 0/n1 10-8-lo-7 to 1o-5-1o-4• We are
`mainly interested in analyzing the buildup of ne and the
`behavior of the spatial distribution of ne during the relaxa(cid:173)
`tion process. We report experimental data on ionization
`relaxation in the range 8 < E/n1 z 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(cid:173)
`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(cid:173)
`tion time for the electron energy distribution function for
`E/n1 ":'; 3 Td is much less than the characteristic ioniza(cid:173)
`tion growth times (tr < 1 #-(B) .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(cid:173)
`ture of the ionized atoms. In what follows, we assume that
`these parameters are known either from published e.li:peri(cid:173)
`mental data or from numerical solution of the Boltzmann
`kinetic equation. 5•6
`
`2~ If we use k to label the excited states (with k
`2
`corresponding to the first excited state), the populations
`
`of the excited states with k > 2 are almost always quasi(cid:173)
`stationary. However, quasistationarity may be violated
`for the lower metastable or resonance 1 evels 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 k""' k±
`1 in the collisional transition kinetics between excited
`states, it is helpful when describing the atomic distribu(cid:173)
`tion over the excited states with k > 2 to use the machinery
`of the modified diffusion approximation (MDA) theory, 4 •~
`which gives the quasistationary populations in terms of
`ne and n2•
`4. Since for E/n1 in the range of interest the ioniza(cid:173)
`tion n~/n~ is much greater than 10-4, we may neglect ra(cid:173)
`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(cid:173)
`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(cid:173)
`tion relaxation using the following system of equations:
`
`(1)
`
`Here n1, ~· and ne are the atomic densities in the ground
`and first excited states and the electron density, respec(cid:173)
`tively; k12 , k21 are the rate constants for the collisional
`transitions 1 ~ 2; f3te and {32e are the rate coefficients for
`
`30
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`0038-5662/83/01 0030-06 $03.40
`
`30
`
`INTEL 1106
`
`
`
`a
`
`~t
`
`k
`
`e - -
`/
`z
`
`r2e
`
`=::>
`VJ;
`
`r,
`
`r;2
`
`direct ionization from the ground state to the first excited
`level; f3st is a generalized coefficient for stepwise ioniza(cid:173)
`tion from the quasistationary levels (k > 2); A21 is the
`spontaneous transition probability 2 ._. 1; 821 is the capture
`fa.ctor for resonance radiation, calculated using the Biber(cid:173)
`man -Holstein equations4; 'i7 r e and 'i7 r 2 are the diffusion
`fluxes of the electrons and excited atoms, and are given
`approximately by 'V'rk ""nkvDk [for a cylindrical geometry,
`VDk "" (2.4)2Dk/R2],
`It can be shown using the MDA theory4 that
`
`(3)
`
`where
`
`IT~= exp (0.2E~/Ei,),
`
`En= (n,/4.5 · 1013
`
`)''• T-;-'1, (n,- incm- 3 • T,- .in'eV),
`
`X"" 0.2, Ry = 13.6 eV, z (x) = 3 ~;; f e-1t'f,dt,
`Q . 4 v21< e•
`1 7 1 o-7
`.r=~= · '
`vmRy'
`To facilitate comparison, we bave retained the nota(cid:173)
`used in Ref. 4.
`
`0
`
`3
`cm/s.
`
`(4)
`
`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 ~ 1 em, n1 ~ 1016 cm-3), the
`radiation capture factor is 821 ""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(cid:173)
`cited atom diffusion times are Tn2 "" 10-3-1o-2 s, so that
`we may neglect 'i7 r 2 compared with n2nef32e in (2).
`The above arguments show that the three-level approx(cid:173)
`imation can be used to describe ionization buildup under
`our assumptions. In dimensionless variables, the equa(cid:173)
`tions for the ionization ldnetics take the form
`BNjih = bNM +eN- dN,
`a.M;a~ = N- bN1vi,
`111l,~o = 1110 , N l,~o = 1,
`111 = 11 2jn,0 , N = ncfn,0 , "'= tn 1k12 ,
`b = n,o~2,/nlkl2• c = ~1,/k12• d = "na/nlk12'
`( 6)
`
`(5)
`
`Equations (5)-(6) easily yield the following relation
`between N and M:
`
`N =1 +M -III+ (1 +c -d) I [(1- bMo)J
`n (1- bM)
`b
`
`0
`
`(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 ""c- d + bMo « 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(cid:173)
`plosive increase in ne• The subsequent increase in ne then
`reaches its maximum value, equal to the rate of excitation
`dN/dT = 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)
`
`The nonlinear system (1)-(2) can be solved numerical(cid:173)
`the coefficients and initial conditions are specified.
`the rate constants given in Refs. 5-8 for the various
`urcJcP.~::~::,,,,_ we can make a series of simplifications making
`to solve (1)-(2) analytically for E/n1 and ne/n1
`range of interest.
`~e "" 10-8-1o-7 cm3/s [Refs. 5, 8], we see from
`(4) that the ratio f32e/ f3 st of the coefficients for
`and stepwise ionization from the first excited level
`on ne and on the electron temperature Te. For
`5•1013 cm-3 and Te ~ 1 eV, the stepwise ionization
`>l:l'!'"~nner from levels with k > 2 is unimportant (f3st « f32e)
`>'>~'·""'~u~;e most of the radiation escapes.
`
`Sov. Phys. Tech. Phys. 28(1). January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`31
`
`
`
`a
`
`'I
`+2
`•J
`·~
`•S
`oS
`
`..t!.
`'I'
`JOO 5
`.
`""-
`"'b
`l3 . ..
`I~ .,
`l:t
`100 ~· ~--""
`
`200
`
`'--:,_"-.
`1-"'
`
`b
`
`FIG, 3, Duration r8 for a discharge in argon, p ~ 11.4 (a) and 3.7 torr
`(b), a: 1) neo = 1,6 • 1010 cm-3
`; 2) 2. 7 ·1010
`; 3) 4,5 • 1010; 4) 6,4 ·toto .
`; 4) 1,6 ·ldu.
`5) 1011
`; 6) 1.6 ·1011
`; b: 1) 9. 7 • 109
`; 2) 3 ·1010
`; 3) 6.8 ·1010
`
`E/n,. Td
`
`E/n,. . 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 n6 (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
`bi11 2/2 + (c
`
`d) ill
`
`t - bMU2- (c- d) 1110 ,
`
`(8)
`
`N
`
`and taking the first arctangent in (9b) equal to rr/2. Equa(cid:173)
`tion (11) shows that the characteristic time for the fast
`stage is T RJ ln (N/N1) « Ts• 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(cid:173)
`havior in ne(t) is quite universal and can be traced to the
`fact that we almost always have {316 « k12•
`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 > ,f\27lii],
`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(cid:173)
`mined by the appropriate rate constant or by the first
`Townsend coefficient, and the "instantaneous ionizationn
`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 T < T1 (M = v"27li)
`after the field is applied, and that the "instantaneous ioni(cid:173)
`zation" approximation does not become valid until times
`t "" T s 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 em and the distance between the electrodes was
`L
`52 em, The gas was preionized by applying a de cur·
`rent ip = 0,5-20 rnA, The parameters of the positive plas·
`rna column were ·calculated using the theory developed
`in Ref, 9 from the experimentally recorded current densitJ,
`and E 0/n1 values, The initial density n60 on the axis varlet:
`in the range 5 •109-4 .1011 cm-3•
`A voltage pulse with rise time (1-2) •10-7 s of positivi
`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 ahd the voltage between the probes were
`100
`recorded by a capacitative divider (C1 = 2 pF and C2
`pF) capable of transmitting rectangular pulses with rise
`time ~ 10-7 s without appreciable distortion, The dis(cid:173)
`charge current was recorded using a zero-inductance
`
`2
`
`A<O,
`(9a)
`(9b)
`
`where A is given by
`
`A
`
`(2/b)
`
`[M0
`
`(c
`
`d)Jbj2
`
`(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 ~ 0, the condition is that bM0 + c > d.
`It can be shown using (6), (7) that the curve M(T) has
`(c d)/b
`0.8/b, after
`an inflection point at M1 = N 1 -
`which M changes slowly. Since M1 differs from the sta(cid:173)
`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(cid:173)
`tion has become stationary:
`
`M = +, -.
`
`In z1
`
`•
`
`(11)
`
`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•
`Equations (9), (11) derived above readily yield expres(cid:173)
`sions for the characteristic times of the slow and fast
`stages. Since we have bM + c- d » bilAl prior to the
`start of the abrupt rise in n6 , the duration Ts of the slow
`stage can be found with sufficient accuracy from (9) by set(cid:173)
`ting the first quotient in the logarithm in (9a) equal to one
`
`32
`
`Sov. Phys. Tech. Phys. 28(1). January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov.
`
`
`
`25
`
`0
`
`a
`
`0
`
`0 1
`+Z
`63
`xlf
`
`b
`
`+
`
`0
`
`~~~~ FIG. 4. Curves for r3 vs. E/n1 for a discharge in krypton; p
`
`~
`
`; 2) 1011; 3)
`10,5 (a) and 6 torr (b). a: 1) neo = 4·1010 cm·3
`; 4) 3.3 •1011
`1.5 •1011
`; b: 1) 3,5. 1010
`; 2) 9.5 ·1010; 3) 1.6.
`1010; 4) 3.2 ·1011•
`
`10
`
`12
`
`1/f
`
`16
`
`_1 _ _______1_______
`18
`1'1
`16
`18 30 22 2'1
`E/n,. Td
`
`26
`
`..... L__L_
`28
`JO
`
`r'esistor connected in series with the cathode-to-ground
`section of the tube, The local dependences ne(t) were re(cid:173)
`<l9rded by observing the plasma emission intensity, which
`'!~proportional to n~ (p
`1). The density of the neutral
`~ffs was monitored using the interferometric technique
`described in Ref.ll and a dual-trace oscilloscope was used
`f!Jr all the measurements. Special experiments were con(cid:173)
`ducted to verify the unimportance of such factors as the
`t{roximity of the shields and grounded objects or the shape
`lind composition of the electrodes (we had pL > 200 em •
`t~rr [Ref, 12]), which do cause appreciable effects during
`breakdown of a cold gas,12 •13 We were also able to repro-
`1d~,ce 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(cid:173)
`~rameters of the gas discharge plasma vary along the tube
`11nd ascertain the importance of various mechanisms in-'
`'valved in discharge formation after an abrupt increase in
`.tile 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(cid:173)
`sion from different regions along the length of the dis(cid:173)
`Qltarge (e, f) show that after a high-voltage pulse is sud(cid:173)
`iqenly applied, the discharge current rises very slowly
`f(!r times t < Ts and the tube voltage remains almost con(cid:173)
`~~~ant, This is followed by a sudden rise in the current, ac(cid:173)
`'?Rmpanied by a voltage drop across the tube, We also see
`(;that the field increases almost simultaneously (to within
`Jo-7 s) over the entire length of the positive column and
`f~en remains constant for t < Ts• The plasma emission
`~iil.tensity from different regions along the length of the tube
`;:~}$o starts to increase almost simultaneously and repeats
`ik~e current trace,
`Our measurements revealed that for t < T s• ne/n1
`;~pcreases by less than a factor of 102, i,e,, we have ne/n1 <
`'19;5 at the end of the slow stage,
`. . . A similar delay in the current increase has been noted
`l>imany other workers when an electric field is suddenly
`:a.Rplied (see e.g., Ref, 12), This lag might be caused by
`l~~~ small velocity of the ionization wave down the tube from
`~!~~high-voltage electrode to the grounded electrode,13,l4
`;~~r,by onset of instability in the uniform quasistationary
`i~~scharge when the field is applied [possibly caused by
`~~(loesses near the electrodes; of, Ref, 15], Taken to-
`.
`the above findings show that in any case, neither.
`tion of an ionization wave nor processes at the
`trodes determine the duration Ts of the slow stage.
`The lag in rapid current buildup has often16•17 been
`
`0 o.z O.!f 0.6 0.8
`r/R
`FIG. 6. Radial distribution ne(r) as a function of time for A > 0 in an argon
`discharge. p 11.4 torr, neo = 1010 em-s. r s = 39 IJS.
`
`attributed to the finite time required for growth of ther(cid:173)
`mal-ionization instability, Interferometric measurements
`show that fort< 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 n10 the growth time for
`thermal-ionization instability is much larger than Ts•
`
`Thus, under our conditions the slow increase in ne
`during the initial stage is due to the combined effects of
`kinetic processes occurring in the bulk of the discharge
`for constant E/n1, quasistationary electron energy dis(cid:173)
`tribution functions, and ne/n1 values < 10-5,
`We now compare the experimental results for Ar and
`Kr with the theory developed above.
`
`The points in Figs. 3, 4 give the experimental values
`as a function of the specific conditions (n1, neo• 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
`
`Sov. Phys. Tech. Phys. 28( 1 ), January 1983
`
`A. A. Kudryavtsevand V. N. Skrebov
`
`33
`
`
`
`I
`
`D
`
`to'
`'IQ
`
`ZD § 1(/.
`€l
`If
`r:J'Z
`p..
`1.0
`
`O.lf
`Q.Z
`
`\.
`....
`
`'
`
`to''
`
`10 17
`
`10 111
`
`'1
`8
`u
`o£
`c:
`
`9
`
`15
`
`39
`
`*5
`
`Z7 33
`Z1
`Ejn,, Td
`FIG. 6. The behavior of ne in the bulk of an argon discharge. 1) neoJTil =
`10" 8
`; 2) 10· 7
`• Stepwise ionization predominates in region I, direct ioniza(cid:173)
`tion processes predominate in region II, and ne does not increase in region
`Ill.
`
`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(cid:173)
`formation of the initial distribution ne0(r) for A < 0 but
`substantial deformation when A > o.
`Experiments (cf. 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 em (R is the tube radius) [Ref, 18]
`and Po
`2-3 torr for R = 1 em [Ref. 19], For discharge
`in helium, the experiments indicate that ionization is uni~
`form for pressures <( 10 torr when R
`1 em.
`
`Our model accounts well for these experimental find-
`ings.
`6 shows calculated curves for a discharge in
`argon. We see that for pR < 1 em • torr, ne cannot in(cid:173)
`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), k12 , f31e,
`f3.ze, vna all have the same order of magnitude, the cor(cid:173)
`responding curves behave similarly for similar values of
`pR. Helium is an exception, since under our conditions
`VDa is much larger and f31e much smaller than for the
`heavy inert gases. The region A < 0 will therefore cor(cid:173)
`respond to larger products pR, and this is also in agree(cid:173)
`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(cid:173)
`ing to equilibrium ionizations n~/n~ » 10-4• 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(cid:173)
`wise ionization processes in the buildup of instabilities
`in self-sustained and externally maintained discharges.11•1n
`Under our conditions, the above equations have the ad(cid:173)
`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(cid:173)
`tion whenever a field is suddenly applied to a weakly
`ionized gas, they must be allowed for when studying emis(cid:173)
`sion mechanisms in pulsed gas lasers, gas breakdown,
`laser sparks, etc,
`
`1A. V. Eletskii, Usp. Fiz. Nauk 125, 279 (1978) [Sov. Phys. Usp. 21, 502
`(1978)].
`zG, V. Naidis, Zh. Tekh. Fiz. £!.. 941 (1977) [Sov. Phys. Tech. Phys. 22,
`562 (1977)].
`3A. A. Belevtsev, Teplofiz. Vys. Temp. 17, 1138 (1979).
`4t. M. Biberman, V. S, Vorob'ev, and I, T. Yakubov, Usp. Fiz. Nauk .!.21•
`353 (1972); 128, 233 (1979) [Sov. Phys. Usp. 12, 375 (1973); ~ 411 (1979)].
`5N, L. Aleksandrov, A.,M. Konchakov, and E, E, Son, Zh, Tekh. Fiz. 50,
`4B1 (1980) [Sov, Phys. Tech, Phys. 25, 291 (1980)].
`6W. L. Nigh an, Appl. Phys. Lett.
`424 (1978).
`7B, M, Smirnov, Ions and Excited
`In Plasmas [in Russian], Atomizdat,
`Moscow (1974).
`6M. G. Voitik, A. G. Molchanov, and Yu. G, Popov, Kvantovaya Elektron.
`(Mosco1~) _!, 1722 (1977) [Sov. J. Ouantum Electron. :I, 976 (1977)].
`
`Ts calculated from
`The solid curves in Figs, 3, 4
`(12) using values for k12 , f3te• and f32e from numerical cal(cid:173)
`culations in Ref, 5.
`Equation (12) shows that for small E/n1, when A c,
`2/b, we have Ts = 1T /f2b. It follows that (n1ne0) 112rs =
`(k12f32e)-i/2 = const for a fixed value of E/n1• For large
`E/n1 we have A- 0, and Ts- 2/(ne0f32e + n1f3te- vna>
`is only weakly dependent on neo•
`
`Figure 3a also gives experimental values of the pa(cid:173)
`rameter (n1ne0) 1/2rs• which for E/n1
`const remains con(cid:173)
`stant to within the experimental error for neo• n1, and Ts
`varying over wide limits. For large E/n1 [Fig. 3b, Fig,
`4b] Ts 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(cid:173)
`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(cid:173)
`tributions ne 0(r) and ~0 (r) are known, This problem is of
`interest in terms of understanding the mechanism re(cid:173)
`sponsible for formation and constriction of the current
`channel in a pulsed discharge,U
`
`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 ""
`2/b). This results in a sudden increase in the magnitude
`of the initial irregularities in ne(r) for t < 7 s• As an il(cid:173)
`lustration, Fig. 5 shows curves giving the time dependence
`of ne(r) calculated from (9b) for E/n1 = 12 Td and neo
`1010 cm-3 for an argon discharge. According to Ref. 9,
`the initial neo and ~0 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 vall
`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. 281 1). January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`
`
`Granovsldi, Electrical Currents in Gases, Steady Currents [in Russian],
`Mosco,,• (1971).
`N. Kondrat'ev and E, E, Nikitin, Kinetics and Mechanisms in Gas(cid:173)
`Reactions [in Russian], Nauka, lv!oscow (1974).
`N, Sl<rebov and A. I. Skripchenko, Teplofiz. Vys, Temp. 19, No, 3
`
`p, Lozanskii and 0. B. Firsov, Spark Theory [in Russian], Atomizdat,
`(1975).
`P. Abramov, P, I. Ishchenko, and I, G. Mazan'ko, Zh, Tekh. Fiz. 50,
`[Sov. Phys. Tech, Phys. ~ 449 (1980)].
`Asinovskii, V, N. Markov, N, S, Samoilov, and A, M. Ul'yanov,
`Vys. Temp. 16, 1309 (1978),
`A leksandrov, R. B. Gurevich, A, V. Kulagina, et al., Zh. Tekh. E'iz.
`105 (1975) [Sov. Phys. Tech. Phys. 20, 62 (1975)].
`
`P, Velikhov, v. D. Pis'mennyi, and A. T, Ral<himov, Usp, Fiz. Nauk
`419 (1977) [Sov. Phys. Usp. 20, 586 (1977)].
`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], N auk a i Tekhnika, Minsk (1970), p. 459.
`19M, N. Polyanskii, V. N. Skrebov, and A, M, Shukhtin, Opt. Spektrosk.
`2B (1973).
`Dautherty, J. A. Mangano, and J. H, Jakob, Appl, Phys. Lett, 28,
`581 (1976).
`
`Translated by A. Mason
`
`Sov. Phys. Tech. Phys. 28( 1), January 1983
`
`0038-6662/83/01 0036-04$03.40
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`36
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