`
`Ionization relaxation in a plasma produced by a pulsed inert-gas
`discharge
`
`A. A. Kudryavtsev and V. N. Skrebov
`A. A. Zhdanov State University, Leningrad
`(Submitted September 29, 1981; resubmitted January 29, 1982)
`Zli. Tekh. Fiz. 53, 53—6l [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/n{’>10““. It is shown that the
`electron density increases explosively in time due to accumulation of atoms in the lowest excited states. An
`approximate analytic solution is found for describing the behavior of the time and spatial increase in its as a
`function of the specific conditions. The proposed model is verified experimentally.
`
`PACS numbers: 52.25.Lp, 51.50. + V, 52.80.Dy
`
`of the excited states with k > 2 are almost always quasi-
`stationary. However, quasistationarity may be violated
`for the lower rnetastable or resonance levels 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 "1 k-1
`1 in the collisional transition kinetics between excited
`states, it is helpful when describing the atomic distribu-
`tion over the excited states with l: > 2 to use the machinery
`of the modified diffusion approximation (MDA) theory,“s5
`which gives the quasistationary populations in terms of
`ne and n2.
`
`4. Since for E/n, in the range of interest the ioniza-
`tion ng/ni’ 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—’?) - 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:
`
`: ninel3la ‘l‘ ""anrp2= ‘i’ ”2".l3st ““ vre»
`
`(1)
`
`0 %
`
`”—[;'l—’ _ nlmlrw
`
`rtgzgkm
`
`Ii,/1,3,,
`
`11,1243“
`
`112/l M02,
`
`Vi‘,
`
`(2)
`
`Here 11,, n2, and tie are the atomic densities in the ground
`and first excited states and the electron density, respec—
`tively; km, kg, are the rate constants for the collisional
`transitions 1 -.+_ 2; file and p28 are the rate coefficients fol‘
`
`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-
`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/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 neg/n1 = 104-10-7 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-
`tion process, We report experimental data on ionization
`relaxation in the range 8 < E/n1 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 /n, increases discontinuously, the relaxa-
`tion time for the electron energy distribution function for
`E/n1 S 3 Td is much less than the characteristic ioniza-
`tion growth times (tr < 1 ;ts).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
`
`130
`
`Sov. Phys. Tech. Phys. 28(l ), January 1983
`
`0038-5662/83/01 003006 $03.40
`
`30
`
`
`
`4
`
`Page 1 ofd
`
`TSMC-1304
`TSMC v. Zond, Inc.
`Page 1 of 6
`
`
`
`
`
`‘eat ionization from the ground state to the first excited
`i 1; 5st is a generalized coefficient for stepwise ioniza-
`from the quasistationary levels (k > 2); A2, is the
`taneous transition probability 2 —> 1; 921 is the capture
`31:01‘ for resonance radiation, calculated using the Biber—
`-—Holstein equations‘; \7I‘e and V P2 are the diffusion
`axes of the electrons and excited atoms, and are given
`T1-oximately by VI‘k :2 nkr/Dk [for a cylindrical geometry,
`is (2.4)2D1;/R2].
`
`R It can be shown using the MDA theory‘ that
`
`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 3 1 cm, n, 3 10“; cm"3), the
`radiation capture factor is 62, 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 "rm 2 104-10”? 5, so that
`we may neglect V F2 compared with n2ne,B,e 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
`
`6N/61:bNM+c1V~dN,
`BM/an 2 N — ENM,
`M: :2,/ngo,
`IV =12,/n,,,,
`b : ncflpfi/nlkl?’
`
`4: = trzlku, M lflo = MD.
`d 2" Va"/Vfllkn.
`5 7: £31»/krz»
`
`(5)
`
`N|1=u=1»
`(6)
`
`Equations (5)—(6) easily yield the following relation
`between N and M:
`
`N 2 41 + M, -_ M + L'“*‘+b"’9 ln
`
`(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/n, <
`300 Td, ne/:11 < 10"5).
`in a steady»-state plasma, we usually
`have Mo €
`1 [Ref. 9].
`
`Using the smallness of b and c, we find from (5)—(7)
`that dN/dM 2 o — 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 :12 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
`LIN/d"r = 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 me 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)
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`31
`
`TSMC-1304 I Page 2 of 6
`
`(3)
`
`(4)
`
`1
`2i
`n,n,l3n=I1,I1fi3‘/:_tg"m
`'
`
`R 3
`
`-We
`_‘L£_;__’
`?'~(T".)
`
`6 e
`
`xp (U.2E‘3[E};),
`
`
`
`(na/4.5 - 10”)’/4T;"/B (n,—- in cm”. T,— .in‘eV),
`V
`4
`.r H J
`13.6 EV,
`S 8 ‘flail,
`
`,
`0,2,
`
`In/§T:g4 N13; , 10-7 Cm:/S‘
`i/ER;/5/’ —
`
`To facilitate comparison, we have retained the nota-
`used in Ref. 4,
`
`The nonlinear system (l)—(2) can be solved numerical-
`the coefficients and initial conditions are specified.
`g the rate constants given in Refs. 5-8 for the various
`sses, we can make a series of simplifications making
`siblo to solve (1)-(2) analytically for E/n1 and ne/n1
`range of interest.
`
`ince 3,8 2 10"3-10“7 cma/s [Refs. 5, 8], we see from
`d (4) that the ratio £28/B St of the coefficients for
`and stepwise ionization from the first excited level
`ds on I19 and on the electron temperature Te. For
`.5- 10” cm"3 and Te 3 1 eV, the stepwise ionization
`el from levels with k > 2 is unimportant (Est << B26)
`31156 most of the radiation escapes.
`
`Sov. Phys. Tech. Phys. 23(1), January 1983
`
`TSMC-1304 / Page 2 of 6
`
`
`
`§§E’.fs‘f’-””~'7”A"3-toulfl-cm"/3 22
`
` FIG. 3, Duration T5 for a discharge in argon. p = 11.4 (a) and 3.7 lon-
`
`a: 1) me. .- 1.6- 101° cm'3: 2) 2.":-10”; 3) 4.5-101°; 4)s.4-101°.
`(b).
`5)1o“; 6) 1.6 v 10“; b: 1) 9.7. 10’; 2) 3.10"’; 3) e.s«1o‘°-. 4) 1.5-mtg
`
`26
`
`Z”
`'Td
`
`73
`E/n,. Td
`
`15
`
`76
`
`15
`
`. mt“, .
`20
`
`E/n..
`
`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 = W”/2 + (c —~ d) M + i — (21113/2 e (c — d) 1710,
`
`(3)
`
`1 : b ~/13,1’;
`X 3mltit::i:”.$T‘%E.‘tZ;:;::::t2%;ll A<gg,»a)
`arotg [ ] — a1~otg[ 1,
`A > 0,
`(9b)
`
`where A is given by
`
`A = (3/5) ~— We + (6 ~ d)/bl”
`
`(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 M1 = N, - (c — d)/b :2 0.8/b, after
`which M changes slowly. Since M, 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:
`
`1
`1
`(11)
`mi.
`214:3, 1:
`(T4.-”F:"dT N.
`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 ng as far as
`the inflection point M = M1.
`
`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/173-l prior to the
`start of the abrupt rise in ne, the duration T5 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 7r/2. Equa_
`tion (11) shows that the characteristic time for the fast
`stage is T rd ln(N/N1) << ‘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 die << km.
`
`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 > @2751},
`
`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 1 < 1', (M = w/275)
`after the field is applied, and that the "instantaneous ioni-
`zation“ approximation does not become valid until times
`t .~c TS after the field was first switched on.
`
`EXPERIMENTAL RESULTS
`
`We studied ionization relaxation in He, Ne, Ar, and Kr
`for initial gas densities in : (0.5~—5) -10” cm'3 by applying
`an additional electric field to the preionized gas. The
`discharge occurred inside a cylindrical tube of diameter
`ZR = 2.5 cm and the distance between the electrodes was
`L = 52 cm. The gas was preionized by applying a dc cur-
`rent ip = O.5—20 mA. The parameters of the positive p1a5'
`ma column were calculated using the theory developed
`in Ref. 9 from the experimentally recorded current dens“).
`and E0/n1 values. The initial density neg on the axis variéii
`in the range 5-109-4 .10“ cm-3.
`
`A voltage pulse with rise time (1~2) - 10'7 s of positivf.
`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 bya capacitative divider (C1: 2 pF and C2 = 100
`pl?) capable of transmitting rectangular pulses with rise
`time ~ 10"7 s without appreciable distortion. The dis—
`charge current was recorded using a zero—inductance
`
`32
`
`Sov. Phys. Tech. Phys. 28(1), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov.
`
`32
`
`TSMC-1304 I Page 3 of 6
`
`TSMC-1304 / Page 3 of 6
`
`
`
`FIG. 4. Curves for T3 vs. E/n; for a discharge in krypton: P =
`10.5 (a) and s torr (b).
`at 1) “C0: 4-10” cm”: 2) .10”: 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“.
`
` o m_
`
` ___|
`
`15
`
`yi=';
`I
`i
`v
`I
`1
`I
`J ,,,_,n
`L
`74
`’/5
`73
`20
`22
`24
`26
`33
`-70
`
`E/n,, Td
`
`iiiesistor connected in series with the cathode—to—ground
`otion of the tube. The local dependences ne(t) were re-
`rded by observing the plasma emission intensity, which
`‘proportional to Hg (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
`all the measurements. Special experiments were con-
`ahcted to verify the unimportance of such factors as the
`proximity of the shields and grounded objects or the shape
`ma composition of the electrodes (we had pL > 200 cm -
`torr (Ref. 12]), which do cause appreciable effects during
`:breakdown of a cold gas .122” We were also able to repro~
`Vduce 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 plasma vary along the tube
`find ascertain the importance of various mechanisms in-:
`lved in discharge formation after an abrupt increase in
`the field strength.
`
`
`
`
`
`Figure 2 shows some typical measured curves. The
`pscilloscope traces of the current (a), tube voltage (b),
`ltage differences between the probes (c, d), and the emis-
`
`FIG. 5. Radial distribution ne(r) as a function of time for A > 0 in an argon
`discharge. p = 11.4 torr, nm = 10"’ cm”, rs = 39 ps.
`
`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 T5.
`
`Thus, under our conditions the slow increase in De
`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 < 10'5.
`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, new E/nx).
`Since in our experiments we had A > 0 [of. (10)], Eq. (9b)
`implies that the duration of the slow stage is given by
`
`m.mg“_¢iT21n!»+<c——d>/bl ”
`
`b/1— % <M,,+ °';'i)2
`
`.
`
`(12)
`
`ant. This is followed by a sudden rise in the current, ac-
`mpanied by a voltage drop across the tube. We also see
`at the field increases almost simultaneously (to within
`'7 s) over the entire length of the positive column and
`en remains constant for t < "rs. The plasma emission
`tensity from different regions along the length of the tube
`so starts to increase almost simultaneously and repeats
`e current trace,
`
`Our measurements revealed that for t < TS, ne/N1
`creases by less than a factor of 102, i,e., we have ne/I11 <
`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
`ied (see e.g., Ref. 12). This lag might be caused by
`e small velocity of the ionization wave down the tube from
`5.’ high—voltage electrode to the grounded electrode,”v”
`y onset of instability in the uniform quasistationary
`Oharge when the field is applied [possibly caused by
`ocesses near the electrodes; cf, Ref. 15]. Taken to-
`‘ther the above findings show that in any case, neither.
`'V pagation of an ionization wave nor processes at the
`éotrocles determine the duration 75 of the slow stage.
`
`The lag in rapid current buildup has often‘5’” been
`
` Sov. Phys. Tech. Phys. 2811). January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`33
`
`TSMC-1304 I Page 4 of 6
`
`TSMC-1304 / Page 4 of 6
`
`
`
`pK,cm-tar:
`
`aweS§-‘§8..
`
`37.5 27
`
`27
`
`.73
`
`.529
`
`‘/5
`
`The form of the radial distribution ne(r) will then be simi.
`lar to the initial distribution and no well —defined plasma
`column is produced. Our model thus predicts slight de-
`formation of the initial distribution 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—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 pH < 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, B16,
`#29, VD), all have the same order of magnitude, the cor-
`responding curves behave similarly for similar values of
`pR. Helium is an exception, since under our conditions
`VD“ is much larger and /316 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 112/n‘,’ >> 10"‘. We conclude
`from a comparison of the experimental spatial and time
`dependences of ne thatthe model is quite accurate. We
`have shown that the increase of ne with time is explosive
`because atoms accumulate in the lowest excited states.
`Our results are important for analyzing the role of step-
`wise ionization processes in the buildup of instabilities
`in self—-sustained and externally maintained discharges.
`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,
`
`17,20
`
`‘A. V. Eletskii, Usp. Fiz. Nauk _1g§_, 279 (1978) [Sov. Phys. Usp. g1_, 502
`(1978)].
`is. v. Naidis, Zh. Tekh. Fiz.g1. 941 (1977) rsov. Phys. Tech. Phys. 22.
`562 (19’7'7)].
`“A, A. Belevtsev, Teplofiz. Vys. Temp. y_, 1138 (1979).
`41.. M. Biherman, V. S. VoIob‘cv, and I. T. Yakubov, Usp. Fiz. Nauk 191.
`353 (1972); _1g§_, 233 (1979) [Sov. Phys. Usp. _1_5_, 375 (1973); g2_, 411 (1979)|~
`5N. L. Aleksandrov, A.,M. Konchakov, and
`E, Son, Zh, Tekh. Fiz. QQ.
`481 (1980) [Sov. Phys. Tech. Phys. E, 291 (1980)].
`‘w. L. Nighan, Appl. Phys. Lett. 3_2_. 424 (1978).
`7B, M, Srnirnov, Ions and Excited Atoms in Plasmas [in Russian], Atomizdah
`Moscow (1974).
`“M. G, Voitik, A, G, Molchanov, and Yu. G. Popov, Kvantovaya Elektron-
`(Moscow) 5 1722 (1977) [Sov.1. Quantum Electron. :71 976 (19'l’l)].
`
`E/rz,,Td
`FIG. 6. The behavior of ne in the bulk of an argon discharge. 1) neg/n1 =
`104-, 2) 10‘7. Stepwise ionization predominates in region 1, direct ioniza-
`tion processes predominate in region 11, and ne does not increase in region
`111.
`
`The solid curves in Figs. 3, 4 give 75 calculated from
`(12) using values for km, file, and £423 from numerical cal-—
`culations in Ref. 5.
`
`Equation (12) shows that for small E/n1, when A 2
`2/b, we have 75 = 1r/\/TIE.’ It follows that (n,ne0)‘/275 =
`(k,2,I32e)“/2 = const for a fixed value of E/n1. For large
`E/n1 we have A ~+ 0, and 75 —> 2/(neg/926 + n,/he -— VDQ)
`is only weakly dependent on neo-
`
`Figure 3a also gives experimental values of the pa-—
`rameter (n1ne0)1/27$, which for E/n1 = const remains con-
`stant to within the experimental error for neg, n1, and 7s
`varying over wide limits. For large E/n1 [Fig. 3b, Fig.
`4b] TS becomes almost independent of neg, as predicted
`by our theory.
`
`We also note that thefunction 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.,0(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/oi = 12 Td and neo =
`10”’ cm"3 for an argon discharge. According to Ref. 9,
`the initial neg and um) distributions in a glow discharge
`at moderate pressures can be approximated by Bessel
`functions, and we took Mo = 0.1. Under these assumptions,
`we see that the theory predicts that the distributions will
`become highly nonuniform at times t 3 ‘('5 after the field is
`turned on,
`
`For large E/n1, A < 0 and the growth of ne during the
`slow stage lcf. (9a)] is determined by direct ionization.
`
`34
`
`Sov. Phys. Tech. Phys. 28H ), January 1983
`
`A. A. Kudryavtsev and V. N. Skrebov
`
`34
`
`TSMC-1304 I Page 5 of 6
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`“E, P, Velikhov, V. D. Pis'mennyi. and A. T. Rakhimov. Usp. Fiz. Nauk
`122, 419 (19‘7’1)[Sov. Phys. Usp. _2_0_. 586 {19'77)].
`"E Napartovich and A. N. Starostin.
`in: Plasma Chemistry, B. M.
`Smirnov, ed., [in Russian] (1979). pp. 6, 153.
`“D, N. Novichkov, in: Top1CS in Low ~Tampe1-ature Plasma Physics [in
`Russian], Nauka i Tekhnika, Minsk (1970), P. 459.
`“M. N. Polyanskii. V. N. Skrehov, and A. M. Shukhtin, Opt. Spektrosk.
`84. 28 (1973).
`“TD. Dautherty, J, A, Mangano, and I. H, Jakob, Appl. Phys, Lett, _2_£}_,
`581 (1976).
`
`Translated by A. Mason
`
`E‘
`Gmnovsldl, Electrical Currents in Gases. Steady Currents [in Russian],
`ukal Moscow (1971).
`N. Kondrat‘ev and E. E. Nlkitin, Kinetics and Mechanisms in Gas-
`ase Reactions [in Russian], Nauka, Moscow (1974).
`N, Skrebov and A. I. Skripchenko, Teplofiz. Vys. Temp. $2. No. 3
`
`
`
`y
`(1981).
`5.1), Lozanskii and O. B. Fxrsov, Spark Theory [in Russian]. Atnmizdat,
`Moscow (1975).
`0‘ P, Abramnv, P. I. Ishchenko. and I. G. Mazan'ko, Zh. Tekh. 1712- _5_0,
`s (1980) [Sov. Phys, Tech. Phys. 29' “:9 (mean.
`0 Asinovskii, V. N. Markov, N. S. Samoilov, and A. M» UVVBROV.
`Ploflyh vyg, Temp. 1_6J 1309 (1978),
`‘Y;._ Aleksandrov, R. B. Gurevich, A. V, Kuiagina, et a1., Zh. Tckh. Fiz.
`05 (1975) [Sov. Phys. Tech. Phys. _2_0, 62 (19'75)].
`
`45
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` Sov. Phys. Tech. Phys. 28(1), January 1983
`
`0038-5662/B3/O1 0035-04 $03.40
`
`35
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