Formation kinetics and parameters of a photoresonant plasma
`
`A. V. EIetskii,Yu.N. Zaitsev, and S.V. Fomichev
`
`(Submitted 28 July 1987)
`Zh. Eksp. Teor. Fiz. 94, 98-106 (May 1988)
`
`The kinetics of a plasma produced by applying to a metal vapor pulsed resonant radiation whose
`frequency corresponds to the energy of a resonant transition in the atom is investigated. It is
`established that the principal ionization mechanism in such a plama, which ensures practically
`total ionization of the vapor at relatively low energy inputs, is stepwise ionization of the atoms by
`electron impact. Under conditions of above-equilibrium occupancy of the resonantly excited
`state, this process takes place at a constant electron temperature Te . The calculated values of Te
`agree with the measurement results. The possibility of producing a nonideal photoresonant
`plasma is analyzed; it is shown that the action of resonant radiation increases the nonideality
`parameter by 1.5 to 2 times compared with an equilibrium plasma. The parameters of the
`equilibrium plasma produced by relaxation of a photoresonant plasma are calculated. These
`parameters (Ne ~ lO‘°—10‘7 cm ‘3, Te ~ 0.4-0.6 eV) are not reached if the plasma is produced by
`traditional methods. It is shown that an equilibrium plasma with arbitrary electron density can be
`produced by the photoresonance method.
`
`1. A photoresonant plasma (PRP) is produced when a
`gas is acted upon by monochromatic radiation whose fre-
`quency corresponds to a resonance transition of the gas
`atom.‘ Even the first experimentsz aimed at producing PRP
`and at investigating its properties have shown that this inter-
`esting physical object differs substantially from a plasma
`produced by traditional methods (gas discharge, charged-
`particle beam, etc. ). Interest in the investigation of PRP has
`increased substantially in recent years in view of the advent
`of high-power tunable visible-light lasers, which are used to
`produce the PRP. Experiments performed with the aid of
`such sources‘ have shown that even relatively low-power
`laser emission sufiices to obtain PRP of Na (Ref. 3), Li
`(Ref. 4), Cs (Ref. 5), Ba (Ref. 6), Mg (Ref. 7) and other
`vapors with quite high ionization. In contrast to a gas-dis-
`charge plasma, electrons in PRP are not subject to the accel-
`erating action of an electric field, so that their temperature
`remains relatively low. By the same token, the resonant ac-
`tion of the radiation on the gas is a special method of produc-
`ing a plasma having extremal physical properties, viz., high
`degree of ionization at a low electron temperature. Produc-
`tion of a supercooled plasma with close to one particle in the
`Debye sphere is reported in Ref. 8. Plasma with such param-
`eters cannot be produced by other methods. The purpose of
`the present study was an investigation of the mechanisms
`and formation kinetics of a PRP and determination of the
`
`best attainable properties of this object.
`2. We shall consider the PRP produced when metal va-
`por of density N is acted upon by monochromatic radiation
`of intensity J. For the sake of argument we consider the con-
`ditions of the experiment of Ref. 8, where Na vapor at a
`density lO'°—4- 10” cm‘3 was exposed to 25-ns pulses of
`resonant radiation of intensity 6107 W/cmz, as a result of
`which PRP was produced with parameters Ne ~10” cm'3
`and Te ~0.2—O.3 eV. The cited paper contains a most com-
`plete description of the PRP parameters.
`In the initial state of PRP production, the main source
`of the charged particles are the collisions between resonantly
`excited atoms, which either lead directly to formation of
`electrons and ions (associative ionization)°:
`
`A’+A'—>Az++e,
`
`or lead to formation of highly excited atomsgz
`
`A‘+A‘—>A"+A,
`
`(1)
`
`(2)
`
`which undergo associative ionization
`A
`(3)
`A“+{ ‘}—>Ae++e
`when colliding with normal or resonantly excited atoms.
`At high substantially above-equilibrium density of the
`resonantly excited atoms, the processes (1)—(3) lead to a
`rapid growth of the electron density in the PRP. Thus, in the
`case of Na the rate constant of process ( 1) is z 3- 10“ ‘ ‘ cm3/
`s (Ref. 9). Therefore the use of resonant radiation of saturat-
`
`ing intensity (J2 102-103 W/cmz) at an Na vapor density
`~1O”’—10‘7 cm” ensures effective ionization at a level
`~10“ cm”/s. Thus, even in the first few tens of nanose-
`conds of the PRP lifetime the electron density in it exceeds
`~10” cm'3. This creates a more effective channel for the
`
`formation of charged particles, due to the stepwise ioniza-
`tion of the atoms by electron impact. In fact, the rate of
`formation of charged particle via associative ionization ( 1)
`does not depend on Ne, whereas the rate of stepwise ioniza-
`tion is proportional to Ne (we are considering the colli-
`sional-kinetics regime typical of the experimental condi-
`tions, when the excited atoms are destroyed mainly by
`electron-atom collisions, and the role of the radiative pro-
`cesses is negligible). The criteria for realization of such a
`regime have been discussed in detail, e.g., in Refs. 10 and 1 1.
`Rougly
`speaking,
`these
`criteria
`are
`of
`the
`form
`Te<fiw, Ne Z l0‘3—1O“‘ crn'3 (ha;
`is the atom-excitation
`potential). The electron density Ne, above which the effi-
`ciency of stepwise ionization exceeds the efficiency of associ-
`ative ionization is determined from the condition
`
`Nel~N1kass/ks!’
`
`where ks, is the rate constant of the stepwise ionization of the
`excited atoms. It will be shown below that ks, ~ 10“7 cm3/s.
`It follows hence that, for example in the case of Na, the main
`
`920
`
`Sov. Phys. JETP 67 (5), May 1988
`
`0038—5646/88/050920-05$04.00
`
`© 1988 American Institute of Physics
`(cid:34)(cid:52)(cid:46)(cid:45)(cid:1)(cid:18)(cid:18)(cid:20)(cid:21)
`ASML 1233
`ASML 1233
`
`920
`
`

`
`mechanism of charged-particle production, even at ioniza-
`tion degrees as low as N,/N 2 10“, is stepwise ionization,
`and the features of its course determine in fact the character
`
`of the formation of PRP and its basic parameters.
`3. Let us analyze the kinetics of PRP formation under
`conditions when the production and neutralization of the
`charged particles are connected with the stepwise ionization
`processes and with the detailed opposite process of triple
`recombination. Under the conditions of interest to us, when
`the main mechanism of destruction of the exciting atoms is
`via inelastic electron-atom collisions, the expressions for the
`rate constants of the indicated processes are of the form"
`
`mm */=
`<2...)
`
`I
`(T T)»
`
`- gg.
`- ;. A
`
`en
`
`610
`
`area : rnl/:T'e/2 '
`
`The dimensionless parameter A = 4 i 0.5 is determined
`here by statistical reduction of a large number numerical
`model calculations, e and m are respectively the charge and
`mass of the electron, and I is the ionization potential of the
`atom. It should be noted that expression (4) is applicable
`not only to a description of the ionization of the atoms in the
`ground state, but also to a description of the ionization of the
`excited atoms (it is accordingly necessary to use in this case
`the ionization potential I1 of the excited atom).
`By virtue of the assumptions formulated above, the oc-
`cupancies of the highly excited states of the atom are con-
`nected with the occupancy of the resonant level by the equi-
`librium Boltzmann relation, with an electron temperature
`T,. To describe the kinetics of PRP formation is suffices
`therefore to consider the balance equations for the density of
`the atoms in the ground (N0) and in the resonantly excited
`(N,) states, and also for the temperature and density of the
`electrons. These equations are
`
`dNo=—JO(No—-EL Ne<N1kqu—N0kexcr)!
`1?
`dt
`g,
`(6)
`
`EV: = Jo (N, — E N.) — 5': — N. (M... — Nok...)
`dt
`g,
`T
`
`—Ne<Nlk —N§arec'
`
`7
`
`‘”V‘= N.<N.k 5.3.’ - a...N.“).
`dt
`
`(8)
`
`3
`(
`3 dT,
`"2". dt —hm(Nikqu‘N0kexc:)~ It+?Te
`
`)
`
`X(Niki(c)1ri)—N:arec).
`
`Account is taken in Eqs. (6) and (7) of the transitions be-
`tween the ground and resonantly excited states under the
`influence of the monochromatic radiation (0 is the cross
`section for such transitions), and also the transitions be-
`tween the indicated states via electron impact:
`
`A+e<->A'+e.
`
`(10)
`
`921
`
`Sov. Phys. JETP 67 (5), May 1988
`
`The constants of processes (10) ar interrelated by the de-
`tailed balancing principle
`
`kexc =kqu exp (_h(1)/Te),
`
`and the constant for quenching the resonant state by elec-
`tron impact under the condition T, <fico is independent of
`Te and is given by”
`
`(12)
`
`_ l).125e’}.”*(2m)"’ ~ 11 N“ [cm3]
`’
`q“ _ 7l""r4:rt
`c
`~ ' T s
`where /l[cm] is the wavelength of the resonant transition,
`r[s] is the radiative lifetime characterizing the considered
`resonant transition. Account is taken in the energy-balance
`equation (9) for the free electrons not only of the processes
`( 10) but also of the stepwise ionization and recombination,
`which influence the change of the average electron energy.
`Relations (4), (5), (11), and (12) are valid for a Max-
`wellian distribution of the electrons in energy, a distribution
`that obtains for the considered rather high degrees of plasma
`ionization. In addition, the approach developed is valid for
`the description of an ideal plasma in which the Debye sphere
`contains considerably more than one particle:
`
`A!35I"D3Ne>>1-,
`
`where r0 = (T,/8n'e2N, ) 1/ 2 is the Debye radius. It should
`be noted in connection with this requirment that the theory
`developed here cannot be used to determine the parameters
`of an ideal PRP if relation (13) is violated. It is possible,
`however, to determine on the basis of this theory the condi-
`tions under which the PRP is no longer ideal.
`4. We have confined ourselves in the analysis of the so-
`lutions of the set of equations (6)—(9) to the most typical
`case of saturating laser-radiation intensity. In this case the
`relation between the occupancies of the ground and resonant
`states is:
`
`No=N1go/gm
`
`and becomes valid before the collision processes come into
`play. Generalization of the results that follow to include the
`case of arbitrary intensity of the resonant radiation leads
`only to a certain complication of the calculations, but not to
`any qualitatively new physical conclusions.
`We analyze first the character of the solution of Eq. (9).
`The initial condition for this equation should be chosen to be
`the value of the electron temperature Teo determined from
`the energy of the electrons produced by associative ioniza-
`tion. It can be assumed that Tea is close to zero. Since the
`ratio N, /N0 in PRP is substantially higher than the equilibri-
`um value, the electrons in such a situation should be heated
`by quenching of resonantly excited atoms [the first term in
`the right-hand side of (9) ]. A rise of T,, causes a rapid in-
`crease of the stepwise ionization intensity, characterized by
`a steeply increasing temperature dependence. Since the step-
`wise ionization process draws energy from the electrons, the
`temperature rise stops at a certain value T9, and further in-
`crease of the electron density takes place at this quasistation-
`ary value of the temperature. This value is a solution of Eq.
`(9) in which it is assumed that dT,,/dt = O and only the
`principal electron heating and cooling mechanisms are con-
`sidered:
`
`Eletskfl era/.
`
`921
`
`

`
`fiwkqu = ([g+’/2Tl1)ki(¢)1r.)(Te1 ).
`
`(15)
`
`We present below the quasistationary values of Te, cal-
`culated on the basis of the solution of (15) with constants
`(4) and (12) for PRP of alkali metals:
`
`Vapor
`T2,.-,v
`
`Li
`0,45
`
`Na
`0.37
`
`K
`0.34
`
`Rb
`0.32
`
`Cs
`0.30
`
`The obtained electron temperature in PRP of Na (0.37
`eV) is close to the measurement results (O.2—0.5, 0.35, and
`0.35 eV in Refs. 7, 13, and 14, respectively) and exceeds
`somewhat the value of 0.2-0.3 eV of Ref. 8. The value
`
`T, = 0.08 eV measured in Ref. 15 seems unjustifiably low. It
`may pertain to the initial stage of PRP formation, when
`quenching of resonantly excited atoms makes no noticeable
`contribution whatever to the electron heating. A strong
`overestimate Te z 1 eV is obtained from the numerical cal-
`culations of Ref. 16, probably due to the neglect of the con-
`tribution of the stepwise ionization to the energy and elec-
`tron-density balance, a contribution decisive under the
`considered conditions.
`We shall use the normalization relation
`
`N,+N,=N
`
`( 16)
`
`(N is the total density of the atoms), where we neglect the
`contribution of all the excited states but the resonant one
`
`(this approximation is justified for an ideal plasma‘ ’ ), trans-
`forming Eqs. (8) and (9), with allowance for the detailed-
`balancing principle, into
`
`1
`to
`———.v, ———k.N
`I.+*/2T.
`dt
`“
`
`at
`g1+go
`
`1-
`
`<
`
`—
`
`“>1
`T.
`
`exp
`
`3 dT,
`2
`dt
`
`1
`I,+“/2T,
`
`1
`
`'
`
`(17)
`
`The solution of this equation under the quasistationary con-
`ditions (dT,, /dt = O) stipulated above is
`
`Ng§,/Veo eXp[ —-——-f “fl/‘:|
`
`ha)
`
`,
`
`(18)
`
`where 7/ = kqu Ng,/(g, + go) . Thus, an exponential growth
`of the electron density in PRP takes place at a constant value
`of Te. The
`characteristic
`time of
`this growth is
`y‘ ’ ~ (kqu N) " and for the typical experimental conditions
`of Ref. 8 (N~ 10” cm‘3, kqu ~10” cm3/s) it does not ex-
`ceed several nanoseconds. It is just in this time that the ener-
`gy of the resonant radiation is converted into the energy of
`the ionized gas. This explains also the repeatedly observed‘
`experimental fact that the gas is rapidly and practically com-
`pletely ionized by the resonant radiation if the transition is
`saturated.
`
`The exponential growth of the electron density (18) at
`constant temperature continues until the ternary recombi-
`nation, which keeps Ne from increasing further, becomes
`substantial. The characteristic time I2 at which this occurs is
`estimated from the expression
`
`N,” (2 )=1v
`2
`
`2g1'( mTe1 )'I'
`g1
`W
`go+gi gl
`2nh2
`
`(
`exp —_
`
`I1)
`Tel
`
`9
`
`(19)
`
`where N, (t) is given by (18). The value
`
`922
`
`Sov. Phys. JETP 67 (5), May 1988
`
`=
`
`ta
`
`(1)
`j-———
`(ZY [1+3/2Te1 )
`
`" [
`ln
`
`N
`
`2g«( mT..
`gt
`-*‘—
`go+gi gr
`Znhz
`
`)1‘
`
`I
`
`>< “P1 ” T:,
`
`(20)
`
`obtained in this manner is substantially larger than 1/7/. At
`t > t2, determined by relation (20), a quasi-equilibrium state
`is established in the PRP. This state is described by the Saha
`relation between the electron density and the density of the
`resonantly excited atoms:
`2g,(mT,
`'I-
`(
`gi 2%,)
`
`= 92
`
`N (t)
`
`.
`
`(21)
`
`N1
`
`exp
`
`'"
`
`1,)
`TO
`
`The N, (t) dependence is expressed here as before by the
`exponential (18), but the electron temperature no longer
`remains constant, but increases with time, since an addi-
`tional electron-heating mechanism enters into the electron
`energy balance and its effectiveness is proportional to the
`square of the electron density. In this regime, the plasma
`becomes fully ionized almost instantaneously. This plasma
`state is of no interest to us, since a noticeable contribution to
`the absorption of the resonant radiation is made in this case
`by mechanism of collective interaction between this radi-
`ation and the plasma electrons, i.e., the PRP acquires fea-
`tures of a laser plasma.
`5. Let us analyze the PRP properties caused by its devi-
`ation from ideal. These properties are characterized by a
`parameter” F = ez/rD T, that determines the ratio of the
`potential energy of the charged-particle interaction to their
`kinetic energy, and is inversely proportional to the number
`(13) of charged particles in the Debye sphere. In a PRP,
`where the electron density is connected by the Saha rela-
`tion“ with the above-equilibrium density of the resonantly
`excited atoms, the value of the parameter F is higher than in
`an equilibrium low-temperature plasma. Using relations
`(14) and (21) and the normalization relation
`
`No+N,-I-N,,=N,
`
`(22)
`
`we determine the maximum attainable values of the param-
`eter 1" in a PRP. First of all, the expression for the electron
`density is of the form
`
`nG [<4N _L1)"’
`e_—2— rLG |
`where
`
`/1]
`
`’
`
`(23)
`
`””‘)"’
`n——(2nfi2
`
`2g‘
`< ")
`exp -T9,, G=g0+g1.
`
`At a fixed atom density N, the parameter F has a non-
`monotonic dependence on the electron temperature and has
`a maximum at a certain value of T, . This maximum F, calcu-
`lated for PRP in Li and Cs vapor, is compared in Table I with
`the maximum value of F for an equilibrium plasma whose
`parameters are related by the equilibrium Saha equation.
`The data for other alkali metals lies in the interval between
`
`the corresponding values for Li and Cs. It can be seen that
`for a given vapor density the values P reached in a PRP are
`higher than in an equilibrium plasma. This is due to the low-
`er value of T, in a PRP having the same electron density as
`an equilibrium plasma.
`6. We analyze now the properties of the plasma pro-
`duced when a PRP relaxes to the equilibrium state when the
`
`Eletskfl et al.
`
`922
`
`

`
`TABLE I. Comparison of the.photoresonant- and equilibrium-plasma parameters for which the
`plasma parameter I‘ IS a maximum. The upper of the bracketed numbers corresponds to PRP,
`and the lower to an equilibrium plasma.
`
`N, cm‘3
`
`F
`
`i
`
`Te.€V
`i N,_,,cm‘3
`I‘
`i
`T._..eV
`| N,,cm"
`
`
`,6
`
`10,.
`,5
`“’
`‘O
`‘O
`10,.
`
`,,
`
`{
`‘i
`i
`i
`{
`
`0,0165
`0.011
`0,041
`0,027
`0.096
`0.066
`0.22
`0.155
`0.46
`0.35
`
`0.27
`0.37
`0.31
`0.42
`0.37
`0.49
`0.44
`0.57
`0.52
`0.63
`
`7.410”
`7.710"
`6.940”
`7.41011
`6.34015
`6.9-10”
`5.410”
`6.240”
`4.310"
`5.310”
`
`0.027
`0.017
`0.066
`0.042
`0.16
`0.10
`0.35
`0.24
`0.73
`0.52
`
`0.19
`0.27
`0.22
`0.31
`0.26
`0.36
`0.31
`0.43
`0.38
`0.51
`
`7.310”
`7.710”
`6.840“-
`7.31011
`6.140”
`6.840”
`5.210“
`6.140”
`4.010"
`5.11017
`
`action of the resonant radiation on the vapor is stopped. The
`characteristic time to establish equilibrium between the res-
`onant and ground states of the atom in such a system is
`~ (Ne kqu ) “ ' and under the typical conditions ofthe experi-
`ment of Ref. 8 it turns out to be ~ 1O‘°—1O‘3 s. As follows
`
`from Eq. ( 15 ), the time to establish ionization equilibrium is
`also of the same order. This time is shorter by 2-3 orders
`than the characteristic time of energy emission from the
`plasma volume, determined by the time of emission of the
`resonant radiation with allowance for strong dragging.
`Therefore the parameters of the PRP and of the equilibrium
`plasma that results from its relaxation are in a one-to-one
`relation, which we determine assuming a plasma-ionization
`degree Ne /N < 1 and neglecting, in accordance with the fore-
`going, the contribution of all but the resonant excited states
`of the partition function of the atom.
`By virtue of the assumptions made, the main energy
`reserve in the PRP is contained in the resonantly excited
`atoms. This energy is equal to
`
`N,fimENfimg./(g.,+g,).
`
`As a result of relaxation of the PRP, this energy goes to raise
`the temperature of the electrons and to additional ionization
`of the plasma. The energy density in an equilibrium plasma
`is Ne (I + (3/2) Te). Equating the bulk reserve of energy in
`the PRP and in a the equilibrium plasma, and using the equi-
`librium Saha equation, we obtain the connection between
`the total vapor density and the electron temperature in an
`equilibrium plasma resulting from PRP relaxation:
`
`N:(é’°+€1)’(£:EE.)2£&("'T=)"'ex(-1)
`g.
`rm
`g.
`2nn=
`‘’ T.‘
`
`(24)
`
`The parameters calculated from (24) for an equilibrium
`plasma that results from relaxation of a weakly ionized PRP
`with saturated resonant transition are listed in Table II. The
`
`resonance level used in the calculation was the state 2P, ,2. As
`seen from a comparison of the data in Tables I and II, decay
`of the PRP results in an equilibrium plasma having a param-
`eter F close to the maximum possible for an equilibrium
`plasma.
`The equilibrium-plasma parameters listed in Table II
`were obtained under the assumption that the initial degree of
`ionization of the PRP is much less than unity. If this assump-
`tion does not hold, relaxation of a PRP results in an equilib-
`rium plasma with a density higher than given in Table II,
`and hence also with a higher electron temperature. Specific
`values of these parameters, as functions of the initial degree
`of ionization of the PRP, can be easily calculated on the basis
`of the relation
`
`N,fi0)+Ne, (I+“/2T,,) =N, (I+"‘/2T,,),
`
`in the PRP is con-
`in which the initial electron density N9,
`nected with the vapor density of the quasi-equilibrium rela-
`tion (21). Thus, it becomes possible to produce a low-tem-
`perature plasma with an arbitrary electron density. This is
`achieved as a result of the relaxation of a PRP in which the
`
`variable parameters are the atom density and the initial elec-
`tron density.
`7. The analysis in this paper has shown that the PRP
`passes in the course of its development through a sequence of
`states, the parameters of each of which can be established
`with sufficient reliability in the limiting case of a saturated
`transition and of high electron density. The first of these
`states is characterized by an exponential growth of the elec-
`tron density at a constant temperature whose values are giv-
`en above. In this state, the energy of the resonant radiation is
`
`TABLE II. Parameters of equilibrium plasma produced by relaxation of a PRP with saturated
`resonant transition.
`
`
`
`10"
`10”
`10”
`10"
`10”
`
`0.0082
`0.021
`0.054
`0.13
`0.32
`
`0 30
`0.34
`0.39
`0.46
`0 55
`
`2.410”
`2.410“
`2.3- 10‘ -"
`2.3-10'“
`2.2310”
`
`0.013
`0.035
`0.088
`0.21
`0.49
`
`0.22
`0.25
`0.29
`0.35
`0.42
`
`25-10”
`2.5- 10”
`2.5- 10”’
`2.4 10'‘
`2.410”
`
`923
`
`Sov. Phys. JETP 67 (5), May 1988
`
`Eletskil er a/.
`
`923
`
`

`
`elfectively converted into plasma-electron energy. The PRP
`passes next into a quasi-equilibrium state in which all the
`excited levels are at equilibrium with the continuous spec-
`trum, and the coupling of the resonant levels to the ground
`state is given by the saturation condition (14). The param-
`eter of this state, which corresponds to the maximum value
`of the parameter F, are listed in Table 1. Through the action
`of the resonant radiation the plasma in such a state is charac-
`terized by an above-equilibrium value of the electron den-
`sity. Finally, when the action of the resonant radiation is
`stopped, the plasma relaxes to an equilibrium state whose
`parameters can be roughly estimated on the basis of (24)
`and are listed in Table II. It appears that the action of reso-
`nant radiation on vapor is the only means of producing an
`equilibrium plasma with so high an electron density at so low
`an electron temperature. The lifetime of such a plasma is
`determined either by the expansion time or by the time of
`emission of the resonant radiation. The latter, with account
`taken of the dragging of radiation, amounts under the condi-
`tions of the experiments of Ref. 8 to ~1O“"—10‘5 s.
`We were able to present here a consistent analysis of the
`PRP development kinetics because, in the considered range
`of plasma parameters (Ne ~1O'5—1O‘° cm”, Te z0.3—0.5
`eV)
`its behavior is determined by collisional processes
`whose characteristics are known with sufficient reliability.
`This pertains primarily to stepwise ionization and ternary
`recombination, as well as to the quenching of resonantly ex-
`cited atoms by electron impact. At the lower electron density
`realized in PRP with low metal-vapor density, a noticeable
`role is assumed in the kinetics by radiative breakdown of the
`excited states, which leads to deviations of the atom distribu-
`tion over the excited state from an equilibrium Boltzmann
`distribution. Description of the plasma behavior in such a
`situation requires a level-by-level kinetic theory whose re-
`
`sults are highly sensitive to rate constants of the collisional
`transitions between the excited states of the atom. The large
`errors to which such constants are subject prevent us from
`describing a low-density PRP with as high an accuracy as
`obtained here for a dense plasma.
`The authors thank B. M. Smirnov and I. M. Beterov for
`
`helpful discussions.
`
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`Translated by J . G. Adashko
`
`924
`
`Sov. Phys. JETP 67 (5), May 1988
`
`Eletskif et a/.
`
`924