Self-consistent modelling of argon microwave discharge sustained by electromagnetic wave in dipole mode

ANELIA ACHANOVA, IVAN ZHELYAZKOV*)

Faculty of Physics, Sofia University, BG.1164 Sofia, Bulgaria

TSVETELINA PETROVA **)

811 Main Street, Gaithersburg, MD 20878, USA

Received 21 August 2003; final version 30 October 2003

Microwave discharges produced by travelling/standing electromagnetic waves have drawn considerable attention in recent years due to their wide application in microelec- tronic industry. Here are presented the axial distributions of the electrons and ions as well as those of excited atoms in an argon microwave discharge at a low gas pressure (400 mTorr) sustained by a 2.45 GHz travelling dipolar electromagnetic wave. The nu- merical calculations are based on a self-consistent kinetic-electrodynamic model and the results obtained are in a good agreement with the experimental data. The model yields also the magnitude of the wave power sustaining the discharge and the axial profiles of all electromagnetic-wave characteristics and the axial distribution of the neutral gas temperature.

PACS: 52.25.Dg, 52.80.Pi Key words: microwave discharges, Boltzmann equation~ axial distributions of charged par-

ticles, atomic excited states

1 In troduc t ion

Over the past two decades microwave discharges produced by travelling elec- tromagnetic waves have been systematically investigated both experimentally and theoretically. Due to the broad range of operating conditions of these discharges, they have received many technological applications such as surface treatments, laser and spectroscopic sources, lamp technology, elemental analysis, sterilization, abatement processes etc. [1]. In most cases the discharge is sustained by a travelling azimuthally symmetric (m = 0, m being the wave mode number) electromagnetic wave. However, in the late 80s of the 20th century Margot-Chaker et al. [2, 3] and Durandet et al. [4] on the basis of experimental studies and some theoretical consid- erations reported on the possibility for sustaining plasma columns by using surface waves in dipolar (m = 1) and quadrupolar (m = 4) modes. The main results of their s tudy is the circumstance that there exists some minimal value of the product f R (wave frequency x plasma column radius) below which it is not possible to produce a stable plasma column. The existence of such a (/R)min ~ 2 GHz cm is associated

* ) F~mail: izh~hys, uni-sof ia. bg **) Current affilliation: National Institute of Standards and Technology, Gaithersburg, MD 20899,

USA

Czechoslovak Journal of Physics, Vol. 54 (2004), No. 2 211

Achanova et al.

with a stability criterion for the discharge [3]. On the other hand, Benova et al. [5] have offered an alternative way for explaining the occurrence of an fR critical value, notably that it is due to the specific dispersion characteristics of dipolar waves - - their (of Benova et al.) (fR)cr = 1.78 GHz cm. It became clear in those years that plasma columns sustained by dipolar electromagnetic waves possess smaller axial plasma density gradients than plasmas produced by azimuthally symmetric modes [6]. Moreover, bearing in mind also that large-diameter microwave plasmas (for fR > 2 GHz cm) can be stably generated only by dipolar electromagnetic waves, it is worth modelling such plasmas on using the most adequate approach.

The circumstances that the wave propagates Mong a plasma column sustained by the wave itself makes the problem rather complex since it is necessary to analyze the plasma behaviour in two aspects: wave and gas-discharge ones. Here, we present a model of an argon plasma column sustained by a travelling electromagnetic wave in dipole mode. The column is contained in a Pyrex tube (with a relative permittiv- ity g d = 4.52) of radius R = 2.35 cm and thickness d = 0.2 cm at a gas pressure of 400 mTorr. The launcher, situated at the one side of the discharge tube, possesses wave frequency f = 2.45 GHz. The supplied power is 120 W and the column length is 20 cm [2]. The model itself consists of two parts: kinetic and electrodynamic ones. The kinetic part is based on a collisional-radiative model of the argon atom [7]. The electrodynamic part yields the wave characteristics along the plasma column and also the axial distribution of the radially averaged electron number density ne (and by means of it, the axial profiles of the excited argon states). The maintenance of the discharge is owing to the wave power Q absorbed by the electrons per unit col- umn length. That power is expended further on for creating new electrons and ions through various inelastic processes. Namely the power Q is the link between the ki- netic and electrodynamic parts of the model. In other words, the model is based on the simultaneous numerical solving of a complete set of kinetic and electrodynamic equations: the electron Boltzmann equation, the electron energy balance equation, a set of particle balance equations for electrons, excited argon atoms, atomic and molecular ions, the gas temperature balance equation, the local wave dispersion relation, and the wave energy balance equation.

The paper is organized in the following way: in the second section we present the kinetic part of the model; the third section is devoted to the electrodynamic part, and results and discussions are the subject of the last, fourth section.

2 Kine t ic par t of th e m o d e l

A detailed modelling of microwave-produced plasmas requires the knowledge of the electron energy distribution function (EEDF), which determines the elec- tron transport parameters, the rates of elementary processes and several other key quantities such as the effective frequency for momentum transfer Veff and the power required for sustaining an electron-ion pair in the discharge 0 [8].

The electron kinetics of gas-discharge plasmas includes a wide spectrum of collisional processes such as elastic collisions (electron-electron, electron-ion, and

212 Czech. J. Phys. 54 (2004)

Self-consistent modelling ...

electron-atom) and several inelastic collisions (excitation, ionization), three-body collisions (which are responsible for the molecular ion formation), recombination, ambipolar diffusion, radiative processes (taking into account the imprisonment of radiation) as well as the impact of the effective wave electric field

E0

Here vc b the electron-neutral collision frequency for momentum transfer, w = 2 r f is the angular wave frequency, u is the electron energy, and E0 is the radially aver- aged wave electric field. Following the condition for stationarity [9], more specifically that the frequency ~ is larger than the energy relaxation frequency, the EEDF may be regarded as time independent.

We consider the Ar(3p54s) configuration in which the four levels (two metastable and two resonance) are treated separately and one lumped Ar(3p54p) block of levels. The energy levels' diagram of the argon atom is shown in Fig. 1.

The Boltzmann equation for the stationary EEDF f0 (normalized with respect to unity) can be schematically presented in the form

(cE + + =

h

(i)

where the left-hand side represents the electron energy (u) derivative of the gain energy fluxes due to the wave electric field, electron-neutral and electron-electron collisions, while the right-hand side describes the energy losses owing to various inelastic processes (excitation from the ground state, excitation~deexcitation be-

m m

Fig. 1. The energy levels of the argon atom.

(eV)

15.755

13.48

12.91

11.828 11.723

11.624 11.548

0 T

Ar +

3p54p

~o 3pS4s

1S o 3p e

Czech. J. Phys. 54 (2004) 213

Achanova eta/.

tween excited argon states, ionization, associative ionization, recombination, diffu- sion, and radiation). (We note that in the numerical code the Penning ionization, associative ionization, recombination, and diffusion are also taken into account in an appropriate way.) The energy fluxes in Eq. (1) are given by

2 e 2 Vc E2u3/20fo (~eln= 2~eu3/2VcfO ' GE - - 3 me V2c + w 2 2 Ou '

Ge~ = -2ue,u3/2 [I'(u)fo + J'(u) O0~] ,

where uc(u), as we have already mentioned, is the electron collision frequency, E0 is the radially averaged wave electric field (see the definition in the next section), M is the mass of ions/neutrals (the other notation is standard); the explicit forms of functions I'(u) and J'(u) are

~oUfOV~ 2 (~oUfOy3/2dy T u3/2 L°°fo ) I'(u) = dy and J'(u) .= -~ dy ,

respectively. The electron-electron collision frequency is

Vee=4~r~,4reom ) \ ~ ' u / nelnAc with Ac = ~ e e ~ , 5 - - ~ ] '

where (u) denotes the mean electron energy, and e0 is the permittivity of free space. The energy dependent collision frequency Vh (u) of various inelastic processes (each one characterized by its specific energy Uh) are calculated from the well-known expression

where O'h(U) is the corresponding cross section. The electron balance equation which accounts for the number of electrons in

the ionized medium can be symbolically presented in the form

k E F?n + + E Fm°¢i°° = Fdi" + F rec , (21 j = 0 m 8

where the left-hand side yields the number of emerged electrons while the right-hand side gives the number of lost electrons. The processes associated with electrons' creation are the direct ionization, stepwize ionization, Penning, and associative ionization [11-13]

Ar(m) + e - . Ar+ + e + e, m -- 1S0, 3P2, 3po, 3p1, 1P1, 4p ,

7 Ar+ + Ar + e, m = 3p2, 3po,

Ar(m) + Ar(m) ~ Ar + + e, m = 3p2.

214 Czech. J. Phys. 54 (2004)

Self-co~,~is~en~ m~deIting...

The loss of electrons is due to ambipolar diffusion and dissociative recombination

Ar + + e ~ Ar(4p) + Ar.

The contributions of the other kinds of recombination processes are negligibly small and are not taken into account. The diffusion frequency is Vdi~ = (#/R)2Da, where Da is the ambipolar diffusion coefficient, and # = 2.405 assuming a Bessel radial number density profile of the charged particles.

The balances of heavy particles (excited atoms and ions) are governed by the following processes: diffusion to the wall, excitation from the ground state [11, 14]

Ar(1S0) + e --* Ar(m) + e, m = 3p~, 3p0 ' 3p1 ' 1p1 ' 4p,

electron impact excitation and deexcitation between excited states [I0,15-i9]

A r ( m ) + e ~ A r ( q ) + e , m,q= aP~,3Po, 3Pt, zPz, 4p,

molecular ion formation [20]

Ar + + 2Ar ~ Ar~,

and radiative transitions with accounting for the trapping of radiation in all al- lowed radiative processes under consideration [7]. The references associated with a particular process give cross sections or rate coefficients. The full set of balance equations can be seen in Ref. [21].

Kinetic equation (1) yields the following electron energy balance equation

k k

E °'= i=O , i< j ]=1 rn

k k k

E E x¢ + E P t °° + Pe-" + + + Prec, (3) i = 0 , i < j j = l i=0

which together with charged particles' balance equations allows us to determine the populations of the excited states and atomic and molecular ions' number densities at a given electron number density ne. Equation (3) yields a detailed information on how the power absorbed by the electrons is dissipated in collisions.

The equation governing the radial change of the gas temperature Tg(r) in a thin column slice is

-rl{d~rr [rx(Tg(r))--'-d-r--rjdTg(r)] } = {pe_n + pc-i)n.(r), (4)

where x(Tg) is the thermal gas conductivity - - for argon it is X = 4.17 × 10-~Tg ~/a W cm- tK -s/a [22]. This equation is solved under the following boundary condi- tions:

dTg(0) dr = 0 and Tg(R) = Tw,

Czech. J. Phys. 54 (2004) 215

Achanova et aL

where Tw is the tube wall temperature that is an entrance parameter, here taken to be equal to 300 K. This value of Tw might seem pretty low for an experimentalist, bearing in mind that at a dissipated power of 120W one expects higher tempera- tures than that. On the other hand, the tube diameter is relatively large compared to much smaller tubes used in producing microwave argon plasma columns by azimuthally symmetric waves and, hence, Tw in (mr case can be of the order of room's temperature or a little bit higher. An increase of the tube wall temperature would not however dramatically change the results obtained since we are calculat- ing radially averaged number densities of charged and neutral particles - - the wall temperature affects mainly the radial distribution of the argon atoms.

By solving Eq. (4) with the corresponding boundary conditions and, for simplic- ity, with a Schotky's radial electron number densi~ profile (as with the diffusion frequency), we get in fact a radially averaged gas temperature Tg according to the expression

Tg = - ~ rTg(r) dr.

3 Electrodynamic part of the model

The electrodynamic part of the model describes the propagation of an elec- tromagnetic wave of a constant angular frequency w in the ionized medium by Maxwell's equations with appropriate boundary conditions. The hf plasma dielec- tric constant ep is derived from the simplest model of a cold, weakly collisional electron plasma with an effective electron-neutral collision frequency for momen- tum transfer ueff < w

W2p + i ue~ Wp 2 ep = 1 w 2 + v~ w w 2 + v ~ '

where Wp = v/nee2/Eome is the electron plasma frequency. The effective electron- neutral collision frequency for momentum transfer [23]

Veff = - g (u) u3/2du

has been introduced since for argon it is energy dependent. Under these assump- tions, one gets the basic equations of the electrodynamic part of the model - - the local wave dispersion relation and the wave energy balance equation. The dispersion equation

(w, k, R, d, ep, ed) = 0 (5)

yields a relation between the electron number density ne and the wave number k at a given axial position - - the so-called phase diagram. The actual expression of Eq. 5 the reader can see in [24] or [6]. The wave energy balance equation reads

dS - - = - q , ( 6 ) dz

216 Czech. J. Phys. 54 (2004)

Self-consistent modelling ...

where S is the wave energy flux along the column (axis z) being a sum of the axial components of Poynting's vector averaged over the wave period 2rc/w and integrated over the plane normal to the plasma column, from the axis to infinity, at a given axial position z:

f2n S = --~-1 Re

too

2/~o ]0 dCJo r (E '× B ) . ~dr

= ~r rS p dr + ccjn rS d dr +

Here superscripts p, d, and v denote plasma, dielectric, and vacuum, respectively, and

S, = I__ (E~.B¢- E•B,.) #o

is the axial component of the Poynting's vector. The wave power per unit column length absorbed by electrons, Q, is

1 t2= tR e2 //eft ,-,2 ,72 Q = ~ w e ° I m ( c P ) j 0 l de jol rlZl 2 dr = 2me , ,2 .j '.,2 7r/% ne/~{~ , (8 )

-- ""ef t

where

E~ = - ~ r(E~ + E~ + E~) dr

is the radially averaged squared wave electric field. The long expressions for S and Q in terms of gas-discharge parameters (wave frequency, plasma radius, tube thickness and relative permittivity) and wave-fields radial structures the interested reader can find in Ref. [6].

The self-consistent link between kinetic and electrodynamic parts of the model is accomplished by the electron energy balance equation: the absorbed power Q should equal the power expended in elastic and inelastic collisions - - this balance equation consequently takes the form

O, = ~rR2n.O, (9)

where 0 is the mean power required for sustaining an electron-ion pair in the discharge. It is calculated in the kinetic part of the model (Sect. 2) as

1 O = - - e 2 ~ ' 2 foo //c(u) us/2dfo(U) du 3 me "~0 go //2 (u) + w 2 du "

Solving the wave energy balance equation (6) together with Eq. (9), giving the supplied power to plasma medium, and the local wave dispersion relation, Eq. (5), we obtain the axial distributions of the plasma density, the wave number (or wave length), and the wave field components. This enables us to derive the axial distributions of all plasma characteristics that have been found as functions of ne in the kinetic part.

Czech. J. Phys. 54 (2004) 217

Achanova et al.

4 Resul t s and d i s c u s s i o n

It is clear that the set of basic equations of the model can only be solved numerically. The argon plasma kinetics and electromagnetic-wave characteristics have been calculated via a three-step procedure as follows: (i) for a number of electron number densities ne (in the expected range of plasma densities, say 10 l°- 1013 cm -3) and a fixed value of Tw = 300 K the kinetic part of the code calculates a discrete set of values for E0, ~eff, and 8. (ii) Those values are necessary in order the electrodyna.mic part of the program to solve together the local wave dispersion relation Eq. (5) and the differential equation (6), which yields the axial distribution of the electron number density n e = h e ( Z ) and all the wave characteristics. (We note that we have resolved the local dispersion equation by assuming a simpler expression for the plasma dielectric function not depending on the Veff, which seems reasonable when Ve2fr << ~v2.) Recall that Eq. (6) is a first-order differential equation and requires one boundary condition - - as such we have used the column length - - the 'building' of the plasma column starts from its end to the position at the wave launcher. (iii) With that n e = h e ( Z ) we enter again the kinetic part of the code - - this provides the spatial evolution of the isotropic distribution function, mean electron energy, transport parameters, gas temperature, various rate constants for electron-heavy particle collisional processes, as well as the axial distributions of atomic and molecular ions alongside the populations of the excited argon states.

Here we present the most important results of our numerical calculations. Figure 2 shows the axial distributions of the plasma density and the wave power S sustain- ing the discharge. The axial profiles of the electron number density ne and the wave power sustaining the discharge are decreasing functions from the launcher position to the column end. We note the good agreement between the experimental and the-

6

5

'E 4 O

O 3

2 ~ 2

0

a)

Fig. 2.

I I I

5 10 15 2 0 0 5 10 15 2 0

z (cm) b) z (cm)

, , , 1 2 0 , , ,

40

21)

0 '

Axial distribution of the a) electron number density he, b) discharge-sustaining wave power S. The experimental points are taken from Fig. 5 in Ref. [2].

218 Czech. J. Phys. 54 (2004)

Sel f-consis tent model l ing . . .

10

8

E

¢-

4

2

0 I I I Fig. 3. Axial variation of the electromagnetic wave 0 5 10 15

length (= 21r/k) . Z (cm) 20

oretical magnitudes of the total wave power sustaining the discharge - - 120 and 116 watts, respectively. It is necessary to point out that we o b t a i n the wave power on using as 'reference quantity ' the experimentally measured column length. The good agreement between the theoretical and experimental values of the wave power al- lows us to claim that the plasma density near the exciter should be nexc ~ 6.0 x 1011 cm -3, while that at the column end is nmin ~ 3.4 X 1011 cm -3, which is lower than the so called r e s o n a n c e plasma density nresonance = 4.11 × 1011 cm -3. (The reso- nance plasma density is defined as a product of ncut off -- w 2 c o r n e / e 2 = 7.44 × 10 l° cm -a and (1 + ed).) The circumstance that the electron number density at the col- umn end turns out to be lower than the nresona,ce is due to the dispersion features of the dipolar (m = 1) electromagnetic wave. On the other side, the axial variation of the wave length, shown in Fig. 3, turns out to be the same as in the case of a plasma column sustained by an azimuthally symmetric (m = 0) wave - - the wave length shortens to the column end.

As can be seen from Fig. 4 the alterations of electric and magnetic field compo- nents are not too strong as, say, in the case of a column produced by an azimuthally- symmetric electromagnetic wave at the same frequency and gas pressure. For a plasma column sustained by a dipolar wave E2 and Be are the dominant wave field components. Their amplitude magnitudes near the exciter of the experimental set- up should approximately be 55 V cm-1 and 40 #T, respectively. The negative values in Fig. 4 mean just an opposite phase of the corresponding wave field component to the phases of the other two components.

The most distinctive result is shown in Fig. 5 - - one observes almost homoge- neous axial distributions of all excited argon states. As seen from Fig. 5, the most populated excited state is that of the 3p2 metastable argon atoms. We note that the arrangement of the argon excited states in a plasma sustained by dipolar wave is the same as that of a plasma produced by azimuthally symmetric mode, however in our case the curves are much flatter (compare, for example, with Figs. 5b and 6 in Ref. [25]). It is rather curious that the places of the two radiative states 3P1 and 1p1 of a dipolar-wave sustained plasma are interchanged compared to the same states in a column produced by an azimuthally symmetric wave according to the

Czech. J. Phys. 54 (2004) 219

Achanova eta/.

60

40

|

E o 20 >

N LU 0

. (> -

LLI

U.j" -20

-40

I i

Ez

E,~

Er

I I

5 15

S 40

30

x Q~ 10

~o d

-10

' i i

Br

Bz

= -20 I n 0 10 20 5 10

a) z (cm) b) z (cm)

l

15 20

Fig. 4. Axial distributions of the wave a) electric and b) magnetic field components at the plasma-dielectric interface.

recent experimental measurements of Lao et al. [26] (see Fig. 7 there). On the other hand, their arrangement of the 3P1 and zPz states at a gas pressure of 200 mTorr turns out to be the same as in our Fig. 5 (see Fig. 6 in Ref. [26]). It is worth noticing that the order of the magnitudes of our theoretically obtained populations of the two metastable argon states is the same as that of experimentally measured ones.

Figure 6 shows the axial profile of the radially averaged gas temperature. It is not surprising that the axial course of the gas temperature Tg follows that of the

~ 1 2

E o 10

o

0

o ~ "-" 6 e -

E ~ 2 z

i

3P 2

3P 1

1P 1

3P 0

4p 0 I I i

0 5 10 15 20 Z (cm)

Fig. 5. Axial distributions of the two metastable 3po, 3p2 and two radiative 1P1,

3Pz argon atom states.

340

335

33O

325 ', t " 4 320

F-- 315

310

3o5

300 0

t t ]

5 10 15

z (cm)

20

Fig. 6. Axial distribution of the radially averaged gas temperature.

220 Czech. J. Phys. 54 (2004)

S e l f - c o n s i s t e n t m o d e l l i n g . . .

Fig. 7. Axial variation of the elec- tron energy distribution function.

ne fo ( ev-3/2 cm-8 )

1012

1010 j ~ ~<~-~:::~;:~t/~ ~

z(cm) 2 ~ 1 5 u(eV)

20

1012

1010

10 8

10 6

10 4

electron number density - - compare Figs. 6 and 2a. The importance of knowing the spatial evolution of the temperature of neutral atoms in microwave-sustained argon plasmas is justified by the fact tha t in a relatively short period of time three theoretical and one experimental articles [27-30] (all for argon plasma columns, sus- tained by azimuthally symmetric microwaves) have been published. In particular, Palmero et al. [29] point out that for obtaining an adequate answer of the question which processes are responsible for gas heating in a low-pressure argon discharge one must use an energy balance equation and a collisional-radiative model of the argon atom. Their model is more precise - - it includes altogether 11 excited argon levels (look at Fig. 2 in Ref. [29]) and yields the radial profile of the gas tempera- ture as a function of radially averaged electron number density. The experimental measurements [30] show that the gas temperature depends on the gas pressure and wave power sustaining the discharge. According to Fig. 3 in the same reference, our theoretical values for Tg - - 336 K near the launcher and 324 K at the column end - - seem to be reasonable.

As we have already mentioned, the numerical code calculates the EEDF along the column length (Fig. 7) and one can conclude that the distribution is non- Maxwellian. The amplitude of the local EEDF decreases in axial direction because of a decrease in the local electron number density while the energy dependence of the EEDF for different axial positions chan~es as a result of electron-electron collisions.

To summarize, our theoretical model of an argon microwave plasma column at low pressure sustained by electromagnetic wave ia dipole mode reveals some impor- tant advantages of such a discharge: relatively low axial plasma density gradient and almost axially flat populations of the excited argon states that can be use- ful in industry applications. Naturally, the modelling might be improved on using a more elaborated, 14-level collisional-radiative model of the argon atom, which would yield the axial distribution of the Ar(3p54p) states and more precise values of the gas temperature.

Czech, J. Phys. 54 (2004) 221

Achanova et al.

This work was supported by the National Council for Scientific Research under Grant No. F-821/98.

R e f e r e n c e s

[1] M. Moisan, J. Hubert, J. Margot, and Z. Zakrzewski: in Advanced Technologies Based on Wave and Beam Generated Plasmas (Eds. H. Schliiter and A. Shivarova), Kluwer Academic Publishers, Dordrecht, 1999, p. 23.

[2] J. Margot-Chaker, M. Moisan, M. Chaker, J. ParaSzczak, and G. Sauv~: in 18 In- ternational Conference on Phenomena m Ionized Gases (Swansea, UK), Contributed Papers, Adam Hilger, Bristol, 1987, Vol. 4, p. 854.

[3] J. Margot-Chaker, M. Moisan, M. Chaker, V. M. M. Glaude, P. Lauque, J. Paraszczak, and G. Sauv6: J. Appl. Phys. 66 (1989) 4134.

[4] A. Durandet, Y. Arnal, J. Margot-Chaker, and M. Moisan: J. Phys. D: Appl. Phys. 22 (1989) 1355.

[5] E. Benova, I. Ganashev, and I. Zhelyazkov: J. Plasma Phys. 45 (1991) 137.

[6] I. Zhelyazkov and V. Atanassov: Phys. Rep. 255 (1995) 79.

[7] Ts. Petrova, E. Benova, G. Petrov, and I. Zhelyazkov: Phys. Rev. E 60 (1999) 875.

[8] C. M. Ferreira and M. Moisan: Phys. Scr. 38 (1988) 382.

[9] R. Winkler, M. Capitelli, M. Dilorando, C. Gorse, and J. Wilhelm: Plasma Chem. Plasma Process. 6 (1986) 437.

[10] L. Vriens and A. M. Smeets: Phys. Rev. A 22 (1980) 940.

[11] M. Hayashi (private communication).

[12] T. Okada and M. Shugawara: J. Phys. D: Appl. Phys. 26 (1993) 1680.

[13] N. B. Kolokolov, A. A. Kudryavtsev, and A. B. Blagoev: Phys. Scr. 50 (1996) 371.

[14] J. Vldek: J. Phys. D: Appl. Phys. 22 (1989) 623.

[15] E. W. McDaniel: Collisional Phenomena in Ionized Gases. Wiley, New York, 1964.

[16] H. W. Drawin: Z. Phys. 225 (1969) 483.

[17] A. Hyman: Phys. Rev. A 18 (1978) 441; 24 (1981) 1094.

[18] I. Yu. Baranov, V. I. Demidov, and N. B. Kolokolov: Opt. Spektrosk. 51 (1981) 316.

[19] I. Yu. Baranov, N. B. Kolokolov, and N. P. Penkin: Opt. Spektrosk. 58 (1985) 268.

[20] J. D. C. Jones, D. G. Lister, D. P. Wareing, and N. D. Twiddy: J. Phys. B: Atom. Molec. Phys. 13 (1980) 3247.

[21] E. Benova, Ts. Petrova, A. Blagoev, and I. Zhelyazkov: J. Appl. Phys. 84 (1998) 147.

[22] Yu. B. Golubovskii and R. Sonnenburg: Soy. Phys. Tech. Phys. 24 (1979) 173.

[23] M. A. Head and C. B. Wharton: Plasma Diagnostics with Microwaves, Wiley, New York, 1965, Chap. 2.

[24] E. Benova, I. Zhelyazkov, and I. Ghanashev: J. Appl. Phys. 71 (1992) 1026.

[25] C. Lao, A. Gamero, A. Sola, Ts. Petrova, E. Benova, G. M. Petrov, and I. Zhelyazkov: J. Appl. Phys. 87 (2002) 7652.

222 Czech. J. Phys. 54 (2004)

S e l f - c o n s i s t e n t m o d e l l i n g . . .

[26] C. Lao, J. Cotrino, A. Palmero, and A. Gamero: Jpn. J. Appl. Phys. 41 (2002) 5659.

[27] J. Henriques, E. Tatarova, F. M. Dias, and C. M. Ferreira: J. Appl. Phys. 90 (2001) 4921.

[28] A. Palmero, J. Cotrino, A. Barranco, and A. R. Gonz~lez-Elipe: Phys. Plasmas 9 (2002) 358.

[29] A. Palmero, J. Cotrino, C. Lao, and A. R. G0nz~lez-Elipe: Phys. Rev. E {}6 (2002) 066401.

[30] A. Palmero, J. Cotrino, C. Lax), A. Barranco, and A. R. Gonz~lez-Elipe: Jpn. J. Appl. Phys. 41 (2002) 5787.

Czech. J. Phys. 54 (2004) 223