J.

I

Phys. D: Appl. Phys. 25 (1992) 401412. Printed in the UK

Computer simulation of microwave and DC plasmas: comparative characterization of plasmas

E V Karoulina and Yu A Lebedev A V Topchiev Institute of Petrochemical Synthesis, USSR Academy of Sciences, Leninsky Prospect 29, 117071 Moscow, Russia

Received 1 March 1991 Abstract. Self-consistentcalculations 01 t h e parameters of self-sustained M I P and OCP in argon and in model gases in the diffusion-controlledregime have been carried out. The calculated parameters of argon microwave produced plasmas (MIP) and direct current plasmas (DcP) are in satisfactoly agreement with experimental data. The use of the proposed invariant group of parameters, namely specific absorbed power Pd,plasma gas pressure p, discharge tube radius R a n d tube wall temperature Twjfor the comparisons of different types of electrical discharges has been grounded. The results of calculations in argon using different models of discharge and the results of calculations in model gas plasmas can be formulated as follows:

(i) For equal Pd, p, R a n d T, the rates of the processes under electron impact, which absorb an essential part of the discharge power, are nearly equal in M I Pand DCP. (ii) The relations between the rate constants (k,) of these processes in M I P and DCP are k!lP(dir.) = k~cP(dir.)[n,DCP/n~lP]

for direct processes and k:lp (step.)= k:cP(step.)[nFP / n ~ l p I I N ~/NWp cp I

for stepwise processes. Here, ne and NMare the electron and excited atom concentrations.

1. Introduction

Up to the present moment there has been a vast body of literature devoted to the physical and chemical characteristics of different types of non-equilibrium gas discharge plasma in atomic and molecular gases. The resulting set of theoretical and experimental data needs to be generalized and systematized in accordance with some general approach. Unfortunately, such an approach has not been finally formulated. The similarity theory for discharges of the same and of different types that has been generally described by Granovsky (1952), Engel (1955), Fransis (1956) and Golant (1958) could become the basis of such an approach. It has been shown, however, that even for the same type of discharge there are some strong limitations on the processes occurring in a plasma for which these similarity laws can be applied. The forbidden processes are, for example, stepwise ionization, inter-electron interaction, super-elastic collisions and so on. It is clear 0022-37271921030401 + 13 $04.50

0 1992 IOP Publishing Ltd

that it is these processes that play a significant role in gas discharge plasma. In comparing discharges of different types, for example, DC with RF or microwave, some great difficulties arise. Besides the usual similarity parameters, the new parameters w/N and E,JN appear in this case (here w is the circular frequency of the alternating electric field and N is the concentration of heavy particles in the plasma). The effective electric field E,, is determined by

where E, is the amplitude of the alternating electric field of frequency w , Y J E ) is the electron-neutral collision frequency for momentum transfer, which is a function of the electron energy E in the common case. This fact is the origin of great difficulties. Ferreira and Moisan (1988) proposed using the parameter o / v , , ( E ) instead of w / N . They supposed that vtr = Y , ~ ( E * )= 401

E V Karoulina and Yu A Lebedev

const where E * is a certain especially determined electron energy. It was assumed that E* should characterize the ‘body’ of the electron distribution function (EDF). For example, Ferreira and Loureiro (1984, 1989) used the values vir," = 2.4 x l O - ’ ~ m - ~ s - ’for N2 plasma and v,,/N = 2.0 x lo-’ cm-3 SKI for Ar plasma. It is clear, however, that for each plasma characteristic depending on E it is possible to select its own & * value so that these characteristics in MIP and DCP are equal, while the other characteristics differ. Despite the obvious difficulties, in almost all the papers concerning the investigation of microwave plasma, the results are compared with those obtained for other types of discharge. The principles of comparison are determined by the authors in an arbitrary manncr. Accordingly, the results of the comparisons and the conclusions ahout the efficiency of the different types of electrical discharge turn out to be arbitrary. In the present paper the possibility and expediency of one of the possible sets of discharge parameters (the invariant group of parameters) being kept equal when the different types of discharge are compared are analysed on the basis of a self-consistent computer simulation of self-sustained argon DC and microwave discharges in the diffusion-controlled regime at reduced pressures. This set of parameters is the absorbed power P,. the gas pressure p , the discharge tube radius R and the wall temperature Tw. An approach to the comparison of discharges and the selection of an invariant group of parameters for comparison is formulated in section 2 along with a description of the model used and the basic equations. In section 3, some results concerning the comparison of argon plasma characteristics calculated and from the literature (the electron energy distributions EDF, the concentrations ne and the mean energy E of electrons, and the rate coefficients of reactions initiated by electrons) are presented in order to prove the conclusions made. The role of the wall temperature is investigated. It is concluded that the set of parameters used allows the formulation of rules for the calculation of the rate coefficients of the energy-capacious pruccsxs undel. electron impact of one type of discharge from those obtained for another type. These rules can be used provided that the concentrations of electrons and excited atoms in both types of discharge are known. The results have been verified for some model gases (different sets of cross sections) and for discharge models with different levels of detail.

2. Description of the

DCP

and

MIP

models

2.1. The selection of an invariant group of parameters

The comparison of the properties of different types of self-sustained discharge plasma can be carried out under equal values of any plasma parameter. The concentrations of the electrons, excited atoms and radicals, the intensities of excited atoms’ radiation, the absorbed 402

power, the rates of plasma chemical etching, polymerization, etc. can be used as well as the usual parameters of similarity theory (E,”, w / N , . . .), but, due to the difficulties mentioned above, the latter do not have any priority. In our opinion, the set of parameters used in the comparison of different plasma types should satisfy certain demands. Firstly, it should completely characterize the plasma. Secondly, it must consists of values that can be easily measured experimentally. The latter facilitates the comparison of experimental and calculated results. The set of external discharge parameters, i.e. the absorbed power per unit volume P,, the gas pressure p , the discharge tube radius R (for cylindrical geometry) and the wall temperature Tw is one of the possible sets. There are some advantages in using the energy characteristic Pd as a parameter. Firstly, it can be measured relatively easily in any type of discharge. This is one of the main electrodynamic parameters to be measured in MIP, for example. Secondly, a knowledge of Pd is necessary because the values of the discharge characteristics per unit absorbed power (the rates of etching or polymerization, intensities of plasma emission and so on) are used in the comparison of the efficiencies of discharges. Ferreira and Loureiro (1989) proposed the use of the parameter Pd/n, in place of P,. But the use of such a parameicr dues nut aiways simplify ihe description of the discharge. The use of P,/n, is possible in a molecular gas plasma where the EDF is practically independent of the degree of ionization (01 = ne,”) up to 01- lo-’. However, in rare gas plasmas, due to the strong influence of inter-electron collisions for 01 3 the EDF and its moments depend on the value of n, and using the parameter P,/n, is not expedient. In the present paper, the set of external parameters P,, p , R and T , are used to model MIP and DCP. 2.2. Plasma models

Ai piese,,t

.I

1,

. . . J

uuc>

_....._. ,,U,

A:r‘?..IA

.---,.- up a

>CEIII UllllLUll L V lllalrc

complete system of master equations describing different types of discharge plasma. The problems usually appear in the solving of such a system, due to the lack of detailed information about the cross sections and the rate constants of the many plasma processes. In so far as the goal of our work is the investigation of the principal differences in MIP and DCP properties, the model used must correctly describe general plasma properties, and at the same time must be sufficiently simple. An infinite, cylindrically symmetrical, spatially homogeneous plasma slab in a diffusion-controlled regime has been considered (Dehye length AD s 2 x cm Q R , effective ion mean free path A,= (Mfi,/e)(3eUk/M)’/’ = cm 4 R , where M and fi, are the ion mass and mobility respectively, Uk= D./ fie is the electron characteristic energy, and D eand fic are the diffusion coefficient and the electron mobility

Computer simulation of

respectively (Ferreira and Loureiro 1984)). The assumption of the spatial homogeneity of the plasma means neglecting the differences in internal structure of MIP and DCP.In the following, we shall assume that all the plasma parameters are the radially-averaged values. The solving of the non-homogeneous problem in a self-consistent way is very difficult and lies beyond the scope of this paper. Plasmas with both types of discharge are supposed to be time independent. This means that, for the case w # 0, we consider plasma conditions with w values high enough to neglect EDF time variations ( W / U P 1). The model presented here differs from that described previously (Karoulina and Lehedev 1987, 1988). The main difference is that besides the combined metastable Ar-level 'P, + 3P0,the resonant level 'PI is taken into account. The processes in which excited argon atoms in resonant states participate were included in the balance equations for charged and excited particles. This is necessary because, for the conditions under which electrical discharges take place, the concentrations of atoms in resonant states is significant. For instance, for an Ar pressure of 0.54 Torr and for discharge currents up to 100 mA, the population of Ar ('PI) atoms is about 25% of the total Ar ('P2) population (Soldatova 1986). The method of gas temperature T, calculation used in this paper also differs from that used in the model of Karoulina and Lehedev (1987, 1988) (see below). A system of equations containing the Boltzmann equation for free plasma electrons, balance equations for charged and excited particles, equations for absorbed plasma power and for gas temperature allows the EDF to be computed, as well as all the averaged quantities of the electron component, the concentrations of electrons and excited particles, the gas temperature and the maintenance electric field. An iterative method has been used. 2.2.1. Boltzmann equation for electrons. The spatially homogeneous Boltzmann equation was used for the calculation of the stationary isotropic part of the electron energy distribution in the well-known Lorentz approximation:

MIP

and

DCP

where

In A = In

k2TeTAf2 e3n:f2

Here f is the EDF, e, m and U are the electron charge, mass and velocity, ucr,U, and u - are ~ the momentum transfer, inelastic and super-elastic cross-sections, U ; is the threshold energy of the mth inelastic process for the ith heavy component of concentration N i , and T, and Tg are the electron and gas temperatures. The stepwise ,processes (excitation and ionization of the 3p54s argon levels) were taken into account by analogy with the direct ones by supposing the admixing of excited gas into a parent plasma gas. The N i values correspond to the concentration of ground state atoms and the summary density of excited 3p54s atoms. The particles taken into account in super-elastic processes were argon atoms in the summary metastable level and in the resonance level (concentrations The EDF is normalized so that

[

& ' / * f ( ~d) &= 1.

(6)

The plasma electron parameters (mean electron energy E, electron diffusion coefficient D,, electron mobilities in direct and alternating electric fields p= and p - , electron impact rate constants k,, and specific absorbed power P d ) are defined by the equations

i[Ni'J:,(E)] dfl 2eEE,[NiU(,(E)]* m u 2 Z J

+

= J,,

+ J,. + J,,

(2) where J,,, Jj, and J,. are the collision integrals for elastic, inelastic and electron-electron interactions respectively:

(9)

1

m

k;

=

(2e/m)1/2

E U ~ ( E ) ~d( eE )

(10)

"G

Ji, =

2

Ni[&u:(&) f (E)

- (E

i

- (E

-

Pd = n,e

+ u',)ub, n

UL)U'-,(E - Ub)f(E - U;)]

(4)

2 2 u',(kkN, - k L W N j c ) . ,,?

(11)

The inelastic processes that determined the EDF were the following: direct excitation of each level of the 3p54s configuration ('P2, 'PI, 'Po, 'PI) and cascades 403

E V Karoulina and Yu A Lebedev

from higher-lying levels to 3p54s, the total excitation of the other levels, stepwise excitation, and direct and stepwise ionization. The major part of t h e calculations was carried out using the argon cross sections given in the paper by Ferreira and Loureiro (1983), but in some cases the cross sections from Kochetov (1977) were used. 2.2.2. Balance equation for charged particles. The balance between loss of electrons to the wall and collisional ionization of the gas in diffusion-controlled plasma with direct, stepwise and associative ionization can be expressed as D,/A2 = w , = Nk:'

+ ( N , + NR)k:'

+ i ( N M+ N , ) 2 k y - M / n ,

(12) where D, is the ambipolar diffusion coefficient, A = R12.405 is the diffusion length of the discharge vessel, w , is the total ionization frequency, and k:', k:' and k y - , are the rate constants for direct, stepwise and associative ionization respectively. 2.2.3. Balance equations for excited particles. The steady-state concentrations of the combined metastable level 3P2+ 'Po and of the resonant level 3Pl of Ar atoms can be defined by

[

=NM D,/NAz +Nk, +N2k,+ N , k r M + N R k Y R

IO

Nn,(k;

+ k ; ) + N M n ,( k = l

k M d p k+ k5 pk

)

(14) The processes that determine the concentrations N , and N R , the corresponding notations, the values of the rate constants used and the corresponding references are listed in tables 1 and 2 . In the equation for N,, an approximate form of the rate of process 6 has been used: k P - , N h = k y - M N , N b , where N L is the value of N M calculated in the previous iteration. An additional complexity appears due to transitions between the configurations 3p54s and 3p54p. The latter consists of 10 levels and each of them is optically connected with at least one of the 4s levels. The transitions between these configurations do not change the combined population in the metastable and resonant levels (because the contribution of stepwise processes from the 4p-configuration is negligibly small) and result only in its redistribution. The partial cross-sections for the 404

4 p 4 s transitions were assumed to be proportional to the corresponding experimental line strengths ujk= ( ( S ) f i k / x k f i k , where (U)is the average cross section for 4 s - 4 ~ transitions measured by Hyman (1978). The radiative branching ratios were obtained according to the transition probabilities a, = Akj/.ZjAkj by analogy with Ferreira e f a1 (1985). The values of Akj and fi, were taken from Wiese and Smith (1969). An effective life-time rCfof the resonant state 'P, was calculated in the following way. Radiative losses from this level are determined by the time of radiation escape from the plasma T ; ~ = wrad = Ag,,,, where A is the probability of the 4s1Su- 4s3P, transition and g,,, is the escape factor, which may be regarded as the reciprocal of the number of emissions and absorptions of an individual unit of atomic excitation prior to its escape from the plasma volume. The escape factor depends not only upon the density of absorbing atoms N and the discharge geometry but also on the resonance-line profile (Holstein 1947). The escape factor was calculated using the formula (Walsh 1959) (15) + gc erf(gcd/gc) where gdand g, are the escape factors for pure Doppler and pure collision broadening, g, is the escape factor under the condition of collision-type emission with Doppler-type absorption and erf(x) denotes the error function. Tie expressions for g,, g, and g,, were taken from Holstein (1951). Note that equation 15 was obtained for resonance line wings because the centre of the resonance line is practically fully absorbed under the typical conditions of low-pressure gas discharges (Mitchell and Zemansky 1971). The calculations of Ferreira and Ricard (1983) have shown g,,, as defined by equation 15, to be independent of pressure for p > 0.5 Torr. It depends only on the collisional process that leads to the maximum broadening. As our estimations have shown, under the considered discharge conditions, this is a process of resonance energy transfer in collisions of excited and uncxciied aiurns gesc = g d

Ar(4s)

exp(-g:d/g:)

+ Ar(lSo)+

Ar(lSu) + Ar(4s)

(16)

The cross sections of resonance-energy transfer processes u ( J J o , J o J ) are high enough and are typically about several orders of magnitude greater than g a s kinetic ones. For example (Sobelman 1963), for the dipole4ipole transition U =z 2 n ( e 2 / f i u ) a i = 104aa (here, U is the relative velocity of colliding particles and a" = fi2/me2is the atomic unit length). This type of broadening is described by the dispersion formula, and the line width is given by y = ~ N ( u u ( J J "JuJ)) ,

= e2/mwuf,J,)V(U+ 1)/Wo

+ 1)

(17)

where wn and fi, are the frequency and line strength respectively, for the J-Ju transition. Under our conditions y = 1.5 x N s-! (for p = 1 Torr and Y =

Computer simulation of 4.5 x lo's-'). The width for Lorentz collisional broadening yL as compared with y is

Y,. = ~ U * N , ~ , , , / G ~1.8 ? Tx =io5 c - i e y . (18) where NA is the Avogadro number. The values of g, calculated in the present work are about two times smaller than those obtained by Ferreira and Ricard (1983). 2.2.4. Equation for absorbed power. The equation Pd = en,fi,E2 (19) where E in MIP means the RMS maintenance electric field, has been used to obtain the radially-averaged plasma density ne. The use of equation 19 in direct current plasma does not lead to any unccrtainty in the value of n, because the electric field E ( r ) does not depend on the radius, and n, and pc represent radially-

MIP

and

OCP

averaged quantities. The situation in MiP is more complicated as E depends heavily on r (because of the skin effect, E increases towards the discharge wall). But the n, profile is practically independent of the ionizationrate profile in the diffusion-controlled regime and does not differ essentially from the Bessel profile. It is also known that the emission of MIP is more homogeneous over the discharge cross section than that of DCP and can often form a plateau (Ricard et a1 1985, Gerasimov and Lebedev 1985). The latter fact was used in our work: a simplified plasma model with homogeneous emission was used in order to describe MiP. In this case, n.(r)E2(r) = constant since the emissive atom concentrations are proportional to n,(r)E*(r) (Moisan eta1 1982). This assumption combined with the assumption that , p e ( r )= constant gives a plasma model with specific power Pd(r)= en,(r)pL,E2(r) = Pd constant over r.

Table 1. List of reactions used in the calculation of the population of the summary metastable level

Ar(3P2+ 3Po)t. Process number

Reactions n

ki (cm3s-')

Reference

.

+e

Ar+e-ArM h?

Ar + e+ (Ar(4p))

ArM + eArM + e

h*

ArM + e

+ 2e

-e

e + Ar(4p)

ArR + e-

A+'

Art

h

e;:

e + Ar(4p) h-6

+e-=sArR + e hs

ArM + ArM hy

Ar+

+ Ar+ e

X A r ; +e

XAr; ++ Are + e

ArM + ArR 'l"" Art ND

ArM2Ar wall

ArM + 2 A r 2 A r p

+ Ar

2.1 x 10-9

I21

1.8 x lo's$

I21

11 x lo-"§

PI

h

ArM + A r 4 2 A r

3x

io-'511

[21

t Electron impact rate constants are calculated using t h e EOF $ In cm-' s-',

5 In cm6 si.

11

Estimation using cross-section value. et a/ 1965; [2] lvanov and Soldatova 1985.

[l] Ferreira

405

E V Karoulina and Yu A Lebedev Table 2. List of reactions used in the calculation of the population of the resonance level Ar(4s3 PI)+.

Process number

Reactions

1

Ar + e

h-q

k, cm3 5s’

Reference

2.35 x lo-’&) 3.1 x 10-7(k-,)

I11 111

2.1 x 10-9

I21

1 .z x

121

ArR + e

ki

2

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

3

ArR + e-

4

ArR +e-

h:

Ar+

3ArR + e

+ 2e

-

e + Ar(4p) *Ar;

k f

Ar

-

40

ArM + e

5

ArR+e-

e + Ar(4p)

hc

k5

ArM + e k-5

kWA

+

6

ArM + ArR&

7

ArRlhetAr+hv

8

ArR + 2Ar&

h

Ar+ Ar + e Ar: + e

A r g ’ + Ar

-

t Electron imoact rate constants are calculated usino the In cms s-I.’ [ I ] Ferreira et a/ 1985; [2] lvanov and Soldatova 1985.

10-39

EOF.

When the computed results were compared with the experimental ones in the modelling of DCP, equation (19) was substituted by the equation for the discharge current density j c = en,pCE.

The model used for M I P (see earlier) leads to Q ( r ) = Q = constant and the solution of equation (20) is

Equations for gas temperature. The gas temperatures

In the modellingi t h e values of the gas temperatures averaged over the distributions (21) and (22) have been used. In our previous papers (Karoulina and Lebedev 1987, 1988), it was supposed that q = 0, but here we suppose that 0 < q < 1 and that the process of elasticelectron-heavy-pdrticl~ collisions is the only process of gas heating i.e.

Tg in both types of plasma were obtained from the heat-conduction equation

(x

where x is the heat-conduction coefficient is a linear function of temperature x = ATg (Zarkova and Stefanov 1Y67)), Q ( r ) = f,(I - q ( r ) ) is the distribution function of the heat sources, and q ( r ) is the fraction of the absorbed power that does not take part in gas heating (plasma emission, for example). I t was assumed that q(r) = constant. Since in DC discharge n,(r) J,)(2.405r/R), taking into account equation (1Y) and the independence of pc, E and q of r (i.e. Q ( r ) = QJl!(r/A)), the solution of equation (20) (Eletsky er a/ 1968) is

-

406

The calculations have shown that, under the typical conditions of glow discharge plasma in argon, the fraction of P,, in Pdranges from 1% to 80% and decreases with decreasing pressure.

Computer simulation of MIP and

OCP

3. Results a n d discussion 3.1. Comparison of the computed and experimental data (verification of the model)

To test the reliability of the conclusions obtained in modelling MIP and DCP, the computed and measured characteristics of argon plasma have been compared. As was noted earlier, two sets of electron cross sections for argon were used. The values of excited argon atom concentrations calculated using the cross sections of Kochetov (1977) were in worse agreement with the measurements and with previous calculated results (see, for example, Ferreira and Ricard 1983, Ferreira et al 1985) than those obtained by means of the cross sections of Ferreira and Ricard (1983). The latter set was used to obtain most of the results presented here. Figures 1-3 show the measured (Soldatova 1986) and calculated values of the electron number densities n,, metastable and resonance argon atom concentrations N M and N, in t h e positive column of directcurrent discharge in the tube of R = 2.4cm for the gas pressures of 0.54Torr and discharge currents of 10, 50 and 100 mA. The calculated curves agree satisfactorily with the experimental dependencies of n,, N , and NR on the discharge current and pressure. We must note that, in spite of this, the calculated values of the maintenance electric field (not shown) are higher than the measured ones and the difference increases with increasing current and pressure.

/',

7t

-.e

/'

J''

/ /'

I

0

.

,

.

,

,

.

,

50

,

,

I

,

,

100 I

lmAI

Figure 2. Populations in the combined metastable level 3P2 t 3P, of argon as a function of discharge current for p = 0.5 Torr (broken curve), 1 Torr (full curve). and 4 Torr

(chain curve). Points: experimental data of Soldatova (1986). Curves 1. 2: calculations using the electron-argon cross sections of Kochetov (1977) and Ferreira (1983) respectively. 0, 0.5Torr; 0, 1 Torr; 0 ,4Torr.

The curves shown in figure 4 represent the measured (Ivanov et al 1976) and calculated electron distribution functions in MIP in argon under the experimental conditions: p = 2 Torr, Pd = 1.6 W ~ m - R ~ ,= 1.5 cm. Two models have been used in the calculations. Curve 2 was obtained using the model described by Karoulina and Lebedev (1987, 19x8) (with argon atoms

't 6

P ITorrl

Figure 1. Plasma densities in direct-current positive column in argon for discharge currents of 10mA (broken curve), 50mA (full curve), and 100 mA (chain curve), and R = 2.4cm as a function of p. Points: experimental data of Soldatova (1986). Curves 1, 2: calculations using electronargon cross sections of Kochetov (1977) and Ferreira 50mA; 0. 100mA. (1983) respectively. 0 , 10mA; 0,

I . , 0

I

I

I

< . , 100# . , I

8

50 (

lmAi

Figure 3. As in figure 2 but for the resonant level 3P,. 407

E V Karoulina and Yu A Lebedev

this paper: the comparative characterization of MIP and DCP.But before that we shall briefly consider one of the external parameters-the wall temperature. 3.2. The influence of wall temperature on the plasma characteristics

10-5

I

i

i

i

i < , ‘, I 6 , , 8 , Ia I I2. . liY I I L

le“,

Figure 6. Calculated electron distribution functions in MIP (curves 1, 3) and DCP (curves2, 4) for p = 2 Torr, R = 1.5“ Jw=300Kinargon, P,,=O.I W ~ m - ~ ( c u r v e s 1 , 2 ) and 1.6 W cm+ (curves 3, 4).

power losses of electrons P, for certain inelastic processes in argon. The inferior index of k, and P, indicates the threshold energy of the process: 11.6eV is the threshold of the metastable ’P2 ’P,, level exci-

+

Table 3. Characteristics of M I P and DCP in argon calculated using different plasma models: (I) basic model; (11) the simplified model (NR= 0);(Ill) the simplified model (NR= 0) with Jgdetermined over the total P,+ Pd= 1.6 W c N 3 (a) and 0.1 W cm+ (b). p = 2 Torr, R = 1.5 cm and Tw = 300 K. (a)

model I

i (eV)

ne, 10” c m 3 NM,10” cm+ NR, 10” k,,.6 cm3 ss’ p11.6/pd (%) k,., cm6 ss’ pl.5/pd

k4.2,

(“/.I

cms s-’

p4.2/pd (%) pl5.6/pd (%)

model II

MIP

OCP

MIP

DCP

2.54 50.2 2.64 1.59 1.11 19.8 1.06 33.2 1.25 1.08 0.32

2.83 45.7 2.62 1.57 1.18 20.5 1.19 32.8 1.59 1.24 0.14

3.03 24.3

4.35 13.1 3.5

2.3

-

-

2.71 28.3 1.23 10.2 1.99 0.46 0.31

5.57 31.5 1.66 11.4 4.06 0.77 0.09

model I

E (eV) ne. 10” cm+ NM,10” cm+ NR,10” k,1.6IO-’’ cm3 s& p11.6/pd

cm6 5.’ PI.S/Pd(“4 k4.2, cm6 SS’

k,.5,

(%I

p4.2/pd p?5.8/pd

(“/.)

model 111

MIP

DCP

2.62 4.79 1.43 0.79 0.51 24.4 1.08 27.4 1.40 1.00 0.022

4.20 2.27 1.87 0.92 1.13 25.6 1.66 25.2 3.75 1.60 0.035

2.93

3.67 ,

2.5

3.00 2.36 1.51

-

1.41 35.1 1.19 10.1 2.07 0.50 0.013

DCP

4.94 1.11 2.37

-

3.12

36.1 1.80 11.4 5.21 0.92 0.02

5.81 1.41 5.83

-

-

8.38 22.9 1.OB 1.49 2.1 0.08 2.97

2.69 28.2 1.88 2.32 6.77 0.23 1.06

model II MIP

DCP

MIP

model 111 MIP

2.73 1.46 1.86

DCP

5.22 0.61 3.08

-

-

2.15 37.5 1.04 6.77 1.82 0.33 0.033

5.62 41.1 1.83 8.30 5.75 0.73 0.017

409

E V Karoulina and Yu A Lebedev

tation from the ground state; 1.5 eV and 4.2 eV are the excitation and ionization thresholds respectively of the metastable level. The chosen processes characterize both the direct and stepwise electron-heavy-particle interactions for a wide range of electron energies. There is a close similarity in the EDF of DCP and M I P (figure 6) and, respectively, in all DCP and MIP plasma parameters (table 3) for the case of large values of the power consumption, which are, however, not typical for DC glow discharges. For P,, < I W c K 3 the plasma parameters are different and the character of these differences looks like that described by Ferreira and Loureiro (1983) and by Karoulina and Lebedev (1987, 1988): the EDF in M I P as compared with that in DCP is enriched in the low-energy region, the mean electron energy is smaller, and the electron density is higher in MIP than in DCP. Note that a large part of the plasma-absorbed power Pd is transferred to the plasma by low-energy electrons (see the values of P,,,/P,, and P,,,/P, in table 3). This is explained by the high density of excited atoms in the plasma and by the importance of stepwise processes. Thus EDF formation occurs due to inelastic collisions of both high-energy electrons and low-energy electrons. In respect of this feature, a noble gas plasma resembles molecular gas plasmas. This is important, in our opinion, because it gives us a basis upon which to reach further general conclusions for the case of a molecular gas and for cases of noble and molecular gas mixtures.

Step (2). Let us now analyse the following questions. Are there any plasma quantities which are equal in MIP and DCP?Is there a possibility that the data known for one type of plasma can he used in the description of another one, and how can this be done? Such quantities, in our opinion, are the power consumptions of the power-capacious processes (the processes occurring' with sufficiently high power consumption) under equal absorbed power P d , pressure p , discharge tube radius R and wall temperature Tw.

Actually, it is seen from table 3 (compare the two columns for model I) that the difference between the values Palp,, in M I P and DCP is small for the powercapacious processes (here, the processes with the thresholds 11.6eV and 1.5eV) and it increases with decreasing absorbed power. Neglecting superelastic collisions, the power consumption P, in equation (11) is determined by the expression n,N'kau; in which the first three parameters depend on the plasma regime. The product of these parameters is just the rate of the a-s process. Thus the above statement about P , can be reformulated as follows: the rates of power-capacious processes are nearly equal in DCP and M I P for equal P d .p , R and

one discharge type if these are known for another type. For direct processes, it is necessary to know the rate constant k, for one discharge type and the values of. the electron densities in both discharges (or their ratio). The relation between k, in MIP and DCP in this case is given by the equation

(24) For stepwise processes, it is also necessary to know the concentrations of the excited particles that interact with electrons (for example, the concentration of metastable atoms) or at least the ratio of these values. The relation between the rate constants of stepwise processes in both plasmas can be described by the expression k!"(dir.)

k!lP(step.)

= kFCP(dir.)[n~cp/n~'P].

= kFcP(step.)

x [fl4CP/n~'P][N~CP/N~"]. (25) In our case, direct processes are represented by the processes with thresholds of 11.6eV and 15.8eV, and stepwise ones are represented by the processes with thresholds of 1.5 eV and 4.2 eV. The error in determining k, using equations (24) and (25) depends on the definition of the powercapacious processes (or on the value of the lower limit of the power for power-capacious processes). This error is due to the substitution of the approximate expression PyIp = P:cp by PYIp = PZCp.Under the considered plasma conditions. the error in determining the rate constants in this way is less than 25% if the contribution of P, to the total absorbed power Pd is greater than 10%. This error increases with decreasing P, and reaches several orders of magnitude at P , = 1%. Thus, although the individual parameters of MIP and DCP are different, some of their combinations can he quite similar. Step (3).Let us now consider the reasoning behind our

earlier statement about the degree of detail of the used plasma model. For this purpose, some calculations have been carried out using two simplified models. One of these models (hereafter designated as modei ii) did not take resonant atoms into account, while in another, the gas temperature was determined by assuming that the total absorbed power Pd is translated into heat (model 111). The results are shown in table 3. It is seen that the transition from the basic model (model I) to models I1 and 111 leads to a change in the absolute values of the plasma characteristics. In particular, the power consumed in stepwise processes increases. But all that we have said so far about the relation between the rates of the processes in DCP and M I P holds good: the power consumption into powercapacious processes in both plasmas differ insignificantly. Thus our main conclusion does not depend on the model used.

Tw .

Resulting from this statement are the rules that allow one to calculate the rate constants of the processes that occur under electron impact in plasmas of 41 0

Step (4). Let us consider the influence of the plasma gas on the relation between the rates of the processes in DCP and MIP.Table 4 shows the calculated ratios of

Computer simulation of

M I P and DCP

Table 4. Ratio of M I Pto oCP characteristics in Ar and in the model gases for different momentum transfer cross sections a,, ({Ar} signifies the argon cross section) calculated using model Ill (see explanations for table 3)).Pressure = 1.3Torr, Pd= 0.3W cm-I, R = 1.5cm and T = 300 K.

Momentum transfer cross section a,,( x

W)

2.0

0.52 4.24 0.38 4.23 39.1 39.4 1.75 5.1 5.6 3.1 0.23 0.44 5.1 3.74

0.79 1.61 0.71 1.73 40.6 43.0 1.23 5.5 5.8 1.5 0.31 0.41 3.75 2.5

the MIP and DCP characteristics in argon and in the model gases which differed from argon in the shape of their momentum transfer cross sections As follows from previous papers (see, for example, Karoulina and Lebedev 1988) the case for U,, corresponds to the case of the 'direct-current analogy' in which all plasma parameters for both types of discharge are quite similar including the EDF and rate constants. I n all other cases, the parameters of DCP and MIP are different: the mean energies of electrons E, plasma densities n,, rate constants of the processes and concentrations of excited atoms in M I P can be higher or lower than those in DCP (for equal values of P,, p . R and Tw respectively) depending on the shape of the momentum transfer cross section. This is explained by the different efficiency of interaction, which depends on u , ~ ( E ) ,of plasma electrons for direct and alternating electric fields sustaining plasma. This is described in detail by Karoulina and Lebedev (1988). It is important that, in spite of the fact that the MIP and DCP parameters are different, all these parameters are self-consistent, so that our earlier statements about the equality of the rates of the processes in both types of plasma are still valid. Actually, table 4 shows that the values of nplP/nPCP and k p f : / k l ' i being compared coincide with good accuracy for momentum transfer cross sections of the same shape. This coincidence agrees with equation (24) for direct processes. For stepwise processes, agreement is observed between the ratio k ~ f P / k ~ : and ' the product of n~"/n,""' and N ~ ' P , " ~ c pin accordance with equation 25. For other direct and stepwise processes the agreement is worse because of lower values of the power consumption in these processes. In particular, for both direct ionization and stepwise ionization, the difference can be as much as several times. To elucidate the influence of the value of the partial

-

=I =I =1

=1 36.1 35.8 =1 4.9 4.9 0.98 0.30 0.30 2.57 2.67

1.27 0.66 1.32 0.59 43.5 39.0 0.82 5.8 5.3 2.42 0.39 0.29 1.97 3.6

cm2)

1.43 0.3 2.5 0.23 47.6 34.9 0.62 6.21 5.01 0.34 0.48 0.21 1.1 5.6

power consumption on the relations between the rate constants in M I P and DCP an additional inelastic powercapacious process (power consumption greater than 90% of P d ) in the low-energy region with a triangular cross section was included in the set of electron cross sections. It was shown, for example, that decreasing the partial power consumption leads to a change for the worse in the agreement for direct process with a threshold of 11.6 eV and to a difference between the rates of ionization (direct and stepwise) of about several orders of magnitude. There is a possibility of using our results to consider some low-capacious processes. The rate constant, as well as the power consumption of the process, under electron impact is determined by equation (10) and depends on both the shape and the value of the cross section u * ( E ) . Let us consider the situation in which there are two processes with nearly equal threshold energies but with different partial power consumptions. Such processes, in our case, are the excitation of the argon levels 'PI and 'PI, + 'P, which differed both in shape and in the absolute value of u ~ ( E )The . former process is low power-capacious because it consumes less then 20% of the power consumed by the latter process. Two cases are then possible. (a) If the EDF in MIP and DCP varies rapidly with E then the rate constants of the processes are determined mainly by the values of the cross sections near the threshold and are practically independent of their shapes. In this case one can apply the above rules for low-capacious processes as well. (b) If in at least one of the discharges, the EDF decreases slowly with E then the rate constant in this case depends on the shape of the cross section and use of the above rules is incorrect. Such a situation was observed, for example, in the model gas with a large cross section in the low-energy region: the rate of EDF 41 1

E V Karoulina and Yu A Lebedev

alteration with E in DCP is twice that in M I P and the power consumption of the process of 'PI excitation in both discharges differs by a factor of two while for the power-capacious 'Po + 'P, excitation the difference is less than 15%. The rules given above, for calculating t h e rate constants for one type of electrical discharge from those obtained for another type, are of general character although they are based on the simulation of argon plasma. These rules d o not depend on the level of detailing of the model. They are correct for rare gases with different momentum-transfer cross sections. Taking into account the considered case with high powerconsumption in the low-energy region, they can be applied to mixtures of inert gases with molecular gases and even to the case of pure molecular gases. Step (5). Naturally, the results need to be verified

experimentally. The experimental data on C H 4 decomposition obtained in a 1% CH, + Ar mixture in DCP and MIP for a pressure p < 10 Torr (Gerasimov et al 1984) can be considered. It has been shown that t h e degree of CH4 decomposition is quite similar in both types of discharge for equal pressure and absorbed power in discharge tubes of equal diameter. As this process occurs due to collisions with electrons, the equality of the degree of composition means that the rates of the pioccssis u n d i i iliitioii impact are equal. The ialculation of the EDF in a C H 4 Ar mixture (CH, content 4%) has shown that this process can consume more than 50% of P+ Thus, the experimental results confirm the conclusions presented here in an indirect way.

+

4. Conclusion

Self-consistent calculations of the parameters of selfsustained M I P and DCP in argon in the diffusion-controlled regime have been carried out. It has been conchded :ha: :hc p:oposcd invariant group of paiaix:cis (absorbed power P,. pressure p , discharge tube radius R and wall temperature Tw) possesses a number of advantages for the comparison of plasmas different types of discharge. The calculated parameters of M I P and DCP are in satisfactory agreement with experimental data obtained with the same external parameters. The calculations have also revealed the influence of Tw on plasma characteristics and the possibility of operating on the properties of a plasma by Tw alteration. The calculation of the plasma characteristics in argon using models with different levels of detail and in model gases with momentum transfer cross-sections having different energy dependences have shown that

412

the rates of power-capacious processes in MIP and DCP turn out to be equal at equal Pd, p , R and Tw. This allows one to formulate rules for the calculation of the constant rates of direct and stepwise processes under electron impact from MIP to DCP and vice versa, provided the concentrations of electrons and excited atoms are known. The results can be useful in analysing the physical and chemical properties of electrical discharges.

References Eletsky A V, Mishenko L G and Tychinsky V P 1968 J . Appl. Spectrosc. 8 425 (in Russian) Ferreira C M and Loureiro J 1983 J . Phys. D: Appl. Phys. 16 1611 - 1984 I . Phys. U: Appl. Phys. 17 I175 - 1989 J . Phys. D: Appl. Phys. 22 76 Ferreira C M, Loureiro J and Ricard A 1985 J . Appl. Phys. 57 82 Ferreira C M and Moisan M 1988 Phys. Scr. 38 382 Ferreira C M and Ricard A 1983 J . Appl. Phys. 54 2261 Fransis G 1956 Handbuch der Physik 22 53 Gerasimov Yu A, Grachova T A and Lebedev Yu A 1984 High Energy Chemistry 18 363 Gerasimov Yu A and Lebedev Yu A 1985 Opt. Spectrosc. 59 704 (in Russian) Golant V E 1958 Usp. Fiz. Nauk 65 39 (in Russian) Granovskii V L 1952 Electric Current in a Gas (Moscow: ._GITTL) (in Russian) U n i c f P i n T i G " 7 E.&.,* C".. i,-r I,-. .\.". - 1951 Phys. Rev. 83 1159 Hyman H A 1978 Phys. Rev. A18 441 lvanov Yu A , Lebedev Yu A and Polak L S 1976 Plasma Phys. 2 871 (in Russian) lvanov Yu A and Soldatova I V 1985 Physico-Chemical Proce.ws in a Low-Temperature Plasma ed L S Polak (Moscow: Institute of Petrochemical Synthesis of the Academy of Sciences of the USSR) pp 5-54 (in Russian) Karoulina E V and Lebedev Yu A 1987 Plasma Chemistry87 ed L S Polak (Moscow: Institute of Petrochemical Synthesis of the Academy of Sciences of the USSR) pp 6-36 (in Russian) - 19881. Phys. D: Appl. Phys. 21 411 - 1989 Proc. Seminar Production, investigation and aoolication of olasma in microwave fields flrkutsk: Isu) p 14 (in Kussian) Kochetov I V 1977 Thesis Moscow Physical-Technical Institute (in Russian) Mitchell A C G and Zemansky M K 1971 Resonance Radiation and Excited Atoms (Cambridge: Cambridge University Press) Moisan M. Pantel R. Ricard A 1982 Can. J . Phvs. 60 379 Ricdrd A,'Hubeit J and Moisan M 1985 Proc. i7th Int. Conf. on Phenomena in Ionized Gases (Budnpest) p 741 Sobelman I I 1963 Introduction in a Theory of Atomic Spectra (Moscow: GIFML) (in Russian) Soldatova I V 1986 Thesis Institute of Petrochemical Synthesis USSR Academv of Sciences (in Russian) von Engel A 1955 Ionized Gases (0xford:'Clarendon'Press) Walsh P J 1959 Phvs. Rev. 116 SI1 Wiese W Land Smith M W 1969 Atomic transition probabilities vol 2 Report NSRDS-NBS 22 Zarkova L P and Stefanov B I 1967 Low-Temperature Plasma (Moscow: Publ. Comp. Mir) p239 (in Russian) 11-.

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