J. Phys. D: Appl. Phys. 22 (1959) 632—643. Printed in the UK
`
`
`
`A collisronal—radiativemodel
`applicable toargondischarges ever a
`widerange ofconditions
`IL Applicationto low-pressure,hollow-
`cathodearcandlow-pressureglow
`discharges
`
`J Vlr“;ek1L and V Pelikani
`t Department of Physics Institute of Mechanical and Electrical Engineering.
`30614 Plzen. Nejedl ého sady 14. Czechoslovakia
`a; Department of Automatic Control Systems. Skoda National Corporation, 31600
`Plzen, Czechoslovakia
`
`Received 12 July 1988, in final form 28 November 1988
`
`Abstract. The extensive collisional—radiative model for an argon atom plasma is
`applied to a low— pressure. hollow— cathode arc discharge and to the positive column
`of a low-pressure glow discharge in order to clarify the mechanisms by which the
`excited levels in these discharges are populated the results being compared with
`experimental investigations in the literature. Computations are carried out for
`various sets of input parameters. such as the electron kinetic temperature Te, the
`atom temperature Ta. the ion temperature T the electron number density ne, the
`ground state atom population n the plasma column radius R and the escape
`factors \m and Am, characterising the non- equilibrium plasmas under
`consideration. The predictions of our model. i. e. the populations in the excited
`levels as a function of the electron number density the effective principal quantum
`number and the discharge current are compared with the experimental results and
`in two cases also with the theoretical results of other authors. it is shown that all
`calculated dependences are fairly close to the corresponding experimental curves
`referring to both discharges The results presented confirm the applicability of the
`so--called analytical top model' of van der Mullen et al and Walshs formula for AV,
`interpreted according to Mills and Hieftje.
`
`1. lntroduction
`
`In the preceding paper (Vlcek 1989). a collisional——
`radiative (CR) model applicable over a wider range of
`conditions than those described in the literature (Gian—
`naris and Incropera 1973. Katsonis 1976. Gome’s 1983.
`van der Sijde er al 1984a. Hasegawa and Haraguchi
`1985) was established for an argon atom plasma.
`Atom—atom inelastic collisions and diffusion losses
`of the metastable states.
`together with the electrom
`atom inelastic collisions and radiative processes usually
`included. are considered in this model.
`taking into
`account 65 effective levels.
`
`Analytical expressions used for the corresponding
`cross sections are in good agreement with currently
`available experimental and theoretical data for argon,
`In particular. the measurements of Chutjian and Cart—
`
`0022-372789050682 4-
`
`'12 $02. 501321989 lOP Publishing Ltd
`
`wright (1981) and the computations of Kimura er a!
`(1985) have been used extensively.
`With the help of the previously established numeri—
`cal method (Vicek 1989). we can calculate the popu-
`lation coefficients determining the populations in all
`excited effective levels. This enables us to study the
`mechanisms by which these levels are populated under
`various conditions in a non-equilibrium argon plasma
`characterised (even in the case of a non-Maxwellian
`electron energy distribution function) (EEDF) by the set
`of parameters TC. Tu. Ti. rig. 111. R. Am” and A,,,.
`The main aim of the present paper is to check
`the reliability of our extensive CR model under the
`conditions in the low—pressure hollow cathode arc.
`studied experimentally by van der Mullen et ai (1978.
`1980) and by van der Sijde er al (1984a. b). and in the
`positive column of the low-pressure glow discharge.
`
`INTEL 1312
`GILLETTE 1412
`
`GILLETTE 1412
`
`

`

`investigated experimentally by Kagan et al (1963b).
`A further motivation for this study is our interest in
`understanding the mechanisms by which the excited
`levels are populated in these two argon discharges of
`practical interest.
`For experimental verification of our CR model, a
`combination of these discharges seems to be suitable
`because the population mechanisms differ appreciably
`in them. Moreover, because the electron energy distri-
`bution function in the hollow-cathode are investigated
`is Maxwellian (Pots 1979) and the radiation trapping
`is negligibly small (van der Sijde er al 1984a,b), the
`calculated excited level populations obtained under
`these conditions cannot be affected by possible inac-
`curacies arising from the solution of the Boltzmann
`equation and from determination of the optical escape
`factors.
`
`2. Results and discussion
`
`Assuming that the quasi-stationary state model can be
`applied (see for example Cacciatore er a1 1976, Biber-
`man er al 1982) and the fundamental mode of diffusion
`, of the metastables to the wall is dominant in the dis—
`charge tube (see for example Delcroix er a! 1976, Fer—
`reira er a1 1985). we obtain a set of coupled linear
`equations
`
`n:.
`2 ant/inn = _6m _ anti”!
`
`(1)
`
`where m = 2 ..... 65. from which the unknown excited
`level populations #1,, may be calculated. provided that
`the coefficients am and (5", are known (Vlcek 1989)
`and the ground state atom population ”1 has been
`determined experimentally.
`Owing to the possibility of investigating the effect
`of the upward ionisation flow of electrons from the
`ground state atom and their downward recombination
`flow from a continuum on the populating of the excited
`levels, the system (1) is solved. in spite of the fact that
`n: is not an independent parameter in our case. in the
`standard form
`
`H” = riff” 1— G§,”ri;
`
`foriz=2.....65
`
`(2)
`
`where the population coefficients riff” and G3.“ represent
`the solutions of (l) with n, = 0 or fit: 1 and on, = 0.
`respectively inserted into their right hand sides.
`The expression (2) can be rewritten as
`
`r... = rirr:
`
`+ 7.2””?
`
`(3)
`
`where ms, and n}? are the corresponding Saha population
`and Boltzmann population, respectively,
`rif“ and rfi,“
`are the so-called CR coefficients relating the actual popu—
`lations n" to Hi and HE. respectively.
`.
`In a general case. the numerical method developed
`allows us to calculate the population coefficients riff”
`and 6:,” as functions of the following input parameters:
`Tc. Ta. Ti. n.3, nl. R. AW, and Am with the possibility of
`neglecting the atom—atom inelastic collisions when their
`
`Collisional—radiative model for argon discharges, ll
`
`Table 1. Data characterising the excited effective levels
`including all actual 4s and 4p states.
`
`Statistical
`Excitation
`Level
`number
`Designation
`energy
`weight
`
`[7
`npqni[K]J
`51:1 (9V)
`9!?
`
`5
`11 .548
`4s[3i’2]2
`2
`3
`11 .624
`4sl3r’2].
`3
`1
`11.723
`45% 2210
`4
`3
`11.828
`4511/2].
`5
`3
`12.907
`4pi1 i2].
`e
`20
`13.116
`4p{3/‘2)1‘2, [51,2123
`7
`8
`13.295
`4p'[3!2]1A2
`8
`3
`13.328
`4p‘[i/2]1
`9
`1
`13.273
`4p[1 r210
`10
`
`it 1 4p’[1 .1210 13.480
`
`
`
`influence on the population mechanism is-studied.
`Under
`a
`reasonable assumption that only the
`reabsorption of the resonance radiation may be impor-
`tant in the discharges investigated. we have used the
`analytical formulae for the escape factors (Mills and
`Hieftje 1984) in which A1,, are dependent only on Ta,
`71; and R, where V: = 3. 512.15.16.17. 20. 21. 26, 27
`and 33 (see table 1 of Vlcek 1989).
`In the special case. when the Boltzmann equation
`need not be solved due to the Maxwellian form of the
`
`EEDF in a plasma. computations become straightforward
`and relatively rapid. Furthermore. the ion temperature
`T, does not then appear among the input parameters
`(Vicek and Pelikan 1985. Vlcek 1989).
`The basic data characterising the excited levels con-
`sidered in our CR model, including all individual 45 and
`4p states, are given in table 1 to help in comparing the
`calculated and experimental results.
`
`2.1. Hollow-cathode arc discharge
`
`Hollow-cathode arc discharges have been widely used.
`for example in the investigation of argon ion laser
`plasmas (Pots er (111978), in fusion-oriented and welding
`technology (Chall and Uhlenbusch 1982).
`in plasma
`centrifuges (Wijnakker er al 1979). or generally in
`plasmas in magnetic fields (Boeschoten er a! 1979).
`Valuable results contributing to the elucidation of
`the mechanisms by which the excited levels in a low-
`pressure, hollow-cathode are are populated have been
`obtained in recent years by van der Mullen et al (1978.
`1980, 1983) and van der Sijde er al (1984a. b).
`In our case. the numerical results for the excited
`
`level populations n,1 and for a CR coefficient rf,” are
`respectively compared with the corresponding values
`determined experimentally by van der Mullen 62 a!
`(1978. 1980) and van der Sijde er al (1984a. b) who
`measured the absolute intensities of numerous Ar 1 lines
`as a function of the electron number density “c for
`known values of T0. T3, in and R in the highly ionised.
`magnetically confined plasma of a low—pressure. hollow-
`cathode arc.
`
`It has previously been verified that the EEDF is
`
`633
`
`

`

`J Vléek and V Pelikan
`
`
`-1
`‘l
`
`”—4
`
`ill
`tl
`
`
`
`
`
`ll
`
`
`
`Flgure 1. The excited level populations nn related to the corresponding Sana
`values n? for various electron number densities na at TE = 40600 K, Ta =
`11600 K, Fl = 1 cm and n, ranging from 2.35 x 10‘3crn“3 for he =
`5 x 1012 cm“3 to 3.62 x 10‘2 cm‘3 for ne = 10‘scm‘3, Present work; curve A,
`n=6; curve B, n: 11; curve C, n=7: curve D, n: 10; curve E, n=8;
`curve F. n = 9 (see table 1); curves G and H, the 5d and 6d groups,
`respectively, with jc = ‘3 (full curves) and jc = % (broken curves). Symbols 0,
`A. 0 show the experimental data of van der Mullen er a/ (1980) for the
`whole 4p. 5d and 6d groups, respectively‘
`
`Maxvvellian (Pots 1979) and that doubly ionised ions
`can be neglected (van der Mullen er a! 1980, Pots 1979)
`under the conditions considered.
`
`In figure 1 we compare the ratios nn/ni measured
`by van der Mullen er a! (1980) for the 4p, 5d and 6d
`groups with the corresponding values calculated by us
`at the plasma parameters Ta, 718 and n1 presented in the
`above-mentioned paper and for Ta = 11600 K and R =
`
`1cm taken from similar experiments (van der Sijde er
`al 1984a) carried out with the help of the same set-up
`According to van der Mullen et al (1978), the accuracy
`in the measurement of n" is estimated to be 50%.
`As can be seen in figure 1, the uniform decrease of
`the calculated values of nn/nfi as n;3 in the so—called
`‘excitation saturation phase’ agrees well with the
`measured dependences, but
`the theoretical values.
`
`634
`
`

`

`Coilisionai—radiative model for argon discharges. ll
`
`
`
`
`
`
`Figure 2. The r“ coefficient as a function of the electron number density ne Theoretical results: curve A. Pots (1979);
`curve B van derpMullen et a/ (1977); curve C van der Siide et al (1984a) (all with T9-— 34800 K); curves D,
`determined by us using the model of Katsonis (1976) at T5 = 35 000 K (full curve) and Te= 40000 K (broken curve);
`present work: curve A‘ n = 3 x to‘ecm‘3 Ta = 3480 K, F?-— 1 cm T8 = 34800 K (chain curve) and T9 = 46400 K
`(dotted curve); B', as torA but at n, = 9 x tolzcm 30’. n1 =3 x103cm‘3 7 =6960K H=1cm. 79:34800K
`(full curve) and T;—— 46400 K (broken curve); D’ as for C’ but at n1 2 9 X 10120m‘3. A, experimental results (van der
`Sijde et al 1984b).
`
`those obtained for the individual 4p states
`except
`denoted by n = 6 and 11. are somewhat underestimated
`compared with the measurements, The values of ne
`relating to the transition of the excited levels considered
`to the regime of the partial local thermodynamic equi—
`librium are in satisfactory agreement with those
`obtained by extrapolation of the measured data in the
`work of van der Mullen et a1 (1980).
`It is known (see for example van der Sijde et al
`1984b) that the values of the CR coefficients r2“, occur-
`ring in (3), may be important for spectroscopic diag-
`nostics of a plasma. e.g. for the determination of the
`electron temperature T2 and the ground state atom
`population n}. if the nth level is in the complete satu-
`ration phase.
`In figure 2 we give our results for the coefficient
`fl”. where n = 9‘ as a function of ”e for comparison
`with the corresponding experimental values obtained
`by van der Sijde et al (1984b) under the following
`conditions in the discharge: 34800 K $ Te S 46400 K,
`3480 K S Ta S 6960 K,
`1013 cm'3 s ne $10H cm’3,
`3 X 1013 cm“3 S n1 s 9 X 1012 cm‘3. As van der Sijde et
`al (1984b) did not give the value for the plasma column
`radius in their work. we have used R = 1 cm again (van
`der Sijde er al 1984a) in our computations illustrating
`the dependence of the coefficient rE,“ on the electron
`
`temperature TS and also on the atom temperature Ta
`and the ground state atom population 11,. Assuming
`that the value of It, changes with 716 in a manner similar
`to the measurements of van der Mullen er a! (1980), we
`have determined the most probable values of rimU at nc =
`1013 cm’3 and 10“ cm‘3. They are denoted by vertical
`arrows in figure 2. Our calculations have proved that
`the plasma investigated is completely optically thin also
`for all radiative transitions excepting the resonance tran-
`sition from the 4511/2], state at n1: 9 ><10‘2crn‘3
`Where we have obtained the value of A15 = 0.46 and
`0.83 for T8 = 3480 K and 6960 K. respectively.
`The numerical results obtained from several CR
`
`models under the assumption that the plasma is com-
`pletely optically thin are also shown in figure 2.
`When we used the extensive 65-level model of Kat“
`
`sonis (1976). in which all important excitations from the
`ground state of an atom to the levels lying above the
`45’ states are omitted, except the excitations to the
`effective levels with n = 15. 16 and 17 (Vicek 1989). the
`model curves for r1,” obtained at Te = 35000K and
`40000K are given. When the simplified models (van
`der Mullen et al 1977, Pots 1979) and the extensive 49—
`level model of van der Sijde er al (1984a) have been
`applied at T = 34800 K. the results for the 4p group as
`a whole are shown. Note thatin the model of van der
`
`635
`
`

`

`J Vléek and V Pelikan
`
`
`
`17,./n‘,",at
`
`4n" ‘m
`
`
`
`t 1
`
`Figure 3. The population factor nn/‘ni — 1 as a function of the effective
`principal quantum number ngqn obtained with the following input parameters: A,
`jc =% and A, jc =§ior Te = 58000 K, Ta = 11600 K, ne = 6.7 x10‘3cm‘3,n.=
`1013 cm”3 and Fi=1cm; V,jc =1; and V‘ jc =% for T9 = 34800 K, Ta =
`11600 K, he = 6.7 x 10‘3cm‘3, n1 = 10’3 cm—3 and H=1cm.., experimental
`results (van der Mullen et a! 1980). The full line x = 6.0 and the broken line x =
`50 represent the slopes predicted on the basis of the analytical top model (van
`der Mullen et a/ 1983).
`
`Slide er al (1984a) the semiempirical formulae of Vriens
`and Smeets (1980) proposed for neutral hydrogen and
`alkali excited states are employed, all 45 states are
`separated and the statistical weights of the f groups are
`increased artificially.
`Figure 3 shows our numerical results together with
`the corresponding experimental results (van der Mullen
`er a1 1980) for the value of nn/rzfi - l as a function~
`of the effective principal quantum number njqn =
`(SP/en)“, where a? and 5,, are the ionisation energies
`for atomic hydrogen in the ground state and for the nth
`level of argon respectively.
`As can be seen. the measured values obtained at
`actual electron temperatures in the range from 34 800—
`
`636
`
`SSOOOK and at 71C = 6.7 X 1013 cm'3' are found to lie
`between the model curves determined by us at these
`two extreme values of Tc. Moreover. the slopes of the
`straight parts of our dependences agree well not only
`with the experimental result but also with that obtained
`on the basis of the so-called analytical top model (van
`der Mullen er a1 1983). Using this model to describe the
`population mechanisms of high—lying excited levels in a
`real. collisionally dominated, ionising plasma with the
`Maxwellian EEDF, one can write the following simple
`power law for non-hydrogenic systems:
`.er
`
`[1
`
`51,5! — l = b(~,(n:q,z)"‘r
`
`where b[1 is a constant and the value ofx is in the range
`
`

`

`Collisional-radiative model for argon discharges. ll
`
`
`
`
`
`
`
`
`153%(011''1)
`
`n1)-
`
`Figure 4. The calculated values of nil” (broken curves) and Gfl-‘m (full curves),
`together'with the measured values (0, O, A, A) of 17,, (van d‘er Mullen et a!
`1978) as functions of He at Ta = 34800 K, Ta = 11600 K, R1 = 1013 cm‘3 and
`F? = 1 cm for selected excited levels (see table 1): curves A, n = 2; curves B,
`n = 3; curves C and A, n= 7; curves D and 0, 5p (jC =%); curves E and A 5d
`(/C = %); F and 0. 75 (jC = %); G. npqn = 10 (jC = is). The dotted curve, 69% for
`"pan 2 .0 (jc = %) but with atom—atom inelastic collisions neglected.
`
`Table 2. The experimental discharge parameters (Kagan er al 1963b) for filling pressure of 5Torr together
`with the estimated values for the atom temperature Ta and the ground state atom population [1,.
`
`Glow
`discharge
`current
`I
`(mA)
`
`Second estimate
`Fir’st estimate
`—-——-——-——-—-——-—— ——-————~—~——
`Electron
`Electron
`Atom
`Ground state
`Atom
`Ground state
`kinetic
`number
`density
`temperature
`temperature
`population
`temperature
`pozpulation
`ne
`7.
`T2"
`nl‘
`7?
`l
`
`(10“ cm—3)
`(104 K)
`(102 K)
`(1018 cm‘3)
`(102 K)
`(1016 cm“)
`
`12.10
`4.00
`16.10
`3.00
`2.50
`2.70
`25
`10.70
`4.50
`15.10
`3.20
`2,20
`6.00
`50
`9.66
`5.00
`13.80
`3.50
`1.80
`12.00
`100
`8.78
`5.50
`12.10
`4.00
`1.60
`22.00
`200
`
`
`
`
`
`
`46.00 1.30 5.00 9.66 6.50400 7.43
`
`637
`
`

`

`J Vléek and V Peiikén
`
`
`
`[7,,Km“)
`
`
`
`
`
`
` m
`
`
`
`2
`
`i.
`
`Figure 5. Populations in the 4s and 4p levels (see table 1) obtained with
`experimental discharge parameters (Kagan et al 1963b) p = 5Torr, R = 1.2 cm.
`I = 25 mA; curves A, theory; A. experimental results; I = 400 mA: curves B,
`theory; 0, experimental; fun curve A. rte = 2.7 x 10’1 cm‘s. Te = 25000 K, Tia” =
`Ti“) = 300 K, n2” = 1.61 X 1017 cm“; broken curve A, T9 = Tim = 400 K,
`Mel-21.21 x 1017 cm ‘3; chain curve B. n3 = 4.6 x 1012 cm“3, Te =13000 K,
`T9) = Tf” = 500 K. n3“ = 9.66 x 1016 cm—S; dotted curve B. rt?) = T5?) =
`650 K. n?’ = 7.43 X 10’5cm“3.
`
`from 5.5 to 6.5; the lowest possible value, x = 5.0,
`may be obtained if ionisation and recombination are
`neglected (van der Mullen et a! 1983).
`In figure 4 we give the calculated values of In?" and
`653511 and the measured values ofnn for selected excited
`levels as functions of the electron number density ne at
`the discharge parameters characterising the experiments
`of van der Mullen et al (1978).
`A comparison of the terms appearing in (2) helps us
`to understand the role played by the upward ionisation
`flow of electrons in the system of excited levels and
`their downward recombination flow in determining the
`excited level populations under the conditions con—
`sidered.
`
`As seen in figure 4. the indicated critical values of
`n5, for which both terms are balanced in (2), decrease
`rapidly with the decreasing ionisation energy of the
`excited levels investigated. When the electron number
`density is higher than these critical values. excited levels
`come into Saha equilibrium where n” is proportional to
`n3. As shown, the results presented are in better agree—
`ment with the measurements (van der Mullen er (211978)
`than in figure 1.
`The effect of the atom—atom inelastic collisions on
`the excited level populations has been found to be
`negligibly small for all sets of discharge parameters
`considered if 116/211 2 1. At lower degrees of ionisation
`this effect is growing. e.g. at ”e = 1012 cm“3 and with
`
`638
`
`

`

`
`
`Collisional—radiative model for argon discharges, ll
`
`
`
`
`
`g.
`
`l
`2
`
`l
`
`l
`1+
`
`l
`
`l
`
`l
`a
`
`i
`
`l
`to
`
`
`
`n,,inn"3J
`
`Figure 6. Populations in the 4s and 4p levels obtained with experimental
`discharge parameters (Kagan eta/1963b) p = 5Torr, Fl = 1.2 cm, l= 50 mA:
`A, experimental results; full curve. theory with he = 6.0 X 10" cm“, T,3 =
`22000 K, Ta = T. = 320 K and n1 = 1.51 x 10‘7cm‘3; broken curve, theory with
`ne = 5.4 X 10‘1 cm’3, Te = 22000 K, Ta = T. = 320 K and 1.51 X 1017 cm‘3:
`dotted curve. theory with r75 = 6.0 X 10‘1 cm‘3, Te = 20 700 K, Ta = T. = 320 K
`and n1 = 1.51 X 10’7 cm‘3; I: 200 mA: 0, experimental results; chain curve,
`theory with n. = 2.2 x 10‘Zcm”3, T. =16000 K, Ta = Ti = 400 K and n. =
`1.21 x1017 cm‘3.
`
`the other parameters determined by van der Mullen er
`a! (1978) it causes positive or negative changes smaller
`than 3.4% and 30.9% in the population coefficients
`n31“) and GER. respectively. The deviations of G S,” are
`strongly dependent on the level number n. For n S 21.
`i.e. when the ionisation energy 8,, 2 0.961 eV (Vlcek
`1989). all changes of Gt,“ are smaller than 3.0%. In
`figure 4, we show the effect caused by neglecting the
`atonpatorn collisions
`in calculating the value of
`szllnj for the excited effective level denoted by n =
`46. which includes all states with the core quantum
`number jc :% and 22”,, = 10. For all the other excited
`levels given in this figure we have registered deviations
`smaller than 1.0% even at I’le = 1012 cm“?
`
`2.2. Positive column of the glow discharge
`
`In recent years numerous studies have been devoted to
`the modelling of a positive column of an argon glow
`discharge in which self-consistent calculations of EEDFS
`and electron excitation and ionisation rates have been
`included.
`
`Effective analytical methods for solving this problem
`have been developed, in particular by Golubovskii er a1
`(1972, 1976), Smits and Prins (1979a. b) and Ferreira
`and Ricard (1983). Many other authors (see for example
`Morgan and Vriens 1980) based their approach on an
`accurate numerical solution of the Boltzmann equation.
`In order to account for the deviations of the actual EEDF
`
`689
`
`

`

`J Vlcek and V Pelikan
`
`
`
` .l
`l
`
`
`
`
`
`
`
`
`n
`
`\Di,l____J___l_..lI
`
`flil/fnlfil
`
`10'5—
`.“our.
`
`6.
`
`04‘7'Wr—r-T-1w
`
`m 1
`
`N.
`
`t
`
`a)
`
`a 192’?
`
`()x
`
`Nmm——
`
`Figure 7. The ratio of the EEDF {(8) to the corresponding
`Maxwellian function Me) for the following input
`parameters. full curve A ne—— 27 X 10“ cm
`T =25000K T“: Tfl=300K nl"=161x10‘7cm
`broken curve A, T97: T?) = 400 K n‘2’—- 1.21 x
`10‘7 cm3; curve B [79—= 60 X 1011 cm‘3 T—- 22000K
`Ta = T— 320K n, = 151 X 1017 cm“3; curve C
`ne=2.2><10‘20m 3 Te=16000K Ta = T-= 400K
`n18=121 x 10‘7cm‘3; curve D_ ne= 4.6 X l3,1012cm“3
`Te=13000K Ta= T-= 500K, H1: 966 X 10‘scm‘3
`
`from the corresponding Maxwellian form in a simple
`way, Vriens (1973) and Morgan and Vriens (1980) pro—
`posed an approximate treatment utilising two groups of
`Maxwellian electrons. representing the body and the
`tail of the distribution, respectively.
`Recently, a realistic self—consistent discharge model
`for a low-pressure argon positive column based on a
`reliable set of electron cross sections (Ferreira and Lou-
`reiro 1983) has been worked out and tested experi—
`mentally by Ferreira et al (1985). The maintenance
`electric field strength together with the populations in
`the individual 45 states has been calculated from a
`set of coupled equations expressing the steady state
`discharge ionisation balance and rate balance for all 45
`states as a function of the gas pressure, the discharge
`current and the discharge tube radius.
`In our case the numerical results for the excited level
`populations are compared with the experimental values
`of Kagan et al (1963b). who investigated the dependence
`of the populations in the four 45 states andin the nine
`
`640
`
`4p states on the fundamental plasma parameters n3. TC
`and the electric field strength E in an argon glow
`discharge with the discharge current I in the range
`25400 mA and the filiing pressure p in the range
`0.18—10 Torr. The discharge tube radius R was 1.2 cm.
`The experimental work of Kagan er al (1963b) has
`been chosen for two reasons. First. a set of the discharge
`parameters measured by these authors is consistent with
`the input parameters in our CR model. Secondly.
`in
`addition to the excited level populations in the 45 states,
`those in higher-lying levels have also been determined.
`in contrast with other studies.
`
`In the same way as Vriens (1973) and Morgan and
`Vriens (1980). who dealt with the applicability of the
`two—electron group model. we use as input parameters
`those sets of quantities which correspond to the pressure
`p =5Torr. As Kagan er al (1963b) did not give the
`values measured for Ta under the various conditions
`and as the estimate Ta = 293 K used by Vriens (1973)
`and Morgan and Vriens (1980) is unrealistic. the most
`probable values of Ta evaluated on the basis of com.
`parison with experimental results obtained under similar
`discharge conditions are utilised in our computations
`(Vlcek and Pelikén 1985).
`The chosen sets of experimental discharge par-
`ameters are listed in table 2 together with those esti-
`mations of Ta and n1 which define the intervals of their
`probable values for the corresponding measurements,
`Under the assumption that Ta = Ti and taking the dis—
`charge tube radius to be R = 1.2 cm, we obtain complete
`sets of input parameters for which the numerical results
`for the excited level populations have been compared
`with measurements.
`
`In calculating the escape factors Am. the pressure-
`broadening term is dominant for all excited levels except
`those with n = 12 and 26 (Vléek 1989). All values of
`A1,, are in the range 4.94 X 10—4—178 >< l0‘3 under the
`conditions investigated.
`Figures 5 and 6 show a comparison between the
`calculated populations 11,, of the excited levels charac—
`terised in table 1 and the corresponding experimental
`data obtained at discharge currents of 25. 50. 200 and
`400 mA. In accordance with Kagan er al (1963a). an
`error of a factor of two was expected in the measure-
`ments of n".
`As can be seen from these figures. the calculated
`populations are in better agreement with the measure-
`ments for the 45 states than for the 4p levels for which
`the behaviour of the functions nn(n) agrees we ll with
`the corresponding experimental dependences. but the
`calculated values of n" are somewhat overestimated
`compared with the measurements. Better agreement
`between theory and experiment is obtained for two
`resonance 45 states than for two 43 metastables in con-
`trast to modelling of the low-pressure argon positive
`column carried out by Ferreira et al (1985).
`In figure 5 a different effect of the two choices of TH
`and n1 given in table 2 on the populations n,, is shown
`for discharge currents of 25 and 400 mA. It may be
`explained by the fact that the tail of the EEDF is much
`
`

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`Figure 8. The calculated ratios nn/nf’, as functions of the effective principal
`quantum number n5“ obtained with input parameters ne = 2.7 x 10“ cm‘s.
`Ta = 25000 K, Ta: 7]: 300 K, n, = 1.61 X 10‘7cm‘3 and H=1cmz A,jc=%:
`A, jc = %; O, jc = l but without atom—atom collisions; O, jc = it but without atom—
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`more dependent on the value of the degree of ionisation
`nc/nl in the case of I = 25 mA than for I = 400 mA. At
`25 mA, a decrease of the ground state atom population
`n1 connected with an increase in the atom temperature
`Ta leads to essentially enlarged values of the rate coef-
`ficients for the excitation by electrons from the ground
`state. which are not compensated for by the lowered
`value of n] in the term Glyn1 determining the popu-
`lations n" of the levels considered under these con—
`ditions. Both effects are approximately balanced when
`l = 400 mA.
`-
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`In the papers of Kagan er al (1963a, b) there are no
`data concerning the accuracy in probe measurements of
`TC and me.
`In figure 6 the deviations of the populations ”n
`
`caused by small changes in the values of Te and me are
`shown for a discharge current of 50 mA. for which the
`largest discrepancies between theory and experiment
`exist. As can be seen. a decrease of the electron number
`density by 10% leads to a drop in the populations nn by
`1842996, whereas a decrease of
`the electron tem-
`perature only to the value of 20700 K, representing a
`reduction in the experimental value (Kagan er al 1963b)
`of 5.9%, results in the lowering of the populations
`investigated by 36—43%. This reduced value of T3 was
`obtained (Morgan and Vriens 1980) with the help of
`the standard Boltzmann calculations at the discharge
`parameters determined experimentally by Kagan er al
`(1963b) for I = 50 mA.
`In figure 7 we give the numerical results for the ratio
`
`641
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`

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`J Vléek and V Pelikén
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`Figure 9. The calculated values of nil” (broken curves)
`and Gilln1 (full curves), together with the measured values
`(symbols with error bars) of n” (Kagan et al 1963b), for
`selected excited levels as functions of the discharge
`current [with the first choices of Ta = T, and n1 given in
`table 2: curves A and O, n = 2; curves B and A, n = 3;
`curves C and V, n= 7; curves D, So (jc = 3); curves E. 5d
`(I: = 31); curves F, 78 (k = i); curves G. npqn =10<ic = it
`
`f (e)/fM(E) to compare the deviations of the realistic
`EEDF f (a) from the corresponding Maxwellian function
`fM(e) at several sets of input parameters listed in table
`2 and used in calculations presented in figures 5 and 6.
`The calculated ratios nnj’nfi as functions of the effec-
`tive principal quantum number 112,11,. are shown in figure
`8 for plasmas with discharge currents of 25 and 400 mA
`and they illustrate a strong tendency to a decrease in
`the overpopulation in higher excited levels compared
`with the Saha values. In addition, the effect of the atom-
`atom inelastic collisions on the excited level populations
`is presented for the case of 1 = 25 mA.
`Figure 9 shows the calculated values of ”(‘01 and
`65,”er and the measured values of r2,2 for several excited
`levels as functions of the discharge current 1 with the
`first choices of atom temperature Ta and ground state
`atom population n1 given in table 2. In contrast to
`the situation in the low-pressure, hollow-cathode are
`discussed in §2.l (see figure 4), the upward ionisation
`flow of electrons from the ground state of the argon
`atom is a dominant populating mechanism for all but
`
`642
`
`the highest-lying excited levels over the whole range of
`the discharge conditions investigated. Compared with
`Gi‘lnl , the terms 21$?) are much more dependent on the
`value of 1 corresponding to the set of plasma parameters
`Tc, Ta. 213 and n1 listed in table 2. When the discharge
`current I grows, the term Gi,”n1 is given by the balance
`between the effects caused by a drop in TE and n1.
`lowering the terms GS’nl, and those caused by a rise
`in ne. enlarging them, whereas both changes in TO and
`ne lead to an increase in the value of the population
`coefficient 722’).
`The measured values of Kagan eta! (1963b) available
`for the selected excited levels over the whole range of
`the discharge current are also shown in figure 9 for
`comparison.
`'
`
`3. Conclusions
`
`The extensive collisional—radiative model for an argon
`atom plasma has been applied to a low-pressure, hollow—
`cathode arc and to the positive column of a low-pressure
`glow discharge in order to clarify the mechanisms by
`which the excited levels in these discharges are popu-
`lated, these having been investigated experimentally by
`van der Mullen er a1 (1978, 1980). van der Sijde er al
`(1984a, b) and Kagan er al (1963b)., respectively.
`Our computations are carried out for various sets of
`the input parameters Te, Ta, Th 22:. n1. R. Am and
`Am which were measured directly in the experiments
`mentioned above or determined from the available
`experimental data.
`From our results the following main conclusions can
`be drawn.
`
`(i) Our model provides a qualitatively good picture
`of the processes determining the populating mechanism
`both in the hollow-cathode are considered. where the
`electron energy distribution function is Maxwellian and
`the radiation trapping is negligibly small, and in the
`glow discharge investigated, where the Boltzmann
`equation for the electron distribution function must be
`solved and the resonance radiation is strongly reab-
`sorbed.
`
`the
`In the low-pressure, hollow-cathode arc.
`(ii)
`predicted values of the populations in all the groups
`considered, i.e. 4p, 5p, 5d and 75. agree well with the
`measurements of van der Mullen er al (1978). but all
`the groups considered. i.e. 4p, 5d and 6d, are somewhat
`underpopulated compared with the experimental values
`of van der Mullen et a1 (1980). In the positive column
`of a l