`
`
`
`J Vléek
`
`Department of Physics, institute of Mechanical and Electrical Engineering,
`306 14 Plzeh, Nejedlého sady 14‘ Czechoslovakia
`
`Received 12 July i988, in final form 28 November 1988
`
`Abstract. A collisional—radiative model with an extended region of applicability is
`developed for an argon atom plasma. Atom—atom inelastic collisions and diffusion
`losses of the metastable states along with the electron—atom inelastic collisions
`and radiative processes are considered in this model, taking into account 65
`effective levels. Among the analytical expressions used for the corresponding
`cross sections. special attention is paid to those determining the set of cross
`sections for excitation by electrons from the ground state, owing to the possibility
`of utilising the formulae recommended in kinetic modelling studies of discharges in
`argon or in mixtures including argon atoms. The numerical method developed
`makes it possible to investigate the mechanisms by which the excited levels are
`populated in a non-equilibrium argon plasma characterised (even in the case of a
`non-Maxwellian electron distribution) by a set ct parameters, such as the electron
`kinetic temperature T9, the atom temperature T2, the ion temperature T,, the
`electron number density ne, the ground state atom population n,_, the discharge
`tube (or the plasma column) radius Fl and the optical escape factors /tmn and Am,
`which are dependent only on the quantities Ta, n, and R in many cases of
`practical interest.
`
`1. Introduction
`
`One of the simplifying assumptions made in basic
`equations describing the extensive collisional~radiative
`(CR) models (see eg. Bates et al 1962, Drawin and
`Ernard 1977, Fujimoto 1979, Biberman er at? 1982, van
`der Sijde er al 1984) is the use of the Maxwellian elec-
`tron energy distribution function (EEDF). However, it
`has been shown by many authors that this assumption
`is unjustified for a wide range of physically interesting
`conditions in various gases.
`In our recent papers (Vlceli and Pelikan 1985,
`1986). we presented a numerical method which enabled
`us to extend the applicability of the existing extensive
`CR models for the argon atom plasma with the Maxwel-
`lian distribution function (Giannaris and Incropera
`1973, Katsonis 1976. Gomés 1983, Van der Sijde et al
`1984. Hasegawa and Haraguchi 1985) to the region
`in which the actual EEDF differs appreciably from the
`Maxwellian form.
`A substantial feature of this method is a numerical
`solution of the Boltzmann equation (Vlcek and Pelikan
`1985) for the EEDF in a non—equilibrium argon plasma
`
`0022-3727/89/050623 + 09 $02.50 © 1989 IOP Publishing Ltd
`
`characterised by a set of parameters, such as the elec-
`tron kinetic temperature T3. the atom temperature T3,
`the ion temperature T3, the electron number density rte
`and" the ground state atom population n1, which are in
`accordance with the usual input parameters of the basic
`equations for the CR models.
`Here. our objective is to present an extensive CR
`model which may be applied to argon discharges in a
`wide range of practically interesting conditions.
`We have used the slightly corrected argon atom
`model of Drawin and Katsonis (1976) which includes
`65 discrete effective levels and refiects the actual atomic
`structure.
`
`In addition to the possibility of using a realistic
`EEDF. two further modifications have been carried out
`in the usual formulation of the basic equations for the
`CR models: the atom—atom inelastic collisions and the
`diffusion losses of
`the two metastable states are
`considered.
`
`In choosing the analytical expressions for the cross
`sections, we have taken into account the most recent
`experimental and theoretical results available for argon
`in the literature. In particular. we have used extensively
`
`623
`
`TSMC-1211
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`TSMC v. Zond, Inc.
`
`Page 1 of 9
`
`TSMC-1211
`TSMC v. Zond, Inc.
`Page 1 of 9
`
`
`
`J Vléek
`
`the measurements of Chutjian and Cartwright (1981)
`and the computations of Kimura er al (1985).
`The local effect of radiation trapping is described by
`introducing the optical escape factors which represent
`further input parameters in our model calculations.
`However, in many cases of practical interest only the
`reabsorption of resonance radiation is important and
`the corresponding escape factors are dependent only
`on the quantities Ta, /11 and R mentioned above (Walsh
`1959, Mills and Hieftje 1984).
`The numerical method developed makes it possible
`to investigate the mechanisms by which the excited
`levels in various practically interesting argon discharges
`are populated. such as low-pressure glow discharges,
`hollow cathode discharges,
`atmospheric or
`sub-
`atrnospheric arcs and ICP discharges.
`
`2. Formal solution of the problem
`
`We will confine ourselves to a quasi-neutral argon
`plasma composed of atoms Ar. free electrons e and only
`singly ionised ions Ar* in the ground state with the
`number density 111 = rte.
`Our computations are based on the argon atom
`model of Drawin and Katsonis (1976). in which we have
`separated the resonance and metastable states originally
`grouped together in the two lowest excited effective
`levels. We consider 64 excited effective levels divided’
`
`into two subsystems, corresponding to either of the two
`core quantum numbers jc = % (‘primed system’) and jc =
`% (‘unprimed system’). Then the idealised model has
`two different ionisation limits referring to the core con-
`figurations 3131,; and 3P3,3, respectively, as the actual
`argon atom. Basic data characterising the 65 effective
`levels considered and numbered according to their ion-
`isation energies are given in table 1.
`The following collisional and radiative processes,
`together with the diffusion losses of the
`bles,
`are considered in our extended version of the CR model:
`CU”!
`
`(i) Ar(m) + e r—" Ar(n) + e
`
`. KR”!
`(ii) Ar(m) + Ar( 1) 7 Ar(n) + Ar(1)
`5”!
`
`(iii) Ar(m) + e ? Ar“ + e + e
`VH1
`
`(iv) Ar(m) + Ar(1) ? Ar‘ + e + Ar(1)
`(IT/\mn)Amn
`
`-T‘
`(v) Ar(m) + hz2,,,,,
`(i_AmjRn:
`
`Ar(n)
`
`(vi) Ar(m)+hz2 ? Ar‘ +e
`BWI
`(vii) Ar(m) -i— Ar(1) + Ar(1) —* Ar; + Ar(1)
`where m = 2 and 4.
`,2, 2
`
`(viii) diffusion of the metastables 4s[3/2]; and 4s’[1/'2]0
`(for which :2 =2 and 4 in table 1, respectively) to
`the wall.
`'
`
`Here, C,,,,, and K,,,,, are the rate coefficients for col-
`lisional excitation by electrons and by ground state
`atoms respectively, F,,,,, and L,,,,, are respectively the
`rate coefficients for the inverse processes (collisional de-
`excitations). Sm and V,,, are the corresponding collisional
`ionisation rate coefficients while O,,, and W,,, are the
`rate coefficients for the inverse three-body recom-
`binations and Rm is the radiative recombination rate
`coefficient. AW, is the transition probability, AW, and
`Am are the optical escape factors for bound—bound
`and bound—free transitions, respectively. B”, is the rate
`constant for the three-body collisions of metastable
`states with the ground state atoms.
`Under conditions allowing the use of the quasi-
`stationary state model (discussed in greater detail by
`Bates er al 1962, Cacciatore er al 1976, Biberman er al
`1982) and calculation of the rate of loss of the meta-
`stables by diffusion to the wall. assuming that their radial
`distribution corresponds to the fundamental diffusion
`mode for the discharge tube (see e.g. Delcroix et ai
`1976, Ferreira er al 1985), we obtain a set of coupled
`linear equations
`6:
`
`2 amnnn = _ am - amlnl
`n=2
`
`, 65, from which the unknown excited
`.
`.
`where m = 2. .
`level populations 21,, may be calculated provided that
`the coefficients a,,,,, and (Sm are known and the ground
`state
`atom population 221
`has been determined
`experimentally.
`The coefficients am" and (Sm are related to the above-
`mentioned processes (i)—(viii) by the expressions
`
`am =neC,,,,,+n1K,,,,,
`
`am = ngF,,,,, +n,L,,,,, +A,,,,,A,,,,,
`55
`
`m>n
`
`m<n
`
`(2)
`
`(3)
`
`am, = —(nes,, +n,V,, +”§1a,,,,, + D,,/A3 +niBt)
`
`m=n
`
`(4)
`
`(5)
`(3m =ngn+(nc0m+n1W,,, +A,,,R,,,).
`Let us recall that in our model the diffusion coef-
`ficient D,, and the three-body rate constant B,,_. appear-
`ing in (4), are non-zero only when n = 2 and n = 4.
`Assuming diffusion in the fundamental mode is
`dominant in a discharge tube of length L. L > R, where
`R is the radius of the tube, and the temperature depen-
`dence of the diffusion coefficients D,, in argon is the
`same as in neon (Phelps 1959), the diffusion term in (4)
`can be rewritten in the form d,, T273/n1RZ, where d,, is
`a constant given by the experimental data for Dnnl
`(Tachibana 1986). The values of B,, are also taken from
`Tachibana (1986).
`The expressions for the rate coefficients referring to
`the inelastic collisions under consideration are obtained
`
`
`
`TSMC-1211 I Page 2 of 9
`
`TSMC-1211 / Page 2 of 9
`
`
`
`Table 1. Data characterising the excited levels considered in the model and the transition-dependent parameters (see
`main text) relating to the cross sections for excitauon by electrons from the ground state. A, optically allowed: P,
`parity-forbidden; S, spin-forbidden transitions.
`Excitation
`Level
`Statistical
`Transition-dependent
`energy
`Nature of
`number
`parameters
`S
`weight
`51,,
`Designation
`n
`transition
`9..
`
`C":-.%=5Y§nf1n and 07 0”‘-.:n= 01' <1’:n
`npqn][K1J
`(ev)
`
`3135
`1?,g00
`4s[3.»-'21.
`. 48
`4s[3=-"21,
`11.623
`4s'{1 521.,
`11.12;
`4s’[1."2],
`11.328
`4p[1 :2].
`12.907
`4PBffln2.fi‘2ka
`13115
`4p’[3.-’2]1_2
`13.295
`4p'[1/2],
`13.328
`4p[1 12]..
`13.273
`4p"[1 /2].,
`13.480
`3d{1 .-''2].,,,, [3212
`13.884
`3d[7:2]3,.,
`13.994
`3d’[23/2}2, 152212,.
`14.29
`5s‘
`14.252
`3d[3:¢2],, [5/2123 + 55
`14.090
`3d'[3./21,
`14.304
`Sp
`14.509
`5;)’
`14.690
`4d -1- 85
`14.792
`4d‘ + 63'
`14.976
`41'
`15.083
`4f
`14,906
`6p’
`15.205
`653
`15.028
`5d’ -1- 7s
`15.824
`5d + 7s
`15.153
`51', g’
`15.393
`51, g
`15.215
`7p’
`15.461
`7p
`15.282
`Gd’ + 88'
`15.520
`6d + 85
`15.347
`108
`6f’, g’, h’
`15.560
`216
`Bf. g, h
`15.382
`12
`Bp’
`15.600
`24
`8p
`15.423
`24
`7d’ + 93'
`15.636
`48
`7d -1' 93
`15.460
`160
`71', g’, h’, 1'
`15.659
`320
`71, g, n, 1
`15.482
`240
`8d’, f’,
`.
`15.725
`480
`8d, f,
`.
`.
`15.548
`320
`9p’, d’, f’,
`15.769
`640
`9;), d, f,
`.
`15.592
`400
`mm = 10
`13:32.1
`800
`1
`. 25
`484
`"W" = ”
`15.243
`968
`1
`. 4
`576
`“W” 7 12
`15.565
`1152
`1
`.857
`676
`"W" = 13
`15.680
`1352
`784
`"nan = *4
`12:33?
`1568
`900
`-
`.87
`1800
`"pen = 1°
`13.705
`1024
`2048
`"mm = *6
`12:33?
`1
`.8
`1156
`“W” ‘ 17
`15.71%
`2312
`1296
`"am = 18
`12:???
`2592
`1444
`1
`. 99
`“W” = 19
`15.?/22
`2888
`
`
`AowmwmmAamaommummawmaommummewmAommumm3$§:5wmVmmhwh4
`
`mbwwdommwmwhmmmmnnanunAnawmmwmmwmwmmmmmmmmmmmaa—A—a_A
`
`Gimmmmmmmmmmmmm
`
`.
`
`.
`
`5.70 X 1072
`1.92 X 1072, 4.00
`9.50 X 1073
`4.62 X 1072, 4.00
`3.50 X 1072
`1.15 X 1071
`3.50 X 1072
`7.00 X 1073
`7.00 X 1073
`3.50 X 1072
`1.50 X 1071
`9.00 X 10'?
`4.20 X 1072
`3.71 X 1073, 4.00
`3.33 X 1072, 4.00
`1.79 X 1072, 2.00
`7.00 X 10'?
`5.00 X 1072
`5.15 X 1072, 1.00
`3.06 X 1072, 1.00
`
`6.50 X 1074, 1.00
`3.69 X 1072, 1.00
`
`2.40 X 10‘2, 1.00
`
`O)~bl\JI\)—‘U'|'\7’\)-33-‘l\3N-4-‘N-{>03-Fl-DV\3O)(X3-bC0l\)-t-C000-l>\lO)<D—*—*(:)mO(A7Cn)v*0)U1-4
`
`128
`
`-hl\.’)|\)—*CD-¥>~l>f\)
`
`llllllllllllllllllllllllllllllll>ll|ll>>l||l>>T"3>>>VW/>1W1717*U'0><I>><I>|
`
`625
`
`TSMC-1211 I Page 3 of 9
`
`TSMC-1211 / Page 3 of 9
`
`
`
`J Vléek
`
`from general formulae (see e.g. Mitchner and Kruger
`1973) on the basis of the reasonable assumption that
`the distribution function for the heavy particles is of
`Maxwellian form in all cases, whereas the distribution .
`of electrons may differ from the Maxwellian function.
`The only component of the EEDF to appear in the
`rate coefficients for the electron—atom collisions in (2)-
`(5) is the isotropic function f(u), where the dimen-
`sionless energy u = 8/kTe (8 being the electron energy
`and k the Boltzmann constant) is introduced.
`The following integral formulae have been used for
`these rate coefficients (Vlcek and Pelikan 1986):
`
`‘
`
`C,,,,, = 8::
`
`5,, = 8.1
`
`F,,,,, = 8n
`
`J’:
`
`f(u)U,,,,,(u)u du
`
`fr f(u)a,,(u)udu
`
`f(u—u,,,,,)0,,,,,(u)u du
`
`R,,=§-’3(i)3 3'“ ix f(/w-e,,,)a*;,(u)V3du
`
`‘~me
`
`23+ 5'”,
`-.
`
`3
`
`(6)
`
`(7)
`
`(V8)
`
`(9)
`
`(10)
`
`electron-electron and electron~ion interactions and to
`the elastic electron—atom collisions. The EEDF is nor-
`malised, so that
`
`2:! (zzkrz-e )3!,’2
`
`J’ u":3f(u) du =1.
`0
`The rate coefficients for the atom—atom collisions
`
`me ,
`
`(ii) and (iv) appearing in equations (2)—(5) are given
`by the following expressions (Bacri and Gomés 1978.
`Biberman er al 1982, Collins 1967):
`
`'2kT * “Z
`flmnl)
`Kmn = 2 <
`a
`
`»'
`5
`b,,_ (_e,,L_+ 7kTa) exp L kn!)
`" mu
`--
`m M
`
`2k?"
`7 flmut
`21)
`V” = ~(
`
`‘*2
`
`b,,(s,_ +2/cr,) exp
`
`"
`
`5
`kn’)
`n
`
`_
`
`Lmn = Knm (gm//grz) eXp(.€mn//kTa)
`
`Wm = V £1
`'" 2 , \2:rmekTe,
`
`3]; exp
`
`\kTa
`
`(12)
`
`(13)
`
`(15)
`
`where mm is the reduced mass of two interacting atoms.
`In (12) and (13) it is assumed that the cross sections
`for the atom—atom excitation (ii) and the atom—atom
`ionisation (iv), Q,,,,, and Q”, respectively, are linear
`functions of energy above the threshold, so that
`
`and
`
`Qmn = bmn(E _ Emu)
`
`Qn =bn(E—€n)
`
`where E is the relative energy of the colliding atoms.
`As has been mentioned above, the local effect of
`radiative absorption is described by the optical escape
`factors A,,,,, and Am. This method, usually used in exten-
`sive CR models (see eg Bates et al 1962, Drawin and
`Emard 1977, Biberman er al 1982, van der Sijde er al
`1984), cannot be considered exact but it should rep-
`resent a reasonable approach which reflects real situa-
`tions well.
`
`In many cases of practical interest only the trapping
`of the resonance radiation is important because of the
`relatively large corresponding transition probabilities
`and high population in the ground state atom.
`In our model calculations, the escape factors A1,, for
`the resonance lines are determined using the formula
`derived by Walsh (1959) for the imprisonment lifetime
`when Doppler and collision broadening of the resonance
`line are present simultaneously. In a cylindrical tube
`of radius R, where axial excitation is dominant,
`the
`following expression (Mills and I-lieftje 1984) holds:
`
`A1»: =goT(R)
`
`(13)
`
`where go = 1.9 for a Doppler—broadened profile and
`go = 1.3 for one dominated by pressure broadening; the
`
`TSMC-1211 I Page 4 of 9
`
`X J1 f u — u,,,)0_,,,(u)u du.
`
`Here, me is the electron mass. c is the velocity of light
`in a vacuum, g- is the statistical weight of the ground
`state of the ions, a,,,,,(_u) and cr,,,,,(u) are the cross sections
`referring to collisional excitations by electrons from the
`nth excited level to the mth level and from the mth
`excited level to the nth level, respectively, whilst 0,,,(u)
`and crf,_(v) are the cross sections for collisional ionisation
`by electrons and for photoionisation of the mth level,
`respectively.
`Under conditions when the distribution function is
`non-Maxwellian,
`the Boltzmann equation for f(u) is
`solved numerically in the form (Vlcek and Pelikan~1985)
`
`Ed; Hm) diff’ + G(u>r<u>) = n1“1M(”)f(u)
`where
`
`(11)
`
`M(u) = O
`
`1w(u) = 0exc(u) + 0101)
`
`usuu
`
`M > N12.
`
`The formulae for the total excitation cross section
`om(u) and the ionisation cross section a1(u) will be
`characterised later.
`
`In (11) the terms H04) and G(u) describe the influ-
`ence of the electron energy gain and loss processes,
`respectively. on the EEDF. H(u) includes the expressions
`corresponding to the Coulomb electron—electron and
`electron—ion interactions, the effect of the axial electric
`field and the thermal motion of gas atoms. G(u) is
`given by the sum of the terms related to the Coulomb
`
`626
`
`TSMC-1211 / Page 4 of 9
`
`
`
`transmission coefficient T(R) is given by
`
`T(R') = TD exp(—:rT%;D,:"4T3'C)
`
`+ TC erf(rr"'2 TCD/‘ZTC ).
`
`Here. TD and TC are the transmission coefficients for
`pure Doppler and pressure broadening, respectively,
`and TCD is
`the coefficient
`for pressure-broadened
`emission and Doppler-broadened absorption profiles
`defined by expressions
`
`TD '5 ikz)R[-T 1n(koR)l1'i2i_i
`
`Tc = (a/'.1“koR)"i‘°
`
`TCD -7 2a/i)V7E[]n(kgR)]1"’2
`
`where kOR is the optical depth pertaining to the line
`centre and a is the damping coefficient (Mills and Hieftje
`1984). In our case, the error function erf(x) (see eg.
`Abramowitz and Stegun 1964) is evaluated with the
`help of the realistic approximation inferred by Hastings
`(1955).
`Note that the formulae for K,,,,, and V,, must be
`multiplied by the factor é when rt = 1 in (2) and (4),
`respectively (see e.g. Mitchner and Kruger 1973). Fur-
`thermore, according to Katsonis (1976), the statistical
`weight g,,, is equal to % or ‘E for m = 1 in (8) and (14)
`when n is the number of the level with }'c equal to 9; or
`‘E, respectively. Similarly, gg is equal to Z or 4 for m 2 2
`in (9), (10) and (15) and rz+ is equal to %ne or En, for
`m 2 2 in (5) when m denotes a level corresponding to
`j: = § or jc = f, respectively.
`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 when excited levels are popu-
`
`lated (Fujimoto 1979), the system (1) is solved, in spite
`of the fact that 711 is not an independent parameter in
`our case, in the standard form
`
`rt, = nifi’ + Gffinl
`
`22 % 2,... , 65
`
`(19)
`
`where the population coefficients n,‘~,'°‘ and Gi,” are
`obtained from equations (1) when we insert
`111 = 0
`or R1 = 1 and 5,” = 0, respectively, in their right-hand
`sides.
`V
`
`. The numerical method developed allows us to cal-
`culate the population coefficients n£,°’ and GS,” as func-
`tions of the following input parameters: Te, Ta, T3, ne,
`nl, R and of the escape factors A,,,,, and A,,,, which are
`given by T3, 71, and R only in a wide range of practically
`interesting conditions. It is based on the extension of
`the method described in our previous papers (Vléek and
`Pelikan 1985, 1986). Its accuracy and reliability have
`been proved in many numerical tests.
`
`3. Cross sections and transition probabilities
`
`3.1. Cross sections for electron—atom inelastic collisions
`
`Collisional~—radiative model for argon discharges. i
`
`energy states. Therefore the effective cross sections for
`excitation by electrons, occurring in (6) and (8), can
`generally be written in the form
`
`amt: = Griz: + 05:1’:
`
`where o-,?,,, and o_§,,, are the cross sections for optically
`allowed transitions, for which AI = :1, A] = :1 with
`the restriction J = 0—> J = 0, and forbidden transitions.
`respectively.
`A lack of experimental values and a need to employ
`a coherent data system require the use of some analytical
`expressions proposed for the cross sections referring to
`the electron—atom inelastic collisions considered (see
`e.g. Drawin 1967. Biberrnan er al 1982, van der Sijde er
`al 1984, Kirnura er al 1985). Of these, we have preferred
`the semi-empirical formulae of Drawin (1967), because
`when their scaling parameters are well chosen,
`they
`agree well with the experimental data available for
`argon in literature. Moreover. Drawin’s parameters for
`numerous optically allowed and parity-forbidden tran-
`sitions between excited levels in argon have been cal-
`culated recently (Kimura er‘ al 1985).
`We have utilised the following formulae for 0;; and
`ajfm, in (20):
`
`Jinn = 4’Ta5
`
`)
`“«5mrz"
`
`fmna/inn
`
`~'
`
`><1n<1.2s 5......
`
`(21)
`
`and
`
`Of
`
`05.» E 05/.» = 4«T0fs<1’5m Cféiil " Uvéi)
`
`(33.
`
`Gin E Gin = 4«‘m§6¥§m U;.3(1 " U513.
`
`(33)
`
`is the ionisation energy for
`51;‘
`Here. U,,,,, = 5,/z-:,,,,,
`atomic hydrogen in the ground state, ac, the first Bohr
`radius of the H atom, f,,_,, is the osciilator strength for
`electric dipole transition,
`o:,_,,,, and 5,,” are transition-
`dependerit parameters, P and S symbolise parity- and
`spin—forbidden transitions,
`respectively. As will be
`shown later, only transitions between the first four
`excited levels are not described by Drawin's formulae.
`The scaling parameters relating to the cross sections
`recommended by us for the excitation by electrons from
`the ground state are given in table 1. We have taken
`into account all transitions studied experimentally by
`Chutjian and Cartwright (1981) and also those found to
`be important by Peterson and Allen (1972). who am»
`lysed the electron impact energy-loss spectra obtained
`for argon.
`As can be seen in table 1, only one term is used in
`(20) for all effective levels regardless of the fact that the
`effective levels denoted by n = 12, 15, 16, 20, 21, 26,
`27 and 33 include both the actual states related to the
`
`ground state by allowed transitions and those related
`by forbidden transitions.
`The evaluation of the cross sections for the excited
`
`In the argon atom model used (see table 1) almost all
`the effective excited levels consist of several actual
`
`states 3d[1_/2] 1, 5s’ [1,-"'2]=,, 3d[3/2]1 and 5s[3/2]1 by means
`of the formulae derived by Peterson and Allen (1972)
`
`627
`
`TSMC-1211 I Page 5 of 9
`
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`
`
`
`J Vléek
`
`and the comparison of the effective cross section depen-
`dence (Chutjian and Cartwright 1981) for the levels
`denoted by n = 12, 15 and 16 with the course of the
`functions (21)—(23) have proved that excitation to the
`effective level with /2 =12 can be described by (23)
`while (21) must be used for the effective levels denoted
`by n = 15 and 16. However, a radiative de-excitation of
`the level with n = 12 characterised by the corresponding
`transition probability is taken into account.
`All optically allowed transitions relating to the
`ground state for which the values of the oscillator
`strength are known (Lee and Lu 1973) are considered
`in our model.
`In the case of the effective levels with n = 20. 21,
`
`26. 27 and 33 lying above the last state (3d'[3/2]1)
`studied experimentally (Chutjian and Cartwright 1981)
`we have utilised the semi-empirical formulae derived
`by Peterson and Allen (1972) for the corresponding
`composite cross sections. Using the oscillator strengths
`(Lee and Lu 1973) for the individual resonance states
`included, we have obtained cross sections for our effec-
`tive levels. The parameters of; have been determined
`at ,8” = 1.00 from comparison of the values of these
`cross sections with the values given by (21) for .9=
`'20 eV.
`No data are available in the literature for the exci-
`tation cross sections of the states 5p and 5p’ which have
`been identified as the last p states in low-energy loss
`spectra of argon (Peterson and Allen 1972). The coef-
`ficients a"; for these effective levels have been obtained
`in the following way. First, the sum of all corresponding
`excitation cross sections recommended by us was sub-
`tracted from the excitation cross section determined
`by Ferreira and Loureiro (1983) for all higher-lying
`optically allowed and forbidden levels (45 and 45' levels
`not included). Assuming the relation between the exci-
`tation cross sections for 5p and Sp’ to be the same at
`5 = 20 eV as in the case of 4p and 4p’ levels, both scaling
`parameters a"f,, have been evaluated at this electron
`energy.
`The electron excitation cross sections proposed for
`the first four excited levels are shown in figure 1 along
`with the experimental results of Chutjian and Cart-
`wright (1981). Our cross sections are in reasonable
`agreement with those (Ferreira and Loureiro 1983,
`Tachibana 1986) determined by making some adjust-
`ments to the experimental cross section data of Chutjian
`and Cartwright (1981) to fit the calculated and measured
`electron swarm parameters.
`The total cross section am for the collisional exci-
`tation by electrons from the ground state is given by
`the sum of all cross sections indicated in table 1. The
`following simplified formula has been used for this cross
`section in (11):
`ill
`
`U€XC =
`
`n=.'l
`
`Oln + 0l.4p 4' ULH
`
`where the cross sections 01,, are characterised in table
`1. amp and 0”, denote the excitation cross sections for
`all 4p and 4p’ states and for all higher-lying levels
`
`628
`
`1.5;
`
`
`
`E ie‘»’l
`
`Figure 1. The electron excitation cross sections for the first
`four excited levels: curve A, 4s[3/212: Curve B, 4s[3/211:
`curve C, 4s'[1/2J0; curve D, 4s’[1/2],. Points and error
`bars, experiment (Chutjian and Cartwright 1981); full
`curves, present work. Also shown is the reduced cross
`section a, X 55 for the ionisation of the ground state by
`electrons: points and error bars, experiment (Rapp and
`Englander-Golden 1965): full curve, present work (§ 3.1.).
`
`considered, respectively, and are expressed by (22) with
`the corresponding threshold energies em}, = 12.907 eV
`1
`and 5-H = 13.884 eV and with the scaling parameters
`a§__.p = 0.234 and ar§’_~,, = 0.480.
`
`‘
`
`
`
`2
`lz
`
`U\
`
`n
`
`E
`
`
`
`1,,,L_._.__.l.-11,1.1,
`
`Figure 2. The total electron excitation cross section for
`argon. Semi-empirical formulae: full curve, present work;
`broken curve, Puech and Torchin (1986); chain curve,
`Eggarter (1975). Deductions from the first Townsend
`coefficient: dotted curve, Ferreira and Loureiro (1983);
`points, Specht er al 1980 (§ 3.1.).
`
`TSMC-1211 I Page 6 of 9
`
`TSMC-1211 / Page 6 of 9
`
`
`
`In figure 2. the total excitation cross section used by
`us is compared with those calculated from a set of
`individual or composite cross sections (Eggarter 1975 .
`Puech and Torchin 1986) or deduced from the analysis
`of the first Townsend coefficient measured in swarm
`experiments (Specht er al’ 1980, Ferreira and Loureiro
`1983).
`The scaling parameters crfifi,‘ at BM = 1.00 and aim
`characterising the cross sections for optically allowed
`transitions from the mth to the nth level. 2 S m <
`n s 45 , and parity forbidden transitions where 2 S m <
`rt s 47 with the exception of the transitions between the
`first four excited levels, respectively, are based on the
`computations of Kimura er al (1985). These authors
`calculated Drav»-'in"s parameters for transitions between
`24 multiplets in two groups referring to the atomic
`core terms 213,; and 2133,’: of argon. Average oscillator
`strengths needed for the determination of the cross
`sections for allowed transitions have been calculated
`(Kimura er a1 1985) on the basis of the (j, K) coupling.
`which excludes the intercombination transitions.
`However, the actual coupling of the rare gases is a
`mixture of (LS) and (j,K)-coupling, termed inter-
`mediate coupling (Drawin and Katsonis 1976, Lilly
`1976, Katsonis and Drawin 1980). By this coupling a
`large part of the oscillator strength set used by us has
`been determined for the transitions from the 4s and 45’
`states (Drawin and Katsonis 1976, Katsonis and Drawin
`1980) and for those between the other excited effective
`levels (Drawin and Katsonis 1976).
`In calculating the parameters mi,“ from the average
`excitation cross sections C7_.x,,1_\,' obtained by Kimura er al
`(1985) for the transitions between the multiplets M and
`we have assumed that the cross sections for the
`various Mm-N,, processes cm are proportional to the
`oscillator strengths for these transitions _f,,_,:. i.e.. om =
`aMNf,,,,,/2,,f,,,_,, When the oscillator strengths were cal-
`culated on the basis of the intermediate coupling
`scheme, the cross section U;-M used by us was given by
`averaging the two cross sections determined by Kimura
`er al (1985) for the corresponding rnultiplets related to
`the core configurations 2P1); and IP33.
`In the case of optically forbidden transitions between
`the first four excited levels, we used the cross sections
`proposed by Baranov er al (1981) for the collisional de-
`excitation by electrons
`from the radiative states
`4s[3_/2], and 4s'[1/'2];
`to the corresponding adjacent
`metastable levels 4s[3/2}; and 4s’[1/Zlu, respectively.
`Owing to the lack of additional information for argon
`in the literature, all intercombination transitions are
`described by the analytical formulae of the same type
`as for the cross section 033(8) but multiplied by a factor
`of 0.1 in a manner similar to the relation between the
`‘average cross sections for de-excitation by electrons
`from the state 4s[3_/2]1 to 4s[3/2]: and from the state
`4s’[1_,.i2]_r, to 4s[3/'2]; in neon (Phelps 1959. Ferreira er al
`1985).
`In the case of the allowed and forbidden transitions
`between the effective levels with 45 $ m < n s 65 and
`
`47 s m < n S 65. respectively, we assume that ozfin =
`
`Collisional-radiative model for argon discharges, l
`
`1.00, fim = 1.00 and a’,§n_,, = aim = 0 (Katsonis 1976).
`respectively in (_21)—('-.23).
`.
`The cross sections for the ionisation by electrons.
`occurring in (7) and (10). can be written in the form
`(Drawin 1967)
`
`0n = 4145-(Ei‘/€n)2«§n€¥n
`
`U,T3(.U}. ""1)1I1(1-25 l3r.U.1)
`
`is the number of energetically
`L3,,
`where U}, = 5/8,,.
`equivalent electrons in shell n and as, and 16” are level-
`dependent parameters.
`Figure 1 shows the cross section for the ionisation
`of the ground state agreeing well at ch = 0.51 and [31 =
`1.00 with the experimental
`results of Rapp and
`Englander—Golden (1965), which are consistent with the
`measurements reported by Kurepa er al (1974) and
`Stephan er al (1980).
`The ionisation cross sections for the first ten excited
`levels are given in figure 3. The following scaling par-
`ameters lead to reasonable agreement between our cross
`sections and those obtained by Hyman (1979) for the
`4s and 4p states: oz,, = 0.35 for 2 s n s 5; a,, = 0.45 for
`72 = 6; 04, = 0.39for7 $ n S 9; an = 0.32 forn =10.,11:
`,8” = 4.00 in all these cases.
`For higher levels there are no data for argon in the
`literature. Hence we have used ct" = 0.67 and fin = 1.00
`in accordance with Katsonis (1976).
`
`3.2. Cross sections for atom—atom inelastic collisions
`
`In the lower energy range. few experimental data on
`atom—atom inelastic collisions are available for argon.
`Harwell and Iahn (1964), McLaren and Hobson (1968).
`Haugsjaa and Amine ( 1970) measured the cross section
`for excitation from the ground level to the 45 states.
`Haugsjaa and Arnrne (1970) also studied the ionisation
`of the ground state experimentally.
`In the case of the optically allowed transitions.
`
`0l10"‘°
`
`(mil
`
`o
`
`5
`
`at
`
`'75
`c lav‘;
`
`2E
`
`'
`
`30”’!
`
`Figure 3. The electron ionisation cross sections for the 4s
`(region A and curve A’) and 4;: states (region B and curve
`B’). The hatched regions correspond to the scattering in
`individual cross sections used by us, while the broken
`curves represent the average cross sections calculated by
`Hyman (1979) (§ 3.1 .).
`
`629
`
`TSMC-1211 I Page 7 of 9
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`
`
`
`J Vléek
`
`
`
`
`
`5.," a’eV)
`
`Figure 4. The b,,,,, coefficient as a function of the threshold
`energy e,,.,,,. The points represent the values calculated by
`the Drawin formula for atom-atom excitation corresponding
`to optically allowed transitions (§ 3.2). Experimental data:
`vertical line with arrows, Harwell and Jahn (1964); open
`circle, McLaren and Hobson (1968); triangle, Haugsjaa and
`Amme (1970).
`
`Drawin and Emard (1973) proposed the following for-
`mula for the excitation cross section:
`
`. 5” “me
`Qmn = 4,105 (figfljmii mg 5“; nm
`/'
`Zmc
`.
`" -3
`><(W»m“1)l1+;n‘‘:;1T(Wmn'1))
`*
`‘
`-'
`e
`.
`r
`
`(24)
`
`where 1‘/,,,,, = E/s,,,,, and mg and mm are the masses of
`hydrogen and argon atoms. respectively. For ionisation,
`(24) is valid if we put f,,,,, = 1.00 (Drawin and Emard
`1973).
`Near the threshold, (24) is in good agreement with
`the corresponding relations (16) and (17).
`The values of the coefficients b,,,,, and b,, appearing
`in formulae (12) and (13), respectively. have been deter-
`mined according to Bacri and Gomés (1978).
`In figure 4 we have plotted the values of b,,,,, as a
`function of the threshold energy em obtained from (24)
`for all optically allowed transitions for which 1 S m S 30
`and m < :1 s m + 10. As a first approximation, the fol-
`lowing formula for b,_,m has been used with respect
`to the experimental values mentioned above for all
`transitions except those between the lowest four excited
`levels:
`
`bm = 8.69 x 10“‘5e;§;~36.
`
`(25
`
`In determining the constant factors referring to the
`transitions between the 4s and 45' states, which replace
`the value of 8.69 X 10'” in equation (25). we have
`utilised the analysis of the decay rates for the metastable
`states 4s[3/2]; and 4s’[1_/"2]Q carried out by Ellis and
`Twiddy (1969).
`Assuming these factors to be the same for both
`transitions with no change in the core quantum number
`f: and for all intercornbination transitions, and having
`used the two-body rate constants of Tachibana (1986)
`
`630
`
`states, we obtain values of
`the metastable
`for
`1.79 X 10”“ and 4.80 X 10"? respectively.
`
`3.3. Cross sections for photo-ionisation
`
`For the photo-ionisation cross sections afl. appearing
`in (9). we have employed the formulae proposed by
`Katsonis (1976) and modified them slightly in the case
`of the 4s and 43' states.
`
`Experimental (Samson 1966, Hudson and Kieffer
`1971) and theoretical results (Amusia er al 1971, Ken-
`nedy and Manson 1972, Chapman and Henry 1972) for
`the cross section of have been reasonably approximated
`(Katsonis 1976) by the following relations expressing
`the cross section of in cm}, as is done in all the equations
`given for 0,‘; below,
`
`’3.5><1O"7
`
`53 shuszett
`
`OP =
`1
`
`,»6n..
`2s><10'16(-1-)
`flu,
`
`hu>2e§*.
`
`in the case of the 4s and 45' states. i.e. when 2sm<5_.
`we use
`
`P
`
`am )2 >< 10*”z>
`£7.91 >< 10~r3ym(npq,,.[)
`
`5.5 s hu < 0.595?
`
`hi» >0.59£:§I.
`
`Here. 545