J. Phys. D: Appl. Phys. 22 (1989) 623-631. Printed in the UK
`
`
`
`J Vléek
`
`Department of Physics, institute of Mechanical and Electrical Engineering,
`306 14 Plzeh, Neiedlého sady 14‘ Czechoslovakia
`
`Received 12 July t988, 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 ot cross
`sections for excitation by electrons trom 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-Maxweliian electron distribution) by a set of parameters, such as the electron
`kinetic temperature T9, the atom temperature T2, the ion temperature T,, the
`electron number density he, the ground state atom population n1, the discharge
`tube (or the plasma column) radius Fl and the optical escape factors Am and Am,
`which are dependent only on the quantities Ta, n1 and R in many cases of
`practical interest.
`
`1. Introduction
`
`One of the simplifying assumptions made in basic
`equations describing the extensive collisionai~radiative
`(CR) models (see e.g. Bates et al 1962, Drawin and
`Erhard 1977, Fujimoto 1979, Biberman er of 1982, van
`der Sijde er a! 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 (Vléeli 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 er a1
`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 (Vicek 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 Ti, the electron number density rig
`and the ground state atom population m, 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 reflects 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
`
`INTEL 1211
`
`INTEL 1211
`
`

`

`J Vléek
`
`the measurements of Chutjian and Cartwright (1981)
`and the computations of Kimura er a! (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 1",, n1 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-
`atmospheric 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 ArT in the ground state with the
`number density )1+ = ”e-
`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 2PM and 3R”, 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 metastables,
`are considered in our extended version of the CR model:
`Cum
`
`(i) Ar(m) + e r: Ar(n) + e
`
`. Kflm
`(ii) Ar(m) + Ar( 1) 1.: Ar(n) + Ar(l)
`5,"
`n
`(iii) Ar(m) + e 0: Ar“ + e + e
`V?”
`
`l m
`(iv) Ar(m) + Ar(1) 1/: Ar‘ + e + Ar(l)
`UTA/unyimn
`
`(v) Ar(m) + hum if:
`(:_Amanz
`
`Ar(n)
`
`(vi) Ar(m)+hz/ R: Ar‘ +e
`8?"
`(vii) Ar(m) 4- Ar(1)+ Ar(1)—> Ar; + Ar(l)
`where m = 2 and 4.
`
`624
`
`(viii) diffusion of the metastables 4s[3/2]3 and 4s’[1/2}0
`(for which :2 =2 and 4 in table 1, respectively) to
`the wall.
`
`Here, Cm and Km are the rate coefficients for col-
`lisional excitation by electrons and by ground state
`atoms respectively, Fm and Lmn are respectively the
`rate coefficients for the inverse processes (collisional de—
`excitations). Sm and Vm are the corresponding collisional
`ionisation rate coefficients while Om and Wm 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. Bm 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 et al 1962, Cacciatore er a] 1976, Biberman et a1
`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 al
`1976, Ferreira er al 1985), we obtain a set of coupled
`linear equations
`6:
`
`2 amnnrt = _ 6m _ amlnl
`n=2
`
`(1)
`
`, 65, from which the unknown excited
`.
`.
`where m = 2. .
`level populations 11,, may be calculated provided that
`the coefficients anm and 6m are known and the ground
`state
`atom population 211
`has been determined
`experimentally.
`The coefficients am" and 6m are related to the above-
`mentioned processes (i)—(viii) by the expressions
`
`amn =neCmn+anmn
`
`am»: =nean+nlen+AmnAmn
`65
`
`m>n
`
`m<n
`
`(2)
`
`(3)
`
`arm 2 _(neSrt +ann + 2 am?! + DH/Az ‘5‘”?8”)
`m=1
`
`m=n
`
`(4)
`
`(5)
`6m =nen+(ne0m+nlwm +AmRm)'
`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 0,, in argon is the
`same as in neon (Phelps 1959), the diffusion term in (4)
`can be rewritten in the form (1,, Tg‘jS/IHIRZ, where d" is
`a constant given by the experimental data for D,,nI
`(Tachibana 1986). The values of B,I are also taken from
`Tachibana (1986).
`The expressions for the rate coefficients referring to
`the inelastic collisions under consideration are obtained
`
`

`

`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 excitation by electrons from the ground state. A, optically allowed: P,
`parity-forbidden; S, spin-forbidden transitions.W
`Excitation
`_
`Transition-dependent
`Statistical
`energy
`Level
`a
`parameters
`Nature of
`weight
`81,,
`Designation
`number
`
`
`
`
`
`n 31/; = 0"}an and B”. or aim, or a?”Wnpqnl [Kb (eV) 9,. transition
`
`356
`45 3.12
`4513.121?
`45115210
`4511.211
`4p[1f2],
`4p[3.»"2].,2, [5.12123
`4013/2112
`4p'[1 .12].
`4p[1 5210
`4p"[1/2]0
`3511,1210,“ [31212
`301732134
`3d'[23/2}2, [512123
`53‘
`sate-2]., [5.12123 + 55
`acne/21.
`Sp
`5p’
`4d + 65
`4d' + 65’
`4f’
`4f
`6p’
`6p
`5d’ 4- 73
`5d + 73
`51’, 9’
`St, 9
`7p’
`Yr:
`6d’ + 88'
`60 + 85
`6f’, 9’, h'
`St. 9 h
`Bp’
`8p
`7d’ + 98'
`7d + 93
`7f’, 9’, h', 1’
`71, g, h, l
`8d', f’
`.
`8d, 1,
`9p’, 0’, f’,
`9p, 0 f,
`.
`n
`_ 10
`W” "
`
`.
`
`.
`
`.
`
`1
`’3
`3
`5
`5
`7
`3
`9
`10
`11
`12
`13
`14
`15
`16
`17
`18
`19
`20
`21
`22
`23
`24
`25
`26
`27
`28
`29
`30
`31
`32
`33
`34
`35
`36
`37
`38
`39
`40
`41
`42
`43
`44
`45
`46
`47
`
`48
`49
`
`50
`51
`
`52
`53
`
`54
`55
`
`56
`57
`
`58
`59
`
`60
`51
`
`62
`63
`
`64
`65
`
`_ 11
`”W” ‘
`
`_ 12
`n
`”‘7” '_
`
`_ 13
`”W" ‘
`
`__ 14
`n
`”‘7” —
`
`_ 1-
`”W" _ °
`
`_ 16
`n
`”q" _
`
`_ 17
`rt
`Pg” '
`
`_ 18
`n
`9‘?” '
`
`_ 19
`n
`W” "
`
`0.000
`11.548
`11.624
`H.123,
`11.828
`12.907
`13.115
`13.295
`13.328
`13.273
`13.450
`13.884
`13994
`14.29
`14.252
`14.090
`14.304
`14.509
`14.690
`14.792
`14.976
`15.083
`14.906
`15.205
`15.028
`15.824
`15.153
`15.393
`15.215
`15.461
`15.282
`15.520
`15.347
`15.560
`15.382
`15.600
`15.423
`15.636
`15.460
`15.659
`15.482
`15.725
`15.548
`15.769
`15.592
`15.801
`15.624
`
`15.825
`15.343
`
`15.843
`15.666
`
`15.857
`15.680
`
`15.868
`15.691
`
`15.877
`15.700
`
`15.884
`15.707
`
`15.890
`15.713
`
`15.895
`15.718
`
`15.899
`15.722
`
`1
`5
`3
`1
`3
`3
`20
`8
`3
`1
`1
`9
`16
`17
`4
`23
`3
`24
`12
`48
`24
`28
`56
`12
`24
`24
`48
`64
`128
`12
`24
`24
`48
`108
`216
`12
`24
`24
`48
`160
`320
`240
`480
`320
`640
`400
`800
`
`484
`968
`
`576
`1152
`
`676
`1352
`
`784
`1568
`
`900
`1800
`
`1024
`2048
`
`1156
`2312
`
`1296
`2592
`
`1444
`2888
`
`;
`A
`S
`A
`P
`P
`P
`P
`P
`P
`s
`s
`P
`A
`A
`A
`P
`P
`A
`A
`—
`—
`-—
`——
`A
`A
`—
`——
`—
`——
`-~
`A
`-—
`—-
`—-
`——
`——
`—
`—
`-—
`-—
`—
`—-
`—
`_
`——
`
`——
`_
`
`—
`—
`
`——
`_
`
`_
`—.
`
`—
`..
`
`_
`——
`
`—.
`—
`
`—
`—
`
`—.
`——
`
`1o 2
`_70
`‘
`6.
`X
`1.92 x 104. 4.00
`9.50 x 10‘3
`4.62 X 10*, 4.00
`3.50 x 10-2
`1.15 x 10-1
`3.50 x 10-2
`7.00 x 10-3
`7.00 x 10—3
`3.50 x 10-2
`1.50 x 10-1
`9.00 x 10'2
`4.20 x 10-2
`3.71 X 10‘3, 4.00
`3.33 x 10-2, 4.00
`1.79 x 10*? 2.00
`7.00 X 10’2
`5.00 X 10‘2
`5.15 X 10‘2. 1.00
`3.06 X 10‘2, 1.00
`-—
`——
`——
`—
`6.50 X 10‘“, 1.00
`3.69 X 10*, 1.00
`—
`.—
`.—
`——
`——
`2.40 X 10‘2, 1.00
`———
`—
`.—
`——
`—
`.—
`—
`—
`—
`—
`—~-
`—
`-
`_
`
`_
`_
`
`__
`_
`
`_
`_
`
`_
`__
`
`_
`__
`
`__
`__
`
`__
`_
`
`__
`_
`
`_
`__
`
`625
`
`.
`
`.
`
`

`

`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 functions
`The only component of the EEDF to appear in the
`rate coefficients for the electron—atom collisions in (2)—
`(5) is the isotropic function flu), where the dimen-
`sionless energy it = 5/ch (a 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:: (:17) j f(u)0nm(u)u du
`
`
`<6)
`
`(7)
`
`5, = 87: (Z) r f(u)on(u)udu
`
`
`
`Fm” = 8x (Seriflfx f(u—umn)om(u)udu
`
`
`87: ”1,, g
`.x
`’71
`R =—-—— ._
`-
`P
`2
`m
`(m) 2g+ i hm” EmJUmUW dv
`1
`3
`
`kn” m,
`h“ a:
`
`X J1 f u — um)0m(u)u du.
`
`('8')
`
`(9)
`
`(10)
`
`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, Omar) and 0mm) 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 am(u)
`and 05,0») 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)
`
`df(u)
`d
`/
`avid) d” + G(u)f(u)) =n1uM(u)f(u)
`where
`
`(11)
`
`M(u) = O
`
`(”(11) = aexeo") + 010‘)
`
`usuu
`
`u > L412.
`
`The formulae for the total excitation cross section
`anew) and the ionisation cross section 01(u) will be
`characterised later.
`
`In (11) the terms H(u) and C(14) describe the influ—
`ence of the electron energy gain and loss processes,
`respectively. on the EEDF. [1(a) includes the expressions
`corresponding to the Coulomb electron—electron and
`electron—ion interactions, the effect of the axial electric
`fieid and the thermal motion of gas atoms. 6(a) is
`given by the sum of the terms related to the Coulomb
`
`626
`
`electron—electron and electron~ion interactions and to
`the elastic electron—atom collisions. The EEDF is nor-
`malised, so that
`,
`
`2:! (2kTe )me ,
`I ul’m‘flu) du =1.
`0
`The rate coefficients for the atom—atom collisions
`
`3/2
`
`(ii) and (iv) appearing in equations (2)—(5) are given
`by the following expressions (Bacri and Gomés 1978.
`Biberman er al 1982, Collins 1967):
`
`
`
`= 2 <L::::;>
`
`
`
`V” = 2 (27:31) L52 ms, +2kTa) eXP ( — 13,)
`
`
`
`(a)
`‘
`
`
`(12)
`
`(13)
`
`(14)
`
`“5)
`
`Ln" = Km (gm/g”) CXPC€mn/kTa)
`
`W = Vm if (5:531)
`
`6"" (577:)
`
`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), QM, and Q”, respectively, are linear
`functions of energy above the threshold, so that
`
`and
`
`an = bmn(E _ Emu)
`
`Qn =bn(-E_—8n)
`
`(16)
`
`(.17)
`
`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 AW, and Am. This method, usually used in exten-
`sive CR models (see eg Bates et a1 1962, Drawin and
`Emard 1977, Biberman et a1 1982, van der Sijde et a1
`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 Hieftje 1984) holds:
`
`Aln =gonR)
`
`(.13)
`
`Where g0 :19 for a Doppler—broadened profile and
`g0 = 1.3 for one dominated by pressure broadening; the
`
`

`

`transmission coefficient T(R) is given by
`
`T(R) 2 TD exp(—:rT%;D,/4T§;)
`
`+ TC erf(it‘"2 TCDfZTC ).
`
`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
`
`To = {kuRlz ln(kOR)]1/2}_~,
`
`Tc = (Whig/€013)“:
`
`TCD = Za/iivff[ln(kflR)]l;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 e.g.
`Abramowitz and Stegun 1964) is evaluated with the
`help of the realistic approximation inferred by Hastings
`(1955).
`Note that the formulae for Km,z and V” must be
`multiplied by the factor it when n = 1 in (2) and (4),
`respectively (see e.g. Mitchner and Kruger 1973). Fur-
`thermore, according to Katsonis (1976), the statistical
`weight gm is equal to l or ‘3‘ for m = 1 in (8) and (14)
`when n is the number of the level with jc equal to 9; or
`4%, respectively. Similarly, g+ is equal to Z or 4 for m 2 2
`in (9), (10) and (15) and 72+ is equal to in, or 'éirze for
`m 2 2 in (5) when m denotes a level corresponding to
`j: = é or jC = 5, 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"!
`n” = rig) + Gillnl
`
`n % 2,... , 65
`
`(19)
`
`where the population coefficients 22,293 and Cl,” are
`obtained from equations (1) when we insert
`111 = 0
`or rt1 = 1 and 5m = 0, respectively, in their right-hand
`sides.
`'
`
`_ The numerical method developed allows us to cal—
`culate the population coefficients 71;?) and GS,” as func-
`tions of the following input parameters: Te, Ta, T3, as,
`m, R and of the escape factors A”, and Am, which are
`given by T3, In 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 (VlCek 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 tor argon discharges, 1
`
`energy states. Therefore the effective cross sections for
`excitation by electrons, occurring in (6) and (8), can
`generally be written in the form
`
`Ohm = US?" + 05m
`
`(20)
`
`where 0-3,” and of," 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. Biberman et a1 1982, van der Sijde er
`a! 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 a}; and
`of“ in (20):
`
`'85 . 2
`a: = mi (——) was. Union. ~11
`\.5m,,/
`
`and
`
`or
`
`05.” 5 05.” = 4Iaga§nnbfhvtiz(1— 021:2)
`
`(22.
`
`at... E as. = was. Una — Us,
`
`(23)
`
`is the ionisation energy for
`1],,m = Siam, all
`Here.
`atomic hydrogen in the ground state, no the first Bohr
`radius of the H atom, fmn is the oscillator strength for
`electric dipole transition, 0.3,”, and 5",” are transition—
`dependent parameters, P and S symbolise parity- and
`spinvforbidden 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 ana—
`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, 55’ [152]}, 3d[3/2]1 and Ssl3/2h by means
`of the formulae derived by Peterson and Allen (1972)
`
`627
`
`

`

`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 (ZN-(23) have proved that excitation to the
`effective level with rz =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 01f), have been determined
`at [31,. = 1.00 from comparison of the values of these
`cross sections with the values given by (21) for 8:
`'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 «‘17,, 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 excl-
`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
`n
`section in (11):
`
`I1=~
`Uexc = Z 01;: + 01.41,) 4‘ ULH
`
`where the cross sections 01,, are characterised in table
`1. amp and a”; denote the excitation cross sections for
`all 4p and 4p’ states and for all higher~lying levels
`
`628
`
`
`as;
`
`
`
`is
`
`.5
`
`.. W 7'
`
`20
`E te‘v’l
`
`"is
`
`30
`
`Figure 1. The electron excitation cross sections for the first
`four excited levels: curve A, 4s[3/2]2; curve B, 4s[3/2]1;
`curve C, 4s'[i/2lo; curve D, 4s’l1 i2]1. Points and error
`bars, experiment (Chutjian and Cartwright 1981); full
`curves, present work. Also shown is the reduced cross
`section 01 x 2—}; 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 51.4}: = 12.907 eV
`1
`and 55H = 13.884 eV and with the scaling parameters
`I afiip = 0.234 and of.“ = 0.480.
`
`
`
`1.. .1,ch..“ 2
`
`12
`
`‘:u
`
`eie‘v’i
`
`
`
`,,cl.c1.11...
`
`U\
`
`n
`
`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 a/ 1980 (§ 3.1.).
`
`

`

`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 a! 1980, Ferreira and Loureiro
`1983).
`The scaling parameters a?“ at 13m = 1.00 and aim
`characterising the cross sections for optically allowed
`transitions from the mth to the nth leveli 2 S m <
`n S 45 , and parity forbidden transitions where 2 S m <
`n 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 Drawin’s parameters for transitions between
`24 multiplets in two groups referring to the atomic
`core terms 2P1; and 2133/2 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 4s’
`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 aim from the average
`excitation cross sections 0M;- obtained by Kimura er al
`(1985) for the transitions between the multiplets M and
`IV we have assumed that the cross sections for the
`various Mm-NR processes 0m are proportional to the
`oscillator strengths for these transitions fm. i.e.. Um =
`aM‘vfmn/anm. When the oscillator strengths were cal-
`culated on the basis of the intermediate coupling
`scheme, the cross section aim.- used by us was given by
`averaging the two cross sections determined by Ki'mura
`et at (1985) for the corresponding multiplets related to
`the core configurations 2PM and 3P3);
`In the case of optically forbidden transitions between
`the first four excited levels, we used the cross sections
`proposed by Baranov er a! (1981) for the collisional de-
`excitation by electrons
`from the radiative states
`4s[3_;"2]1 and c-ts'[1‘/2]2 to the corresponding adjacent
`metastable levels 443/2}; and 4s’[1;/2]0, 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 tie-excitation by electrons
`from the state éls[3f2]1 to -—’ls[3,/2]z and from the state
`4s’[1;"2]9 to “[3!"th in neon (Phelps 1959‘ Ferreira er a1
`1985).
`In the case of the allowed and forbidden transitions
`between the effective levels with 45 S m < n S 65 and
`
`47 S m < n S 65, respectively, we assume that egg" =
`
`Collisional-radiative model for argon discharges, l
`
`1.00, [3m = 1.00 and 01;“ = 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 = 4Jag(£li/j5n)Z‘EnanU;2(_Un “- 1) 1110-25 11.11”)
`
`is the number of energetically
`15”
`where UH = 5/8”,
`equivalent electrons in shell n and or” and 16” are level-
`dependent parameters.
`Figure 1 shows the cross section for the ionisation
`of the ground state agreeing well at a1 = 0.51 and [31 =
`1.00 with the experimental
`results of Rapp and
`EnglandersGolden (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: or” = 0.35 for 2 S n S 5; an = 0.45 for
`n = 6; an = 0.39f0r7 S 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 a" = 0.67 and 3,, = 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 Jahn (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 Amme (1970) also studied the ionisation
`of the ground state experimentally.
`In the case of the optically allowed transitions.
`
`zof—
`
`_
`
`I
`
`a110"“
`
`(m2) '5
`
`{ec‘v“;
`
`25
`
`27-
`
`30
`
`Figure 3. The electron ionisation cross sections for the 43
`(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 f),
`
`629
`
`

`

`J Vicek
`
`
`
`
`
`
`
`a.“ ieV)
`
`Figure 4. The bmn coefficient as a function of the threshold
`energy em". 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
`Amine (1970).
`
`Drawin and Emard (1973) proposed the following for-
`mula for the excitation cross section:
` H 3
`’1
`
`eri = 41(1fj(
`Eur
`E]
`) mA, t1
`2m,
`\Smm» mH
`me+mAr
`
`rrm
`
`_’7
`
`"'
`2mC
`,
`"
`<24)
`>< (Wm 1) (1 + m, +mi, (Wm 1))
`where Wm = 5/5"", and mi and mAr are the masses of
`hydrogen and argon atoms. respectively, For ionisation,
`(24) is valid if we put fm = 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 bnm and [9,, 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 < n s m + 10. As a first approximation, the fol-
`lowing formula for bmn has been used with respect
`to the experimental values mentioned above for all
`transitions except those between the lowest four excited
`levels:
`
`17,,m = 8.69 x 10‘155;,§;35.
`
`(25
`
`In determining the constant factors referring to the
`transitions between the 4s and 45' states, which replace
`the value of 8.69 x 10’18 in equation (25), we have
`utilised the analysis of the decay rates for the metastable
`states 4s[3/2]3 and 4s’[1/"2]0 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
`fC and for all intercombination 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‘20 and 4.80 X 10—22, respectively.
`
`3.3. Cross sections for photo-ionisation
`
`For the photo-ionisation cross sections 0;. 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 a1 1971, Ken-
`nedy and Manson 1972, Chapman and Henry 1972) for
`the cross section a}? have been reasonably approximated
`(Katsonis 1976) by the following relations expressing
`the cross section of in cmz, as is done in all the equations
`given for of}, below,
`
`P ._
`01—
`
`Issue-17
`
`x H-
`51
`28X10'16(Z—)
`\ V,
`
`s
`
`ash/s25?
`
`hu>2ett
`
`In the case of the 4s and 45' states. i».e. when 2Sm<5,
`we use
`
`0:1:
`
`,_
`
`q
`
`[2 x 10‘1’371ml'npqn,l)
`(.91, milil’mquwfilal
`
`"€45
`
`2‘5
`
`Isl-I
`
`5,5 sbys0595‘?
`(rt)hp>0.59EE=L-I.
`
`3
`
`Here, a}, is the mean ionisation energy of all 45 and 45'
`states and ym(npq,,. l) is the level-dependent parameter
`given by a coefficient for the separation of the hydro-
`genic photo‘ionisation cross sections according to the
`orbital quantum numbers (Burges