J. Phys. D: Appl. Phys. 24 (1991) 290-300. Printed in the UK
`
`Electron and heavy-particle kinetics in
`the low pressure oxygen positive
`
`column
`
`G Gousset'|', C M Ferreirazt, M Pinheiroi, P A Sat, M Touzeaut,
`M Viallet and J Loureirozt
`’rLaboratoire de Physique cles Gaz et des Plasmas (Associated with the CNFtS)
`Université de Paris-Sud, Centre d’Orsay, 91405 Orsay Cedex, France
`t Centro de Electrodinamica da Universidade Técnica de Lisboa (INIC), lnstituto
`Superior Técnioo, 1096 Lisboa Codex. Portugal
`
`Received 11 April 1990. in final form 2 October 1990
`
`Abstract. A kinetic model for the low-pressure oxygen positive column is
`presented and discussed. The model is based on the electron Boltzmann equation
`and the rate balance equations for the dominant heavy-particle species. which are
`solved simultaneously in order to take into account the coupling between the
`electron and the heavy-particle kinetics. The effects of vibrationally excited
`molecules, dissociated atoms and metastable states on the electron kinetics are
`analysed in detail. The predicted populations of O2(X3E), O2(a‘A), O(3P), and O’
`are shown to agree satisfactorily with previously reported measurements. A
`combination of this kinetic model with the continuity and transport eqriations tor the
`charged species e. O‘, and O; is shown to provide characteristics for the
`maintenance field that agree reasonably well with experiment.
`
`1. Introduction
`
`This paper is concerned with the analysis of a kinetic
`model for the most populated reactive species present
`in the classical positive column of oxygen at low press-
`ures, specifically O2(X32) and O2(a'A) molecules,
`O(3P) atoms and 0‘ ions. The present analysis is an
`extension of previous work in which firstly the species
`concentrations were measured by VUV absorption
`spectroscopy [1] andsecondly a simplified model was
`developed [2] using a selected number of reactions
`along with electron transport and collisional data
`derived from swarm experiments in O2[3]. Although
`reasonable agreement was obtained between theor-
`etical predictions and experiment,
`the observed dis-
`crepancies seemed, nevertheless, sufficiently important
`to justify further investigation.
`A significant improvement achieved in the present
`paper is a consistent treatment of both the electron and
`heavy particle reaction kinetics.
`In fact,
`the electron
`Boltzmann equation is solved here taking into account
`not only the excitation and ionization of O2(X) mol-
`ecules but also of O(3P) atoms, as well as the effects
`of electron superelastic collisions with metastables and
`with vibrationally excited molecules. Such processes,
`which are absent
`in a swarm experiment, can have
`a significant
`influence on the electron transport and
`collisional data under discharge conditions [4]. On the
`
`0022-3727/91/030290 + 11 $03.50 © 1991 IOP Publishing Ltd
`
`other hand, these electron data are necessary in order
`to predict the populations of the various species in the
`discharge. Therefore, the Boltzmann equation must be
`solved simultaneously with a system of rate balance
`equations for the various heavy particle species.
`A consistent treatment of this type is also necessary
`to improve currently available theories of positive col-
`umns in electronegative gases, as we have shown in
`a recent paper [5]. Indeed, we have found that the
`application of
`the theory to the oxygen positive
`column, using electron transport and collisional data
`from swarm experiments, failed to predict accurately
`the maintenance field for the positive column. A poss-
`ible explanation for this fact could be the changes in
`the electron energy distribution function caused by the
`various processes referred to above, as stated in [5].
`This fact provides additional motivation for the analysis
`carried out in the present work.
`In section 2 we present the Boltzmann analysis, and
`we investigate the effects caused by the presence of
`dissociated atoms and by superelastic collisions. These
`problems are discussed on a general basis by using
`the fractional concentrations of the various dominant
`
`In order to investigate the
`species as parameters.
`effects of vibrationally excited molecules, O2(X3E, u),
`a characteristic vibrational temperature is also used as
`a parameter.
`Section 3 is concerned with the heavy particle kin-
`etics, specifically discussing the main reactions that det-
`ermine populations of the principal species.
`
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`TSMC-1216
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`

`
`Kinetics in 02 positive column
`
`In section 4 the predicted populations as obtained
`by simultaneously solving the Boltzmann equation and
`the rate balance equations are compared to the exper-
`imental populations previously reported [1].
`In section 5 we combine the present kinetic model
`with the continuity and the transport equations for the
`charged species e, O‘, and 02+ discussed in [5] in order
`to re-examine the problem of the maintenance field for
`the oxygen positive column.
`Finally, in section 6 we present the principal con-
`clusions of this work.
`
`collisions of electrons with neutrals of type S; T3 is the
`gas temperature in Kelvin) and to rotational excitation
`of molecules. The latter process is treated here in the
`continuous approximation [6]; B = 1.792 X 10"‘ CV is
`the rotational constant for O2 and 00 = Srrqzafi/15,
`where an is the Bohr radius and q = 0.29 is the electric
`quadrupole moment in units of eafi.
`The operators J,_, on the ans of (1) represent the
`effects of inelastic and superelastic collisions of the
`electrons with the heavy species 3. The explicit form
`of these terms is
`
`2. Electron kinetics in the oxygen positive column
`
`2.1. Boltzmann equation
`
`J.-. = 2 6..-[(u + v.-.-)oJ..(u + v.-.-mu + V.-.->
`
`- uoi.(u)f(u)] + 2 5., nu — v.,-)o;,-(u — v.,-)
`l.j'
`
`The fractional concentration of dissociated atoms in
`
`X f(u — V.-1) " Mai,-(u)f(u)l
`
`(2)
`
`the oxygen positive column can reach very high values
`(-10%) under the conditions of interest here [1].
`Therefore, we must treat the problem of the electron
`kinetics in a mixture of O2 molecules and O atoms.
`Moreover, each of these species can be found in a
`variety of different quantum states
`such as,
`for
`example, O2(X 32, u), O2(a 'A), 02(b 12), O(3P),
`O(1S), O(‘D), etc. Let N denote the total gas density,
`NM and NA the total number densities of molecules
`and atoms, respectively, and N,,», with s = M or A the
`number density of particles of species s in the quantum
`state j. Let also 6 represent a fractional population,
`defined relatively to the total density N. Then, we
`obviously have
`
`2N,=NM+NA=N;2N,,»=N,;
`.r
`i
`
`E6,=1;6M+a,,=1;
`
`2 6:} = 65; E '55,)" =
`I
`SJ
`
`Keeping the above considerations in mind, we can
`write the homogeneous electron Boltzmann equation
`as derived from the classical two-term spherical har-
`monic expansion as follows:
`
`d i—“-<£>%<2a.eo.>
`77 3(E5,a,) N du
`. M.
`
`KT d
`
`xuz (f+ eg£)+4<5MBanuf]=EJ,,_,.
`
`(1)
`
`where f(u) is the electron energy distribution function
`(EEDF), normalized such that ff‘? f(u)u‘/2 du = 1, and
`u = mvz/2e is the electron energy expressed in elec-
`tronvolts.
`
`in
`The three terms on the LHS of (1) represent,
`order, the energy gain due to the applied field of inten-
`sity E and the energy losses due both to elastic col-
`lisions of the electrons with heavy particles of mass M,
`(0, denotes the momentum transfer cross section for
`
`where cr’,,- is the electron cross section for the excitation
`from state i to state j > i; V), is the energy threshold
`(in eV) for this process; and 0;,» is the cross section for
`the reverse (superelastic) process.
`As seen from equations (1) and (2), the electron
`kinetics is strongly coupled to the heavy particle kin-
`etics if and when the fractional concentrations of dis-
`sociated atoms and of excited molecules or atoms
`
`become important. This is precisely the case with the
`oxygen positive column since large relative populations
`of atoms and O2(a 1A) metastable molecules have been
`experimentally detected under such conditions [1].
`Incidentally,
`the concentrations of vibrationally
`excited molecules O;(X32‘., u) and metastable states
`O2(b‘2), O(‘ D), 0(‘ S) can also be sufficiently high
`so as to play a non—negligible role in the electron
`kinetics. For this reason we shall carry out below an
`analysis of the influence of the populations in various
`states on the electron kinetics. For the moment, this
`analysis will be performed using the fractional popu-
`lations as independent parameters. This procedure is
`instructive since it enables us to identify the most
`significant processes and to evaluate their effects. Once
`this goal
`is achieved, we will be able to construct a
`self—consistent kinetic model that couples the electron
`and the heavy particle kinetics together (section 4).
`
`2.2. Electron processes and cross section data
`
`The inelastic and superelastic processes taken into
`account in the present work are listed in table 1 along
`with the pertinent references on cross section data. The
`cross sections for excitation of O2()( 32, l _<. u S. 4) and
`O2 electronic states and for ionization from O2(X 32,
`v = 0) are the same as proposed by Phelps [3]. This
`set
`is a rather complete one and provides, when
`inserted into the Boltzmann equation, excellent agree-
`ment between theoretical and experimental electron
`swarm parameters. However, additional processes
`must be considered under discharge conditions. For
`example, we also included in the model vibrational re-
`excitation of O2(X, u) from ls U $4 to upper-lying
`
`291
`
`TSMC-1216 I Page 2 of
`
`TSMC-1216 / Page 2 of 11
`
`

`
`Table 1. Inelastic and superelastic collision processes considered in the
`Boltzmann equation, and corresponding references on cross section data.
`
`Reference
`
`Phelps [3]
`vi
`1|
`
`n nuuu
`
`Hall" and Trajmar [25]
`(see text)
`(see text)
`
`Electron processes
`
`Molecular oxygen
`(1) e + O2(X, V) :2 e + O2(X. w)
`(2) e + O2(X. v = 0) :2 e + O2(a‘A)
`(3) e + o2(x, v = o) 2 e + 02(1) 12)
`(4) e + O2(X. v = O) —> e + O2(4.5 eV)
`(5) e + o2(x. v = o) —> e + 02(6.0 eV)
`(6) e + o2(x, v = o) —> e + o2(5.4 eV)
`(7) e + O2(X, v = 0) —> e + O2(9.97 eV)
`(8) e+O2(X,v=0)—re+e+O{
`(9) e + o2(x, v = 0) ~+ e + oz(14.7 eV)
`(10) e + O2(a ‘A) =2 e + O2(b‘E)
`(11) e + O2(a‘A)-> e + e + O;
`(12) e + O2(b‘2)—> e + e + 02*
`
`Atomic oxygen
`
`(13) e + O(3F’) 4——* e + O(‘D)
`(14) e + O(3P) .2 e + o(‘s)
`Stone and Zipf [27]
`(15) e + 0(3P) -> e + o(3s)
`Henry er al [26]
`(16) e + O(‘D) ¢—_*e + O('S)
`Fits and Brackmann [28]
`(17) e+0(3P)—>e+ e+O*
`Drawin [7]
`(13) e+O(‘D)—>e+e+ 0+
`
`(19) e+0(‘S)——>e+e+O+
`
`Henry .9! al [26]
`
`G Gousset et al
`
`vibrational levels w S u + 4. The cross sections for such
`
`processes are unknown and they were assumed here to
`he
`the
`same
`as
`those
`for
`the
`transitions
`
`02(X, 0-) w — v) but with the threshold appropriately
`shifted due to the anharmonicity of the molecular
`vibration. The reverse processes, i.e. superelastic col-
`lisions producing vibrational de-excitation, were also
`taken into account. The model also includes the fol—
`
`superelastic de—excitation of O2(a IA) and
`lowing:
`O2(b’2) to the ground state as well as transitions
`between these states and their ionization (assuming the
`cross section to be the same as for ionization from
`
`the ground state but with the appropriate shift in the .
`threshold); excitation of the atomic states O(1D),
`O(‘S), O(3S) and ionization from ground state O(3P)
`atoms; ionization from O(‘D) and O(‘S) (using cross
`sections calculated according to Drawin’s formula [7]);
`transitions between O( 11)) and O(‘S); and superelastic
`de—excitation of both of these states to the ground state
`O(3P). All the cross sections for superelastic processes
`have been determined from those for the direct pro-
`cesses by detailed balancing.
`We note that the above list of electronic processes
`implies that the solutions to the Boltzmann equation
`depend on the fractional populations, 65.}, of the fol—
`lowing species: O2(X 32,
`O s. u s 8);
`02 (a ‘A);
`O2(b '2); O(3P); O(‘D); O('S). In practice, for lack
`of data we cannot take into account electronic exci-
`
`tation and ionization of 02 from O2(X 32, 0 > 0).
`Therefore, we have dealt with such processes as if they
`only occurred from the v = 0 level but assuming in this
`case that all the 0204’. 32) state pop-uiation is in that
`level. In other words, the distribution of O2(X 32) mol-
`ecules among various vibrational levels only was taken
`into account when dealing with vibrational excitation
`
`292
`
`or de-excitation processes within the ground electronic
`state. Since no attempts were made in this work to
`sh :_'t assi_i.rne
`model the vibrational kinetics of O1
`in the following that the vibrational distribution can
`be characterized by a vibrational temperature whose
`meaning is explained below.
`
`2.3. Vibrational distribution function of O2(X*"E, 0)
`molecules
`
`We represent the intermolecular potential by an anhar—
`BY
`monic Morse oscillator whose ener
`levels are given
`by
`
`5., = fiwel(v + t) — xctv + -“$21
`
`(3)
`
`where for O2(X32), we: 1580.19 cm" and wcxe=
`11.98 cm” [8,9]. We assume that the vibrational dis- -
`tribution function has the form proposed by Gordiets
`et al [10], namely
`
`NU:N.,exp{—v[KIi(,U~(v—1)f]}
`
`xcfl
`
`AE
`
`USE
`NU=NU..——
`v
`
`u*SuS.u**
`
`os_u*
`
`(4)
`
`(5)
`
`v > 0”. Here, Nu
`for
`and ND ~ exp(--AE),/KTE)
`denotes the number density of molecules in level
`:2;
`AEL0 = fiwc(l —2xC) is the energy difference between
`the levels 0 = 1 and v = 0; I9 = fiwe/K = 2270-73 K; Tv
`is the characteristic vibrational temperature; and 12* is
`the vibrational quantum number corresponding to the
`
`TSMC-1216 I Page 3 of
`
`TSMC-1216 / Page 3 of 11
`
`

`
`Kinetics in 02 positive column
`
`Table 2. List of cases considered in the parametric study of the solutions to the Boltzmann equation of
`section 2 (see main text for notation).
`
`TV
`
`0359
`(K)
`<51-.11x
`5Ma
`51116
`5A1:
`¢5Ao
`5A3
`
`0
`0
`0
`0
`0
`1
`300
`A
`0
`0
`0.10
`0
`0.18
`0.72
`300
`B
`0
`0
`0.10
`0
`0.13
`0.72
`2000
`C
`0
`0
`0.20
`0
`0.1 6
`0.64
`2000
`D
`1.0 x 10-5
`2.6 x 10-5
`7.0 x 10-2
`4.65 X 10-3
`4.65 x 1072
`0.379
`300
`E
`
`
`
`
`
`
`
`300 0.765 0.121 1.3 x 10-2 0.10 1.0 x 10"F 1.0 x 10-5
`
`minimum of the Treanor distribution [11} (equation
`(4)) given by
`
`0* ==;(1+ AEL0 Tg
`
`KTV XEB
`
`(6)
`
`For the conditions considered here (TV s 3000K;
`Tg < 700 K) only the Treanor-like region (equation
`(4)) and the plateau region (equation (5)) have any
`relevance in what concerns the effects on the electron
`
`kinetics. In the following we shall assume TE = 300 K
`and we shall investigate the effects of the parameter
`TV by considering two extreme situations: TV = 300 K
`and TV = 2000 K. However, the experimental values of
`TE will be used in the calculations of section 4.
`
`2.4. Electron transport parameters and rate coefficients
`
`The various combinations of independent parameters
`TV and 65,- considered in the present analysis are listed
`in table 2, where the subscripts to 6 refer to the fol-
`lowing molecular and atomic states: ém, O2(X3E);
`6M3: 02(3 115); 6Mbs 0207 ii); 6A1’: 0(3P); (SAD: OCD);
`6A5, O(1S). In all cases Tg = 300 K.
`Case A corresponds to the so-called molecular cold
`gas approximation in which the electrons can be
`assumed to collide only with ground state molecules,
`O2(X 32, u =- 0). This is the case considered by Masék
`er al [12] and Laska er al [13] in their previous analysis
`of the oxygen positive column, and also by Phelps [3]
`in his Boltzmann analysis of electron swarms in 02.
`We have checked that our Boltzmann code reproduces
`Phelps’s macroscopic data (transport parameters and
`rate coefficients) to within a few per cent, in this case.
`Case B is intended to analyse the effects caused by the
`presence of O(3P) atoms and by superclastic collisions
`with metastable O2(a IA) molecules, using realistic val-
`ues for the relative concentrations of these species [1].
`In case C we additionally consider
`the effects of
`vibrationally excited molecules, assuming a relatively
`high degree of vibrational excitation. Present exper-
`imental evidence indicates that TV ~ Tg (see below) so
`that case C should be regarded only as an extreme
`situation of vibrational excitation. Case D is also an
`
`extreme situation in what concerns the presence of
`dissociated atoms. Finally, cases E and F investigate
`
`
`
`i-I
`
`(GVJ
`
`Figure 1. Electron energy distribution function for E/N =
`3 >< 10“6Vcm2 (l) and 10“‘5Vcm2 (II), and for the
`following cases considered in table 2: A, full curve; C.
`broken curve; D, chain curve.
`
`the effects caused by collisions with the various mol-
`ecular and atomic metastable states, using realistic val-
`ues for their relative concentrations. The latter two
`
`cases should be regarded as the most realistic ones.
`Figure 1 shows the EEDF obtained in cases A, C
`and D, for E/N = 30 and 100 Td (1 Td =10"‘7Vcm2).
`We note that there is a small enhancement in the tail
`
`of the distribution in cases C and D compared with
`case A. This enhancement is principally caused by sup-
`erelastic collisions at low E/N and by the presence of
`O(3P) atoms at high E/N. The computed EEDF for case
`A can be compared to Langmuir probe measurements
`by Rundle er a1’ [14] in a discharge tube of 1.26 cm
`diameter
`for pressures 0.54 torr and currents 1-
`10 mA. Under such experimental conditions the c0n~
`centrations of atoms and excited molecules are indeed
`too small
`to affect the EEDF. The measurements of
`
`Rundle et al [14] were found to agree well with the
`calculations of Hake and Phelps [15] which in turn also
`agree with our calculations for case A (apart from small
`differences caused by re-adjustments made in cross
`section data from [15] to [3]).
`
`293
`
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`

`
`G Gousset er at
`
`2.2
`
`20
`
`‘.i
`
`DFHFTVELOCITYinjur-1‘) I
`
`0.8
`
`
`
`
`
`0
`
`I0
`
`5
`
`a
`
`mi i10"7vcm2)
`
`“)0
`
`130
`
`Figure 2. Electron drift velocity as a function of E/N for the
`same cases A, C and D as in figure 1.
`
`
`
`' o
`
`50
`
`E/Nriciuvcmzi
`
`100
`
`no
`
`Figure 3. Electron mean energy (I) and characteristic
`energy (II) as a function of E/N for the same cases as in
`figure 2.
`
`Figures 2-5 show various electron transport par-
`ameters and rate coefficients obtained in the same three
`
`cases as above. The effects of superelastic collisions and
`dissociation on the electron drift velocity (figure 2), aver-
`age energy and characteristic energy (figure 3) are small
`but tend to increase with E/N. Such effects are, however,
`more important at low E/N for the excitation rate coef-
`ficients shown in figure 4. Figure 5 shows the total rate
`coefficient for ionization, including in cases C and D the
`contributions of ionization from O2(X 32), O2(a 1A) and
`O(3P). These contributions, which are weighted accord-
`ing to the relative populations in these states, are shown
`in figure 6 for case C. Compared with case A, the total
`ionization rate increases by about one order of magnitude
`
`294
`
`rem’i‘)
`
`0
`
`30
`
`60
`E/N ( i6"v cm‘:
`
`90
`
`120
`
`
`
`RATECOEFFICIENTS
`
`Figure 4. Electron rate coefficients as a function of E/N for
`the same cases as in figures 2 and 3: excitation of the
`states a ‘A and b ‘)2. and 6.0 and 8.4 eV energy loss
`processes.
`
`icn-\:'94]
`
`IONISATIONCUEFFICIENI
`
`.9
`
`ion
`so
`E/N uci”'vcm’i
`
`use
`
`Figure 5. Total electron rate coefficient for ionization as a
`function of E/N for the same cases as in figures 2-4.
`
`at the lower E/N values and by a factor of 1.5-2 at E/N =
`100 Td.
`
`Figure 7 shows the electron percentage energy
`losses as a function of E/N in cases A and C. In order
`to evaluate the respective effects of O2(a‘A) and
`O(3P), also shown in this figure are curves obtained
`by setting 6,“. = 0 and keeping the ratio [O2(a 'A)]/
`[O2(X 32)] the same as in case C.
`A detailed numerical analysis of cases E and F
`reveals that the presence of the small (but realistic)
`populations in the states 02(b l2), O(‘S) and O(1D)
`considered in table 2 has a negligible effect on the
`electron kinetics. Therefore, it appears that only the
`presence of vibrationally excited molecules (at high
`TV), O2(a ‘A) metastables and O(3P) atoms may affect
`the electron kinetics in the oxygen positive column.
`
`TSMC-1216 I Page 5 of
`
`TSMC-1216 / Page 5 of 11
`
`

`
`Kinetics in 02 positive column
`
`2
`IO
`
`50
`
`75
`
`100
`
`125
`
`($1
`
`
`FRACHONALPOWERrnnnsren
`TRANSFER(1)
`
`FRACTIONALPOWER
`
`25
`
`I)
`
`rem“
`
`
`
`IONISATIONCOEFFICIENT
`
`0
`
`50
`E/N <io"’v cm’:
`
`100
`
`130
`
`Figure 6. Weighted electron rate coefficient for ionization
`rom: O2(X 32), dotted curve; O2(a ‘A), chain curve; O(3P),
`broken curve: in case C of table 2. The full curve is the
`..otal ionization rate.
`
`\
`
`3. Heavy-particle kinetics
`
`3.1. Basic reactions
`
`The most populated heavy particle species under con-
`ditions investigated in our previous work [1,2]. are
`020132), Ogia 'A), O(3P) and 0'. (gas pressure p =
`. 0.2-2 torr; tube radius R = ().8cm; discharge current
`/ = 5-80 mA;
`gas
`temperature
`'I’_L. = 3()(l—7(l() K;
`reduced electric field E/N= 10-80 Td). Using avail-
`able compilations of reactions and rate coefficients in
`oxygen [16]. one can select reactions 1-11 listed in
`table 3 as the most important ones in determining the
`,nopulations of the above species. We note that pro-
`l
`-
`«
`.
`.
`cesses 4 and 10 form other species. namely 03 and
`Ozlb 2). respectively. For mathematical convenience
`,we also include in the kinetic model the processes 12-
`22 involving these two species in order to ensure that
`‘the complete system of master equations corresponding
`, to this kinetic model has a non-zero steady—slate solu-
`tion for a given total gas density.
`’l‘herefoi'e.
`the
`inclusion of these latter processes is merely dictated by
`. mathematical reasons and in so doing we do not expect
`to describe accurately the kinetics of C); and O3(b '2).
`‘BCSidcs. these two species have much smaller popu-
`_ lzitlons than the dominant ones referred to above and
`
`play no significant role in the basic kinetics of the latter.
`We also note that the dissociative attachment pro-
`. cesses from O3(X 32) and Oziii 'A) included in table 3
`(reactions 1 and 2) have not been accounted for in
`Wlving the Boltzmann equation (see table I). since the
`numerical code employed requires conservation of the
`number of electrons. Nevertheless. the rate coefiicieiits
`
`’ for these processes have been determined using the
`i;i~.l)l" calculated from the Boltzmann code and the
`
`for
`attachment cross sections taken from Phelps
`‘()2(X 32), and from Fournier [17] for O3(a lA). Such
`
`25
`
`so
`
`75
`E/Ntio"'vcm2)
`
`100
`
`125
`
`Figure 7. Percentage electron energy losses by various
`collisional processes as a function of E/N in case A (lUll
`curve), C (chain curve) and in the case (SAP = 0. Onx :_0-8v
`om = 0.2, TV = 2000 K (broken curve): (a) 1, O2. elastic: 23
`total ionization; 3, vibrational excitation; 4, total excitation
`from O(3P); 5, 8.4 eV energy loss: (b) 6, rotational
`excitation; 7, b ‘X; 8, a ‘A; 9, sum of the 4.45, 9.97 and
`14.70 eV energy losses; 10, 6.0 eV energy loss.
`
`a procedure introduces no significant errors in the coni-
`putation of the attachment rates in the range of E/N
`considered here.
`.
`V
`Below we present some continents on the kinetic
`model iornied by the reactions of table 3-
`
`3.2. Electron and negative ion deiisities
`
`For a given discharge current. I. we dcI‘iVC lllc INCH“
`radial value of
`the electron density.
`Ii...
`front
`the
`equation I/JTR3 = <'iiL.i>.,. The drift velocity is calculated
`from the Boltzinann code (see section 2) using exper-
`imental values oi
`[3 and N. The gas density, N.
`is
`determined from the nieasured pressure and gas tern-
`pei'ature [1] using the ideal gas law. As shown in [2]
`the iiL. values so derived are in siitislactoiy agreenient
`with ineasureineiits by probe and inierowave cavity
`techniques.
`We note that no attempts were made in this work
`to include a detailed radial description in the kinetic
`
`
`
`TSMC-1216 /2l35age 6 of 11
`
`TSMC-1216 / Page 6 of 11
`
`

`
`G Gousset et al
`
`Table 3. List 01 basic reactions considered in the kinetic model of section 3, and references used for cross section and rate
`coefficient data (note that the references indicated for electron processes are for cross section data: the rate coefficients are
`calculated in the present work as explained in the main text).
`
`
`
` N0 Reaction Rate Reference
`
`
`
`
`
`1
`2
`3
`4
`5
`5
`7
`8
`9
`10
`11
`12
`13
`14
`15
`15
`17
`
`K, = f(E/N)
`K2 = f(E/N)
`K3 = 1.4 X 10“° cm3 3“
`K, = 3 x 10*“ cm3 s“
`K5 = f(E/N)
`Ks = l’(E/N)
`K-, = 200 VT,/300 s-1
`K5 = l‘(E/N)
`K9 = f(E/N)
`K10 = f(E/N)
`K1, = 0.4 S"
`K12 = f(E/N)
`K13 = f(E/N)
`K14 = f(E/N)
`K15 = 400 S"
`K15 = 2.1 x 10-34 e345/Ts cm“ 5'1
`
`o2(x 9:) + e e 0- + 0
`02(a ‘A) + e —> O‘ + 0
`O‘ + O —+ O2(X32) + 9
`0‘ + O2(a ‘A) --> 03
`o2(x32) + e —> 0 + o + e
`0213113) + 9 —* 0 + 0 4“ 9
`o + wall is ioaix 3:)
`O2(X 32) + e —> Ogle 1A) + e
`O2(a"A) + e —> O2(X 32) -+- e
`Og(a1A) + e --—> O2“) ‘2) + e
`02(a ‘A) + wall —> O2(X 32)
`O2(b 12) + e —> O2(a 1A) + e
`o2(x 3:) + e we 020312) + e
`O2(b ‘2) + e —> O2(X 32) + e
`020) 12) + wall —> O2(X 32)
`0 + 0 + o2(x 3):) —> 0, + 0
`O + O2(X 52) + O2(X 32) —> 03 -1-
`O2()( 321)
`K17 = 6.4 X 1045 em/Ts crrifi s“
`03 + O —> 02(a‘A) + O2(X 32)
`18
`K15 = 1 X ‘l0_11e_230n/Ts cm6S"1
`03 + O —> 2O2(X 32)
`19
`K19 =13 X 10-116-2300/TE Cmfi S-1
`0g(l312) + 03 —> 2O2(X 32) + O
`20
`K20 = 1.5 ><10“‘crn3 s“
`K21 = 52 X 10‘“ e'2s"'”/Ta cm3 s‘1
`02(a‘A) + 03 —) 202 + O
`21
`03 + e —» o + 02 + e
`22
`K22 = 5K5 Cma S21
`
`Phelps [3]
`Fournier [17]
`Eliasson [161
`
`Phelps [3]
`Fournier [17]
`Gousset er al [2]
`Phelps [3]
`Fournier [17]
`
`Wayne [29]
`Fournier [17]
`Phelps [3]
`
`Wayne [29]
`Eliasson [16]
`
`in particular concerning the radial density dis-
`model,
`tributions of the dominant charged species e, O‘, and
`02*. Such a description would require a formulation
`similar, for example, to that presented in a previous
`paper [5] and would have the advantage of providing
`also the characteristics of E/N against NR for the posi-
`tive column (see [5] for details). Since this is not
`attempted here, these characteristics cannot be derived
`from the present model and we make use of the exper-
`imental E/N values. Nevertheless, we shall further dis-
`cuss this problem in section 5.
`In our experimental conditions the 0‘ ions are prin-
`cipally created by the dissociative attachment reactions
`1 and 2 (table 3). As shown in previous works [5, 18],
`the 0* ions are trapped in the space-charge potential
`well, and virtually none reach the tube wall. Thus,
`these ions are principally destroyed in the volume by
`detachment collisions with O atoms (reaction 3) and
`02(a 'A) metastables (reaction 4).
`
`3.3. Atoms
`
`As shown in [2], dissociation by electron impact on
`O2(X 32) and O2(a 'A) molecules (table 3, reactions 5
`and 6) constitutes the dominant process of atom
`creation. Although atoms are also produced by reac-
`tions 1 and 2, these processes are much less important
`in the creation of atoms. We note that in the present
`model we only consider ground state atoms. The two
`reactions
`
`e + O2(X 32)» e + O(3P) + (O3P)
`
`and
`
`296
`
`e + o,(x 32)—>e + O('D) + o(31>)
`
`usually referred to in the literature as 6.0 and 8.4eV
`energy loss,
`respectively, are considered here as a
`single reaction leading to the creation of two atoms,
`with a total rate coefficient given by the sum of the
`rates for the elementary processes above. Such an
`approximation cannot be avoided, as long as a detailed
`study of the populations of excited atoms in the positive
`column has not been carried out in order to include
`excited atoms in the kinetic model as well. Such a
`
`study is already being made and a first report on the
`populations of the O(‘D) and O(‘S) states can be found
`in [19].
`The principal destruction reaction for O atoms
`under the present conditions is recombination at the
`wall, which occurs with a probability y ~ 5 X 104, here
`assumed to be nearly independent of the gas tem-
`perature [2]. The recombination frequency K7 is
`related to the probability 1/ by the expression
`U
`K7 = %
`
`where L": is the average oxygen atom velocity. For a gas
`temperature Tg = 300 K and R = 0.8 cm this expression
`yields K,-200 5*‘,
`thus K-,~200(Tg/3O0)1/2 s", as
`given in table 3.
`
`3.4. Metastable molecules
`
`The singlet metastable O2(a1A), which is highly popu-
`lated under the present conditions [1], is mainly excited
`
`TSMC-1216 I Page 7 of
`
`TSMC-1216 / Page 7 of 11
`
`

`
`Kinetics in 02 positive column
`
`current for various pressures. The agreement is in gen-
`eral quite satisfactory except for the lower pressure
`(p = 0.38 torr) and higher currents, in which case the
`predictions are about 50% lower than the measure-
`ments. Figure 9(a)—(d) compares theoretical and exper-
`imental populations of O2(X 32), O2(a IA) and O(3P)
`respectively as a function of I for four values of p. We
`find very good agreement for the O2(X 32) and O(3P)
`populations but we can see that the kinetic model fails
`to predict correctly the experimental behaviour of the
`O2(a IA) population at low currents. Moreover, we can
`note that at low currents this disagreement increases
`with pressure as shown in figure 10.
`This fact suggests that an additional quenching reac-
`tion of O2(a IA) should be considered, involving some
`species whose concentration increases linearly with
`pressure and thus with [O2(X 32)]. Since we could not
`find in the literature any reaction that could explain
`the observed behaviour of O2(a IA), a simulation has
`been made including also in the model a reaction
`
`02 (aIA) + O2(X 32) —> products
`
`whose rate coefficient has been taken as a fitting par-
`ameter. The inclusion of such a reaction with a rate
`coefficient of 6 ><10‘I° cm3 s‘I results in excellent
`
`agreement between predictions and measurements as
`shown by the solid curves in figures 9 and 10. However,
`the value 6 X 10‘I6 cm3 s‘I is about 200 times larger
`than reported values for the quenching of O2(a IA)
`by O2(X32) [16], which seems to exclude O2(X 32)
`molecules participating directly in the above reaction.
`A plausible
`explanation is
`the quenching of
`O2(a IA) by
`the
`states O2(c I2), 02(A 32)
`and
`O2(C 3A) for which the rate coefficient is 6 X 10‘”-' cm‘
`s’I [16}. In the positive column these states are mainly
`excited by electron collisions from the ground state
`(4.5 eV loss) and mainly quenched by oxygen atoms
`and O2(a IA) molecules [19]. At low currents the con-
`centration of these latter two states increases linearly
`with the electron density, so that the concentration
`of O1(c I2), 01 (A 32) and O2(C 3A) should be nearly
`proportional to that of O2(X 32). In order to explain
`our observations, the relative concentration of these
`states should be, then, of the order of 10“I. This value
`seems plausible for states whose excitation energies lie
`in the range 4-4.5 eV since the relative concentrations
`of O2(a IA) and O2(b I2), situated at nearly 1 and 2 eV
`above the ground state, are of the order of 10”I and
`10‘3—1O“’-, respectively. Further investigations are nec-
`essary, however,
`in order to test
`the above inter-
`pretation.
`The predicted concentrations of 0' ions relative to
`electron concentrations are shown in figure 11 as a
`function of gas pressure and discharge current. These
`predictions
`are
`qualitatively in
`agreement with
`measurements of 0’ concentration using a laser photo-
`detachment technique [21].
`
`297
`
`TSMC-1216 I Page 8 of
`
`50
`
`”7ELECTRONDENSITY«to
`
`cm“:JOha
`
`0
`
`
`
`D =0.lII) ton’
`
`60
`40
`20
`DISCHARGE cunnem «mm
`
`so
`
`Figure 8. Mean electron density as a function of discharge
`current. Curves, model results; data points. experiment of
`[1]-
`
`from the ground state by electron impact (reaction 8).
`The reverse reaction, i.e. the superelastic collision of
`electrons with O2(a IA) (reaction 9) and the re-exci-
`tation to the upper—lying O2(b I2) metastable state
`(reaction 10) constitute the principal destruction mech-
`anisms of O2(a IA) by electron collisions. Destruction
`of O2(a IA) at the walls and by various other volume
`processes are also included in the model, but they play
`a small role in determining the concentration of these
`metastables. However,
`it will be shown in section 4
`that some other reaction not included in table 3 must
`be involved in the kinetics of this state.
`
`4. Results of the kinetic model and discussion
`
`In this section we present the predicted populations of
`electrons, O2(X 32), O2(a IA), O(3P) and O", and a
`comparison with measurements [1] is made. The theor-
`etical results were obtained by simultaneously solving
`the system of master equations corresponding to the
`reactions of table 3 and the Boltzmann equation dis-
`cussed in section 2. In solving the Boltzmann equation
`we have assumed here that 6A5 = 6,“, = 0 (that is, we
`have only considered atoms in the ground state O(3P))
`and that TV = T3, with TE taken from experiment [1].
`The latter assumption is justified by recent measure-
`ments of the vibrational distribution of O2(X 32,0)
`under the present discharge conditions using CARS
`(coherent
`anti—Stokes Raman spectroscopy). These
`measurements have revealed that TV is always close to
`the gas
`temperature [20] which can possibly be
`explained by the high V—T