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'l', C M Ferreirai, M Pinheiroi, P A star, M Touzeau’r,
`M Vialle’r and J Loureirozt
`tLaboratoire de Physique des 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), Instituto
`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 02(X32), 02(a‘A). 0(3P), and O’
`are shown to agree satisfactorily with previously reported measurements. A
`combination of this kinetic model with the continuity and transport equations for 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 02(X32) and 02(a'A) molecules,
`0(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] andgsecondly a simplified model was
`developed [2] using a selected number of reactions
`along with electron transport and collisional data
`derived from swarm experiments in 02B]. 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 02(X) mol—
`ecules but also of 0(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, 02(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.
`
`GILLETTE 1216
`
`GILLETTE 1216
`
`

`

`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, 0‘, 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 3; 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“l eV is
`the rotational constant for 02 and 00 = anzafi/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 RHS 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
`
`I.-. = Z are + mot-(u + vnflu + V.)
`
`— uoi.(u)f(u)] + 2 5.,- [(u — mag-(u — V.)
`L}
`
`The fractional concentration of dissociated atoms in
`
`x K“ — V,,) " u0§;(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 02 molecules and O atoms.
`Moreover, each of these species can be found in a
`variety of different quantum states
`such as,
`for
`example, 02(X 32, u), 02(a 'A), 02(b12), 0(3P),
`0(18), 0(1D), 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 5 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.,=N5;
`.r
`j
`
`Ea,=1;6M+aA=1;
`
`2 5:} = 65; 2 6;,1' =1
`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:
`
`(fizfldzfiszflm)
`”
`di
`‘5 2426.0.) N d”
`.
`”’5
`
`
`KT (:1
`
`X ”2 (f+ e gfi) +46MBanuf] = EJg—x
`
`(1)
`
`where flu) is the electron energy distribution function
`(EEDF), normalized such that IE? f(u)u”2 du = 1, and
`u = mvz/Ze 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 oi,- 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 02(a 1A) metastable molecules have been
`experimentally detected under such conditions [1}.
`Incidentally,
`the concentrations of vibrationally
`excited molecules 02(X32, u) and metastable states
`02(b12), O(1 D), 0(1 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 02(X 32, l s o S. 4) and
`0: electronic states and for ionization from 02(X 3E,
`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 02(X, u) from IS 0 $4 to upper—lying
`
`291
`
`

`

`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 + 02(X. v) :2 e + 02(X. w)
`(2) e + 02(X. v = 0) :2 e + 02(a‘A)
`(3) e + 02(X, v = 0) a e + 02(b ‘2)
`(4) e + 02(X. v = O) —> e + 02(4.5 eV)
`(5) e + 02(x. v = 0) —> e + 025.0 eV)
`(6) e + 02(X, v = 0) —> e + 0203.4 eV)
`(7) e + 02(X, v = 0) —> e + 049.97 eV)
`(8) e+02()(,v=O)—re+e+02+
`(9) e + 02(X, v = 0) -> e + 02(141 eV)
`(10) e + 02(3 ‘A) :2 e + 02(b12)
`(it) e + 02(a‘A)~—> e + e + 02+
`(12) e + 02(b12)—) e + e + 02+
`
`Atomic oxygen
`
`(13) e + 0(3P) .—_> e + 0(1o)
`(14) e + 0(3P) 2 e + 008)
`Stone and Zipf [27]
`(15) e + 0(3F’) «e e + 0(38)
`Henry of at [26]
`(16) e + ovo) as + cos)
`File and Brackmann [28]
`(17) e+0(3P)—>e+ e+O+
`Drawin [7]
`(18) e+0(‘D)—>e+e+ 0+
`
`(19) e+0(‘S)——>e+e+0+
`
`Henry .9! a! [26]
`
`G Gousset er a!
`
`vibrational levels w s v + 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 — 0) 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 rte—excitation of 02(3 1A) and
`lowing:
`02(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 0(1D),
`0(18), 0(38) and ionization from ground state 0(3P)
`atoms; ionization from O{‘D) and 008) (using cross
`sections calculated according to Drawin’s formula [7]);
`transitions between C(11)) and C(18); and superelastic
`tie-excitation of both of these states to the ground state
`OPP). 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: 02(X 32,
`0 ._<_ v S 8);
`02 (a 1A);
`02(b l2); 0(3P); O(‘D); O('S). In practice, for lack
`of data we cannot take into account electronic exci-
`
`tation and ionization of 02 from 02(X 3E, o > 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 02(X 3E) state popuiation is in that
`level. In other words, the distribution of 02(X 32) mol-
`ecules among various vibrational levels only was taken
`into account when dealing with vibrational excitation
`
`292
`
`or ole-excitation processes within the ground electronic
`state. Since no attempts were made in this work to
`model the vibrational kinetics of 01 we shall assume
`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 02(X3E, 0)
`molecules
`
`We represent the intermolecular potential by an anhar-
`monic Morse oscillator whose energy levels are given
`by
`
`E. = fiweKv + t) - xctv + W]
`
`(3)
`
`where for 02(X32), we: 1580.19 crn‘l and Lodge:
`11.98 our1 [8, 9]. We assume that the vibrational dis- -
`tribution function has the form proposed by Gordiets
`er a! [10], namely
`
`NU=NDexp{—v[
`
`
`AE
`
`w—(c—DX—H
`
`E6
`
`oSu‘“
`
`*
`
`U
`NU=NU..—v— M‘suét)”
`
`(5)
`
`v > 0’”. Here, NI,
`for
`and ND ~ exp(-AEv/KTE)
`denotes the number density of molecules in level 0;
`AELO = fiwcfl —2xc) is the energy difference between
`the levels 0 = 1 and v = 0; 6 = hone/K : 2270.73 K; TV
`is the characteristic vibrational temperature; and 0* is
`the vibrational quantum number corresponding to the
`
`

`

`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
`
`Case
`(K)
`(511x
`5Ma
`51116
`(SAP
`6AD
`5A3
`
`0
`o
`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
`o
`1.0 x 10-5
`2.6 x 10-5
`7.0 x 10-2
`4.65 x 10-3
`4.65 x 10‘2
`0.879
`300
`E
`
`
`
`
`
`
`
`300 0.765 0.121 1.3 x 10-2 0.10 1.0 x 10-4F 1.0 x 10-5
`
`—z
`10
`
`a
`
`.1o.
`
`-4
`10
`
`-5
`
`1O
`
`-3
`
`
`
`ttuiiav’31
`
`minimum of the Treanor distribution [11] (equation
`(4)) given by
`
`u*=%(l+
`
`
`AE
`T
`
`1'0 g).
`
`KTV X129
`
`(6)
`
`For the conditions considered here (TV $ 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 Tg = 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: 6m, 02(X3E);
`6M1” 02(3 1‘5); 5114b, 020) ‘2); (SAP: 0(3P); 5A!) 0CD);
`(5A5, 0(18). 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,
`02(X 32, u = 0). This is the case considered by Masék
`et 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 0(3P) atoms and by superelastic collisions
`with metastable 02(a 1A) 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 ~ It"g (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
`
`‘1 19"]
`
`Figure 1. Electron energy distribution function for EN =
`3 X 10‘16ch2 (I) and 10“”ch2 (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 =10'17ch2).
`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
`0(3P) atoms at high E/N. The computed EEDF for case
`A can be compared to Langmuir probe measurements
`by Rundle et a! [14] in a discharge tube of 1.26 cm
`diameter
`for pressures 0.541 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 er a! [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
`
`

`

`G Gousset et at
`
`-1
`1
`
`l
`
`DFHFT
`
`VELOCITY[flDrill
`
`(cm:a")
`
`e
`
`30
`
`60
`EN (IGWchzi
`
`90
`
`120
`
`
`
`RATECOEFFICIENTS
`
`' c
`
`In
`
`so
`
`EN t10 ch )
`47
`2
`
`Inn
`
`130
`
`Figure 2. Electron drift velocity as a function of E/N for the
`same cases A, C and D as in figure 1.
`
`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.
`
`
`
`' 0
`
`so
`
`EINHO‘UVCMZ)
`
`100
`
`130
`
`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 02(X 32), 02(a 1A) and
`0(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
`
`Innis“)
`
`IONISATIONcosmonaut
`
`In
`
`50
`
`I00
`.n
`2
`EIN (l0 ch t
`
`150
`
`
`
`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 02(a‘A) and
`0(3P), also shown in this figure are curves obtained
`by setting (SAP = 0 and keeping the ratio [02(a 'A)]/
`[02(X 3'Z)] 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 ‘2), C(18) and 0(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), 02(3 ‘A) metastables and 0(3P) atoms may affect
`the electron kinetics in the oxygen positive column.
`
`

`

`10-10
`
`2
`l0
`
`Kinetics in 02 positive column
`
`10-12
`
`..O.
`
`"1
`35.1)
`
`[ONISATIONCOEFFICIENT(C
`
`0
`
`50
`E/N oo'"v cm?)
`
`100
`
`130
`
`Figure 6. Weighted electron rate coefficient for ionization
`rom102(X 32), dotted curve; 02(a 1A), 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 cort-
`ditions investigated in our previous work [1.2]. are
`03(X 32), Ozfa ‘A), 0(3P) and 0'. (gas pressure p =
`. 0.2—2 torr; tube radius R = 0.80m; discharge current
`/ = 5—80 mA;
`gas
`temperature
`'11. = 300—700 K:
`reduced electric field E/N= 10—80’I‘d): Using avail-
`able compilations of reactions and rate coefficients in
`‘oxygen [16]. one can select reactions l—ll
`listed in
`table 3 as the most important ones in determining the
`.nopulations of the above species. We note that pro—
`cesses 4 and 10 form other species. namely 03 and
`03(1) '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—state solu-
`tion for a given total gas density.
`’l‘heref'orc.
`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 O; and 03(1) '2').
`'Besides. these two species have much smaller popu—
`_ lations 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—
`'
`, ccsses from 03(X $2) and 03(21 'A) included in table 3
`(reactions 1 and 2) have not been accounted for in
`solving the Boltzmann equation (see table 1). since the
`numerical code employed requires conservation of the
`number of electrons. Nevertheless. the rate coefficients
`
`' for these processes have been determined using the
`l;l~.l)l" calculated from the Boltzmann code and the
`
`attachment cross sections taken from Phelps [3] for
`‘()3(X 32), and from Fournier [17] for 03(a IA). Such
`
`25
`
`50
`
`75
`
`100
`
`125
`
`(“CJ
`
`
`FHACiIONALPOWERTRANSFER
`TRANSFER(s)
`
`FRACTIONALPOWER
`
`
`
`2
`.11
`E/N (IO ch )
`
`Figure 7. Percentage electron energy losses by various
`collisional processes as a function of 5N in case A (full
`curve), C (chain curve) and in the case oAp = 0, <th :08.
`0M3 = 0.2. TV = 2000 K (broken curve): (a) 1, 02, elastic; 2,
`total ionization; 3, vibrational excitation; 4, total excrtatlon
`from 0(3P); 5, 8.4 eV energy loss: (b) 6, rotational
`excitation; 7, b ‘2; 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 com-
`putation of the attachment rates iii the range of E/N
`considered here.
`Below we present some comments on the kinetic
`model tormed by the reactions 01' table 3.
`
`3.2. Electron and negative ion densities
`
`For a given discharge current. I. we derive the mean
`radial value of
`the electron density,
`lie.
`from the
`equation l/JrR3 : Flirt)”. The drift velocity is calculated
`front the Boltzmann code (see section 2) using exper—
`imental values of [f and N. The gas density. N.
`is
`determined from the measured pressure and gas tem—
`perature [1] using the ideal gas law. As shown in [2]
`the IiL. values so derived are in satisfactory agreement
`with measurements by probe and microwave cavity
`techniques.
`We note that no attempts were made in this work
`to include a detailed radial description in the kinetic
`
`295
`
`

`

`G Gousset et at
`
`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).
`
`
`
` No Reaction Rate Reference
`
`
`
`
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`1o
`11
`12
`13
`14
`15
`15
`17
`
`K, = Its/N)
`K2 = f(IE/N)
`K3 = 1.4 X 1th—10 cm3 s“
`K4 = 3 x 10““ cm3 s‘1
`K5 = f(E/N)
`KE = f(E/N)__
`K, = 200 V 15/300 5-1
`K, = l‘(E/N)
`K9 = f(E/N)
`K10 = firs/N)
`it11 = 0.4 3"
`K12 = fiE/N)
`l<13 = fiE/N)
`K14 = f(E/N)
`KS 2 400 5"
`K”, = 2.1 x 10-34 6345/7} cm6 5-1
`
`02(x a2) + e a o- + 0
`02(81A) + 9—) 0‘ + 0
`0‘ + O-> 020(3):) + 9
`0‘ + 02(a ‘A) «—> 03
`ozixas) + e —> 0 + o + e
`02(a 1A) + e —> O + O + e
`o + wall ._, l02(X 32)
`02(x 3:) + e —> 02(a 1A) + e
`02(a‘A) + e —> ozix 3:) + 9
`02(a 1A) + e «a Ozib 1:) + 6
`02(a ‘A) + wall —> 02(x32)
`02(b ‘2) + e —> 02(a 1A) + e
`02(x 3:) + e ml 020: 12) + e
`02(b 12) + e —> 02(x32) + e
`020312) + wall —> 02(X 32)
`o + o + ozlx 3)2) —> o, + 0
`o + oaix 32) + ozix 3:) —> 03 +
`KW = 6.4 X 10’35 e663”: cm6 s‘1
`02(X 32)
`K15 = 1 x 10—11e‘23m/Tgcm65“1
`03 + o —> 02(a‘A) + 020(32)
`18
`K19 = 1.3 x 10*11 e—mO/Tg cm6 5'1
`03 + O —> 202(X 32)
`19
`K20 = 1.5 x 10-11 cm3 s.-1
`oalbtz) + 03 —> 202(x32) + 0
`20
`K21 = 5.2 X 10—11 e'zm/Ta cm3 s"1
`02(a‘A) + 03 —) 202 + 0
`21
`
`
`03+e—>0+02+e22 K22=5Kscmask1
`
`Phelps [3]
`Fournier [17]
`Eliasson [161
`
`Phelps [3]
`Fournier [17]
`Gousset et at [2]
`Phelps [3]
`Fournier [17]
`
`Wayne [29]
`Fournier [17]
`Phelps {3]
`
`Wayne [29]
`Eliasson [1e]
`
`in particular concerning the radial density dis-
`model,
`tributions of the dominant charged species e, 0‘, 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 01 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
`02(X 3E) and 02(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 + 02(X 32)*~> e + 00]?) + (03?)
`
`and
`
`296
`
`e + o,(x 3Z)—>e + O('D) + 0013)
`
`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 C(11)) and OCS) states can be bond
`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 10—3, 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
`
`yr}
`K7:2rt
`
`where L7 is the average oxygen atom velocity. For a gas
`temperature Tg = 300 K and R = 0.8 cm this expression
`yields K7 ~ 200 5”,
`thus K3. ~ 200(Tg/300W2 s“, as
`given in table 3.
`
`3.4. Metastable molecules
`
`The singlet metastable 02(a1A), which is highly popu—
`lated under the present conditions [1], is mainly excited
`
`

`

`
`
`
`
`
`
`to,ELECTRONDENSITY(10cm“)
`
`D =0.l||) torr
`
`50
`
`:-a
`
`
`
`60
`do
`20
`DISCHARGE cunnEN'r thI
`
`50
`
`47-0
`
`0
`
`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 02(a LA) (reaction 9) and the re-exci-
`tation to the upper-lying 02(b ‘2) metastable state
`(reaction 10) constitute the principal destruction mech-
`anisms of 02(a 1A) by electron collisions. Destruction
`of 02(a 1A) at the walls and by various other volume
`processes are also included in the model, but they play
`a small role in determining the c0ncentration 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, 02(X 3El), 02(a 1A), 0(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 (5A5 = 5A1) = 0 (that is, we
`have only considered atoms in the ground state 0(3P))
`and that TV = Tg, with Tg taken from experiment [1].
`The latter assumption is justified by recent measure-
`ments of the vibrational distribution of 02(X 32,:2)
`under the present discharge conditions using CARS
`(coherent anti-Stokes Raman spectroseopy). These
`measurements have revealed that TV is always close to
`the gas
`temperature [20] which can possibly be
`explained by the high V—T rates associated with col-
`lisions of 02(X 32, u) with O(3P) atoms. We note that
`under
`the
`conditions
`Tv = I!"g
`the
`presence of
`vibrationally excited molecules has an almost negligible
`effect on the electron kinetics, so that only the presence
`of 02(a 1A) and 0(3P) in high concentrations play a
`significant role under the present circumstances.
`Figure 8 shows a comparison of theoretical and
`experimental values of E, as a function of the discharge
`
`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 02(X 3E), 02(a1A) and 0(3P)
`respectively as a function of 1 for four values of p. We
`find very good agreement for the 02(X 32‘.) and 0(3P)
`populations but we can see that the kinetic model fails
`to predict correctly the experimental behaviour of the
`02(a 1A) 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 02(a 1A) should be considered, involving some
`species whose concentration increases linearly with
`pressure and thus with [02(X 32)]. Since we could not
`find in the literature any reaction that could explain
`the observed behaviour of 02(a ‘A), a simulation has
`been made including also in the model a reaction
`
`02 (alA) + 02(X 3E) —> 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‘16 cm3 5‘1 results in excellent
`
`agreement between predictions and measurements as
`shown by the solid curves in figures 9 and 10. However,
`the value 6 X Ill—‘6 cm3 s"1 is about 200 times larger
`than reported values for the quenching of 02(a1A)
`by 02(X3E) [16], which seems to exclude 02(X 32)
`molecules participating directly in the above reaction.
`A plausible
`explanation is
`the quenching of
`02(a 1A)
`by
`the
`states 02(c ‘2), 02(A 3E)
`and
`02(C 3A) for which the rate coefficient is 6 X 10—12 cm3
`s" [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 02(a 1A) molecules [19]. At low currents the con-
`centration of these latter two states increases linearly
`with the electron density, so that the concentration
`of 02(0 ‘2), 02 (A 32) and 02(C 3A) should be nearly
`proportional to that of 02(X 32). In order to explain
`our observations, the relative concentration of these
`states should be, then, of the order of 10“. This value
`seems plausible for states whose excitation energies lie
`in the range 4—4.5 eV since the relative concentrations
`of 02(a 1A) and 02(b 12), situated at nearly 1 and 2 eV
`above the ground state, are of the order of 10‘1 and
`10—3—10‘2, 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