MODELLING OF METAL VAPOUR NOBLE GAS DISCHARGES IN THE TRANSITION REGION

Transcription

1 Philips J. Res. 37, , 1982 R1059 MODELLING OF METAL VAPOUR NOBLE GAS DISCHARGES IN THE TRANSITION REGION by M. J. C. VAN GEMERT*), O. P. VAN DRIEL**) and J. MEZGER Abstract Modelling of metal vapour noble gas discharges has been performed using a hypothetical gas system consisting of metal atoms with a low ionization potential and noble gas atoms with a much higher ionization potential. Both the metal and the noble gas atoms are assumed to have only a ground state and an ionization level. Numerical calculations of the electric fielddischarge current (E-I) curves reveal the existence of a low and a high field region. The intermediate region can either be multivalued or single valued in E, depending on whether or not the metal vapour ionization rate is much larger than the nobie gas ionization rate. 1. Introduetion Electrical discharge characteristics showing multivalued behaviour have long been known in the literature 1-5). For example a maximum for the discharge current exists for a low-pressure mercury arc 1,2) while low-pressure discharges in mixtures of Cs-Ar 3 ), and of Na-Ne 4 ) show characteristics multivalued in the voltage. For the latter type of discharges modelling has so far been limited to metal vapour ionization only"). Even in such a restricted model the electric field is calculated to be multivalued; because of metal vapour depletion a maximum in the current is found, since noble gas ionization is not taken into account. When noble gas ionization is taken into consideration the electric field-discharge current (E-1) characteristics consist of a low field region at low currents, where metal vapour ionization predominates, and a high field region at higher currents where noble gas ionization is predominant. A current maximum is then not found. The purpose of this work is to present an analysis of the E-1 characteristic in the intermediate region, where both metal vapour ionization and noble gas ionization play a part. To this end a simple discharge model is developed in sec. 2. The numerical results are presented in sec. 3. A discussion in given in *) Present address: Department of Medical Technology, St. Joseph Hospital, Eindhoven, The Netherlands. **) Present address: Elcoma Division, Nijmegen, The Netherlands. Philips Journalof Research Vol.37 Nos 5/

2 M. J. C. van Gemert, O. P. van Driel and J. Mezger sec. 4. An appendix is added giving details of the physical parameters used in the model and some of the mathematical aspects of it, both analytical and numerical. 2. The model energy rev) 1 The model, shown schematically in fig. 1, is based on a hypothetical metal atom with a low ionization potential and a noble gas atom having a much higher ionization potential. For the metal vapour the physical processes considered are: (1) elastic losses, (2) ionization, (3) ambipolar diffusion and (4) wall recombination and neutral back diffusion Dal o KI f7;,) N--'-- Fig. 1. The model; M denotes the metal atom with ionization energy U" ionization rate KI(T.), am bipolar diffusion coefficient DOl and neutral diffusion coefficient Dl. The noble gas atom is represented by N, with ionization energy U2, ionization rate K2(T.), and ambipolar diffusion coefficient D02. The wall density of the metal vapour is determined by the saturated vapour pressure which corresponds to the wall temperature. For the noble gas: (i) elastic losses, (ii) ionization, (iii) ambipolar diffusion and wall recombination. The number. density of the noble gas is determined by the gas filling pressure. The density ofthe noble gas is taken constant over the tube cross section. Particle densities are about m-a for the metal atoms and m-a for the 266 Phlllps Journal of Research Vol.37 Nos5/6 1982

3 Modelling of metal vapour noble gas discharges in the transition region noble gas. The gas temperature is taken independent of the radial coordinate and equal to the wall temperature. It may be noted that a similar approach was followed by Polman 7), who studied the relaxation of the electron velocity distribution in a metal-atom noble gas discharge. In what follows, the subscripts 1, 2 represent the metal vapour and the noble gas, respectively. For simplicity, we assume that the ion mobilities u, of the metal and of the noble gas are the same, which means that the ambipolar diffusion coefficients D«are also equal. Thus,uil =,ui2 =,ui, Dal = Da2 = D«, (1) where D; =,ui k Tele and Te is the electron temperature. The following set of continuity equations can now be written down to represent stationary discharge behaviour in a cylindrical tube with radial coordinate r and tube radius R Da 'ij2 nil(r) + KI(T.) n.(r) M(r) = 0, D; 'ij2 ni2(r) + K2(T.) ne(r) N = 0, Dl 'ij2 M(r) - K1(T.) ne(r) M(r) = 0, ne(r) = nil (r) + ni2(r). (2) (3) (4) (5) The boundary conditions used are for r = 0, for r=r, dnil/dr = dni2/dr = dmldr = 0, nil(r) = nï2(r) = and M(R) = Mw. (6) (7) Note that the assumption of charge neutrality is taken into account by eq. (5). In the above equations, 'ij2 is the Laplace operator, n. is the ion number density, ne the electron density, and M, N are the number densities of the metal vapour and the noble gas, respectively; KI,2(T.) are the rate coefficients for ionization ofthe metal vapour, and the noble gas, respectively. Dl is the diffusion coefficient of the metal atoms in the noble gas. This set of equations is completed by the electron energy equation..as in ref. 6 the integrated form is used, and T. is taken independent of r. Thus and R 5 (dn.) EI =!dr [21tr ne(pin.l + Pel)] - - k TeD; -- 21t R, o 2 dr R R 1= «s] dr21t r neu«, o (8) (9) Phillps Joumnl of Research Vol. 37 Nos 5/

4 M. J. C. van Gemert, O. P. van Driel and J. Mezger where E,1 are the electric field strength and the discharge current, respectively; Pisa.P«arethe inelastic and elastic losses per electron per second, and u, is the electron mobility. The rate coefficients are taken as l = 1,2, (10) where BI is a constant cross-section, me is the electron mass, and UI is the ionization energy. The rate coefficients (10) follow from a Maxwellian electron energy distribution and a cross-section for ionizing collision of the form al(e) = 0, e ~ UI; al (e) = bi (1 - Ulle), e > UI. (11) The calculations are performed for R = 1 cm. The numerical solution of this set of equations requires a coordinate transformation and an iteration procedure. Details of the model, the numerical procedure involved, and the physical parameters used, are presented in the Appendix. 3. Numerical results Figs 2 and 3 show typical E-l curves obtained with the present model. The curves consist of two parts with, respectively, a low and a high electric field. The lower E-field region can be considered as the metal vapour part (A = 0 in eq. 22) while the upper part is due to a noble gas discharge disturbed by metal vapour processes close to the wall (where metal vapour ionization remains the dominant discharge process). In the case 1-+0 the radial depletion of M(r) disappears and the model reduces to a Schottky model. For the ionization of the metal vapour tends to occur very close to the wall only, and the model reduces again to a Schottky model but now for the noble gas. The intermediate part of the E-I curve can either be multivalued or single valued in the electric field, depending on whether the ratio of K, and K 2 is, respectively large or small. In fig. 2 the KlIK2 ratio is changed by variation of B 2 in eq. (10). Using BI = lo-19 m", UI = 5.14 ev and U2 = 21.6 ev the curves are multivalued for B2 < 3 X m", and single-valued for B 2 > 3 X lo-22 m". In the latter situation a positive characteristic is calculated. Similar results are shown in fig. 3 where U2, the ionization energy of the noble gas, has been varied. For 268 Phllips.Journal of Research Vol.37 Nos 5/6 1982

5 Modelling of metal vapour noble gas discharges in the transition region E {V/m} t P I{A} Fig. 2. The electric field, E, as a function of the current, J, for B, = m", B2 = 10-21, and m 2 ; M; = 0.13 xl0 21 m- 3, U, = 5.14 ev, and U2 = 21.6 ev. The points P, Q, Rand S correspond to a radial depletion M(O)/Mw of 8 xl0-2, 2.8 xlo- 6, 3.5 xl0-12 and 2.3 xl0-20, respectively (see fig. 5). E {V/m} r I {A} Fig. 3. The electric field, E, as a function of the current, J, for parameters U2 = 13.0, 15.2 and 21.6 ev and B2 = m", Other parameters as in fig. 2. Phllips Journalof Research Vol. 37 Nos 5/6 198Z 269

6 M. J. C. van Gemert, O. P. van Driel and J. Mezger U2..$15.2 ev the E-I curves are also single valued, using UI = 5.14 ev, BI = m 2 and B2 = m". It is emphasized here that a multivalued curve is also predicted by van Tongeren's model") (B 2 = 0, hence KI/K2-00). 9 8 c;' 'e: ~ 7.Q..._ ê: 6 c:'" o o IrA) Fig. 4. The electron density on the axis, neer = 0), as a function of the current J. U 2 = 21.6 ev, B2 = m". Other parameters as in fig. 2. Fig. 4 shows an example of n.(o) vs I in case the intermediate part of the E-I curve is multivalued. At small currents ne(o) increases up to a maximum and then decreases to a minimum corresponding to the point in the E-I curve where I starts to increase again. Results of this type are well-known to occur in radially depleted discharges 3,6) and will not be discussed here. Fig. 5 shows some results of ne(r) and M(r) for several values of M(O)/Mw. These curves refer to the points P, Q, Rand S in the E-I curve, as indicated in fig. 2. In fig. 5a a Bessel function for ne(r) is shown for comparison for a radial depletion of M(O)/Mw z 8 X 10-2 Figures 5b, 5c show the situation where ne(r) tends to become flat, due to the strong depletion; M(O)/Mw is 2.8 X 10-6 in fig. 5b and 3.5 x10-12 in fig. 5c. In fig. 5d the depletion is 2.3 x10-20 The ne(o) value has increased again due to the fact that ionization of the noble gas tends to become the dominant process, except close to the wall, where sufficient metal atoms are still present. 270 Phlllps Journni of Research Vol.37 Nos 5/6 1982

7 ~ ~ Modelling of metal vapour noble gas discharges in the transition region 'r ,7.6 Br , M ni ,4 5 5 al 4 O.B 3 0,6 2 O'~-~-~-~-~-~ o 0, , r/r ', ,7.4 4 n, 170"m t 2 bi J o 0, O.B W O'~-~-~-~~~-~O _r/r 'r ,7,4 O.B M O.B 3r- ~n.~ _ n. 170"m' ,6 cl ~~~OL.2~-O~.4--0~.6--J0'~8-~ -r/r dl O~ L- L- L- ~~O o O.B 7,0 -r/r Fig. 5. The electron density n; and the neutral atom density M as a function of rl R, Parameters as in fig. 4. Figs Sa, b, c and d correspond to the points P, Q, Rand S in fig. 2, respectively. 4. Discussion The results of figs 2 and 3 show that the occurrence of voltage-controlled 5) E-1 curves depends strongly on the ionization rates of metal vapour vs noble gas. In the present simple model the ratios of B2/ BI or U2/ UI (fig. 1) are sufficient to indicate multi-valued or single-valued electrical behaviour. Voltage controlled behaviour results when B2/BI is sufficiently smallor U2/UI is PhllIps Journal of Research Val.37 Nos 5/

8 M. J. C. van Gemert, O. P. van Driel and J. Mezger sufficiently large. Physically this means that a voltage-controlled characteristic results when the elastic losses dominate over the inelastic losses for most of the radial coordinate, over a sufficiently large range of re values. Using the experimental information that Na-Ne 4 ) and Cs-Ar 3 ) discharges are multivalu~d in the voltage, whereas Na-Hg8) is single-valued, voltage-controlled behaviour can be predicted for various combinations of metal vapour-noble gas discharges. For example, we expect that all combinations of Cs and Na with the noble gases except perhaps Xe show voltage controlled behaviour, because the ratios of the ionization energies of Cs and Na, and the metastable state energies of the noble gases are large. Moreover, no other noble gas ionization channels are known that might correspond to an enlargement of B2 In the case of Hg the situation is different. Although He and Ne are expected to show multi-valued behaviour, we believe that the other noble gases have metastable states which are close enough to the Hg-ionization level that the transition from Hg ionization to noble gas ionization is too gradual to yield a voltage-controlled characteristic. It has mathematically been shown by de Bruijn 9) that a solution of the model equations can exist under certain conditions only. One of these requirements is T 2 < 2.405, (12) which is equivalent to (eqs 14 and 29) (13) This implies that T. < re,max, where Te,max is the Schottky limit for the noble gas discharge. If the electrons lost energy only to the noble gas atoms, it would follow that E < Emax. However, close to the wall (r:::::: R) inelastic losses to the metal atoms can be considerable (see fig. 6) and therefore it remains possible that E> Emax for certain values of the current 1. In any case, E - Emax for The present model is an extension of previous work by Bouwkamp 10) who solved the rate equations without noble gas ionization. In fact, eqs (2)-(7) were reduced by him to an eigenvalue problem of one differential equation in cj)(r) = (Da/D1)(ne/M) and cj)(r) was calculated for several values of the eigenvalue Kc = klmw/da. No attempt was made to calculate the corresponding E vs 1curves. Nevertheless, when computed also those E-l curves are multi-valued 11), similar to the model results of van Tongeren 6). 272 Phlllps Joumnl of Research Vol.37 Nos 5/6 1982

9 Modelling of metal vapour noble gas discharges in the transition region ~ 15 'e: :;::: It) S2.._ c0 i- a; c ~.. c: ~ ~ t total losses 5 -r/r Fig. 6. The elastic and the inelastic losses as a function of rl R, Parameters as in fig. 4. M(O)/Mw'" Acknowledgement The authors are indebted to J. de Groot of our laboratory and to J. K. M. Jansen and G. W. Veltkamp of Eindhoven University of Technology for discussions on the numerical aspects of this study. Appendix I. Physical parameters used The actual relations used for Pel, Pinel and «, are those used by van Tongeren (eqs (2.5), (2.6) and 2BA of ref. 6). The data used for the metal atoms and the noble gas refer to, respectively, Na and Ne. Philips Journal of Research Vol. 37 Nos5/

10 M. J. C. van Gemert, O. P. van Driel and J. Mezger IJ. Mathematical details Equations (2)-(7) are transformed in the following way. Firstly, define A = N K2(T.) R2/Da, B = s,(te) R2/Da, C=KI(Te)R2/Db D=Mw. (14) (15) (16) (17) Then, using in (2)-(7) the transformations x= rlr, t=xt, ne(x) = P U(x T), M(x) = Q V(x T), (18) (19) (20) (21) where T, Pand equations Q are constants to be fixed later, yields the following set of d 2 U 1 du (A BQ) V U=O dt2 t dt T2 T2 ' (22) d 2 V 1 dv CP UV=O dt 2 t dt T 2 ' t = 0; duidt = dvdt = 0, t = T; U(T) = 0, V(T) = D/Q, (23) (24) (25) Consider now the following, related initial-value problem: d 2 U 1 du (1 +V) U = 0 dt 2 t dl' ' (:l2v.r dv -+---UV=O dt 2 t dt ' (26) (27) t = 0; duidt = dv/dt = 0; U(O)= a, V(O) = p. (28) Integration of the latter system up to the first zero of U(t), at t = T, gives the solution of (22) to (25) provided that a and p are correctly chosen and that 274 PhllIps Journalof Research Vol.31 Nos 5/6 1982

11 Modelling of metal vapour noble gas discharges in the transition region B=A/Q C=A/P D = Q V(T) = Mw. (29) (30) (31) (32) The set of eqs (26)-(28) has been solved numerically using a procedure described by Bulirsch and Stoer 12). The result is the functions U(t) and V(t) and T, the first zero of U(t). When T has been determined, A is known by (29), and thus Te by (14). Then Band C can be determined from (15) and (16), and thus Q and P from (30) and (31). An iterative method for the determination of a and p is still required to make sure that D = M; eq. (32). In practice we found it convenient to present P and to change a until (32) was obeyed. Fig. 7 shows an a vs P curve obtained by iteration. Finally, ne(r) and M(r) are loga log (J 8 Fig. 7. An iterated a versus p curve. Parameters as in fig. 4. obtained from (20) and (21); (8) and (9) then yield E and J. It should be men- -tionedthat the numerical procedure breaks down for extremely small values - of p, i.e. for very small values of M(O)/Mw In that case the function U(t) can Philips Journalof Research Vol. 37 Nos 5/

12 M. J. C. van Gemert, O. P. van Driel and J. Mezger be approximated with reasonable accuracy by the zeroth order Bessel function Jo(t), which is a solution of (26) with.y = 0, for a certain range t < ti < T. The numerical approach is then applied to the range ti ~ t ~ T. Philips Research Laboratories Eindhoven, September 1982 REFERENCES 1) A. W. Hull, Trans. Am. Inst. Electr. Eng. 53, 1435, ) J. E. Allen and P. C. Thonemann, Proc. Phys. Soc. London, 867, 768, S) J. H. Waszink and J. Polman, J. Appl. Phys, 40, 2403, ) M. ~. C. van Gemert and T. G. Verbeek, Appl. Phys. Lett. 31, 500, ) B. K. Ridley, Proc. Phys. Soc. London 82, 954, ) H. van Tongeren, Philips Res. Repts. Suppl. 3, ) J. Polman, Physica 54, 305, ) T. G. Ver beek, unpublished results, 1974; private communications to the authors. 9) N. G. de Bruijn, Philips J. Res., 36, 229, ) C. J. Bouwkamp, Siam Rev. 10, 114, ) M. J. C. van Gemert and J. Mezger, unpublished results, ) R. Bulirsch and J. Stoer, Num. Math. 8, 1, Phlllps Joumol of Research Vol.37 NosS/6 1982