.Chemical Physics 20 (1977) 417-429 0 North-Holland Publishing Company

A THEORETICAL CALCULATION OF DISSOCIATION RATES OF MOLECLILAR HYDROGEN IN ELECTRICAL DISCHARGES

M. CAPITELLI, M. DILONAHDO Centro di Studio per la Chimica dei PIasmi de1 C.N.R.. Diparrimento di Ckimica, University di Bari. Via Amendola 173. 70126 Ban-. Italy

and

E. MOLINARI Dipartimento di Chimica, Universiti di Rona, Piazzale delle Scienze. OOIOO Roma. Italy

Received 30 September 1976

Dissociation rates of molecular hydrogen in electrical discharges have been calcclated at different electron (Te) and gas (Tg) temperatures (10000 G Te c 23000 K, 500 c Tg =G 4000 K), at different pressnresp (5 Gp i 50 torr) and electron number densities ne (0 =G ne < lOI* cn-?).

The results have been obtained by solving a system of master equations. including V-T (vibration-translation), V-V (vibration-vibration) and e-V (electron-vibration) microscopic processes.

The results obtained at me + 0 show a “laser-type mechanism” in the dissociation of molecular hydrogen in electrical dis charges. In particular one notices a strong increase of dissociation rates with decreasing gas temperature and pressure.

The results show that this mechanism is as important as the mechanism of direct dissociation by electron impact.

I_ Introduction

The dissociation of molecular hydrogen in electrical discharges has received considerable theoretical and ex- perimental interest [l-l]. Two theories have so far been proposed to explain the process. The first (model 1) tries to explain the dissociation process by direct electronic impact

e+H2 3 e +2H, (a)

the rate of which is ud = tZe&.$2ki (ne andNH, are the electron and hydrogen number densities respectively)_ In this case the more probable reaction path is the ex- citation of the hydrogen molecule in its ground state (lx) to the first repulsive excited state (3X), followed by dissociation.

The second theory (model 2), recently proposed in our laboratory [4,5], invokes binary encounters be-

tween vibrotational excited molecules, the concentra- tion of which is higher than the corresponding Boltz- mann concentration at the gas temperature Tg, due to the electric field.

In general both mechanisms act at the same time and it has been possible to evaluate the relative importance of each mechanism, depending on the particular system and conditions of the discharge [443. The aim of the present work is to understand, on a microscopic scale, . the role of vibrational excitation in the dissociation process. To this end we present a theoretical treatment of the dissociation of molecular hydrogen in an elec- trical discharge, which represents an extension of the methods’used by several authors for describing the dis- sociation of moIecular species in a thermaI bath [6,7] - The extension consists in the inclusion of vibrational excitation by electron impact.

Basically, one solves a system of u’ + 1 differential eqrations (u’ is the number of vibrational levels in the

418 M. Capitelii et al-/Hz dissociation rates in electrical discharges

ground state hydrogen molecule), each of which de- scribes the evolution of a given vi’orational level under theaction ofvibration-vibration (V-V), vibration- translation (V-T) and electron vibration (e-V) energy exchanges. The last of these equations (u’ f 1) simu- Iates the dissociation process. After a transient period, a pseudo first order rate constant ki can be defined for the dissociation process, which can be compared with the corresponding Icz value of model 1 and with the experimental rate constant of ref. [4] _

The accuracy of the calculations will be discussed in connection with the adopted model and with the choice of the relevant cross sections.

2. Method of calculation

The temporal evolution of the density N, (cme3) of the uth vibrational level belonging to the ground state hydrogen molecule (‘ZI) can be written as Fohows r&91 : aN”lat

=P v+l,vNH, IN”+1 - exp I-&+1 ‘- ~J/~~,lN,I -P v,u_1NH2 CrV, - ewd-(Ev - E~_lYkTgl~u_13

- C P;$f l {N"N~ w=o - exd-_(Eu +E, -Ewe1 - ~u_lY~~gl Nu_lN,Hl 1

+ne wFo K&,CN, - exp[-_(Ew --EJ/W,1~,>,

whereP”,t v, P, U-l ,Pr+-t$“, Puw;‘_y~l and A’L repre- sent the ratk coefficients ot- the f&owing microscopic processes:

Hz + Hz@ + 1) 2 Hz + Hz(u), V-T

Hz + Hz(u) Z H, + Hz@-I), I

H2(u + 1) + H2(w - I) ” Hz(u) + Hz(w), (b) v-v

H*(u) + Hz(w) Z H&I - 1) + H&w + I),

H&v) f e 2 Hz(u) + e. e-V

(NH* is the total number of hydrogen molecules and Te is the electron temperature.)

It should be noted that only sir&e quantum jumps‘ have been considered for V-T and V-V processes, while multiple quantum jumps have been considered for e-V processes.

The dissociation process has been simulated by con- sidering a pseudolevel (u’ + I) for dissociated molecules_ This level is populated according to the following equa- tion:

It should be noted that this model does not consider atom recombination. Moreover, to avoid extrapolation of transition probabilities of the pseudoleve1 lying in the continuum, it was decided to merge the pseudo- level on the uppermost vibrational level. In this case the dissociation rate can be considered as an upper limit to the true dissociation rate. Dove and Jones have how- ever shown, in the case of thermal dissociation of H,, that this procedure gives errors not larger than 1.5% as compared with the case in which the pseudolevel is lo- cated in the continuum [6] _

To soIve the problem, one requires the vibrational enere diagram of H, and the rate coefficients appear- ing in eqs. (1) and (2)

An anharmonic Morse oscillator has been used for calculating the vibrational energy:

E, = El0 Ku i-9) - No i- -1-)*I 9 (3)

withE,, = 8.726 X lo-l3 erg and 6 = 0.0278_ The rate coefficients for electronic excitations have

been calculated by numerical integration, over a max- wellian distribution function of the electrons f,, of the experimental cross sections [lo] &, for the transi- tions O-l, O-2,0-3,

Kg-“=1 f,cQ Ue(E) &“@I dE. (4) 0

The results are reported in fig. 1. The rate coefficients For 0-u electron transitions

(u > 3) have been obtained by extrapolating the experi- mental Kz_U values, while those for h-u transitions

M. Capitelli et al./H2 dissociation rates in electrical discharges 419

I

D-

,

1-

Selected values of V-T and V-V cross sections are re- ported in fig. 2 as a function of vibrational quantum _ number u.

Rate coefficients for reverse processes have been ob- tamed via detailed balancing (also present in eq. I) in the form

The system of 16 differential equations (1.5 for the vibra- tional levels and one for the dissociation level) has been integrated with the numerical algorithm described in appendix B. Most of the results have been obtained by imposing at t = 0 the following initial condition:

N”(f=O)=O foru#O, N&c = 0) = NHz _ (7a)

It should be noted that according to our model without recombination, the distribution of levels at t = 00 will he:

NU(t=m)=O forO<u<14, NH = 2NH,. (7b)

3. Results Fig. 1. Rate coefficients for e-V processes (cm3/s) as a func- tion of electron temperstire.

(h > 0) have been put equal to the corresponding O-u transitions (i.e., K& = Ky_2 = Ki_3, and so on).

Rate coefficients for V-V and V-T processes have been calculated by means of SSH theory as modified by Keck for V-T processes [ll] and by Bray for V-V processes [ 121. The corresponding equations are reported in appendix A. These equations have been used to obtain the dependence of rate coefficients on the vibrational quantum number and on the gas tem- perature. They have been matched to the Kiefer and Lutz correlation data for V-T relaxation times 1131,

@%12-H2 = 3.9 X 10-lo exp(lOO q”3) atm s, (Sa)

and to the Kiefer’s correlation data for V-V relaxation times [14]

(pr)‘)v_V = 4.7 x 10-7 5 112 atm s. Gb)

The phenomenological dissociation constant k, of molecular hydrogen can be obtained by writing eq. (2) in the form [6J,

After a transient period, kd(t) settles to a constant vahre ki, the socalled “steady-state dissociation con- stant” 161. The transient period is obviously dependent on the particular conditions studied. Fig. 3 shows ty- pical kd(t) plots.

A “true” steady-state dissociation rate constant can not be achieved in our model because one neglects the recombination process, and this yields ka(r = -) = 0. ‘Ibe ki values reported in the present work are there- fore those corresponding to plateaus or maxima of kd(f) plots (see fig. 3).

Values of log ki, calculated in the present work,

420 M. Capitelli et al&i2 dissociation rates in electrical discharges

2 6 10 34 2 6 f0 14

VIBRATIONAL QUANTUM NUMBER

Fig. 2. Rate coefficients for V-T and V-V processes (cm3/s) as a function of vibrational quantum number.

-5 -4 -3

log t Fig. 3. Temporal evolution of k&)-

have been plotted in fig. 4 as a function of I/Tg for n = IO’* cme3 and T = 20000 K. In the same figure w: have also reported fog ki values obtained under the same conditions but neglecting V-V terms in the mas- ter equations. Also included are values at );2e = 0.

The trend of log ki vahes versus I/T6 calculated inserting V-T, V-V and e-V processes IS quite inter- esting, showing two-completely different mechanisms, depending on the temperature range considered.

At high temperature (T > 3000 K) the ki values calculated with V-T, V-$, and e-V processes are close to those obtained at ne = 0. Moreover V-V processes have practically no effect on ki. This means that, in this temperature range, thermal dissociation (practical- ly the V-T terms) is the most important reactive chan- nel for hydrogen dissociation. An apparent activation energy of 99.6 kcal/mole is obtained in this case.

At low gas temperature (500 K < 2000 K) the ks values are strongly dependent on V-V and e-V pro- cesses. Moreover one notices an increase of the disso- ciation constant with decreasing gas temperature, when V-V, V-T, and e-V processes act at the same time. This yields negative apparent activation energies.

M. Capitelii et al./H2 dissociation rates in electricai discharges 421

4L

2-

O-

-z-

-4-

-S-

9

E -a-

-lO-

-l2-

-14-

-,6-

+,a-

Fig. 4. VaIues of Iog ki and of Iog(k3e) as a function of l/T (-) normal V-V and V-T cross sections; (---) (V-V) X L6, W-T)IlO; <x)fV-V) = 0; (0) values obtained considering dis- sociation frozn each vi%rationaI level (see text))_

This behaviour should be attributed, in particular, to the presence of V-V processes, as can be appre- ciated by comparing the dissociation constant calcu- lated with and without V-V processes_ The reported trend of ki with l/T gas temperature, the ?

is linked to the fact that, at low or-ward V-V processes are not

compensated by the backward ones, since the Boltz- mann factor which relates forward and backward V-V rate constants (see eq. 6b) is small. At higher gas tem- perature forward and reverse V-V processes practical- ly cancel, because the BoItzmann factor approaches 1. As a consequence, the higher vibrational levels are more populated at low temperature than at high tempera- ture.

The importance of e-V processes at low gas tem- perature should also be stressed (compare the com- plete calculation with the ne = 0 case).

The dissociation mechanism at low T can there- fore be ascribed to the simultaneous a&n of V-V and e-V processes in overpopulating the higher vibra-

tional levels. This mechanism which is responsible for the Iasing of some eIectricaI molecular lasers (e.g., CO), is essentially due to the anharmonicity of vibrational levels 181.

In this connection we note that in the electrical CO laser, as described in fig. 7 of ref. [S] , the partial in- version plateau disappears at 300 < T < 500 K. Simi- lar results should be applied to N, mo ecules, gi due to their similarity with CO_ This could explain the fact that PoIak [15J, who considered the dissociation of molecular N2 at Ts =: 750 K, did not fmd the popula- tion inversion present in our results.

It should again be pointed out that the population inversion is due to the anharmonicity of vibrational levels. This in fact enters in the forward and reverse V-V processes (for example for the non-resonant pro- cess d) through the factor:

exp(-AE/kT& = exp(-2El,,Gv/kTg). (9)

The product El026 is much larger for H2 than the cor- responding values for CO and NZ, so that the inversion for molecular hydrogen occurs at higher temperatures than those for N, and CO.

Fig. 5 compares the population density of the vibra- tionaI quantum states during the tempo& evohrtion. for different gas temperatures. One can note that at low Tg (e.g., Tg = 500 K) an inversion in the population density occurs at IJ = 8,9, as a consequence of the pre- ferential pumping of these levels. This brings the higher vibrational levels to be overpopulated; as a consequence we observe the high dissociation constants of fig_ 4.

At high temperature (7’s =_3000 K) the evolution passes through different quasrBoltzmann distributions; the high vibrational levels are populated by V-T pro- cesses, a situation typical of thermal dissociation. The situation at Tg = 1000 K is closer to that observed at T = 500 K, even though a true population inversion d8es not occur in this case.

Fig. 6 shows the behaviour of Iog ki versus 11’Ts for the same conditions as of fig. 4 except for Te = 10000 K. The results are very similar to those reported in fig. 4, since the rate coefficients for the excitation of vibrational Levels by electron impact do not change appreciably in the electron temperature range tC1000 G T, < 20000 K (see fig. 1). Again one observes an inversion in the population density at 7’ = 500 K, which is responsible for the high dissocration rates at this temperature.

422 M. CapiteE et al/H diwocintion rates in electrical dischaqes --. \ \ .-w%ec \ \ \ \ T3-i.i~ : \ \ \ \ \ \ \ w’sec

10‘'see-4 \ \ ‘a

= 500°K \

24 66 10 12 2 4 6 8 $0 12 2 4 6 8 10 12

VIBRATIONAL QUANTUM NUMBER

Fig. 5. Population densities (logNu) as a function of vibrational quantum number at different timer

F& 6; Val~ej of log ka and of log (k&) as a function of l/T-

The pIots of fig. 7 show that the dependence of ki on n, is quite strong for 500 < Tg G 2000 K (kia n3*l). As already pointed out, dissociation constants b&ome practically independent of ne at T > 3000 K. In figs. 8 and 9 we report population .densi%es of differ- ent vibrational levels, at a given time, for different eIec- tron densities ne_ At T = 1000 K the vibrational Ievels practically follow a Bo%zmann distribution for ne = 1010 cm-3 , while at n, = 101* cme3 this distribution is significantly altered, but RO population inversion becomes evident. A population inversion is on the con- Wary always present at Ta = 500 K for rze ranging from 1010 to 1012 cm-3.

Fig. IO shows the dependence of k$ on the pressure for a typical discharge condition (T = 500 K, T, = 20000 K and ne = 1012 cmW3). Thegdecrease of ki with Increasing pressure can be understood by considering that a pressure increase increases the importance of V-T processes with respect to V-V processes, and this destroys the “‘laser mechanism” responsible for the high dissociation rates at low Tg. In the same figure we re-

- : ____

AZ. Capitelli et &Hz dissociation rates in electrical discharges 423

ELECTRON DENSITY n,km-“1

Fig. 7. Values of logk% as a function of electron density (cni3) at different gas temperatures: (-_) normal V-T cross sec- tions; (---I W-T)/25).

port ki as a function of pressure for purely thermal conditions (n, = 0). In this case ki increases linearly with pressure, since ki, & defined by eq. (S), is a pseudo-first order kinetic constant.

4. Dependence of ki on cross sections

The distributions of the densities of vibrational levels of figs. 5,8 and 9 suggest that these distributions, as well asks, strongly depend on the assumed set of V-V, V-T and e-V cxoss sections. This point is of particular importance in our calculations, since the ab- solute values of V-T and V-V cross sections are known to an order of magnitude only. Therefore we have re- calculated the ki values for some of the conditions previously reported, scalingwith arbitrary factors all cross sections. We note (see table 1) that an increase of all V-V cross sections by a factor 10 leads to an in- crease of k$ by 80 times and, similarly, a decrease of the V-T values by l/3 is reflected by an increase of

NH*= 9.65*10’%m-” T, = 20000°K

i ’ i VlBRATlONAL QUANTUM NUMBER

Fig. 8. Population densities NU as a function of vibrational quantum number at a given time for different electron densi- ties ‘Tg = 1000 K).

ki of 60 times. The fact that ki is mare sensitive to the V-T processes than to the V-V ones is a conse- quence of the strong dependence of ki on the tail of theIV, distribution, which, in turn, is dominated by V-T processes (see fig. 11 and ref. 1161).

The dependence of ki on V-T cross sections has also been studied at different n,,(T = SO0 K, Te = 20000 K, NHz = 9.65 X 1016 cm-q. The strong in- crease of the dissociation constant obtained by decreas- ing the V-T cross sections by a factor 25 is apparent from fig. 7, particularly in the low electron density range. Nv distributions obtained with the new cross sections at ne = lOlo cmm3 (see fig_ 12)present typi- cal population inversion, not observed in fig. 9, because V-T processes have now lost their importance in de- stroying the V-V pumping effects. Fig. 4 shows the ef-

424 M. Capitellli et al. f& dissochtihrion rates in electrical d&charges

- .I

z 4 E 8 10 12 VlBRATlONAL QUANTUM NUMBER

Fig. 9. Population densitiesi?‘, as a function of vibrational quantum number at a given time for different electron densi- ties (Tg = 500 K):

Fects on kg obtained by a 10 times increase of the V-V cross sections or by decreasing the V-T cross sec- tions by the same factor. One should distinguish among different effects, depending wz the n, values. In the therms! region (tz, = 0) the V-V processes have prac-

Table I Values ofk$ (<I) cakulatcd with different assumptions on V-V and V-T cross sections CTg =_750 K, Te= 20000 K, rxe = IO” cc3, NH., = 9.65 X 10’6 cm 3,

p 31 d

k; W kS c) d

2.2 16.8.0 126.0

a) Normal values of V-V and V-T cross sections. b, V-V cross sections increased by a factor IO. ‘) V-T cross sections decreased by a factor 3.

i0 20 30 40 50

PRESSURE (torr)

Fig. IO. VaIues of log “2 as a function of prekure (torr>.

tically no influence on ki, while the V-T ones affect directly the term P,, U,+l in eq. (2), without consider- ably &ahangingiVU,. Ii this regime V-T and ks values are therefore proportional. Experimental dissociation rate constant in the purely thermal regizzze (Tg > 2500 K), as collected in ref. [ 171, fall between the full and dotted line of fig. 4. The situation is reversed for te + 0, when both V-V and V-T processes strongly influence kg primarily by acting on the population derz- sities (in particular on IV,.). In this case ki increases with decreasing V-T cross sections.

The sensitivity of ki on the rate coefficients of e-V processes depends on the particularprocess con- sidered. A strong dependence on the rate coefficient is in fact observed for the process (see table 21

e+HZ(u=0)+e+H2(u= 1). (4

The remaining electron vibrational process play, on the contrary, a minor role. This has been tested numerical- ly by equaling to zero all e-V transitions but 0-l (see table 2).

It is evident that the strong dependence of ki on the microscopic processes populating and depopulating

NV’s can not be easily explained, because of the great number of processes which create the IV,, distribution.

M. Capitelli et al/H2 dissociation rates in electrical discharges 42s

VISRATIONAL QUANTUM NUMBER

J

Fig. 11. Populationdensities N, es a function of vibrational quantum number at a given time: (--_) normal V-T and V-V cross sections; (---I (V-V) X 10; (-. -. -) (V-n/3).

Table 2 Values ofki (<I) calculated with different aswmptions on the toM number ofvIbrational levels and on e-V cross sections (~_;jSX_l K, Te = 20000 K, PZ~ = 1Ot’ ~I@,NH, = 9.65 X lof6

k; a) kib) kS c) d

kS a) d

18.9 41.0 17.0 11.0

a) Nom-d dues of e-V cross sections. b) Kz_l increased by a factor of 1.5. c, Ke = 0 for all transitions but O-1. d) ;yL = 17 instead of 16.

However some considerations can be made in order to 3NU,,/at = NUN&$*’ +N~P~~~~:-NuI.INH2Pvtl,“. understand qualitatively the results reported above. (W

I-

1 JIBRATIONAL QUANTUM NUMBER

Fig. 12. Population densitiesN” as a function of vibrational quantum number at a given time for different electron densi- ties CW-T)/25).

I ’ 2 4 6 8 70 72

One can distinguish three ranges of !he N, distribu- tier,: (i) the low levels, up to u = 4; (iij. the “plateau”, 4 < u =G 10; and (iii) the tail, u > 10. An attempt to ex- plain the tail of theNv distribution, dhich strongly af- fects ki, can be made by assuming that the populating rates of the tail come from the non-resonant process.

IIz(u = 1) f H2(u) -, H2(u = 0) + H&J + l), Cd)

and near-resonant V-V processes

Hz(u) + H,(u) + “Ju + 1) + H2(u - l), (e)

whiie the depopulating rates come from the V-T pro- cesses (see for example ref. [ 16) )

H2(U + 1) i- H, + Hz(U) + H, - Q

The simplified master equation for the (n f 1)t.h level can be written as

426 M. Capitelli et al./Hz dissociation rates in eIectrical discharges

This equation clearly shows the pumping effects of the V-V processes as well as’ the depletion effects of the V-T ones, the prevailence of the different terms being determined by the cross sections. In particular at high u the depopulating effects of the V-T processes be- come very important due to an increase in the corres- ponding cross sections (see fig. 2).

On the other hand the near-resonant V-V proces- ses, of the same order of magnitude of the correspond- ing resonant V-V reported in the same iigure, play an important role in affecting the distribution of the tail. Similar arguments apply to the non-resonant V-V, due to the large population of the first level.

The interplay of these energy exchanges is respon- sible of the resulting tail in the distribution.

5. Accuracy of the results

As shown in section 4, the accuracy of the present calculations is strongly affected by the assumed set of V-T, V-V and e-V cross sections. Other factors, ai- fecting the accuracy of the results, are: (1 j the pro- posed model for the dissociation level; and (2) the use of a maxwellian distribution function for the electrons.

As for point (1) we remember that the recombina- tion process is neglected in the present model as well as the V-T exchanges of hydrogen atoms with the cor- responding molecules. Some justification for this lies in the fact that kd settles to the stationary value, when the concentration of hydrogen atoms does not exceed 2%.

In the present mode1 one furthermore neglects both multiple V-T exchanges and the direct transition from each vibrational Ievel to the continuum. This last point has been found to be very important by Johnston and Birks [17], who used the following set of cross sections:

cu = (u + l)l(u’ + 1)Z exp [-(DO, - EU)IkTg] , (11) where 2 is the gas kinetic collision frequency and Di is the dissociation energy of the molecule.

Using these cross sections and inserting the micro- scopic processes

H, + X2(u) + H, •r- H2(u’ + 1) (g)

in the relevant master equations, we have estimated the importance of these transitions on k: _ The results re- ported in fig. 4, show that the above processes, relevant

at high temperature, can however be neglected in the T region characteristic of the discharges (300 d Tg B . 1500 K). AS for the multiple exchange transitions between bound Ievels such as

HZ(u) * H2 + ‘fi2(w) + H,, Au=w -tG=i, (h)

these are generally lower than the Au = 1 process. As for the location of the dissociation level, we have

already justified in section 2 the identification of this level with the last bound level of the molecule. Data re- ported in table 2 show the effect of a different location of the (u’ f 1) level.

The correct approach to handle point (2) should be to solve an appropriate Boltzmann equation forf, taking into account elastic and inelastic processes. The calculated fe should then be used in the calculation of the electronic rate coefficients. The use of a maxwellian distribution function for f,, made in this work, can therefore modify the absolute vahres of Kz, and there- for of ki, without, however, altering the qualitative be- haviour discussed in the previous pages. Large devia- tions from a maxwehian distribution are not expected in H, plasmas.

It is interesting to compare the present results with those obtained by the direct electronic mechanism e + H, + e f 2H. This mechanism is essentially due to the impact of electrons on the ground vIbrational state of hydrogen molecule (’ Z), exciting the target to the repulsive (3Z) molecular state, which dissociates to atoms.

Values of kz have been obtained by numerical in- tegration of the experimental cross sections of Corrigan [18] over a maxwellian distribution function forfe (see eq. 4), The results (cme3 s-I) have been reported in fig. 1 as a function of eIectron temperature. The cor- responding values kzn, (s-l) are reported in figs. 4 and 6 and compared with the present /ci values.

It should be noted that the value of kzn,, which is independent of the gas temperature, lies always above. our ki values for T, = 20000 K, tze = lOI cme3 and for T < 3000 K and aIways below the ki values of the same$igure, calculated with different V-T or V-V cross sections.

The comparison at T, = 10000 (fig. 6) shows that kztz, 5 ki for T, 5 750 K. These results indicate that in an electrical discharge the dissociation of molecules can occur along two different paths, the relative im- portance of which depends on the particular conditions

M. Capitelli etal.l./Hz dissociation rates in electrical discharges 427

Table 3 A comparison of theoretical and experimental (ref. [4] 1 ka vahs (in cl)

ExperimentaI conditions s a kd texp.) kd k%ne

p (ton) T,(K) Tg (K) fze (cln-3)

10 15500 510 1.3 x 10” 59 68 3 10 23000 640 1.4 x 10” 133 53 28 20 15500 700 3.1 x 10” 114 60 7 20 16500 860 1.4x 10” 68 5 5 20 21000 910 1.9 x IO” 149 13 24

a) Values obtained by increasing the V-V cross sections by a factor of 10 and by deaeasin, D the V-T cross sections by a factor of

of the discharge [i.e., on Te, Tg, ne (fig. 7) and pres- sure (fig. la)] _

A comparison between experimental and theoretical values of kz is faced, at present, wilh the uncertainties involved in both the theoretical calculation and in the evaluation of discharge parameters. In the range of the gas temperatures characteristic of the electrical dis- charges investigated (300 < Tg < 1500 K) calculated values of ki are, as discussed, strongly dependent of the selected values of V-V, V-T and e-V cross sec- tions, as well as on n, and on NH~ _ Experimental con- ditions estimated in ref. [4} have been collected in table 3 together with the corresponding rate constants kd(exp). Calculated vaiues of ki using “normal” cross sections are lower than thk experimental ones by more than two orders of magnitude. Values of kzne, repor- ted in table 3, are also weU below the experimental ones. However, a reasonable variation in the absolute values of V-T and V-V cross sections can lead to sig- nificant improvements. A factor of 10 in the V-V cross sections can be justified by the fact that the cor- relation data of ref. 1141 for V-V relaxation times (see eq. 5b) could be reasonably decreased 10 times (increasing V-V cross sections by the same factor) (see fig. 11 of ref. [ i4] ). A factor OF 3 in the V-T cross sections is, at Least, the incertitude involved in the equations used. Values of ki calculated with this new set of cross sections have been collected in table 3.

Scaling of the e-V cross sections should not exceed a factor of about IS, since the experimental data of ref. [lo] for the O-l e-V transition do not differ by more than this factor from those measured by other

workers [ 19,20]_ However, one should consider the possibility of the following mechanism for populating the first vibrationai level

e+H2(u=0,j=O)+e +H2(u=0,j=2), 6)

e+H2(U=0,j=2)~e+H2(u= I,i=O), 6)

where j is the rotational quantum number. This mecha- nism (or a similar one involving other vibrotational sta- tes), due to the large cross sections of process(j) [L9] , could increase the electronic rate coefficient of level 1, because of the smaller energy gap of process (j) as com- pared with

e+H2(u=0,j=O)-+e+H2(u= l,i=O)_ (0

This contribution shoutd increase with increasing popu- lation of the i> 0 levels, i.e., with increasing rotational temperature TR, hence with increasing T , which re- presents the minimum possible value of %

It should be appreciated that a variation of 12~ by a factor of 2 entrains a variation of kz by a factor of 10 at n, = 10IL cmm3 and T = 750 K (fig. 7). This strong dependence of k: on ,Iegmakes the comparison very sensitive to the actual value of ne in the discharge. This value is, however, affected by errors particularly as.a consequence of the radial constriction of the plasma column [21] ; values of ,ze reported in table 3 are esti- mated radial averages.

The dependence of k$ on both tze and the e-V cross sections KS, (table 2) is such that the calculated values of ki can be brought into coincidence with the experi- mental ones by very reasonable variations of Kzu and/or

%-

418 I#_ CapitelIi et al./Hi dissociation rates in electrical discharges

The present results refer to conditions in which the effect of diffusion of excited molecules to the walls can be neglected. Diffusion effects should however be taken into account, when dealing with laboratory plasmas. According to ref. [4] the diffusion effects should become important at pressures around 5 torr. For a correct comparison with experiments one should therefore include diffusive fluxes in each master equa- tion. Such computations are however beyond the pur- pose of the present paper.

6. Conclusions

The results reported in the previous pages clearly show the importance of a ‘Yaser-type mechanism” in the dissociation of molecular hydrogen in an electrical discharge *_ This mechanism acts at low gas tempera- ture as a result of a Rumping of higher vibrational levels by e-V and V-V processes. The V-T processes tend, on the contrary, to create a Boltzmann d.istribution along& at Tg, so decreasing the dissociation rate in the temperature range 500 K G Ti < 3000 K.

A thermal dissociation mechamsm is indeed impor- tant for T > 3000 K; in this region V-V and e-V processes f oose their importance as compared with V-T processes_

Besides the “laser-type mechanism”, a direct disso- ciation mechanism (e -t Hz = 2H) is an important re- active channel. The two mechanisms operate simul- taneously; however, it is possible, in principle, to de- fine the electrical discharge conditions, for which one or the other mechanism prevails.

The comparison wi+h+he experimental rate constants sari lead to satisfactory agreement, but its significance is, at present, limited by the uncertainties involved in both the cross sections of V-V and V-T processes and in the experimental discharge parameters.

Work is now in progress in this laboratory for im- proving the accuracy of the present results, even though we believe that the qualitative behaviour discussed in the previous pages w‘fi be left unchanged.

* The terminology ‘laser-type mechanism” is used in this pa- per for indicating the effect of V-Vpumping on the H2 dii sociation. V-V pumping, however, is not the only possibility for a dissociation mechanism induced by lasers,

Acknowledgement

The authors thank the experi,mental group .of our laboratory (Professors F. Cramarossa; R. d’Agostiuo and P. Capezzuto) for suggesting and discussing the problem, professor D. Trigiante for the numerical algorithm of Appendix B, and professor E. Ficocelli Varracchio for stimulating discussions on the relevant cross sections. Drawings are by U.Farella.

Appendix A: Rate coefficients for V-T and V-V pro- cesses

The following set of equations has been used for cal- culating the rate coefficients of V-T and V-V proces- ses,

P ","_I =qJPllTgw(l -~v)lF(r,u_,>~

q;:i* =Z11Q11T'b/U -au)1 2

x wu - 6 WI nY;;y 1, where Z1 1, the Hz-H2 collision number (cm3/s), is given by

o11 = *Hz-Hz = 2.93 A, and

F(x)=+]3 - exp(-2x1311 exp(-2x/3),

X v,v_l = t+J3’2(6;,/Tg)1’2U - 25u),

y;;‘r = 2S(:)3’2(0;,/Tg)1~2(u -w) >- 3

%I. = 3.48 X IO6 K,

and where PII and Q,, have been obtained by matching the equations to the correlation data of relaxation times reported in section 2.

Appendix B

The system of differential equations typified by eqs. (1) and (2) can be written in vector form as;

adat = j(n).

&f. Capitelli er al&z dissociation rates in ekWrica1 discharges 429

The following numerical algorithm has been used to solve it 1221 ,

n(t + h) = n(t) f (I- :hJ + h-h2 J2)-%f(n(f)),

where I is the identity matrix: 3 is the jacobian of the system calculated at time t and k is the step size. The method assures the absolute stability of the solutions and the error is of the order of h-‘.

Several tests were made in order to choose the best value for h. It was found that a step size of lo-’ s was to be used in the early stages of solution (up to t = 10W6 s), while longer step sizes can be used from 10m6 s on.

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