P H O T O D I S S O C I A T I O N OF D I A T O M I C M O L E C U L E S

U N D E R N O N E Q U I L I B R I U M C O N D I T I O N S

M. N. S a f a r y a n UDC 541.141.1

It has recent ly become possible to study photochemical p rocesses at high light intensit ies. In the presence of laser radiation the known laws of p r imary photochemical p rocesses , which do not take into ac- count the nonlinear effects of the interaction of light with an absorbing medium, may become inaccurate . Specifically, the reaction may be accompanied by the breakdown of s tat is t ical equilibrium, and therefore by the breakdown of the l inear law of light absorption. The e lementary react ion of molecular photodissocia- tion [1] is an example of this . In this paper simple methods are given for obtaining a nonequilibrium quasi- s ta t ionary function for the distribution of molecules in vibrational levels and, consequently, an expression for the photodissociation constant of diatomic molecules which takes into account the breakdown of the Boltzmann distribution if the equilibrium value of the dissociation constant is known.

Let us consider a gas at constant tempera ture T and p ressu re p, with an initial concentration N of absorbing molecules . Exposed to light of frequency v and intensity I a molecule will pass, with probability wig), f rom the i - th vibrational level of the ground electronic state to one of the upper s tates; if the excited state corresponds to a repulsive potential curve, photodissociation takes place. Molecular rotation is not considered, and periods of t ime t > T r r (Trr is the rotational relaxation time) are taken into account.

The photodissociation constant can be written in the form

1

where n is the total molecular concentration and f i is the concentrat ion in the i - th level, summation being ca r r i ed out over those levels f rom which light absorption takes place. The equilibrium value of the photo- dissociation constant is therefore

k o 1 o

i i

where f0 is the Boltzmann distribution over the vibrational levels, ~0 and cr~ are , respect ively, the total c ro s s section and the c ross section for the absorption of light f rom the i - th level in the l inear region of ab- sorption, and S = 1 / h u is the photon flux.

It follows f rom (1) and (2) that

m

w h e r e ~, ~ f , l~ , y ' f, = n, 1=o

then (3) takes the form

~) 0

l (1

- ~ - = n V w f ~ o ~ ' X.~ i ' i p

i

m

2 f~ = N. If absorption takes place mainly f rom one level, i .e. , f~w~ >) ~.V fkwk, i=o ~=~

(3)

Institute of Prob lems in Mechanics, Academy of Sciences of the USSR, Moscow. Transla ted f rom Teore t icheskaya i l~ksperimental 'naya Khimiya, Vol. 8, No. 3, pp. 322-326, May-June, 1972. Original art icle submitted March 15, 19 71.

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260

k__p__ N kO -- q~ ~- ~ 1 --O~ (4) .p

and the difference between the photodissociation constant in nonequilibrium conditions, and its equilibrium value is unequivocally related to the breakdo~z of the Boltzmann distribution over the vibrational levels.

To determine ~i, let us wri te a sys tem of kinetic equations for equilibrium under conditions of s ingle- quantum vibrational t ransi t ions of molecules ar is ing f rom interaction with the surrounding medium:

df i dt -- (P/,i-z @ P/.I+L)):] + Pi+1,/fi+I + P1-1,1[i-1 - - wifi (1 = O, I . . . . . m), (5)

where Pj , j+I is the probabili ty of a molecule passing f rom the j - th to the (j + 1)-th level due to collisions with gas par t i c les . We will assume that Tp = kp I >> ~-v, where ~'v is the vibrational relaxation time, the study being res t r i c ted to time periods t > T v. In this case the determination of 0i is grea t ly simplified.

In the sys tem under considerat ion, where t > T v, a quasis tat ionary condition is established in which the distribution function depends on t ime only in t e rms of the total molecular concentrat ion. This is ex- plained by the fact that for Tp >> ~'v at each moment of t ime t > ~'v the vibrational relaxation is able to "follow" the concentrat ion change. This situation is typical of s imi la r sys tems with negative sources , and is repeatedly discussed when considering thermal dissociation in s tat is t ical nonequil ibrium conditions. Note that an overal l descr ipt ion of the kinetics in our sys tem has a formal analogy with thermal d i ssoc ia - tion, except that in this case the negative source of par t ic les originates f rom light absorption and can be situated at any level, but in the case of thermal dissociation it is at the last level: the vibrational r e l axa - tion mechanism i tself is identical, i .e., thermal .

The quasis ta t ionary solution of Eq. (5) can be put in the form f j = n~j, where ~j is independent of

df i ~ dn 51. "~ t ime. Therefore , ~-[ = [ - - ~ - - - n ~ w J J i and, for values Cj = 1 - (N/n)~pj, (5) takes the form

(1 - - (D /) ~ w~ ( l - - ~ ) i )~ ~ = (P j , /+= .-[- P M - I ) O / - - P l d + z O i + l - - P i , i - l O i -1 - - w / ( 1 - - 0 i ) . (6)

For cases of prac t ica l in teres t sys tem (6) can be l inearized, i .e. , we can put

:" ' ~ w~r~-- V,w~:~O~, (7) (1 - % ) ) Z ~ , tl - o , 1 / ~ (1 - % 1 Z ~~ ~0 i r i

and ~i is then determined from the usual solution of a sys tem of l inear nonuniform algebraic equations. In

m 0 pract ice it is sufficient in this sys tem to consider l algebraic equations, i < l << m, ~ :: ~(, 1, assuming in

the approximation that e -zh~ ~ 1 q)z ~ 0 (/~m is a quantum of vibrat-ional energy).

If i >> 1, the following simple calculation method, which Was used previously for the thermal d i ssoc ia - tion problem [2], can be used to determine the nonequilibrium distribution function. Let the photodissocia-

m

tion take place mainly f rom one i- th level, and in addition ~f~ (~ i; in this approximation it can be a s - sumed (as in [2]) that :~t

p o dn - - v = Pi.i_1~iiq)i - - /_1,if/_:pi_ I, v = - - d"-T " (8)

that Taking the condition of detailed balancing into account, i .e. , Pj , j - l f j = P j - l , j f~ - l , it follows from (8)

i

- - V v p o

i = l + 1

= % - - r

f rom which, after multiplying by f~l and summing over l f rom 0 to i, and in the same approximation putting v = w i f i , we have

N wj~N-1Ao]-I, n % = ii + (9)

261

w h e r e

A0=E r: E ' P i,j-lf~ /=0 /=lq-I

(lo)

(Eq. (9) with (10) c o r r e s p o n d s to a s i m i l a r equat ion in the nonequ i l ib r ium t h e o r y of t h e r m a l d i s soc ia t ion , if ins tead of wi we put Pkd, the p robab i l i ty of t h e r m a l decay.)

Note that a s i m i l a r me thod can a lso be applied when i - 1 [1], but with li o0/kT << 1, so that f~ << 1; such high t e m p e r a t u r e s a re highly unl ikely in pho tod issoc ia t ion , and t h e r e f o r e when i = 1 and i << m, it is m o r e use fu l to d e t e r m i n e r f r o m (6) o r (7).

The r e s u l t (8) can a l so be obta ined without us ing the r e s t r i c t i o n that the v ib ra t iona l t r an s i t i on s m u s t be s i ng l e -quan t um t r ans i t i ons , p rov ided the condi t ion 1i w~ kT<< 1 is fulfi l led (it is suf f ic ient tha t 1 - e-bOo/kT ~_ lic0/kT). In this c a s e , ins tead of s y s t e m (5) the F o k k e r - P l a n c k diffusion equat ion m u s t be used fo r the d is t r ibut ion function f (s t) for a v ib ra t iona l e n e r g y s and it mus t be taken into account that the q u a s i s t a t i o n a r y flow of m o l e c u l e s in the space ~ is equal to the pho tod i ssoc ia t ion r a t e , i .e . ,

0~ = v, v = Wfo (e~, t) (11) --Bf~ (B is the "diffusion" coef f ic ien t [3], and w f ~ (s t) = w i f e ) . In tegra t ing (11) ove r de f r o m 0 to s i, mu l t ip ly -

ing the r e s u l t by f0 (s and in tegra t ing again, then, in the approx ima t ions t" f~ ,~.N and I fde ----. n o 0

(9) and (10), we have

N 8 --~- q~ ( t, t) = [I Jr" wf~ (ei)N-'Al] -1,

, i n s t e a d of

(12)

where

e l el

�9 B f ~ ( s ) " 0 �9

(13)

I t can ea s i l y be seen f r o m (9) or (12) tha t if wif~N-1A0 << 1, then cPiN/n -~ 1 and kp - k~. On the o the r hand, fo r w i f ~ 0 >> 1 we have

i .e . , the pho tod i ssoc ia t ion cons tan t is e n t i r e l y d e t e r m i n e d by the d i v e r g e n c e of the d i s t r ibu t ion function f r o m the Bol tzmann f o r m and is independent of the light i n t e n s i t y .

I t a l so fol lows f r o m (6) and (9) that the effect of nonequ i l ib r ium is m o r e impor t an t fo r p h o t o d i s s o e i a - t ion f r o m uppe r l eve l s . Al though this p r o c e s s is u sua l ly un impor tan t , when l a s e r rad ia t ion is u sed it can b e c o m e s igni f icant and the light in tens i ty at which the b reakdown of s t a t i s t i ca l equ i l ib r ium should be taken into account can be obta ined f r o m the r e l a t ionsh ip w i f ~N-1A0 ~- 1, i .e . , S ~ S o = gr~176 on the o ther hand, when S << S 0 this f a c t o r can be neg l ec t ed . Fo r the ca lcu la t ion it is use fu l to a s s u m e that fo r a mode l h a r m o n i c o sc i l l a t o r mo lecu l e when i >> 1

f 8iho~/kT elBe/kT

i=I

Molecu la r photod issoe ia t ion unde r nonequ i l ib r ium condi t ions is o b s e r v e d e x p e r i m e n t a l l y when ruby l a s e r l ight is p a s s e d th rough iodine vapor ( t ransi t ion IZ~ -~ A3IIm) fo r t < r v. Let us eva lua te ~ i for this r e a c - t ion us ing w i f r o m [1]. (In ca lcu la t ing w i it was n e c e s s a r y to r e s t r i c t the potent ia l c u r v e for the II1u s ta te taken f r o m [5], so that the r e l a t i ve va lues of the F r a n c k - C o n d o n f a c t o r s given in [I] fo r the t r ans i t i ons

2 6 2

o0I f r o m levels i = 0, 1, 2, 3, and 4 a r e only approximate . ) We now have ~)0---- , w i / w 0 = 1; 0.7;

w~ feN_,

i

0.57; 0.14; 0.56 .(i = 0, 1, 2, 3, 4); ~0 = 1.5 �9 10 -i9 c m 2, hv = 14,400 c m -1. In o r d e r to fulfill the condit ion T v << Tp -- ( I a ~ - l , it is n e c e s s a r y that 0.5IT v << 1 (I in W / c m 2, T v ill sec) , i .e . , for ~'v - 10-s sec [4] the in tens i ty of a con t inuous ly opera t ing l a s e r {t > T v) m u s t be in the reg ion I << l0 s W / c m 2. In p r inc ip le the pho tod issoc ia t ion r eac t ion e a n b e o b s e r v e d e v e n w h e n I ~ W/cm2, * provided T v << t < T r , w h e r e ~'r is the r ecombina t ion t ime , including r ad ia t ive r ecombina t ion . Final ly , if w0~- V - 0.2, then fo r r (0 -< i < 5) we find f r o m (7) tha t ffi < if0 -~ 0.1.

1. 2. 3. 4. 5.

L I T E R A T U R E C I T E D

M. N. Sa fa ryan , Opt. i Spek t rosk . , 3__00, 767 (1971). E . V. Stupochenko and A. I . Osipov, Zh. F iz . Khim. , 3._~2, 1673 (1958). E . V. Stupochenko and M. N. Sa fa ryan , T e o r . i ~ s p . Khim. , 2, 783 (1966). N. A. Genera lov , G. I . Kozlov, and V. A. Masyukov, Zh. l~ksp, i T e o r . F iz . , 58, 438 (1970). L . Math ieson and A. Rees , J . Chem. Phys . , 2_.55, 753 (1956).

Amount m i s s i n g in Rus s i a n o r ig ina l - Consul tants Bureau.

263