Vibrational relaxation of an anharmonic oscillator

V I B R A T I O N A L R E L A X A T I O N O F A N A N H A R M O N I C O S C I L L A T O R

M . N. S a f a r y a n a n d N . M. P r u c h k i n a UDC 536.45

We have d e t a i l e d knowledge of v i b r a t i o n a l r e l a x a t i o n fo r d i a t o m i c m o l e c u l e s in an i n e r t g a s if the m o l e c u l e can be r e p r e s e n t e d as a h a r m o n i c o s c i l l a t o r [1]. One e x p e c t s that a n h a r m o n i c i t y w i l l not g r e a t l y a f fec t the m a c r o s c o p i c v i b r a t i o n a l r e l a x a t i o n t i m e , s ince v i b r a t i o n a l a n h a r m o n i c i t y i s i m p o r t a n t only f o r the h i g h e r v i b r a t i o n a l l e v e l s , w h o s e p o p u l a t i o n s p l a y a p a r t only at r e l a t i v e l y high t e m p e r a t u r e s , w h e r e v i b r a t i o n a l r e l a x a t i o n cannot be c o n s i d e r e d as s e p a r a t e f r o m o t h e r p r o c e s s e s , e .g . , d i s s o c i a t i o n . Howeve r , t h e r e can be m a r k e d e f f ec t s on the r a t e of a t t a i n m e n t of the e q u i l i b r i u m d i s t r i b u t i o n and the e q u i l i b r i u m m e a n e n e r g y , so it i s of i n t e r e s t to e x a m i n e the e f f e c t s of a n h a r m o n i c i t y on the r e l a x a t i o n . A n h a r m o n i c i t y n a t u r a l l y b e c o m e s m o r e i m p o r t a n t i f the s y s t e m con ta in s h igh ly e x c i t e d m o l e c u l e s p r o d u c e d by c h e m i c a l r e a c t i o n .

D e t a i l e d q u a n t u m - m e c h a n i c a l c a l c u l a t i o n s [2-4] * have been made fo r a d i a b a t i c c o l l i s i o n s of m o l e c u l e s wi th i n e r t - g a s a t o m s , which a l low one to e v a l u a t e the e f f ec t s of v i b r a t i o n a l a n h a r m o n i c i t y at r e l a t i v e l y low t e m p e r a t u r e s : e -hc0 /kT << 1 (hw i s a quan tum of v i b r a t i o n a l e n e r g y and T i s t h e r m o s t a t t e m p e r a t u r e ) . At h i g h e r t e m p e r a t u r e s such tha t hc0/kT << 1 (in p r a c t i c e , fo r e - h w / k T ~ 1 - h w / k T ) , one can c o n s i d e r the r e l a x a t i o n wi th in the f r a m e w o r k of c l a s s i c a l s t a t i s t i c s ; in p a r t i c u l a r , the d i f fus ion a p p r o x i m a t i o n can be u s e d . H e r e we u s e the d i f fus ion t h e o r y wi th s o m e a s s u m p t i o n s about the d i f fus ion coe f f i c i en t to e s t i m a t e the e f f ec t s of a n h a r m o n i c i t y on the v i b r a t i o n a l r e l a x a t i o n t i m e and on the d e t a i l s of the p r o c e s s .

C o n s i d e r a s y s t e m c o n s i s t i n g of a s m a l l p r o p o r t i o n of a d i a t o m i c s p e c i e s in an i n e r t ga s (a t h e r m o s t a t at t e m p e r a t u r e T). At the s t a r t , the s t a t e of the m o l e c u l e s r e p r e s e n t s e q u i l i b r i u m at t e m p e r a t u r e T 0. W i t h - out spec i fy ing how t r a n s l a t i o n a l e n e r g y i s c o n v e r t e d to v i b r a t i o n a l o r v i ce v e r s a , we a s s u m e that the m o l e - c u l e s i n t e r a c t only w e a k l y wi th the t h e r m o s t a t a t o m s , so the r o o t - m e a n - s q u a r e i n c r e m e n t e in the v i b r a - t i ona l e n e r g y i s s m a l l r e l a t i v e to the e n e r g y r a n g e wi th in which f ( e , t) v a r i e s a p p r e c i a b l y . We can then use the F o c k e r - P l a n c k equa t ion [5] a s the k i n e t i c equa t ion fo r f ( e, t) :

(--~-8 - - Olnf~ , af a {B of at de

w h e r e B = <(A~)'2>/2T, A r i s the change in the e n e r g y of a m o l e c u l e a s a r e s u l t of c o l l i s i o n wi th a t h e r m o - s ta t ; ~- i s the t ime t aken by a m o l e c u l e to t r a n s v e r s e a m e a n f r e e path; ( . . . } de no t e s a v e r a g i n g o v e r a l l c o l l i s i o n s of a m o l e c u l e ; f 0 ( e ) i s the e q u i l i b r i u m d i s t r i b u t i o n c o r r e s p o n d i n g to a t h e r m o s t a t t e m p e r a t u r e T. The "d i f fus ion coe f f i c i en t " B i s dependen t on the f o r m of the i n t r a m o l e c u l a r and i n t e r m o l e c u l a r i n t e r - a c t i ons ; the r e s u l t s a r e a s fo l lows [7, 8] r e s p e c t i v e l y fo r a h a r m o n i c o s c i l l a t o r and a M o r s e o s c i l l a t o r :

B =: ~lkTe, (2)

B = 2~IkYD l'- 1 - - e/D (1 - - i / 1 - - e/D}, (3)

w h e r e D i s the m o l e c u l a r d i s s o c i a t i o n e n e r g y and ~? (the coe f f i c i en t of f r i c t i o n [6]) i s t aken a s independen t of the i n t e r n a l s t a t e of the m o l e c u l e , i . e . , of 5. Th i s i s we l l j u s t i f i e d fo r n o n a d i a b a t i c c o l l i s i o n s of the m o l e c u l e s wi th the t h e r m o s t a t a t o m s (in p a r t i c u l a r , i t fo l lows f r o m [5, 8]), but i t i s an a s s u m p t i o n , a s in

*In r e f e r r i n g to [4], we have in mind r e s u l t s f r o m an unpub l i shed c a l c u l a t i o n on th i s topic .

I n s t i t u t e of C h e m i c a l P h y s i c s , Moscow Reg ion Branch , A c a d e m y of S c i e n c e s of the USSR. T r a n s l a t e d f r o m T e o r e t i c h e s k a y a i E k s p e r i m e n t a l ' n a y a K h i m i y a , Vol . 6, No. 3, pp. 306-310 , M a y - J u n e , 1970. O r i g i n a l a r t i c l e s u b m i t t e d M a r c h 19, 1969.

�9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $]5.00.

249

/,2

1,o

o,8

o,5

0,4 i/ / I

0,2

7 e

! uncate d g~ armonic

oscillator 0,4,

0,2 %=30

'8 /2 ' ~ '

Fig. i Fig. 2

Fig. 1. Dependence of ~-v/~-i on the rmos ta t t empera tu re . Solid l ines,

, ..L , ~

I !

2 3 f = _ t _ rs

no allowance for dissociat ion; broken l ines, d issocia t ion incorporated.

Fig. 2. 1 and 2) Relaxat ion of Morse osc i l l a to rs : 1) without allowance for dissociation; 2) with dissociat ion; 3) re laxat ion of harmonic osc i l l a - to rs .

[6, 7], fo r col l is ions of another type. If the molecules a re harmonic osc i l l a to r s , ~? = AE/kT , where AE is the average energy t r a n s f e r r e d to an unexcited osc i l l a to r in unit t ime, which can be calculated c lass ica l ly o r quantum mechanical ly.

The re is a well known solution to (1) with the coefficient of (2) ( relaxat ion of ha rmonic osci l la tors) in the absence of chemical react ion, which impl ies that:

a) the dis tr ibut ion for harmonic osc i l l a to r s r e l axes while retaining a Bol tzmann dis tr ibut ion f (~, t) = e - s where 0 = T - ( T - T 0) e-t/~'t ;

b) the mean molecu la r energy E v a r i e s as follows, no m a t t e r what the initial distr ibution:

dE _ E e --- E (4) s dt ~1

where T t = ~-1 is the v ibra t ional re laxat ion t ime for harmonic osc i l l a to r s and E e is the equi l ibr ium E. As r 1 is governed solely by ~, the effects of v ibrat ional anharmonic i ty can be examined without specifying the nature of the coefficient of friction.

Equations (1) and (3) desc r ibe the v ibra t ional re laxat ion of Morse osc i l l a to r s with the equi l ibr ium function

D e - - e , k T . e - e / a T

_ _ _ _ dE, fo (e) = A (r) V]--~-~ e-TD - ' A (7")- .,i V1 - - e/D 0

and the conditions

e=O-'=- : ! g--8/kTo of of =- o; f (8, o) = Os ~ ~=D A (To) ]t/i ~ e/D

Numer ica l (computer) solutions have been obtained. The v ibra t ional re laxat ion t ime ~'v is defined fo rmal ly a s

E (~v) = Ee - - (Ee - - Eo) e - ' (5) D

(E (t)= :~ ~ f ( ~ , t) d~: is calculated, while E 0 is initial energy). The symbols used below a re x=~ : /D , 0

250

2

{ =0,2X f "

/ / , I I f I I

,:1.

a:_3o I I / I I

2

r r i r , _ 1

o 0,2 0,4 o, fi 0,6 e b x=~

Fig . 3. R e l a x a t i o n of the r e l a t i v e d i s t r i b u t i o n s fo r M o r s e o s c i l l a t o r s [~0~ ix, i)l and h a r m o n i c ones [%h (x, t)] fo r : a) m o l e c u l a r a c t i va t i on ; b) m o l e c u l a r deac t i va t i on .

•! a~,=5, o=30 ~ / /

I %= 7, o=20

O i i i i , , , a

0 0,2 0,4 0,6 x=z D

Fig . 4. E f f e c t s of i n i t i a l and e q u i l i b - r i u m m o l e c u l a r e n e r g i e s on ~M (x, ~)/%h (x, ~').

-t = b? = t / T 1, q~ = f / f o , a = D / k T , a 0 = D / k T 0. F i g u r e s 1-4 g ive the r e s u l t s , which a r e a s fo l lows .

At h igh t e m p e r a t u r e s (a o r a 0 ~ 20), r v d i f f e r s f r o m "q (Fig . 1), and -r v/~'1 i n c r e a s e s s l i g h t l y a s a d e c r e a s e s (a < a0}: a t a = 5 by about 30%. Subsequen t ly , ~'v/~] < 1 fo r a < 3, but ~'v fo r e =D r e m a i n s g r e a t e r than the r e l a x a t i o n t i m e f o r t r u n c a t e d h a r m o n i c o s c i l l a t o r s . The t r e n d in " rv / ' r I wi th a i s not m o n o - ton ic b e c a u s e E e / k T h a s a s i m i l a r t r end , which i s d i r e c t l y r e l a t e d

to the f o r m of f ~ fo r M o r s e o s c i l l a t o r s , a s i t b e g i n s to i n c r e a s e f o r e / D > 1 - 1 / 2 a . Thi s f e a t u r e h a s l i t t l e p r a c t i c a l e f fec t , on accoun t of d i s s o c i a t i o n , w h i c h m u s t be t a k e n into accoun t fo r a <- 7, and wh ich c a u s e s E(t) fo r t ~ ~'v to r e l a x not to E e but to s o m e va lue E* < E e that c o r r e s p o n d s to b a l a n c e b e t w e e n the e n e r g y inf lux f r o m the t h e r m o s t a t and the l o s s f r o m d i s s o c i a t i o n [7]. F i g u r e 1 shows that the v i b r a t i o n a l r e l a x a t i o n t i m e with d i s - s o c i a t i o n i s a l w a y s g r e a t e r fo r a M o r s e o s c i l l a t o r than i t i s fo r a t r u n c a t e d h a r m o n i c o s c i l l a t o r , but on the whole i t is l e s s than when d i s s o c i a t i o n i s ne g l e c t e d . That t i m e i s de f ined by (5) wi th E e r e p l a c e d b y E * , whi l e E* i s found f r o m n u m e r i c a l so lu t ion of the ana logous p r o b l e m wi th the b o u n d a r y cond i t ion f ( D , t) = 0.

If we s p e c i f y the f ina l e q u i l i b r i u m s t a t e ( i .e . , a) and v a r y the i n i t i a l s t a t e ( i .e . , a 0), Tv then i n c r e a s e s wi th E 0. F o r i n s t a n c e , ~-v/~-i v a r i e s f r o m 1.1 to 1.3 a s a o g o e s f r o m 15 to 1 i f a =30 ( i .e . , E0/D g o e s f r o m 0.07 to 0.23). If the i n i t i a l and f ina l s t a t e s c o r r e s p o n d to high t e m p e r a t u r e s , Tv i s s o m e 10-15% l a r g e r than when a l o w - t e m p e r a t u r e s t a t e i s i nvo lved (a < a 0 ~ 30 o r a 0 < a ,-~ 30); if a and a 0 ~ 20, ~'v e x c e e d s ~'1 by not m o r e than 5-10%.

F i g u r e 2 shows tha t the t i m e d e p e n d e n c e of the m e a n m o l e - c u l a r e n e r g y d i f f e r s f r o m (4); but i f in (4) we r e p l a c e ~'l b y the c o r r e s p o n d i n g T v g iven by (5), the u s u a l e x p o n e n t i a l law c l o s e l y d e s c r i b e s a l s o the v i b r a t i o n a l r e l a x a t i o n of M o r s e o s c i l l a t o r s (in fac t , t h i s s e r v e s a s an a p p r o x i m a t i o n to the c a l c u l a t e d curve) . The M o r s e o s c i l l a t o r r e l a x a t i o n i s a l so only s l i g h t l y dependen t on the f o r m of the i n i t i a l d i s t r i b u t i o n . F o r i n s t a n c e , ~-v/~-t fo r f (x, 0) ~ 5 (x-x0) d i f f e r s only s l i gh t ly f r o m the va lue found fo r an i n i t i a l B o l t z m a n n d i s t r i b u t i o n , w h i l e (5) s e r v e s a s a good a p p r o x i - mat ion .

The a n h a r m o n i c i t y h a s an a p p r e c i a b l e e f fec t on the b e h a v i o r of the d i s t r i b u t i o n du r ing the a p p r o a c h to e q u i l i b r i u m . R e l a x a t i o n c a u s e s a d e v i a t i o n f r o m the i n i t i a l B o l t z m a n n f o r m , and t h i s d e v i a - t ion i s l a r g e s t fo r h a r m o n i c o s c i l l a t o r s at high e n e r g i e s (e ~ D);

f o r a 0 > a ( T o < T), the d e v i a t i o n i s i m p o r t a n t fo r t < ~'l, wh i l e i t i s i m p ~ fo r t > "r 1 i f a 0 < a (T 0> T) (Fig . 3). T h i s d i f f e r e n c e i s m o r e p r o n o u n c e d fo r d e a c t i v a t i o n (T O > T) than fo r a c t i v a t i o n (T o < T); in the region ~ kT, eM/%h i.

If the i n i t i a l t e m p e r a t u r e i s h igh (a 0 ~ 7 a n d a 0 < a , e s p e c i a l l y fo r a 0 << a) , q~0 M r e l a x e s wi th c o n s i - d e r a b l y m o r e r e d i s t r i b u t i o n in the u p p e r l e v e l s ( i .e . , m o r e s lowly) than i s the c a s e fo r h a r m o n i c o s c i l l a t o r s ; fo r i n s t a n c e , q~0M/~P h - 102 in the r e g i o n e ~ 0.7 D, 2~- 1 -< t s 5~- 1 f o r a 0 =7 and a =30. T h i s e f fec t i n c r e a s e s w i th the i n i t i a l e n e r g y of the m o l e c u l e and a s the f ina l equa l e q u i l i b r i u m e n e r g y d e c r e a s e s (Fig . 4). In p r i n c i p l e , d e v i a t i o n f r o m a B o l t z m a n n d i s t r i b u t i o n could be d e t e c t e d by e x p e r i m e n t ; but the e f fec t in the u p p e r l e v e l s fo r M o r s e o s c i l l a t o r s is s u b s t a n t i a l l y r e d u c e d by the change in the i n t r a m o l e c u l a r p o t e n t i a l on account of ro t a t i on ; r e c o m b i n a t i o n a l so t ends to r e d u c e the ef fec t .

The c a l c u l a t i o n s show that the a n h a r m o n i c i t y e f fec t so l e ly c o n c e r n s the u p p e r l e v e l s (e ~ D), a s would b e expec t ed . The m e a n e n e r g y i s d e t e r m i n e d l a r g e l y by the p o p u l a t i o n s in the l o w e r l e v e l s (~ ~ kT), so

251

allowance for anharmonici ty does not affect the o r d e r of magnitude of the re laxat ion t ime, nor does it cause an apprec iable deviation f rom exponential relaxation. On the other hand, it may be n e c e s s a r y to take account of this effect in r e s e a r c h involving the upper levels, because the dis tr ibut ion for Morse osc i l l a to r s in that region differs substant ial ly f rom that for ha rmonic osc i l la tors . At high t e m p e r a t u r e s ( D / k T ~ 7), v i b r a - tional re laxat ion is coupled to dissociat ion; anharmonici ty plays a l a r g e r pa r t for deact ivat ion and r e c o m - bination.

These r e su l t s have been obtained within the f r a m e w o r k of the phenomenological diffusion theory [6] and are applicable to a sys tem whose molecules in terac t weakly with the the rmos ta t a toms. Fo r such a sy s t em we should have ((As) 2) << (kT) 2, o r using the express ion for ((Ar 2> , we have ~?T =1"/~" 1 <<1, which appl ies to many s y s t e m s over wide t e m p e r a t u r e ranges. A m o r e accura te condition is ~D" << To/T << 1 if T o <<T o r T <<To; but there is an o rde r -o f -magn i tude difference between E and kT only for a very short t ime, so it is v i r tual ly sufficient to have 7~ << 1).

These r e su l t s a re also of in te res t as a solution to the F o c k e r - P l a n c k equation with a nonlinear depen- dence for the diffusion coefficient.

We a re indebted to P r o f e s s o r E. V. Stupochenko for valuable discuss ions .

1.

2.

3o

4. 5. 6. 7. 8. 9.

LITERATURE CITED

E. V. Stupochenko, S. A. Losev, and A. I. Osipov, Relaxation P r o c e s s e s in Shock Waves [in Russian], Moscow, Nauka (1965). N. W. Bazley, E. Montroll, R. L Rubin, and K. E. Shuler, J. Chem. Phys. , 2_~8,700 (1958); 2___9, 1185 (1958). E. E. Nikitin, DAN SSSR, 124, 1085 (1959). C. E. T reanor , I. W. Rich, and R. G. Rehm, J. Chem. Phys. , 48, 1798 (1968). M. N. Safaryan and E. V. Stupochenko, ZhPMTF, No. 1, 93 (1965). H. A. K r a m e r s , Physica , 7, 24 (1940). T. A. Bak and K. Andersen, Mat. Fys. Medd. Dan. Vid. Selsk., 33, No. 7 (1961).

I E. V. Stupochenko and M. N. Safaryan, TEKh, 2, 784 (1966). M. N. Safaryan and E. V. Stupochenko, KhVE (in p ress ) .

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