4
VIBRATIONAL RELAXATION OF AN ANHARMONIC OSCILLATOR M. N. Safaryan and N. M. Pruchkina UDC 536.45 We have detailed knowledge of vibrational relaxation for diatomic molecules in an inert gas if the molecule can be represented as a harmonic oscillator [1]. One expects that anharmonicity will not greatly affect the macroscopic vibrational relaxation time, since vibrational anharmonicity is important only for the higher vibrational levels, whose populations play a part only at relatively high temperatures, where vibrational relaxation cannot be considered as separate from other processes, e.g., dissociation. However, there can be marked effects on the rate of attainment of the equilibrium distribution and the equilibrium mean energy, so it is of interest to examine the effects of anharmonicity on the relaxation. Anharmonicity naturally becomes more important if the system contains highly excited molecules produced by chemical reaction. Detailed quantum-mechanical calculations [2-4] * have been made for adiabatic collisions of molecules with inert-gas atoms, which allow one to evaluate the effects of vibrational anharmonicity at relatively low temperatures: e-hc0/kT << 1 (hw is a quantum of vibrational energy and T is thermostat temperature). At higher temperatures such that hc0/kT << 1 (in practice, for e-hw/kT ~ 1 -hw/kT), one can consider the relaxation within the framework of classical statistics; in particular, the diffusion approximation can be used. Here we use the diffusion theory with some assumptions about the diffusion coefficient to estimate the effects of anharmonicity on the vibrational relaxation time and on the details of the process. Consider a system consisting of a small proportion of a diatomic species in an inert gas (a thermostat at temperature T). At the start, the state of the molecules represents equilibrium at temperature T 0. With- out specifying how translational energy is converted to vibrational or vice versa, we assume that the mole- cules interact only weakly with the thermostat atoms, so the root-mean-square increment e in the vibra- tional energy is small relative to the energy range within which f(e, t) varies appreciably. We can then use the Focker-Planck equation [5] as the kinetic equation for f( e, t) : (--~-8 - - Olnf~ , af a {B of at de where B = <(A~)'2>/2T, Ar is the change in the energy of a molecule as a result of collision with a thermo- stat; ~- is the time taken by a molecule to transverse a mean free path; (...} denotes averaging over all collisions of a molecule; f0(e) is the equilibrium distribution corresponding to a thermostat temperature T. The "diffusion coefficient" B is dependent on the form of the intramolecular and intermolecular inter- actions; the results are as follows [7, 8] respectively for a harmonic oscillator and a Morse oscillator: B =: ~lkTe, (2) B = 2~IkYD l'- 1 -- e/D (1 -- i / 1 - - e/D}, (3) where D is the molecular dissociation energy and ~? (the coefficient of friction [6]) is taken as independent of the internal state of the molecule, i.e., of 5. This is well justified for nonadiabatic collisions of the molecules with the thermostat atoms (in particular, it follows from [5, 8]), but it is an assumption, as in *In referring to [4], we have in mind results from an unpublished calculation on this topic. Institute of Chemical Physics, Moscow Region Branch, Academy of Sciences of the USSR. Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 6, No. 3, pp. 306-310, May-June, 1970. Original article submitted March 19, 1969. 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

Vibrational relaxation of an anharmonic oscillator

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Page 1: 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

Page 2: Vibrational relaxation of an anharmonic oscillator

/,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

Page 3: Vibrational relaxation of an anharmonic oscillator

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

Page 4: Vibrational relaxation of an anharmonic oscillator

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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