Kinetics of vibration-translation exchange of diatomic molecules—Anharmonic oscillators in a rare gas medium

KINETICS OF VIBRATION--TRANSLATION EXCHANGE OF DIATOMIC

MOLECULES --ANHARMONIC OSCILLATORS IN A RARE GAS MEDIUM

M. N. Safaryan and O. V. Skrebkov UDC 541.124

The role of anharmonicity in vibrational relaxation of diatomic molecules, in particular inkinetics of vibration--translation (V--T) exchange, was considered in a number of investiga- tions. This was definitely indicated by experimental results on vibrational relaxation in rarefied gas flows, and a significant difference was observed between vibrational relaxation times measured in shock-wave and nozzle experiments [1-3].

The role of anharmonicity in the kinetics of V--T exchange was treated theoretically in two limiting cases of oscillators interacting with the thermal reservoir: in the quantum- mechanical variant for the adiabatic case ($o >> !, ~o is the adiabaticity parameter) [4, 5], andwithin scattering theory for nonadiabatic case (to § 0) [6] interactions. Intermediate case in the adiabaticity interaction parameter occur in optical of chemical pumping of highly excited states in rarefied flows and shock waves. The nature of this interaction depends not only on the gas temperature and pressure, but also on the degree of oscillator excitation.

The diffusion kinetic equation obtained in [7] for the distribution function of anharmonic oscillators allows us to study in detail the role of anharmonicity in the kinetics of V--T exchange for arbitrary values of the adiabaticity parameter. The results of such a study for a wide interval of ~o and degrees of initial molecular excitations (for initial Boltzmann and inversion distributions) are given in this paper.

Statement of the Problem

To describe vibrational relaxation of diatomic molecule oscillators under conditions of weak interaction with the thermal reservoir we use the diffusion kinetic equation for the distribution function f(e, t) in vibrational energy e, which has the following form (for details see [7]):

~t - - rll ~ k T D 4n ~ [o . L n = l

Here T is the temperature of the thermal reservoir, f~ is the Boltzmann function at temperature T, D is the dissociation energy of the molecule, m(e) is the cyclic vibrational frequency of the oscillator having energy ~, mo is the fundamental vibrational frequency, B =

~o/~/2D, ~ is the reduced mass of the oscillator, ~ r n = ( l / 2 ~ ) ~ ~ r ( z ) e x p ( . i n z ) d z ; z = ~ t + ~i �9 is

the vibrational phase, r(t) is the trajectory of vibrational motion of a noninteracting oscillator, ~o = ~oTint, Tin t is a characteristic interaction time of the oscillator with particles of the thermal reservoir, and ~-~(e, ~c,) is the adiabaticity function [7]

T ~ ' . = ~ , F ,~\ ' F~ = (t) d t , Fr.o (t) d t , \I ho I ~ Zer --oo

(2)

Moscow. Translated from Fizika Goreniya i Vzryva, Vol. ii, No. 4, pp. 614-623, July- August, 1975. Original article submitted September 20, 1974.

�9 1976 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

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where F(t) is the force acting on the oscillator as a result of collisions, the angular brackets denote averaging over all collision parameters, qx is a parameter independent of (a calculation within classical mechanics gives nx = (~/2~n)<lFhoI2/~kT >, and rn is the oscillator mean free time in the gas.

One of the basic properties of Eq. (I) is the transition at m = mo to the well-known diffusion equation for harmonic oscillators with a diffusion coefficient line at e. At the same time the quantity q~ is uniquely related to the vibrational relaxation time of harmonic oscillators T:(qx E r7 x) and may not be specified in determining the role of anharmonicity.

In deriving Eq. (i) assumptions were used, typical of evaluating V--T exchange for weak interactions. The change in the oscillator vibrational energy is treated in first-order per- turbation theorY. It is assumed that this change does not affect the motion trajectory of a thermal reservoir particle, and that the force F(t) is independent of the vibrational coordinate of the oscillator. These assumptions imply a large error in the energy range r - D, and Eq. (6) below should be considered, strictly speaking, as an extrapolation at e ~ D.

Further, ~-~ and IBrn[ 2 in (i) should he determined. The calculation of ~-~ is very simple for the dynamic picture of collisions used in the Schwartz--Slavskii--Herzfeld theory. The result is of the form [7, 8]

~.= r (rid/r (~c); (3)

~P (z) = z ~ [ exp (-- y)ch ~ (z/~fy) dg; (4) 0

~ : ~C,(D (E)/(00, ~0 = ~0~C5 -I (M/kT) '1~, (5)

where M is the reduced mass of the oscillator and a thermal reservoir particle, and u is an interaction potential parameter of exponential shape.

For short'range forces the quantity ~'~ depends weakly (much more weakly than <IFnl2>) on the shape of the intermolecular interaction potential W and on the collision dynamics. We point out that for Go >> 1 it follows from (3), (4) that

while for three-dimensional collisions and with the use of the Lennard-Jones potential we obtain for Go >> i, on the basis of [9],

'~" ~" (r176176176176 exp '[ '~ ~2/m i l , L --I\-~O]O ~,2/m]Ij).~ )

This is, obviously, the maximum difference associated with a choice of W, and it drops sharp- ly with decreasing Go.

To determine I Brni 2 the form of the intramolecular potential should be specified. We use the Morse potential for the latter. As a result, we obtain an equation written in the following convenient form for numerical interpretation*:

a e x p ( - - a x ) Ocp O x . e x p ( - - a x ) ,~(X,~o)...~" ; 1" 1 --x Or, ~ " ~ I" l - - x

(6)

1 (x, ~o) = B/B~ .o . ~ X

tT.eX n - I 2 - - 2 s n -c- 2 s

An- 22n_ 2 CtsnX s , , S ( 7 ) �9 ash = �9 2s ~- n

�9 For An [see M. N. Safaryan and A. M. Berezhkovskii, Preprint of the Institute of Problems of Mechanics, Academy of Sciences of the USSR (1975), p. 55], a simpler equation can be written:

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D/~ r I 5 I0 20 ' 40 60 80 lOC

/ 2 3 4 5 6 7 8 9 ,~0~,,.̂ ( D / k : ) ~z

Fig. i. Dependence of $o on the thermal reservoir temperature (D/kT) for pairs of colliding particles, i) HF--Ar, HBr--Ar, HI--Ar; 2) HCI--Ar; 3) Fa--Ar, N2-- Ar, No--Ar, HF--He; 4) H2--Ar, CO-- Ar, 02--Ar; 5) HCI--He, HBr--He, HI--He, Ha--He; 6) CI2--Ar; 7) N=-- He, F~--He; 8) NO-He, CO-He, O~-- He; 9) Bra--Ar; i0) l~--Ar, Cla--He; ii) Br~--He, l~--He.

~o=O I - - ]

0 2 4 6 ,~

Fig. 2. Shapes of g(~) : ao = 7, a=40.

Here x=e/D; T-----t/~1; ~p~H[~ a=D/kT; :~-~ is given by (3) with the use of (4), (5); ~ = ~o~ -- x; and ~(x, ~c) equals the ratio of diffusion coefficients for Morse oscillators (B) and harmonic oscillators (Bho). It should be pointed out that the use of the diffusion equation (6) assumes satisfaction of the conditions

kT (i)[ ~ o) ~ o~

SO Chat f o r M/~ << 1 t he d i f f u s i o n a p p r o x i m a t i o n i s p r a c t i c a l l y a p p l i c a b l e f d r any v a l u e s o f t o , w h i l e f o r ~/M ~ 1 t he weak i n t e r a c t i o n c o n d i t i o n (8) i s s a t i s f i e d o n l y f o r r ~ 1 , and t he more so t h e l a r g e r r i s , i . e . , t h e more a d i a b a t i c t h e i n t e r a c t i o n i s .

E q u a t i o n (6) was s o l v e d w i t h t he b o u n d a r y c o n d i - t i o n s

x. exp ( - - ax). ( 1 - - x ) - V 2 N (x, ~.) ~-~-~x 1 x-,o = O. (9)

Two distribution functions were used as initial conditions:

i) a Boltzmann distribution with temperature To (shock wave, nozzle) ,

A (T) D'A(To)]/~f-'~'exp (-- a~ ~1 cp (x, O) = A--F~o) e(~-~,)~, (f(x, O) = ; ( 10 )

2) a strong inversion distribution (chemical or optical pumping),

~.A (T) V i - x (f (x, O) = c. e-p(~-x.,P). ( l l ) qO (X, O) = exp [-- ax + p (x -- Xm)~] '

!

Here A(T) = f e x p ( - - ax)(1 -- x)- ' /~dx;ao=D/kT~); p>>l; c i s d e -

l

re t ra ined f rom t h e c o n d i t i o n :ff(x,O)dx= 1;. and x m i s 0

the coordinate of the distribution peak.

Below we determine, on the basis of numerical solution of (6), (9)-(11), the nature and degree of deviation of kinetic laws from those well known for harmonic oscillators.

General Remarks on the Effect of Anharmonicity on the Process Kinetics

Several qualitative estimates of the role of anharm0nicity can be made without solving (6) but starting from the dependence ~(x, ~ , ~0) (7) To estimate ~(X, ~0) it should be

taken into account that for to > > i ~'i~ exp [3~/3i]- 3F-- , p l--x)], while for to << I,~-~I. The quantity ~=B/Bho deviates most from unity for ~o >> I, while at the same time~(~ 50)>>I already at x~ 0.2. In the opposite case (~0~I) ~(x, ~0)<l in the main x region, and~(~ ~0) <<l only for x % i. Therefore, anharmonicity should mostly affect the process kinetics in case of an adiabatic interaction. For to << 1 anharmonicity can occur already in the region x << i, while for $o < 1 anharmonicity can be disregarded in this energy region. Further, since ~(x,~)>>l for ~o >> i and ~(x, ~o)<I for $~i i, anharmonicity should lead in case of an adiabatic interaction to an increase in the rate of vibrational relaxation, and to its decrease in case of a nonadiabatic interaction. Correspondingly, a ~o value should exist, for which these effects partially cancel, so that the effect of anharmonicity becomes minimal.

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We also point out that the function ~(x)iis monotonic for ~o >> i and ~o < l,and for intermediate ~o (2~0~I0)~ values it depends on x nonmonotonically: it has a maximum, which is shifted with increasing ~o in the range of higher excitation energies. Such behavior of ~(x, ~0) is related to the fact that with increasing e the nature of the interaction can change from adiabatic [m(e)Tin t >> 1 for x << i], to almost nonadiabatic [m(e)Xin t < 1 for x ~ i].

It follows from all this that the value of ~o is quite important in estimating the role of anharmonicity. Figure 1 illustrates ~he range of variation of ~o for real molecules in a wide interval of D/kT values (~-~ = 0.2 A was taken in Fig. i). It is seen that ~o depends linearly on (D/kT) I/2:

~o---~(D~T) ~, ~=~-"(M/~)%, (12)

and y depends only on the molecular shape and on the thermal reservoir particles. For more accurate evaluation of anharmonicity at x <<i among the three possible spectroscopic con- stants D, mo, and meXe it is necessary to use as initial ones ~o and meXe; i.e., the quantity D in (12) corresponds to the valuer (h*mo)a/4meXe (meXe is the anharmonicity constant in cm-*).

It also follows by general considerations that the effect of anharmonicity increases when highly excited states, for which anharmonicity of vibrations is appreciable, become more significantly populated. It is exactly for this reason that anharmonicity should be differ- ently manifested in processes of vibrational activation and deactivation. It should be more influential in deactivation kinetics, since the process begins with significant population of highly excited states, while in the activation process such states are not very important for the kinetics of the initial nonequilibrium stage, and close to the transition the system is around the equilibrium state.

Relaxation of Oscillator Mean Vibrational Energy l

One of the main characteristics of a system is its average vibrational energy ~ = ,fide. 0

For harmonic oscillators the dependence is exponential independently of the shape of the initial distribution for e

(t) ----ep-- (ep--eo) e~p (--t/Tl). (13)

~o and ~p correspond to ~ and the relaxation time t = 0 and t § =) is a constant quantity.

To describe the function ~(t) in case of anharmonic oscillators we introduce, similarly to (13), the following quantity:

= ~-- T =--. (14)

Here ~k(T), the relaxation time "at each moment," formally determines the relaxation time at each moment of time T. At the same time relation (13) with ~k = g*T1 instead of ~i formally describes the time variation of the average energy of a system of anharmonic oscillators. A constancy in time of the quantity g implies that the exponential law for ~(t) is satisfied and, conversely, deviation of g(T) from a constant implies that this law is broken.

Usually the value t Tk, at which the initial deviation from equilibrium changes by a factor e, is used as characteristic (or effective) relaxation time; here T k = g(1)*T1. Below we present results of a calculation based on the initial Boltzmann distribution (i0), as the relaxation of ~(t) is most interesting for gasdynamic experiments.

Typical forms of g(T) for the deactivation process are shown in Fig. 2. It is seen that g(T) is a varying quantity: g(T) increases from small value at T = 0 (remaining less than unity) if the collisions are adiabatic, and decreases from some value g(0) ~ 1 (remaining larger than unity) if the collisions are nonadiabatic, and for t § = it reaches an asymp- totic value g(T) # i; i.e.,_during the nonequilibrium stage of the process ~(t) is not exponential. At T ~ 5, when s = Cp, g(T) ~ const. This corresponds to a well-known fact: a near-equilibrium process is described by an exponential law, but its characteristic time [Tk(~) = g(~)'rl] depends on ~o.

tHere and later, h* = h/2~.

524

In case of vibrational excitation g(T) decreases with time at to >> I from values close to unity,_and, on the contrary, increases for ~o ~i. The value of g for T § ~ depends neither on (co --ep) nor on co; g(=) < i for to ~ 3 and is somewhat larger than unity for ~o < i.

Since the effect of anharmonicity on g(T) is opposite in limiting cases of the interaction, g(~) > 1 for ~o << I and g(r) < i for to >> i, it is natural to expect that there exists such a ~o at which this effect is minimum [g(~) = i]. In the given case for ~o = 2, g(~) is almost constant during the whole process: for deactivation (a = 40) g(T) = i, and for excitation (a = 7) g(T) = 1.15.

It should be pointed out that a dependence similar to g(T) was also considered in [5], where the authors show that a behavior of ~k(t) close to that evaluated is observed in shock- tube experiments (02 + Ar).

Variation of the initial vibrational temperature and calculations with an initial inversion distribution show (similarly to [i0]) that unlike the limiting (T >> i) near-equilibrium value of the relaxation time, which depends only on T and ~o, the quantity ~k(t) depends strongly (particularly for T'<<_I) both on the amount of initial deviation from equilibrium (so -- Cp), and on the value of so determining the initial population of highly excited states. Generally, during the whole process the deviation of ~k(t) from ~k (~) is determined by the quantity [~(t) -- ~p]. This law is, obviously, dominant in the nature of the gasdynamic process. This can definitely explain the differences in data of various authors measuring the ratio of relaxation times for processes of excitation (T A) and vibrational deactivation (Td). We dwell on this in some detail.

Experimentally, ~A/Td was obtained for the CO--Ar system; according to [i-3], l~A/~d~! 5. This result can be explained by taking into account that the quantity really measured was YA/~d, which depends on the amount of system deviation from equilibrium at the moment of mea- surement. Thus, in experiments with "rigid" gasdynamic conditions, i.e., for significant deviations from equilibrium, the measured T k correspond to our calculated Yk values at �9 << i. On the contrary, under "soft" gasdynamic conditions, i.e., at small deviations from equilibrium, the measured quantity ~A/Td corresponds to ~k calculated for T >> i. In [3] results are given of measuring the quantity TA/T d on a shock tube-nozzle instrument, allowing the ob- taining of results for varying amounts of deviation from equilibrium .From the figure given in [3] for the dependence of the vibrational temperature on the velocity of shock-wave reflection it follows that with increasing velocity (i.e., with increasing initial deviation from equilibrium) the measured TA/T d values vary from i to 2-5. Our calculation for a = 40 and to = 12, which approximately corresponds to the experimental conditions of [3, 5], gives i <~'YA/fd~: 3 (T >> i, ~A/~ d = i, r << I,TA/f d = 3). In comparing various experimental data it is useful to use the quantity Tk. The calculation shows (for details, see [i0]) that the value of T k = g(1)'T1 describes more or less correctly the behavior of s(t). Therefore, in describing anharmonicity corrections to ~(t) one may use (13) with TI replaced by T k. Values of Tk/~1 are shown in Fig. 3. Anharmonicity can lead to a deviation of T k from ~ by a factor of 2-3. The value of ~A/Td can be easily determined from Fig. 3 (T A ~ T k in the process of vibrational excitation, ~d ~ Tk in the process of deactivation). For the CO--Ar system TA/~d varies from 1.4 to 1.7 for a temperature variation from 8000 to 2500~

Consider further the relaxation of s for the initial inversion distribution (ii): the laws mentioned above do not change. The behavior of g(~) is given in Fig. 4, from whose com- pariso~ with Fig. 2 it follows that the deviation of g from unity is somewhat larger due to large eo (~o = xmD). The nature of g(T) and (Tk) is determined both by to and by x m. It can be seen from Fig. 4 that for to ~ 5, ~k = g-T~ and it decreases with increasing x m (this is related with the conclusions of [4]); for ~o ~2 the dependence of x m (or ~o) is the opposite. Variation of the temperature at a larger than 30 is related only to the quantity g(~), but its variation is not large (not more than 10%) and can be neglected.

In conclusion, we provfde some useful analytic relations for g(r), which can be obtained in case of an adiabatic interaction. For to >> i, Eq. (i) simplifies in the range of rela- tively low excitation energies (x < ~/3):

§ a f . (15)

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O,S

O,4 o2-L

.~I l ~

: l /- i . . . . . . LI

!0 ?0 SO ~0 7 ~/i~,

Fig. 3. The ratio ~k/Y l as a function of the thermal reservoir temperature. i) y = 0.31 (Br2--He); 2) y = 2.3 (CO-- Ar); 3) y = 4.9 (HCI--Ar). i) Solid line) deactivation, ao = 7 (T k --- r d) ; 2) Broken line) activation, ao = 43

(~k = TA)"

' . j

. . . . . .

l, Oq "q 2

" ./C ro"

[1 ~ - - - T - - ~ - 1 - - - - ~ . . . . . .

o 2 " ~ "~

Fig. 4, The behavior of g(r) for a strong initial inversion distribution (ii), a = 40. Solid line) Xm = 0.3; broken line) x m = 0.7.

;; T,,I ~,~ o,,, o.8,. ~ o .~ o , o~8 ,,o

~ ~--r:~.~l o,, - ,.-o i 3,0

1 i . . . . I - i " I - 2 I ,~ , , 1 2 1 o oA 0,8 i,o ..~ o 0,4. o,8.~,o

F i g . 5 . . D e p e n d e n c e o f T k / T on t h e amount of molecular excitation for various Y [2) 4.9; 4) 2.3; ii) 0.31] in the case of deactivation (a = 7, a -- 40). (The curve numbers correspond to Fig. i.)

Equation (15) is similar to quantum-mechanical treatments with equidistant level conditions and single-contact oscillator transitions, where anharmonicity corrections were included in the expressions for the transition probability [4, 5]. Multiplying (15) by x and integrating with respect to dx, we obtain (accurately up to terms of the

order of e -a e -a~

dx Xp - - "x

dT g ('0 ' x = e / D , xp = e p / D ~--.a - l ,

e (~ ) =

a -! --x

(16)

/ ~2/S ) i a -I .~ exp (x~02/3) f dx + I ~a0-~-- 1 x exp (x[~ 3)[dx 0

Relation (16) allows us to determine the quantity g from the well-known average energy e = x-D. Di- gressing from the properties of the distribution

function, we formally take for f(x, r) a Boltzmann shape with an effective vibrational temperature Tk*, corresponding to e(t) [see (19) and (20) below].f Then

g(~) = [;(~).Bgz_ i]~, f < ~-~/~. (17)

Using (14) to find g(T) from (16), we obtain

g = [ [ x p - - (Xo - - xp) exp ( - - x / g ) . ~/3 _ 1}~. (18)

Comparison of the result of (18) with that obtained from solving (6) shows a wider applicability region of this equation. Thus, for to ~ 5 (18) gives for ~(T)_an error not larger than 10-15%. Assuming in (18) T = i, we obtain T k for given to, xo, Xp.

Effect of Anharmonicity on Relakation of the Distribution Function

The largest effect of anharmonicity is on relaxation of the distribution function. discuss below only the main features of this effect (for details, see [i0]).

We

fThis does not imply an assumption about the existence of a vibrational temperature.

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o o.4 0.8 1,v

,o" . . . . . . . . . . . . . . . ~- . . . . . . . . . . . . . :o j _ ~ . i ~o=O ~ 7eJ '

~- ~ o _g,>__--1

0 0,4 0,8 ~,0 a3 0 0,4 0,8

Fig. 6. x dependence of ~P/~P* for various $o in case of deactivation (~ = 7, ~ = 40).

/ EO -

r~'/?m ~0 = O i i

I ~ 1 7 6 ~_ . . . . . . . ~ - , [

0 2 4- 6"r

Fig. 7. Time dependence of the ratio of fm = f(xm, T) to the same quantity for harmonic oscillators for a = 40, x m = 0.3.

Unlike harmonic oscillators the initial Boltzmann shape of the distribution of anharmonic oscillators is not conserved during the relaxation process. This is il- lustrated in Fig. 5, where the quantity Tk(x, T)/T is shown for the case of deactivation. Tk(x , T) has a pure- ly formal meaning (x > 0)

Tk T{ l@ln[~(O, '~) /~(x ,z )] /ax} - I , T* ~(~) (19)

The absence of a single value of Tk for all x values is a consequence of the breakdown of the Boltzmann distribution shape, and the degree of this deviation can be obtained by the difference between ~ and ~*, where (p* is formally determined, similarly to harmonic oscillators, by means of the equation

= ( r / r ; ) exp [ a x ( 1 - r l r ;) ] (20)

[we recall that for harmonic oscillators Tk* = Tk(~) = g(T)/k, and ~(r) is given by Eq. (13)]. The ratio~/~* for the case of deactivation is shown in Fig. 6. The deactivation process of anharmonic oscillators gives at ~o ~ i0 significant underpopulation of highly excited states, initially generated, and at ~o < 2 some repopulation of them, occurring at T ~ i. The process of vibrational excitation (see [I0]) gives repopulation of excited states at to >> i, and less significant underpopulation at ~o < !, while the deviation of ~ from ~* is largest initially (but is smaller than in the deactivation process). In case of vibrational excitation T k also differs from Tk* , but this deviation, and particularly that of ~ from ~*, is significantly smaller than for deactivation.

Consider further the effect of anharmonicity on relaxation of the inversion distribution (ii). The process occurs in two stages: first, the peak of the original distribution is "smeared out," and then the process becomes similar in nature to relaxation with an initial Boltzmann distribution.

Most interesting is the effect of anharmonicity on the change of population fm = fm(Xm, T) of states with x = x m. It is seen from Fig. 7 that the relative effect of anharmonicity on fm varies in time and that for some value T = rm it is maximum. The values of ~m and of the difference between fm and fho (f~o = fm for harmonic oscillators) depend on to and even more strongly on x m (see [i0~). T m decreases with increasing $o and for $o = 30, Xm~0.3 , we have ~m ~ 0.5. For to >> i, rm practically corresponds to the fade-out time of the original inversion.

We point out, in conclusion, the main features of the effect of anharmonicity on the kinetics of V--T exchange. This effect depends on ~o and x and increases with them. The nature of deviations from harmonic oscillator laws is opposite for adiabatic and nonadiabatic interactions. While qualitatively the interaction can be assumed adiabatic for ~o ~ 3 and nonadiabatic for ~o ~ 2, in determining the deviation an interaction with go ~ i0 should be taken as adiabatic, while one with to ~ 1 should be considered nonadiabatic. The largest effect of anharmonicity is on relaxation of the distribution funs the concept of vibrational temperature becomes inapplicable. The exponential law for c(t) no longer holds, but it can serve as a useful approximation if in (13) ~i is replaced by rk; ~ for ~o ~5 values of ~k/T1 are given by Eq. (18).

Anharmonicity effects are most important in the change kinetics of the inversion distribution. During t ~ T:/2 the effect of anharmonicity is dominant in the process kinetics; for $o >> i, anharmonicity can decrease by an order of magnitude the existence time of the original inversion. The most concrete characteristic role of anharmonicity in the kinetics of V--T exchange was given in [i0].

527

All results above were obtained for classical oscillators. To estimate their possible extension to a quantum system we performed a quantum-mechanical calculation for single-quantum oscillator transitions (Go >> i) with expressions for the probability transition taken from [ii]. By comparing with results of the present calculation it follows% that the treatment performed above is fully satisfactory for h*mo/kT~l; such conditions at D/kT~45 are satisfied by,the 02, N2, NO, CO, Br2, 12, and C12 molecules. For the relative quantities ~A/rd and f/fno (at T <i) there is practical coincidence with the quantum-mechanical calculation (with an error not exceeding 20%) for any realistic h*mo/kT. Values of ~ = f/fo and g(T) are more sensitive to this parameter. For example, at h*mo/kT = 4, Go = 30, ~qu differs from the classical ~ at T = 0.2 by a factor of 2-3, but with increasing T this difference decreases quickly; gqu differs from gel by 30-50% on the side of enhanced anharmonicity effect.

It follows from the discussion that in using the results of the solution of Eq. (6), the condition h*mo/kT << i is not required to be satisfied rigidly. The results given above have a wider range of applicability.

LITERATURE CITED

i. S. A. Losev, Fiz. Goreniya Vzryva, 9, 767 (1973). 2. T. I. McLaren and J. P. Appleton, J. Chem. Phys., 53, No. 7, 2850 (1970). 3. C. W. Rosenberg and R. L. Taylor, J. Chem. Phys., 54, No. 5, 1974 (1971). 4. E. E. Nikitin, Dokl. Akad. Nauk SSSR, 124, 1085 (1959). 5. S. A. Losev, O. P. Shatalov, and M. S. Yalovik , Dokl. Akad. Nauk SSSR, 195, No. 3, 585

(1970). 6. M. N. Safaryan and N. M. Pruehkina, Teor. Eksp. Khim., 6, No. 3, 306 (1970). 7. M. N. Safaryan, Preprint, Inst. Probl. Mekh, Akad. Nauk SSSR, No. 41, Moscow (1974). 8. M. N. Safaryan, Zh. Prikl. Mekh. Tekh. Fiz., No. 2 (1974). 9. H. K. Shin, J. Phys. Chem., 77, 1666 (1973).

i0. M. N. Safaryan and O. V. Skrebkov, Preprint Inst. Probl. Mekh. Akad. Nauk SSSR, No. 42, Moscow (1974).

ii. J. Keck and G. Carrier, J. Chem. Phys., 43, 2284 (1965). 12. M. N. safaryan and O, V. Skrebkov, Preprint Inst. Probl. Mekh. Akad. Nauk SSSR, No. 56,

Moscow (1975).

#Further comparison of results of classical and quantum-mechanical calculations is given in [12].

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Kinetics of vibration-translation exchange of diatomic molecules—Anharmonic oscillators in a rare gas medium