Collisionless dissociation of SF6 in an intense IR field

Chemical Physics 41 (1979) 103-I 11 0 North-Holland Publishing Company

COLLISIONLESS DISSOCIATION OF SF6 IN AN INTENSE IR FIELD

Received 30 October 1978

A model for multiphoton dissociation of polyatomic molecules which has been previously discussed by a number of authors is carefully investigated. It involves the coherent escitationnf the molecule within the lower discrete energy levels, “leakage” into the quasicontinuum of levels, the incoherent escitation within the quasicontinuum and the dissociation itself. A closed set of equations for the populations is derived assuming quasistationary populations of the lower levels. An approximate solution to these equations is given in the post-threshold region where they can be reduced to a simpler system of two equations for the relative concentrations of activated and nonactivated molecules. In this approximation the problem becomes mathematically equivalent to the classical problem of thermal dissociation with the only difference that the rates of activation and deactivation are functions of the field intensity, I. A relatively slow (power-like) rise of dissociation yield, IV,with increasing I in the post-threshold region is shown to occur under some conditions. The results of the numerical solution to the initial set of equations for SF6 are reported and compared with the predictions of the approximate theory and with esperiment. The theory presented explains well the dependence of IV upon I in the post-threshold region provided that the field frequency does not satisfy the mutliphoton-resonance condition, agreement between the calculated and observed abso!ute values of IV being quite good as well in spite of the very low populations of the excited vibrational states. There is, however, disagreement between theory and experiment concerning the dependence of IVon the pulse duration 7 at a fLyedenergy fluence UJ= Ir. The vibrational heating of the molecules is also wlculated and compared with experiment.

1. Introduction

to the field intetisity,

I. This model predicts

the occur-

rence of a threshold, ‘Pth) in the energy fluence, + = 17, In the present work a quantitative treatment of the collisionless dissociation of SF6 in an intense IR field

(T is the pulse duration) but it does not explain why the dissociation yield, W, goes on growing in a powerlike

[l-3] is developed in the framework of a model which combines several ideas discussed previously by a number of authors. In refs:[4-61 the coherent excitation ofN(=D/fiw) equidistant or nearly equidistant levels has been considered using a set of equations for the amplitudes of the wavefunction (D is the dissociation limit, w is the frequency of the IR field). The main drawback of this model is that the assumption of coherence preservation during the laser pulse justified in general only for the low-lying levels is extended to the strongly excited ones belonging to the quasicontinuum region. In refs. [7-91 the incoherent excitation of N equidistant levels is investigated by solving the rate (balance) equations for populations where the transition rates between adjacent levels are assumed to be proportional

fashion with increasing (1)above threshold; namely, it predicts rapid saturation, IV+ 1, at CD> (Dth_ Much attention has been given in the literature to the problem of the vibrational anharmonicity of the lower levels. Taking into account the anharmonicity, the positions of several low-lying levels of the triply degenerate u3 mode of SF6 and the corresponding average populations under coherent excitation have been calculated in ref. [IO]. Among the levels obtained there are four equidistant ones corresponding to u = 0, 1,2,3, with energy spacing being equal to one quantum of the v3 mode, ftw,,l. Consequently the purely resonant excitation of the molecule up to the lower boundary of the quasicontinuum corresponding to u=4 [l] is, in principle, possible. According to refs. [ 1I-141 the u = 3 level can also be populated

nance conditions.

efficiently

under multiphoton

reso-

104

ES. nl~d~cdev/Co[[isiorlless dissociariorr ofSF6

However, as follows from experiment [l], the resonance conditions are, in general, not necessary for efficient excitation of molecules because the latter has been observed within a wide frequency range so that possible detunings are not small:L.etokhov and Makarov [12] have proposed the following explanation to this phenomenon_ With no resonance and at moderate field intensities the population of the u=3 level will be very low but “leakage”, i.e. transitions from u = 3 to the quasicontinuum, can result in efficient excitation of the molecules provided that the pulse is sufficiently long. Recently, experimental data implying the occurrence of this effect have been reported in ref. [IS]. However, in ref. [12] coherent excitation within the quasicontinuum has been assumed. A similar idea has been advanced in refs. [16-191, where light absorption by the v3 mode has been considered and the “leakage” effect has been taken into account by introducing damping due to intermode interaction. It was shown [16,17] that at large field intensities SF6 efficiently absorbed the field energy within a wide frequency range about the resonance frequency. These models, however, predicted rapid saturation in the post-threshold region as was explicitly demonstrated by the numerical calculations in refs. [18,19]. From the physical point of view, the most acceptable model seems to be one which combines the coherent off-resonance excitation of the lowest yibrational states of the v3 mode, u f 3, with the “leakage” to the quasicontinuum, u = 3 *u= 4, and the subsequent incoherent resonant light absorption. The necessity of considering both coherent and incoherent effects has been emphasized in refs. [3,i 1,1420]. A rigorous solution of this problem would require the derivation of an equation for the density matrix of the system of which the density of states and the rate of the coherence relaxation steeply increase with energy [20]. The method used in the present work is less rigorous but it is physically clear and makes it possible to obtain a comprehensive solution of the problem.

2. Basic equations We consider first the coherent excitation of the unequally spaced discrete levels u= 0, 1, ___, M neglecting transitions into the quasicontinuum. The wavefunction is written in the form

M $(t) = u5 b”(t) e-ivw*$v,

(0

where $, is the time-independent eigenfunction of the molecular hamiltonian corresponding to the eigenvalue E,, w is the laser frequency. The molecule-field interaction will be assumed to have nonvanishing matrix elements only for adjacent levels, (vi Vlu t 1) = -d & CDSwt ,

(2)

where d is the dipole matrix eIement independent of u, e is the field strength. Neglecting oscillating terms in the Schriidinger equation one obtains the usual equations for b,(t): ib, = A, b, - i ~R(bD-I + b,,r ) , fiA,=E,-vz2w.

(3)

+=d&/fi

(3

(here b_l g 6,+, = 0 ) which is to be solved under the initial conditions 6”(O) = c:‘Q”,o t

(5)

where co is the initial population of the ZJ=0 level. The solution to eq. (3) can be written as

b,(t) = Ci12 sFo asuexp (-ies t) ,

(6)

where es and asu are, respectively, the eigenvalues and eigenvectors of the coefficient matrix on the rhs of eqs. (3) [6,10,16]. Below we shall be interested in the population of the u=M level. From eqs. (3)-(6) the following expression for ashf can be derived (7) It is seen from (6) that the populations lP,(t)12 oscillate with the frequencies IE, - ~~31.In general, all these frequencies do not vanish. Moreover, at typical values of the parameters the longest oscillation period is small compared to all time parameters involved *. Therefore in what follows we shall consider only the average populations, lb,(t)P = COP”, * This condition can be violated at multiphotonresonanuz

(8) [14].

105

ES. Med~~edev~Collisiorlless dissociatiotlof SF6

where pU is the vibrational distribution normalized to unity,

where K,, is the dissociation rate from the level FZ at n >N,, N1 is the first level above the dissociation limit, k n,rl+l

(9)

&=I. u

w

v=o

hl

c

(wR/2)*”

M

l-I (Es- g-2

s=oI;=0

(11)

.k+s

as results from (C?)-(8). Let us include now transitions to the quasicontinuum. Due to energy conservation in the system “molecule + field” only such states will be populated whose energies are E,, t ~lliw, 0 < 11
(13

(u is

the cross section of the quasicontinuum) are low compared to the oscillation frequencies of the amplitudes (B), s#s’.

k
(144

~o=-k,,~Mco+klOcl,

c, = kol plMco - ($0 + $z)q

2
-

k,,+l ,,z = (Ilfiw) 0;

1

(W

Q(O) = 1 , ISn
In the derivation of eqs. (14a) and (14b) it was taken into account that the time dependence of the total population of the discrete levels, co(f), results only from the transitions W= 0 (u=M) 211 = 1 and the population of the level u = M is plcrco as follows from (8). The dissociation rate K,~ increases rapidly at n > IV, . The total number of levels, N, wllich can be populated with a considerable probability during long time periods can be determined by the condition that depopulating of the N+ 1 level is much more efficient than its populating, i.e. (17)

KN+~ 3. kN,,v+l 2 KN .

Then the infinite set of equations (14) can be truncated at FZ = N and one can set cN+l = 0 in the last equation. With no collisions the dissociation yield during the pulse and after it is off is given by

(18)

(13)

Then the transitions n =O (u = M) 2 n = 1 will not affect the distribution (9) and will result only in changes in the total population of the discrete levels, co, which is now a function of time. Thus, in this approximation the average populations are given as before by eq. (8) withp, from eq. (9) and co depending on r. In the model considered every energy state of the quasicontinuum absorbs and emits incoherently, which is described by the balance equations for-the populations c,*, 1 Q II SN. Those are to be completed by the balance equation for co. The whole set of equationi is

i;, = SI-1,&*-l

(rlfiw) 43

q,(O)= 0,

Specifically pn =

=

$ie are the absorption and emission cross sections. The initial conditions are

+ k21c2 >

(14b)

(kn,n-l •I-k,Z,n+l i- x&n fk,+lJznC,+l~ (14c)

and N (19) respectively, where the c,~(T)are the populations at the end of the pulse. The total dissociation yield is N~-l IV= Fc’dCwa=1 - X0

c,(r).

(20)

The average vibrational excitation of the molecules which have not dissociated is calculated from the formula

ES. Medvedev/Collisionless dissociarion ofSF6

106

Eqs. (14) completely describe both light absorption and dissociation in the framework of the present model. In the next section the physical consequences of the model will be analyzed using an approximate

solution

to eqs. (14) and they will be compared with the numerical results in section 4.

eliminating the induced emission. Expression (26) has been used in ref. [2] to estimate CJ.1r1refs. [9,14,?0] fth has been estimated assuming oi = u,“,= u, which gives obviously the very rough upper limit, Ith z_ N%w/or. At the intensity (26) the dissociation yield is approximately Wth = c,&)

3. The physical consequences of the model 3. I. Preliminay consideraiions Before studying eqs. (14) it will be useful to discuss the simpler problem of incoherent excitation of N equidistant levels. The corresponding set of equations is derived from (14) by omitting eq. (14a) and the terms with /co1 and /cl0 in eq. (14b); the initial conditions (16) are replaced by c,(O) = 6,,,~, 1
(22)

=&l&l

(where g,! is the density of states near the level II) then for polyatomic molecules the inequality o;
(23)

will hold because g,, increases rapidly with n. Therefore, the induced emission can be neglected in calculating the threshold intensity It,,_ Replacing C$ with an average cross section D one easily gets the following expression for the population of level N at time r: CJT) = e-eO”-l/(N-

I)! ,

(24)

IOi/iiW

(25j

_

From (24) the threshold 0 th z N and hence fth = lVfiGJ/Ui,

condition*

is obtained as

WI

which gives in fact the lower limit to Ith because of

+ The experimental deftition

Of fth contains an uncertainty depending on the accuracy of the measurements. In this work the threshold intensity, Ith. at a given pulse duration T is defmed by the condition of equalizing the populations of the upper and lower quasicontinuum levels.

.

(27)

The second essential feature of the above model, the incoherent excitation of an N-level system, is the rapid saturation, W + 1, at I>Ith with the pulse duration fixed. Indeed, let 1, be the saturation intensity. In the model considered there is only one parameter u on which both Ith and 1, depend. It is clear, therefore, that it must be Z,, = Is. Now we proceed to the discussion of the present model when excitation and dissociation of molecules are described by eqs. (14). Evidently, in this case the threshold intensity is given as before by eq. (26) because by. the above stated definition,l,h depends on the properties of the quasicontinuum only. A new feature of the model is the “bottleneck” between the discrete levels and the quasicontinuum which results in a much lower dissociation yield at I= It,, than that of eq. (T7). The transition rate from the highest discrete level into the qtiasicontinuum, II = 0 (u =M) + II = 1, is reduced by a factor pin as compared to the transition rates between quasicontinuum levels, see eq. (14a). Recall that phf is the relative average population of the u =M level under coherent excitation and depends on the ratio between the field broadening, de, and the mean detuning, the latter being of the order of the anharmonicity constant, fin,. We introduce the quantity I, = (c/477) (tlAa/d)2 .

where

e=

= (bN)-112

m

Early theoretical studies on the molecular dissociation in an IR field have suggested that Za was the threshold intensity but the intensities observed have proven to be much smaller [I]. More recent studies [10,13] treat f, as the saturation intensity, fs. As will be shown in section 3.3 the present model results in I,
(2%

holds where the exponent v GM (see eq. (11)) and depends on the detuning A = wol - W.

107

ES. hledvedev/Collisiorrlessdissocintion of SF6

lt is clear from the above that if the condition (30)

Ith &la

levels the time variation of which is governed by Amin. Now eqs. (14) are easily reduced to a set of two equations in co and c. The first equation is simply eq. (14a) and the second one results from the addition of eqs. (14b) and (14~). Using (32) one gets

holds, i.e. if the effect of “bottleneckir$” does not vanish after the threshold is surpassed then a relatively slow (power-like) increase of W caused by this effect has to occur in the region

E, = -41cu

Ith
E’41Cl-J -(42+4+,

(31)

In this region an approximate solution to eqs. (14) can be obtained because at I>Irh the totality of the quasicontinuum levels behaves as a single level and the problem becomes mathematically equivalent to the classical “two-level” problem of thermal dissociation [?_I]. On this fact is based the approximate method used in the next section to solve eqs. (14). 3.2. Approximate solution to eqs. (14) Eqs. (14) are characterized by two different transition rates: the average transition rate between quasicontinuum levels, k, and the n = O(u=M) + tz = 1 tran-

sition rate, ~,~k. The right-hand side inequality (31) results in pbr < 1, so that these rates are essentially different and if the rate k is large enough, the left-hand side inequality (31), then a quasistationary distribution among the quasicontinuum levels must be established during the pulse period. This distribution can be obtained as follows. Due to the presence of the small parameter pnl the minimum eigenvalue, Xmin - pMk, of the coefficient matrix of eqs. (14) will be much smaller than all the other eigenvalues, X, m k (1
ldn
(32)

where the g,Zare constant coefficients normalized by the condition

&“=I, and c is the total population of the quasicontinuum

+qzc,

(344 (34b)

where

43 =

,~v,Kn~n -I-kN,N+lsN -

It is seen that eqs. (34) are identical to the rate equations for the concentrations of activated and nonactivated molecules appearing in the classical theory of

dissociation in its simplest form [2 11. Here co and c are the concentrations of nonactivated and activated molecules, respectively, and ql, q2, q3 are the rate constants of activation, deactivation and unimolecular decay, respectively. In fact one has to set 6 = 0 within the quasistationary approximation considered because in eq. (32) c is a constant proportional c=

to co_ According to eq. (34b)

(36)

L&(42+4~33co~

The rate and the yield of dissociation are given by qdiss E -~O/cO ‘q1’?3/(qz ‘43).

(37)

and w = 1 - exp (-qd&r)

,

(38)

respectively. in the derivation of (38) the contribution from the levels n 2 1 has been neglected inasmuch as c,r
The range of validity of the above formulae is given by the quasistationarity condition which has been assumed above to coincide with (31). Now this condition can be made more precise. As in the classical theory [21], the quasistationarity is set in when the rate of activation is much smaller than that of deactivation, 41<42 -

(3%

In this case eqs. (34) have two eigenvalues, Xmin=qdiss and X,, = 42 + Q , and Xmin4 X,,. It is clear that the quasistationary state where the variation with time of

ES. hledvedes/CoUision!essdissociation of SF,

108

the populations is governed only by the eigenvalue X,.,,, will set in at times t 9 l/&rau. Therefore, in addition to (39) another condition,

whatever conditions and depends on numerical parameters of the system under consideration. In the following it will be assumed that

Y&&=(q2+q$@

rt,~r,,r,

1

(40)

must be satisfied. The inequalities (39) and (40) are the precise conditions for the quasistationary approximation to be valid. We rewrite them in a more suitable form making some simplifications. Firstly, the rates k ,1,,1+1will be assumed to depend insignificantly upon n and will be replaced by the mean value (12). Then all a’s in eq. (32) are approximately of the same order of magnitude and, by (33), &-- l/N-

(41)

Secondly, in making a rough estimate the first term in eq. (35b) can be neglected inasmuch as according to (17) it is not, in any case, larger than the second one. Then the conditions (39) and (40) can be written in the form I,,
(42)

9

(46)

because in this case will IVincrease relatively slowly from the threshold value up to saturation in the range Ith
-

(47)

Within the interval (47) expression (38) reduces to (48) Considering (35) and (29): two important conclusions can be drawn from expression (48): (1) within the range (47) W(I) behaves as a power-like function with an exponent changing from v+ 1 to v with inrecasing I; (2) absolute values are FV- q17 - pnllir and hence even at small values of PM the dissociation yield can take on appreciable values if kr % 1. The latter is really the case in the post-threshold region because according to (12) and (26) one has (49)

where lo = I&/&#V

(43)

e-

The inequalitites (42) determine the range of validity of the approximate solution to eqs. (14) obtained in this section and specifically of expression (38). This range is much narrower than has been assumed earlier (inequalitites (31)).

4. Numerical calculations for SF, In this section the numerical solution of the starting equations (14) will be given and treated in the framework of the approximate

theory of section 3.

3.3. Dissociation yield versus field intensity

The discrete energy levels were approximated by (fi=l)

It is clear from the above that at I > lo the dissociation yield will reach saturation rapidly, W+ 1, because the rates of activation and deactivation become of the same order of magnitude, 9I = q2_ However, the saturation can be set in formerly, within the range of (42) where (38) holds. Obviously, in this case the saturation intensity 1, is determined by the condition

E, = uwol + X33u(u - 1) ,

Q&..7 = 1 -

(44)

In making a rough estimate ofl, one can assume that q2- + and therefore qdiss = ql. Using (355, (26), (29) and (12) one gets from (44) r, -Ia(lt&V~a)1~(*i)

.

(45)

The value of I, relative to It,., and I, is not restricted by

(W

where u = 0, 1, ...M. At v=4 the density of vibrational states of SF6 amounts to 200 states/cm-l [l] and therefore this level can be considered to belong to the quasicontinuum. Thus we set M= 3. The following values of the parameters have been adopted for SF6 : ool = 948 cm-l, Xj3 = -2.8 cm-l [22],Aa=12X331=5.6cm-1,d=0.388D [23].The absorption cross section has been assumed to be independent of n*, oi = u, and the emission cross section o”, has been evaluated from (22). The numerical values * The expressionfor 0: given in ref. [7] has also been used, which did not change the results essentially.

109

ES. hledvedev/Collisiottlessrlissociatiotlof SF,

of the density of states, g,L,and of the detailed dissociation rate, K,z,have been taken from ref. [8]. The latter had been calculated in ref. [8] using the RRKM theory with the dissociation limit D = 32100 cm-l = 34Rw. The number of quasicontinuum levels, N, has been chosen in such a way that a further increase in N would leave the results unchanged. This was the case at N=40. The experimental value of the threshold energy density, (I+,, = Ithr, amounts to 1.4 J/cm2 [2]. Substituting this into eq. (26) gives 0 = 5 X lO-I9cm2. The usual method to solve the linear differential equations i = Ax involves the investigation of the eigenvalue problem for the matrix A. However, in practice it is too inconvenient because it becomes time con: suming with increasing order of the matrix. We have used a simpler and more reliable algorithm which directly calculated the matrix BO= exp (At,-,) for any fixed moment ro_ It goes as follows: (1) The series of points tk = fo/2/’ (X-=1,2, .... m) is taken.

(2) The number tn is chosen such that r, < where A,, is the maximum eigenvalue or the l/&,, maximum matrix element of A. .(3) The matrix B,, = exp (~0,~~)is calculated by series expansion. Rapid convergence of the series is ensured by the above choice of tn and therefore a few number of terms,p, has-to be retained. (4) The matrices BL-I= _ (Bk)2 are calculated at k = ttz, m - 1, .._which gives eventually B. _Thus the complicated eigenvalue problem for a matrix of large order is avoided. In our calculations m = 7-12 and p = 5 has been used. increasing tn orp leaves the results unchanged. in figs. 1 and 2 the full lines show the dissociation yield versus the energy density Cp= li as calculated from expression (20) with N1 = 35 and C,,(T)obtained by the numerical solution of eqs. (14) and (3). Different curves in fig. 1 correspond to different detunings A = ool -w, with T fued and in fig. 2 they correspond to different 7 with A fiied. Dashed lines in fig. 1 are the populations of the u =3 level, p3 (eq. (1 l)), and in fig. 2 these are N


1

30

IO

Fig. 1. The dissociationyield W(full lines) and the populations of the 0 = 3 level, ps (dashed lines), versus energy density as obtained by the numerical solution of eqs. (14) and (3) at a puke duration r = 100 ns. The detuniugs A = wol - w are: (1) 0 cm-‘, (2) 1.4 cm-‘, (3) 2.8 cm-‘, (4) 4.2 cm-‘. Dotted lines represent the experimental data for IV:(5) w = 951.2 cm-l [l], (6) w = 947.7 cm-’ [l] ,(7) w = 944.2 cm-’ [2] _

Fig. 2. The dissociation yield W(full lines) and the average number of photons stored by the molecules which had not dissociated, 01) (dashed lines) versus energy density as obtained by the numerical solution of eqs. (14) and (3) at A = 4.2 cm-‘. The values of 7: (1) 100 ns, (2) 10 ns, (3) 0.5 ns. At 7 = 100 ns tn) is very small and is not indicated. Dotted lines are the experimentaldata of ref. [3] for (II): (4) 7 = 100 ns, (5) 7 = 0.5 11s.

110

E.S. Medvedev/Collisiotdess

the average numbers of photons stored by the molecules which had not dissociated, eq. (21). f..et us compare the numerical results of fig. 1 with the predictions of the approximate theory of section 3. To do this one must estimate Ia, 1, and 1,. Here it is convenient to use the corresponding energy densities at fixed r = 100 ns. With the above values of the parameters eq. (28) leads to @a= 200 J/cm2.
5. Co.mparison with experiment and conclusion In fig. 1 the dotted lines represent the experimental data for w=951.2 and 947.7 cm-l [I] and w = 944.2 cm -t [2] at r = 100 ns. It is seen that for the long pulse the theory well reproduces the observed behaviour of the dissociation yield within the post-threshold region. It is essential to note that the predicted absolute values of W are large enough to agree satisfactori!y with the experimental ones while the values of pg (dashed lines) are very small. When the two-photon resonance occurs (curve 3) the magnitude of p3 [and W) grows appreciably but at the same time the region of slow growth of W disappears and W rapidly reaches unity within the pest-threshold region. It has been noted earlier [3] that the “bottleneck” effect will result in a strong dependence of iVupon r at fixed density. According to fig. 2 the theory predicts an increase of Wby several orders of magnitude when T changes from 100 to 0.5 ns, however, observations [3] reveal only slight (=530%) increase of W. The results of fig. 1 show the dependence of W upon the detuning A which is in qualitative agreement with experiment. Note the peculiar behaviour of curve 3 (the two-phonon resonance, 20 = wo2) as compared with the other curves. Surely, this difference will be

dissociation of SF6

diminished when taking into account the width of the resonance due to intermode interactions. As for the three-photon resonance, the approximation used here seems not to be applicable at all in this case because the lapse of time needed for the stationary populations of the discrete levels to set in (which increases rapidly with the multiplicity of resonance) becomes of the order of the pulse duration [14]. Consideration of these effects can become important when dealing with a more realistic energy spectrum [IO,241 and consequently with a larger number of two- and three-photon resonances. Lt has been discussed in the literature whether the dissociation of polyatomic moIecules in an intense IR field is a statistical thermal process [3,7J. In ref. [3] it has been observed that considerable vibrational heating occurs at @ > (Pth with both long and ultrashort pulses (see fig. 2, dotted lines), which makes it possible to interprete the experimental dissociation yield in the framework of the usual statistical theory. On the other hand the present calculations of the average number of photons stored by the molecules which had not dissociated (fig. 2, dashed lines) show that this model predicts the due vibrational heating to occur with the ultrashort pulses only when there is no “bottlenecking”, the heating being much less prominent at 7 = 10 ns and vanishing at T = 100 ns (the latter case is not shown in fig. 2). Here we do not discuss the interesting experimental results on SF6 dissociation by two lasers [25,26] because it is beyond the scope of the present paper. In conclusion, the investigation of SF6 dissociation in an intense IR field is carried out in the framework of a modei separate aspects of which have been discussed previously by a number of authors. The main feature of the present work is that for the first time the overall process is treated on the whole. Using some general assumptions a closed set of equations has been derived which describes the coherent excitation of molecules within the lower discrete levels, “leakage” into the quasicontinuum, the incoherent light absorption within the quasicontinuum, and the dissociation itself. This approach enabled us to compare the theory with the available experimental data. This comparison has shown that along with doubtless merits the model considered has some shortcomings and does not explain all the experimental facts cited above. Nevertheless, the basic ideas of this model seem to be fruitful and it is

,

E.S. ~ledvedev/Collisiot~lessdissociation o~SFS

expected that further development of both theory and experiment will bring them in a closer agreement.

[lo] C.D. Cantrelland [ll j 1121

Acknowledgement 1131

I would like to thank Professor V.I. Osherov, Drs. A.I. Voronin and E.B. Gordon for valuable discussions and Dr. A. Krestinin who turned my attention to the advantages of the simple numerical algorithm used in this work.

1141 [15]

[16]

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S. Mukamel, Phys. Rev. Letters 38 (1977) 1131. [4] V.M. Akulin. S.S. Alimpiev, N.V. Kylov and [5]

[6] [7]

[8] 191

L.A. Shelepin, Zh. Experim. i Teor. Fir. 69 (1975) 836. V.M. AkuBn, S.S. Alimpiev, N.V. Karlov, B.G. Sartakov and L-A. Shelepin. Zh. Experim. i Tear. Fiz. 71 (1976) 454. S. Mukamel and J. Jortner, J. Chem. Phys. 65 (1976) 5204. E.R. Grant,P.A. Schulz, Aa.S. Sudbo, Y.R. Shen and Y.T. Lee, Phys. Rev. Letters 4G (1977) 115. J.L. Lyman, J.Chem. Phys. 67 (1977) 1868. A-V. Eletsky, V-D. KBmov and V.A. Legasov, Dokl. Akad. Nauk SSSR 237 (1977) 1396.

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