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Kinetic model for laser-assisted reactions
Volume
117, number
KINETIC Joseph
CHEMICAL
6
MODEL
PHYSICS
FOR LASER-ASSISTED
5 July
LETTERS
1985
REACTIONS
J- BELBRUNO
Deporrmenr
Received
of Chemistry,
7 February
A bimolecular
Darrmourh
College.
1985; in final form
expression
16 April
has been developed
laser fields
The result is a function
parameters
which
deline
Hanover.
for the predlction
potential.
USA
1985
of the laser power
the internuclear
NH 03755,
density.
or reactive rate conswnts
the electronic
The final form
is based
transiuon
and cross seclions
momenL
the spectral
on a phenomenological
model
in non-resonant detunmg
treaung the reaction a~ a unimoleeular process. Available data are discussed in terms of a van der Waals potential correlated a.~a funcrion of the long-range altractlon. Agreement wirh published resulls is very good. The model is used predictions
or branching
experiments
IS presented.
ratios
in several
systems
and
1. Introduction Laser-assisted collisional processes are currently the focus of both experimental and theoretical efforts. The characteristic feature of these processes is the absorptron (or emissron) of a photon not resonant with any state of the reactants or products. The interest enters on the possibility of observrng the transrtion from reactants to products, i.e. the “transition state”, as well as the potential for populating inaccessible atomic and molecular levels and creating new laser systems. Some of the recently published results are summarized below. This listing is by no means complete and is intended merely to illustrate the possibilities. Various laboratories have reported findings mvolvmg energy transfer (e.g., ref. [I]), associative ionization [2] and Penning ionization [3] III atomic collisions. In addition, a timedependent perturbation approach [4] which successfully predicts these experimental results has been developed.Moknrlar laser-assisted processes are relatively unexplored. The opening of a closed reaction channel
a discussion
of
the
molecular
properties
and the
and is the result ol
requisite
and
are
to make to successful
tween CO molecules have also been examined [9]. Considerable theoretical work has been done on Hz + F, among other systems [lo] and a laser-induced harpooning model has been proposed as a gtude in the selection of suitable experimental systems [l l] . However, the selection of systems for experimental observation remains a difficult process. The Hz + F research is too detailed for routine applicatron, whiIe the harpoon model is restricted to a certain class of molecular reactions. We have employed an effective two-body “collision theory” model with general applicability to non-ionic reactions, since many of the systems of chemical interest belong to this category. The model successfully reproduces the results of many of the experiments described above and offers a simple means of choosing experimental systems. Moreover, it Illustrates the physical properties necessary to improve the probability of observing the effect we have just described. In particular, the final result depends upon the strength of the long-range attractive term in the potential function describing the entrance channel.
in the K + HgBr, reaction [S] and emission from Na* due to the decomposltron of a photoexcited transient complex in the reaction 163 of K with NaCl have been
Unfortunately,
reported. Three laboratories [7,8] have reported experiments involving the reaction between Xe and Cl,. (Unfortunately, there is no agreement on the interpretation of the results.) Energy transfer collisions be-
tion moment, laser power density and spectral detuning are also shown to be crucial parameters. The latter two are, of course, easily controlled, while the former does not vary si@kantly for the reactions reported
592
this feature leads to a failure of the
model for systems best described ionic curve crossing mechanisms.
0 009-2614/85/S (North-Holland
by very-long-range The electronic transi-
03.30 0 Elsevier Science Publishers B.V. Physics Publishing Division)
Volume 117, number 6
CHEMICAL
PHYSICS
in the literature. A similar correlation has been employed in the study of collision-induced changes [12] For those collisional processes, the cross section was expressed as a function of collision duration, polarizability or well depth, with the latter the most successful. The problem has been approached as the unimolecular decay of an excited transient complex. The concentration of these species is calculated via the radial distribution function (to obtain the total quasi-molecule concentration) and first-order petiurbatlon theory (to obtain the excited species concentration). Statistical decay into products is assumed, but not critical to development of rhe model. The result of this analysis is a rate expression resembling that for a bimolecular process. A description of this model is given m section 2. In section 3, the model is applied to systems under the influence of van der Waals-type forces and a discussion of the molecular parameters is presented. Section 4 compares the model calculations to the available experimental results and speculates on future developments in this field of chemical dynamics.
2. Model
plexes in molecular
beam
scattering
experiments
[ 141.
We envision a process m which a quasi-molecule forms, absorbs a photon and, subsequently, decays to products. treats these events independently
since
assume that they occur sequentially. The reaction proceeds through a transient complex as:
we
A + B + [A-B]
+
IW +
[A-B]
* + products,
either or both of the reactants may be molecular species. A schematic representation of the mechanism is shown in fig. 1. The rate of the laser-assisted process is given by where
rate = kdNCA_-B)*,
(I)
5 July 1985
#_,
PRODUCTS
A--B
Fig. 1. Schematic representation of the unimolecular reaction mor?el for laser-assisted mo1ecuJa1chemistry. An atom-molecule system is depicted. Reactants are initially in their respectIve ground states and inftitely separated. As they approach to within a distance r,-,,, absorption by a quasi-molecule is passible. This transition is detuned from a resonant atomic transition by AU. The excited state of the quasi-molecule is above Ea, the minimum energy for product formation, and the systern decays into product channels,
where the rate constant, k,, is that for the unimolecular decomposition of the activated quasi-molecule. All competing processes involving the excited complex are assumed 10 be slower than the laser-assisted reaction, i.e. the mechanism
A photoactivated, unimolecular decay mechanism is used to estimate the rate of the overall laser-assisted reaction. The procedure is identical to that employed for the determination of the rate of an “ordinary” unimo!ecular reaction [13], except now the reactant is a transient quasi-molecule. This type of analysis has been applied to the study of long-lived collision com-
Our description
LETTERS
requires
that we operate
at low
back-
molecular beam or low-pressure flow reactors are required. The number density of collision pairs,NA_B, depends upon the internuclear potential. As is the usual procedure for fluids which may be treated classically and are well represented by paiMnse interactions, this concentration is obtained via the radial distribution function [15] _For a single A moIecule at a fixed position, the probability of observing a B molecule at a distance between r and r + dr isN,@(r)dr, where gc2)(r) is the radial distribution knction and r is the intermolecular separation. For the low-pressure amditions required by the nature of these experiments, the radral distribution function may be approximated [ 151 as exp [-V(r)lkT] , where V(r) is the intermolecular potential evaluated at the internuclear separation, r. Taking into account the concentration of A molecules, the differential number density of transient paus is given by ground
pressures.
mA_B
= 4flAiVB
The total number
This conditicn
exp[-V(r)/kT]p density
of
A-B
implies
that
dr.
quasi-molecules
is 593
Volume 117, number 6 the integral of eq. @a}.
The limits of this integraI are related to the distance at which the laser-assisted interaction is expected to occur. As a practical matter, we must define the ‘bond length” in the transient complex. Previous reports 13,101 have pointed out a potential source of difficulty. The radiative coupling can be significant over a large range of internuclear separations, so that many calculations of laser-assisted rate constants are lower limits. This is especially true when the laser-assisted rate is greater than the fieldfree value_ However, the distance we require has been customarily defined as that which results m a classical orbitmg collision [ 16,171 The orbrting process increases the probability of a quasi-molecule lifetime of sufficient length to permit absorption of a photon. The value Of ‘orbiting may be calculated from the effective potential and the initial kinetic energy [16,18]. Eq. (2a) is then integrated over a suitably small interval about r orbitirtg- We defer further discussion until the model is apphed to a particular potential, but note that the range of r values should be restricted such that the corresponding kinetic energy interval is within the limits of the experimental distribution_ For the present, we wnte the integral form of eq. (2a) with indefinite linuts N A-B = 4nNAN8 J r3- exp[-F(r)/kT]
5 July 1985
CHEMICAL PHYSICS LFXTERS
dr.
(2b)
Anticipating the final result, the required quantity is formally given by the product of eq. (2b) (the concentration of complexes) and the probability of the quasimolecule absorbing a photon. For the weak-moderate field case, this probabfity may be expressed in the usual manner as the result of a perturbation on a stationary system [ 19]_ The development follows that used UI the computation of the probability of any molecular dipole transition. One may write the Hamiltonian in terms of a time-mdependent part and a perturbation (the molecule/photon interaction) The perturbation couples the ground and excited states of the transrent complex. The transition probability IS then obtained by evaluating the coupling integral between limits r’ = 0 and r’ = t. For short interaction times, the result of this procedure is an expression wluch is a function of the transition moment and spectral detunmg, but independent of time. The final expression for the rate of the readon is
rate = 161T4p21 ch 2(Aw)2
NANBka
Jr’
exp[-V(r)/kT]
dr,
(3)
where ao (=w~ - G)~,,,.,.,~) is the detuning, p is the transition moment of the complex at the frequency wres and all other variables have their usual meaning. Eq. (3) has the form of a rate law for a bimolecular process, with a laser-power-dependent rate constant. We may identify this bimolecular rate constant as 167r4p21
k,=k,
_r r2 exp[-v(r)lkT]
ch2(Aw)’
dr.
(4)
The remaining task is to defme the decomposition rate constant, k,. For molecular systems, an RRKtype term 1s acceptable, while atomic reactions are most easily handled by an analogous rate constant which reflects the density of product states [20] _ In either case, the rate constant will contain a term which is best represented as the probability of drssociation into the specified products_ A unimolecular rate constant has units of inverse time. A second factor [ 121 must be included to obtain the proper folm for k, _This factor is the interaction time, 7. This is typically a few picoseconds or less and may be estimated as rorbitin$(v~, if unknown. Inserting the resultmg expression for kd into eq. (4) yields the final expression for the rate constant k
b
’ 6T4p2r ch2(Aw)2
P 7-l Jr’
exp[-V(r)/kT]
dr_
d
(54
A bimolecular cross section may be written by employing the usual k = u(v) relationship, where (u) is the average relative velocity for the chemical collision partners_ The procedure described above is related to the quasi-static theory (QST) of line broadening employed so fruitfully by Gallagher and co-workers [21]. Both models
rely on a calculation
of the absorption
proba-
bility in a collision complex. However, the QST is typically used to extract potential enerw curves from the observed intensity of the emission or absorption from a collision complex. The work described in this report is concerned with the direct estimation of cross sections for laser-assisted product formation. As such, it requires not only the fraction of excited quasi-mol-
Volume
117, number 6
5 July 1985
CHEMICAL. PHYSICS LETTERS
ecules as in QST, but also the probability of decay of the collision complex into specific product channels. A statistical theory, RRK, is applied. As shown above, a completely kineric approach has been taken to arrive at the final result.
term. The location of the maximum centrifugal barrier, rm , for a potential of tbis type is (2c6/@16, where E is the initial kinetic energy. For a suitable small interval, Ar, about r,.,.,, the number density of transient @rs is obtained via eqs. (2) as N
3 _Application In this section, we discuss practical matters such as the form of the intermolecular potential, the transition moment, etc. After making appropriate substitutions, we arrive at the final expression and employ it in an attempt to systematize the experimental data. We shall arbitrarily restnct our discussion to neutral-neutral reactions, since systems involving ionic species are usually best described by very-long-range ionic curve crossings. This crossing point 1s typically at intemuclear distances which are considerably larger than rorbi~ and the model will, therefore, underestimate the cross section. The laser-assisted harpooning calculation [l l] is applicable to these reactions. The transition moment and spectral detuning may be considered jointly. The required transition moment, p, is that for excitation of the quasi-molecule. This is
A--B
exp [--V(r,,,)/kTl
= 4flANBr~
b-
(6)
Eq. (Sa) becomes k, = 1.379
X 1O23 j~&+
(=I
after making these substitutions, setting Ar equal to
05 A and assuming the classical value for the average initial kinetic energy. The value of Ar is arbitrary, but is a conservative value that corresponds to an energy uncertainty of 30 meV_ Treating the final result as the product of the density of transient pairs, the excitation probability and the decomposition probabdity; the effect of varying these parameters may be examined. Fig. 2 is an estimate of the number density of transient pairs per unit interval width under two sets of euperimental conditions_ Clearly, those systems in which the long-range attraction is greatest hold the most promise for observing laser assisted chemistry. Increasing the pressure of the reactants also produces a larger density
an unknown quantity_ However, since the transition
occurs at a large internuclear separation (see below), this may be approxrmated as the transkon moment for the atom (molecule) A, i.e. pA _This approximation -may incorporate an error in the final result, but at the present time an easily applicable alternative is lacking. A similar approach is taken in most attempts to model the laser-assisted process [l l] . The resonance denominator is then given by the detuning of the experimental wavelength from the resonance frequency of the unperturbed atom/molecule A as AmA = o - w*_ The simplest choice for the potential function that still retains the physics of the interaction IS the van der Waals form. While some of the systems that have been studied cannot be accurately described as isotropic, the approximation is adequate_ Since the process occurs at large internuclear separation we are only concerned with the attractive part of any other chosen potential and, for the vdW form, set V(r) = x6/+ _ An obvrous acceptable alternative is to represent the potential as a Lennard-Jones interaction where the C, coefficient is obviously related to the well depth of the U function and we require only the attractive, I/+
56
1
I
57
58
59
-log
ce
Fig. 2 Plot of the differential number density of collision complexes as a function of the sbngth of the long-range attraction. Data are presented for the low- (1 Torr) and high(100 TonI pressure cases. Increasing the number densty results in ilarger experimental signal
Volume
CHEMICAL
117, number 6
PHYSICS
of quasi-molecules and a larger signal, but even at 100 Torr, our assumption that competing processes were insignificant is no longer valid sin= collisional destruction of the transient pair will occur at a substantial rate. The importance of the long-range attraction to laser-assisted chemistry 1s evident in the results of Brooks and co-workers [5,6] _The collision systems described in these reports are known to invoive especially long-lived complexes and along with the Xe + Cl2 reaction, which may proceed through a stable XeC12 molecule [7,8], are the only examples of reactive, laser assisted chemistry_ The excitatron probabihty is written in terms of a transltion moment and a spectral detuning. Since all of the transitions in section 4 are allowed, the electronic transltion moment does not vary significantly for the systems we model. In general, however, one would clearly desire to choose a reaction which involved a strongly allowed electronic transition. (The rate constant increases linearly with & _) Fig. 3 presents the dependence of the rate constant on the spectral detuning, an experimentally controllable variable for a given reaction system. As anticipated, the rate constant increases as the laser frequency approaches the value WA _Again, an examination of the available data reflects this point. For those experiments in which the
LETTERS
5 July 1985
cross section was iecorded as a funciion*of the laser frequency, the cross section is largest for the value closest to the known resonance [l-3] _We now apply the model to systems which have been observed ‘experimentally, as well as for the prediction of branching ratios in future experiments
4. Results Figs. 2 and 3 are useful as aids in selecting experimental systems, but the model is inappropriate if it does not reproduce the reported results within reasonable limits. We have calculated the cross sections of several atom-atom and atom-molecule reactionsSome of the required physical properties were unknown and were estimated based on available data. The reaction systems and input data are shown in table 1. A comparison of the model and measured data is made in table 2. The agreement is very good, especially in l&t of the uncertainty m some of the reported results due to the diffkult nature of these
experiments. The Ar* + Ca system has been included as an indication of the magnitude of the reported data, as well as to demonstrate the previously described difficulty in handIing reactions involving ionic potential curves. The magnitude of this cross section is an indication that the overestimation of the Sr* + Ca value is not unrealistic. The first two entries in table 2 are more than adequately predicted by the model. The experimental result for the second reaction (Ca* + Sr) appears to differ with the first (reported by the same authors) only by virtue of the smaller laser field. The model results are clearly influenced by that factor, but also the spectral detuning. The Ca* + Sr experiments were not as extenavely detailed as the reverse reaction and it is difficult to determine ifthe detuning effects were observed in the laboratory. The last three entries in table 2 involve atom-molecule laser-assisted reactions and it is this type of system that most interests chemists. The smaller cross sections which seem to be characteristic of this type of process are more ‘tificult to predict. Agreement within an order of magnitude and
__/_I___/_ 0
1 log
2
3
(filu) - -_-
Fig. 3. Plot of the rate constant for excited state production as a function of the detutig from resonance A malkr detuning yields a larger rate constant_
596
correct relative ordering of the results is certainly acceptable_ This has been achieved for the Xe+C12 and K + NaCl reactions. The apparent failure for the
K + HgBrz system is unfortunate,
but the authors of
5 July 1985
CHEMICAL PHYSICS LEITERS
Volume 117, number 6 Table 1
Input parameters Reaction
104(“1
I
AW
G=m-?
0V cm-l)
(cm-l )
41 a)
1035,u2 (erp cm’)
rorb (A)
Pd
b)
5.93 d) 5.75
0.45 0.72
Sr* + Ca
9.09
1 x 1010
Ca* + Sr
9.09
3 x 109
397
b)
Ar*+ca Xe + Cl1
5.17
1 x 10’
1122
b)
7.69
0.98
3.80
8 x 10’
1774
b)
4.68
e)
2 x 106
3822
556
5.94
0.28
3 x 103
2109
5.56
4.95
0.39
5.87
K + HgBr*
13.8
K + NaCl
=)
a) All energy dirferences were obtained using the data in ref. [20] _ b, The transitiondipole moment has not been measured. Therefore, since the transition is allowed, it has been estimated to be 2 x 10-3s erg cm6_ c, Ref. [22]. ‘) The van der Waals coeffkients were calculated using the Slatcr-Kirkwood approximation. See ref. 1231. e, Not applicable: ref. [7] asserts that the decomposition is bimolecular.
Table 2 Comparison of model and experimental results Reaction
amodel (A’)
aexp (A’)
Sr* + Ca
4380
1000 a)
Ca* + Sr
22 01
AI*+&
0.03
Xc + Cl, K + HgBr2
10-9
K + NaCl
lad
100 a) 6200 b, 0.01=) 0.1 d) 104
e,
a) Ref. [l]. b, Ref. [3]. c, Several measurements have been reported_See refs. [7.8] ‘1 Ref. [5]. e, Ref. [6].
_
experimental study report that they have not been able to observe this process in a new cw apparatus due to interference from side reactions_ Therefore, it is im-
the
possible at this time to determine why, or if, the model has failed in this instance_
An advantage of treating the reactions in the manner deskibed above is the ability to predi’ct branching ratios for the products. The result will be dependent upon the definition-of the dissociation probability, but by statig
the assumptions employed in the calcu-
lations comparisons may be made among different reac-
tions. We have chosen a statistical model and apply it to the calculation of relative branching ratios for the related reactions: K + NaCl + KCI + Na* and Rb + NaCl + RbCl + Na* [24] _The cross section for the latter is smaller, even though the long-range attraction is greater and the collision complex lifetime is longer [25]. The cause of these effects is the dissociation probabitity,Pd . An alternate product channel for both reactions is the formation of an excited state of the entrance channel alkali atom. K* and Na’ differ in energy by approximately 1000 cm-l. Although the formation of K’ would be favored on statistical grolinds, the ratio, Na*/K*, is 0.65. However, Rb’ lies approximately 4000 cm-l below the energy of Na*. The statiskal ratio for the formation of Na* relative to Rb* is GO.1, assuming experimental conditions similar to those in the K + NaCl case. The cross section for tbe Rb reaction should be approximately a factor of 5-10 smaller, after accounting for the larger probability of complex formation, due to the predominance of the alternate channel. This is the type of effect observed in preliminary experiments. Finally, it is possible to use eq. (6) to correlate the experimental data. Fig. 4 is a plot of tbe experimental cross section per unit laser power density, u/MW cm-z, as a function of rorbiting_ The orbiting radius is a function of the vdW C6 uxfficients and the initial kinetic energy. Tlus is a more accurate reflection of the strength 597
Volume
CHEMICAL
117, number 6
PHYSICS
LEmRS
5 JuIy.1985
ployed it to explain some unexpected features in those results caused by ~favorabIe branching ratios, The associated correlation are a useful guide to the selection of experimental reactions. Iaboratoj work has begun on react;lon systems which satisfy all of the above critena.
Acknowledgement This research was supported Corporation.
by a grant from Research
log r
rig. 4. Log-log plot of the reduced cross section versus the orbiting radius for the reactions discussed in the text and shown in tables 1 and 2.
of the long-range interaction. The result is a straight increasing the range of the internuclear attraction mcreases the cross section. Plots such as this are useful in the process of selecting experimental candidates. For a given laser power, usually limited by the available apparatus, and a given detuning, where the pumping wavelength must be as close to a real transition as possible while satisfying the energy reqmrements of the reaction, the experimental&t would like to be as far along this curve as possiile. This IS identical to the conclusions reached m the study of collisional induced state changes [ 121. In summary, we have shown that the laser-assisted chemical reaction involving neutral species may be adequately represented by a mechanism involving the statistical decay of a photochemically activated quasimolecule. The conditions necessary to maximize the cross section for this process are: (I) as large a laser power density as is feasible without causing m~tiphoton effects to become competitive with the reaction of interest; (2) the energy defect supplied by the laser photon shouid be nearly resonant with an allowed transition of one of the reactant species; (3) the interaction should be governed by a strongly attractive potential; and (4) the product channel should be statistically favored in the decomposition of the collision complex. We have shown that the model successfiIl.ly reproduces a variety of reported experiments and have em-
line;
598
References 111 W.R. Green, J. Lukxik.
J-R_ W-n, M.D. Wright. J.F_ Young and SE. Hanis, Phys. Rev. Letters 42 (1979) 970. 121 P. PoIak-Dingels, J.F. Delpech and J. Weiner, Phys. Rev. Letters 47 (1981) 1888. VI JH. Goblc, W_E_ HoUingsworth and JS. Wmn, Phys. Rev. Letters 47 (1981) 1888. r41 SE. Harris and J.C. White,lEEE J. Quantum Electron. QE-13 (1977) 972; SE_ Ha&s and D-B. Lidow. Phys. Rev. Letters 33 (1974)
674. Ul P. Hexing, P.R. Brooks, R-F_ Curl Jr., R.S Judson and P.S. Lowe, Phys- Rev. Letters 44 (1980)
687.
161 T-C. Maguire, P.R. Brooks and R.F. Curl Jr., Phys. Rev. Letters 50 (1983) 1918. 171 RE. Wilcomb and R_ Burnham, J. Chem. Phyr
74 (1981) 6784. 181 H.P. Gneneisen, Xue-Jmg and K.L. Kompa, Chem. Phys. Letters 82 (1981) 421; JX. Ku, G. Inoue and D.W. Setser, J. Phys Chem. 87 (1983) 2969. [91 J. Lukasik and S.C. Wallace, Phyr Rev: Lettezs 47 (1981) 240. T-F. George, I H. Zimmexman. P-L_ DeVrics,, J-M. Yuan, K S. Lam, J.C. BeIIum. H-W Lee, M-S.. Slirt&y and J-TLin, in: Chemical and biochemical appLications of lasers, Vat 4, ed. C-8_ Moore (Academic Press, New York, 1979). fll] J. Weiner, 3. Chem. Phys 72 (1980) 5731.‘ [12] H.N. LIn, M. Seaver, K-Y. Tang, ABSII. Kr&ht and C-S. Parmenter, J. Chem. Phys 70 (1979) 5442. [ 131 W. Frost, Theory of unimolecuIax reactions (Academic Press, New York, 1973). [ 141 W-B. Miller, %A. Safron and D.R; Herschbach! !. Chem. Phys. 56 (1972) 3581. [ 151 T.L. Hi& An introduction to stalistic thermodynamics (Addison-Wesley, Reading, 1960). [ 161 SJ. Riley and D.R;Herschbach. J. Chem Phys 58 (1973) 58.
iw
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117. number 6
CHEMICAL.
PHYSICS
1171 J.P. Toennles, W. Welz and G. Wolf, J. Chem. Phys. 71 (1979) 614. [ 181 R.E. Weston and HA. Schwa, Chemical kinetics (Prentice-Hall. Englewood Cliffs, 1972). [ 191 A. Yariv. Ql’zatum electronics, 2nd Ed. (Wiley, New York. 1975). [20] CE Moore. Atomic energy levels, NSRDS-NBS 35 (NatL Bur. Std.. Washington. 1971). [21] RE.M. Hedges, D L. Drummond and A. Gallagher, Phys Rev. A6 (1972) 1519.
LEITERS
5 July 1985
1221 W.L. Wiese and G_A. Marcin, Wavelengths and transition probabMties for atoms and atomic ions, NSRDS-NBS 68 (Natl_ Bur. Std , Washington, 1980). [ 231 H.L. Kramer and D.R. Herschbach, J. Chem. Phye 53 (1972) 2792. [ 241 R.F. Curl and P.R. Brooks, The In temational Congress 6f Pacific Basin Societies, Honolulu, Hawaii (1984) Abstract 5BO43. Symposium on the Frontiers of Spectsos[25]
copyW B. Miller, S-A. Stiron and D.R. Herschbach, Discussions Faraday Sot. 44 (1967) 108.
599
117, number
KINETIC Joseph
CHEMICAL
6
MODEL
PHYSICS
FOR LASER-ASSISTED
5 July
LETTERS
1985
REACTIONS
J- BELBRUNO
Deporrmenr
Received
of Chemistry,
7 February
A bimolecular
Darrmourh
College.
1985; in final form
expression
16 April
has been developed
laser fields
The result is a function
parameters
which
deline
Hanover.
for the predlction
potential.
USA
1985
of the laser power
the internuclear
NH 03755,
density.
or reactive rate conswnts
the electronic
The final form
is based
transiuon
and cross seclions
momenL
the spectral
on a phenomenological
model
in non-resonant detunmg
treaung the reaction a~ a unimoleeular process. Available data are discussed in terms of a van der Waals potential correlated a.~a funcrion of the long-range altractlon. Agreement wirh published resulls is very good. The model is used predictions
or branching
experiments
IS presented.
ratios
in several
systems
and
1. Introduction Laser-assisted collisional processes are currently the focus of both experimental and theoretical efforts. The characteristic feature of these processes is the absorptron (or emissron) of a photon not resonant with any state of the reactants or products. The interest enters on the possibility of observrng the transrtion from reactants to products, i.e. the “transition state”, as well as the potential for populating inaccessible atomic and molecular levels and creating new laser systems. Some of the recently published results are summarized below. This listing is by no means complete and is intended merely to illustrate the possibilities. Various laboratories have reported findings mvolvmg energy transfer (e.g., ref. [I]), associative ionization [2] and Penning ionization [3] III atomic collisions. In addition, a timedependent perturbation approach [4] which successfully predicts these experimental results has been developed.Moknrlar laser-assisted processes are relatively unexplored. The opening of a closed reaction channel
a discussion
of
the
molecular
properties
and the
and is the result ol
requisite
and
are
to make to successful
tween CO molecules have also been examined [9]. Considerable theoretical work has been done on Hz + F, among other systems [lo] and a laser-induced harpooning model has been proposed as a gtude in the selection of suitable experimental systems [l l] . However, the selection of systems for experimental observation remains a difficult process. The Hz + F research is too detailed for routine applicatron, whiIe the harpoon model is restricted to a certain class of molecular reactions. We have employed an effective two-body “collision theory” model with general applicability to non-ionic reactions, since many of the systems of chemical interest belong to this category. The model successfully reproduces the results of many of the experiments described above and offers a simple means of choosing experimental systems. Moreover, it Illustrates the physical properties necessary to improve the probability of observing the effect we have just described. In particular, the final result depends upon the strength of the long-range attractive term in the potential function describing the entrance channel.
in the K + HgBr, reaction [S] and emission from Na* due to the decomposltron of a photoexcited transient complex in the reaction 163 of K with NaCl have been
Unfortunately,
reported. Three laboratories [7,8] have reported experiments involving the reaction between Xe and Cl,. (Unfortunately, there is no agreement on the interpretation of the results.) Energy transfer collisions be-
tion moment, laser power density and spectral detuning are also shown to be crucial parameters. The latter two are, of course, easily controlled, while the former does not vary si@kantly for the reactions reported
592
this feature leads to a failure of the
model for systems best described ionic curve crossing mechanisms.
0 009-2614/85/S (North-Holland
by very-long-range The electronic transi-
03.30 0 Elsevier Science Publishers B.V. Physics Publishing Division)
Volume 117, number 6
CHEMICAL
PHYSICS
in the literature. A similar correlation has been employed in the study of collision-induced changes [12] For those collisional processes, the cross section was expressed as a function of collision duration, polarizability or well depth, with the latter the most successful. The problem has been approached as the unimolecular decay of an excited transient complex. The concentration of these species is calculated via the radial distribution function (to obtain the total quasi-molecule concentration) and first-order petiurbatlon theory (to obtain the excited species concentration). Statistical decay into products is assumed, but not critical to development of rhe model. The result of this analysis is a rate expression resembling that for a bimolecular process. A description of this model is given m section 2. In section 3, the model is applied to systems under the influence of van der Waals-type forces and a discussion of the molecular parameters is presented. Section 4 compares the model calculations to the available experimental results and speculates on future developments in this field of chemical dynamics.
2. Model
plexes in molecular
beam
scattering
experiments
[ 141.
We envision a process m which a quasi-molecule forms, absorbs a photon and, subsequently, decays to products. treats these events independently
since
assume that they occur sequentially. The reaction proceeds through a transient complex as:
we
A + B + [A-B]
+
IW +
[A-B]
* + products,
either or both of the reactants may be molecular species. A schematic representation of the mechanism is shown in fig. 1. The rate of the laser-assisted process is given by where
rate = kdNCA_-B)*,
(I)
5 July 1985
#_,
PRODUCTS
A--B
Fig. 1. Schematic representation of the unimolecular reaction mor?el for laser-assisted mo1ecuJa1chemistry. An atom-molecule system is depicted. Reactants are initially in their respectIve ground states and inftitely separated. As they approach to within a distance r,-,,, absorption by a quasi-molecule is passible. This transition is detuned from a resonant atomic transition by AU. The excited state of the quasi-molecule is above Ea, the minimum energy for product formation, and the systern decays into product channels,
where the rate constant, k,, is that for the unimolecular decomposition of the activated quasi-molecule. All competing processes involving the excited complex are assumed 10 be slower than the laser-assisted reaction, i.e. the mechanism
A photoactivated, unimolecular decay mechanism is used to estimate the rate of the overall laser-assisted reaction. The procedure is identical to that employed for the determination of the rate of an “ordinary” unimo!ecular reaction [13], except now the reactant is a transient quasi-molecule. This type of analysis has been applied to the study of long-lived collision com-
Our description
LETTERS
requires
that we operate
at low
back-
molecular beam or low-pressure flow reactors are required. The number density of collision pairs,NA_B, depends upon the internuclear potential. As is the usual procedure for fluids which may be treated classically and are well represented by paiMnse interactions, this concentration is obtained via the radial distribution function [15] _For a single A moIecule at a fixed position, the probability of observing a B molecule at a distance between r and r + dr isN,@(r)dr, where gc2)(r) is the radial distribution knction and r is the intermolecular separation. For the low-pressure amditions required by the nature of these experiments, the radral distribution function may be approximated [ 151 as exp [-V(r)lkT] , where V(r) is the intermolecular potential evaluated at the internuclear separation, r. Taking into account the concentration of A molecules, the differential number density of transient paus is given by ground
pressures.
mA_B
= 4flAiVB
The total number
This conditicn
exp[-V(r)/kT]p density
of
A-B
implies
that
dr.
quasi-molecules
is 593
Volume 117, number 6 the integral of eq. @a}.
The limits of this integraI are related to the distance at which the laser-assisted interaction is expected to occur. As a practical matter, we must define the ‘bond length” in the transient complex. Previous reports 13,101 have pointed out a potential source of difficulty. The radiative coupling can be significant over a large range of internuclear separations, so that many calculations of laser-assisted rate constants are lower limits. This is especially true when the laser-assisted rate is greater than the fieldfree value_ However, the distance we require has been customarily defined as that which results m a classical orbitmg collision [ 16,171 The orbrting process increases the probability of a quasi-molecule lifetime of sufficient length to permit absorption of a photon. The value Of ‘orbiting may be calculated from the effective potential and the initial kinetic energy [16,18]. Eq. (2a) is then integrated over a suitably small interval about r orbitirtg- We defer further discussion until the model is apphed to a particular potential, but note that the range of r values should be restricted such that the corresponding kinetic energy interval is within the limits of the experimental distribution_ For the present, we wnte the integral form of eq. (2a) with indefinite linuts N A-B = 4nNAN8 J r3- exp[-F(r)/kT]
5 July 1985
CHEMICAL PHYSICS LFXTERS
dr.
(2b)
Anticipating the final result, the required quantity is formally given by the product of eq. (2b) (the concentration of complexes) and the probability of the quasimolecule absorbing a photon. For the weak-moderate field case, this probabfity may be expressed in the usual manner as the result of a perturbation on a stationary system [ 19]_ The development follows that used UI the computation of the probability of any molecular dipole transition. One may write the Hamiltonian in terms of a time-mdependent part and a perturbation (the molecule/photon interaction) The perturbation couples the ground and excited states of the transrent complex. The transition probability IS then obtained by evaluating the coupling integral between limits r’ = 0 and r’ = t. For short interaction times, the result of this procedure is an expression wluch is a function of the transition moment and spectral detunmg, but independent of time. The final expression for the rate of the readon is
rate = 161T4p21 ch 2(Aw)2
NANBka
Jr’
exp[-V(r)/kT]
dr,
(3)
where ao (=w~ - G)~,,,.,.,~) is the detuning, p is the transition moment of the complex at the frequency wres and all other variables have their usual meaning. Eq. (3) has the form of a rate law for a bimolecular process, with a laser-power-dependent rate constant. We may identify this bimolecular rate constant as 167r4p21
k,=k,
_r r2 exp[-v(r)lkT]
ch2(Aw)’
dr.
(4)
The remaining task is to defme the decomposition rate constant, k,. For molecular systems, an RRKtype term 1s acceptable, while atomic reactions are most easily handled by an analogous rate constant which reflects the density of product states [20] _ In either case, the rate constant will contain a term which is best represented as the probability of drssociation into the specified products_ A unimolecular rate constant has units of inverse time. A second factor [ 121 must be included to obtain the proper folm for k, _This factor is the interaction time, 7. This is typically a few picoseconds or less and may be estimated as rorbitin$(v~, if unknown. Inserting the resultmg expression for kd into eq. (4) yields the final expression for the rate constant k
b
’ 6T4p2r ch2(Aw)2
P 7-l Jr’
exp[-V(r)/kT]
dr_
d
(54
A bimolecular cross section may be written by employing the usual k = u(v) relationship, where (u) is the average relative velocity for the chemical collision partners_ The procedure described above is related to the quasi-static theory (QST) of line broadening employed so fruitfully by Gallagher and co-workers [21]. Both models
rely on a calculation
of the absorption
proba-
bility in a collision complex. However, the QST is typically used to extract potential enerw curves from the observed intensity of the emission or absorption from a collision complex. The work described in this report is concerned with the direct estimation of cross sections for laser-assisted product formation. As such, it requires not only the fraction of excited quasi-mol-
Volume
117, number 6
5 July 1985
CHEMICAL. PHYSICS LETTERS
ecules as in QST, but also the probability of decay of the collision complex into specific product channels. A statistical theory, RRK, is applied. As shown above, a completely kineric approach has been taken to arrive at the final result.
term. The location of the maximum centrifugal barrier, rm , for a potential of tbis type is (2c6/@16, where E is the initial kinetic energy. For a suitable small interval, Ar, about r,.,.,, the number density of transient @rs is obtained via eqs. (2) as N
3 _Application In this section, we discuss practical matters such as the form of the intermolecular potential, the transition moment, etc. After making appropriate substitutions, we arrive at the final expression and employ it in an attempt to systematize the experimental data. We shall arbitrarily restnct our discussion to neutral-neutral reactions, since systems involving ionic species are usually best described by very-long-range ionic curve crossings. This crossing point 1s typically at intemuclear distances which are considerably larger than rorbi~ and the model will, therefore, underestimate the cross section. The laser-assisted harpooning calculation [l l] is applicable to these reactions. The transition moment and spectral detuning may be considered jointly. The required transition moment, p, is that for excitation of the quasi-molecule. This is
A--B
exp [--V(r,,,)/kTl
= 4flANBr~
b-
(6)
Eq. (Sa) becomes k, = 1.379
X 1O23 j~&+
(=I
after making these substitutions, setting Ar equal to
05 A and assuming the classical value for the average initial kinetic energy. The value of Ar is arbitrary, but is a conservative value that corresponds to an energy uncertainty of 30 meV_ Treating the final result as the product of the density of transient pairs, the excitation probability and the decomposition probabdity; the effect of varying these parameters may be examined. Fig. 2 is an estimate of the number density of transient pairs per unit interval width under two sets of euperimental conditions_ Clearly, those systems in which the long-range attraction is greatest hold the most promise for observing laser assisted chemistry. Increasing the pressure of the reactants also produces a larger density
an unknown quantity_ However, since the transition
occurs at a large internuclear separation (see below), this may be approxrmated as the transkon moment for the atom (molecule) A, i.e. pA _This approximation -may incorporate an error in the final result, but at the present time an easily applicable alternative is lacking. A similar approach is taken in most attempts to model the laser-assisted process [l l] . The resonance denominator is then given by the detuning of the experimental wavelength from the resonance frequency of the unperturbed atom/molecule A as AmA = o - w*_ The simplest choice for the potential function that still retains the physics of the interaction IS the van der Waals form. While some of the systems that have been studied cannot be accurately described as isotropic, the approximation is adequate_ Since the process occurs at large internuclear separation we are only concerned with the attractive part of any other chosen potential and, for the vdW form, set V(r) = x6/+ _ An obvrous acceptable alternative is to represent the potential as a Lennard-Jones interaction where the C, coefficient is obviously related to the well depth of the U function and we require only the attractive, I/+
56
1
I
57
58
59
-log
ce
Fig. 2 Plot of the differential number density of collision complexes as a function of the sbngth of the long-range attraction. Data are presented for the low- (1 Torr) and high(100 TonI pressure cases. Increasing the number densty results in ilarger experimental signal
Volume
CHEMICAL
117, number 6
PHYSICS
of quasi-molecules and a larger signal, but even at 100 Torr, our assumption that competing processes were insignificant is no longer valid sin= collisional destruction of the transient pair will occur at a substantial rate. The importance of the long-range attraction to laser-assisted chemistry 1s evident in the results of Brooks and co-workers [5,6] _The collision systems described in these reports are known to invoive especially long-lived complexes and along with the Xe + Cl2 reaction, which may proceed through a stable XeC12 molecule [7,8], are the only examples of reactive, laser assisted chemistry_ The excitatron probabihty is written in terms of a transltion moment and a spectral detuning. Since all of the transitions in section 4 are allowed, the electronic transltion moment does not vary significantly for the systems we model. In general, however, one would clearly desire to choose a reaction which involved a strongly allowed electronic transition. (The rate constant increases linearly with & _) Fig. 3 presents the dependence of the rate constant on the spectral detuning, an experimentally controllable variable for a given reaction system. As anticipated, the rate constant increases as the laser frequency approaches the value WA _Again, an examination of the available data reflects this point. For those experiments in which the
LETTERS
5 July 1985
cross section was iecorded as a funciion*of the laser frequency, the cross section is largest for the value closest to the known resonance [l-3] _We now apply the model to systems which have been observed ‘experimentally, as well as for the prediction of branching ratios in future experiments
4. Results Figs. 2 and 3 are useful as aids in selecting experimental systems, but the model is inappropriate if it does not reproduce the reported results within reasonable limits. We have calculated the cross sections of several atom-atom and atom-molecule reactionsSome of the required physical properties were unknown and were estimated based on available data. The reaction systems and input data are shown in table 1. A comparison of the model and measured data is made in table 2. The agreement is very good, especially in l&t of the uncertainty m some of the reported results due to the diffkult nature of these
experiments. The Ar* + Ca system has been included as an indication of the magnitude of the reported data, as well as to demonstrate the previously described difficulty in handIing reactions involving ionic potential curves. The magnitude of this cross section is an indication that the overestimation of the Sr* + Ca value is not unrealistic. The first two entries in table 2 are more than adequately predicted by the model. The experimental result for the second reaction (Ca* + Sr) appears to differ with the first (reported by the same authors) only by virtue of the smaller laser field. The model results are clearly influenced by that factor, but also the spectral detuning. The Ca* + Sr experiments were not as extenavely detailed as the reverse reaction and it is difficult to determine ifthe detuning effects were observed in the laboratory. The last three entries in table 2 involve atom-molecule laser-assisted reactions and it is this type of system that most interests chemists. The smaller cross sections which seem to be characteristic of this type of process are more ‘tificult to predict. Agreement within an order of magnitude and
__/_I___/_ 0
1 log
2
3
(filu) - -_-
Fig. 3. Plot of the rate constant for excited state production as a function of the detutig from resonance A malkr detuning yields a larger rate constant_
596
correct relative ordering of the results is certainly acceptable_ This has been achieved for the Xe+C12 and K + NaCl reactions. The apparent failure for the
K + HgBrz system is unfortunate,
but the authors of
5 July 1985
CHEMICAL PHYSICS LEITERS
Volume 117, number 6 Table 1
Input parameters Reaction
104(“1
I
AW
G=m-?
0V cm-l)
(cm-l )
41 a)
1035,u2 (erp cm’)
rorb (A)
Pd
b)
5.93 d) 5.75
0.45 0.72
Sr* + Ca
9.09
1 x 1010
Ca* + Sr
9.09
3 x 109
397
b)
Ar*+ca Xe + Cl1
5.17
1 x 10’
1122
b)
7.69
0.98
3.80
8 x 10’
1774
b)
4.68
e)
2 x 106
3822
556
5.94
0.28
3 x 103
2109
5.56
4.95
0.39
5.87
K + HgBr*
13.8
K + NaCl
=)
a) All energy dirferences were obtained using the data in ref. [20] _ b, The transitiondipole moment has not been measured. Therefore, since the transition is allowed, it has been estimated to be 2 x 10-3s erg cm6_ c, Ref. [22]. ‘) The van der Waals coeffkients were calculated using the Slatcr-Kirkwood approximation. See ref. 1231. e, Not applicable: ref. [7] asserts that the decomposition is bimolecular.
Table 2 Comparison of model and experimental results Reaction
amodel (A’)
aexp (A’)
Sr* + Ca
4380
1000 a)
Ca* + Sr
22 01
AI*+&
0.03
Xc + Cl, K + HgBr2
10-9
K + NaCl
lad
100 a) 6200 b, 0.01=) 0.1 d) 104
e,
a) Ref. [l]. b, Ref. [3]. c, Several measurements have been reported_See refs. [7.8] ‘1 Ref. [5]. e, Ref. [6].
_
experimental study report that they have not been able to observe this process in a new cw apparatus due to interference from side reactions_ Therefore, it is im-
the
possible at this time to determine why, or if, the model has failed in this instance_
An advantage of treating the reactions in the manner deskibed above is the ability to predi’ct branching ratios for the products. The result will be dependent upon the definition-of the dissociation probability, but by statig
the assumptions employed in the calcu-
lations comparisons may be made among different reac-
tions. We have chosen a statistical model and apply it to the calculation of relative branching ratios for the related reactions: K + NaCl + KCI + Na* and Rb + NaCl + RbCl + Na* [24] _The cross section for the latter is smaller, even though the long-range attraction is greater and the collision complex lifetime is longer [25]. The cause of these effects is the dissociation probabitity,Pd . An alternate product channel for both reactions is the formation of an excited state of the entrance channel alkali atom. K* and Na’ differ in energy by approximately 1000 cm-l. Although the formation of K’ would be favored on statistical grolinds, the ratio, Na*/K*, is 0.65. However, Rb’ lies approximately 4000 cm-l below the energy of Na*. The statiskal ratio for the formation of Na* relative to Rb* is GO.1, assuming experimental conditions similar to those in the K + NaCl case. The cross section for tbe Rb reaction should be approximately a factor of 5-10 smaller, after accounting for the larger probability of complex formation, due to the predominance of the alternate channel. This is the type of effect observed in preliminary experiments. Finally, it is possible to use eq. (6) to correlate the experimental data. Fig. 4 is a plot of tbe experimental cross section per unit laser power density, u/MW cm-z, as a function of rorbiting_ The orbiting radius is a function of the vdW C6 uxfficients and the initial kinetic energy. Tlus is a more accurate reflection of the strength 597
Volume
CHEMICAL
117, number 6
PHYSICS
LEmRS
5 JuIy.1985
ployed it to explain some unexpected features in those results caused by ~favorabIe branching ratios, The associated correlation are a useful guide to the selection of experimental reactions. Iaboratoj work has begun on react;lon systems which satisfy all of the above critena.
Acknowledgement This research was supported Corporation.
by a grant from Research
log r
rig. 4. Log-log plot of the reduced cross section versus the orbiting radius for the reactions discussed in the text and shown in tables 1 and 2.
of the long-range interaction. The result is a straight increasing the range of the internuclear attraction mcreases the cross section. Plots such as this are useful in the process of selecting experimental candidates. For a given laser power, usually limited by the available apparatus, and a given detuning, where the pumping wavelength must be as close to a real transition as possible while satisfying the energy reqmrements of the reaction, the experimental&t would like to be as far along this curve as possiile. This IS identical to the conclusions reached m the study of collisional induced state changes [ 121. In summary, we have shown that the laser-assisted chemical reaction involving neutral species may be adequately represented by a mechanism involving the statistical decay of a photochemically activated quasimolecule. The conditions necessary to maximize the cross section for this process are: (I) as large a laser power density as is feasible without causing m~tiphoton effects to become competitive with the reaction of interest; (2) the energy defect supplied by the laser photon shouid be nearly resonant with an allowed transition of one of the reactant species; (3) the interaction should be governed by a strongly attractive potential; and (4) the product channel should be statistically favored in the decomposition of the collision complex. We have shown that the model successfiIl.ly reproduces a variety of reported experiments and have em-
line;
598
References 111 W.R. Green, J. Lukxik.
J-R_ W-n, M.D. Wright. J.F_ Young and SE. Hanis, Phys. Rev. Letters 42 (1979) 970. 121 P. PoIak-Dingels, J.F. Delpech and J. Weiner, Phys. Rev. Letters 47 (1981) 1888. VI JH. Goblc, W_E_ HoUingsworth and JS. Wmn, Phys. Rev. Letters 47 (1981) 1888. r41 SE. Harris and J.C. White,lEEE J. Quantum Electron. QE-13 (1977) 972; SE_ Ha&s and D-B. Lidow. Phys. Rev. Letters 33 (1974)
674. Ul P. Hexing, P.R. Brooks, R-F_ Curl Jr., R.S Judson and P.S. Lowe, Phys- Rev. Letters 44 (1980)
687.
161 T-C. Maguire, P.R. Brooks and R.F. Curl Jr., Phys. Rev. Letters 50 (1983) 1918. 171 RE. Wilcomb and R_ Burnham, J. Chem. Phyr
74 (1981) 6784. 181 H.P. Gneneisen, Xue-Jmg and K.L. Kompa, Chem. Phys. Letters 82 (1981) 421; JX. Ku, G. Inoue and D.W. Setser, J. Phys Chem. 87 (1983) 2969. [91 J. Lukasik and S.C. Wallace, Phyr Rev: Lettezs 47 (1981) 240. T-F. George, I H. Zimmexman. P-L_ DeVrics,, J-M. Yuan, K S. Lam, J.C. BeIIum. H-W Lee, M-S.. Slirt&y and J-TLin, in: Chemical and biochemical appLications of lasers, Vat 4, ed. C-8_ Moore (Academic Press, New York, 1979). fll] J. Weiner, 3. Chem. Phys 72 (1980) 5731.‘ [12] H.N. LIn, M. Seaver, K-Y. Tang, ABSII. Kr&ht and C-S. Parmenter, J. Chem. Phys 70 (1979) 5442. [ 131 W. Frost, Theory of unimolecuIax reactions (Academic Press, New York, 1973). [ 141 W-B. Miller, %A. Safron and D.R; Herschbach! !. Chem. Phys. 56 (1972) 3581. [ 151 T.L. Hi& An introduction to stalistic thermodynamics (Addison-Wesley, Reading, 1960). [ 161 SJ. Riley and D.R;Herschbach. J. Chem Phys 58 (1973) 58.
iw
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CHEMICAL.
PHYSICS
1171 J.P. Toennles, W. Welz and G. Wolf, J. Chem. Phys. 71 (1979) 614. [ 181 R.E. Weston and HA. Schwa, Chemical kinetics (Prentice-Hall. Englewood Cliffs, 1972). [ 191 A. Yariv. Ql’zatum electronics, 2nd Ed. (Wiley, New York. 1975). [20] CE Moore. Atomic energy levels, NSRDS-NBS 35 (NatL Bur. Std.. Washington. 1971). [21] RE.M. Hedges, D L. Drummond and A. Gallagher, Phys Rev. A6 (1972) 1519.
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1221 W.L. Wiese and G_A. Marcin, Wavelengths and transition probabMties for atoms and atomic ions, NSRDS-NBS 68 (Natl_ Bur. Std , Washington, 1980). [ 231 H.L. Kramer and D.R. Herschbach, J. Chem. Phye 53 (1972) 2792. [ 241 R.F. Curl and P.R. Brooks, The In temational Congress 6f Pacific Basin Societies, Honolulu, Hawaii (1984) Abstract 5BO43. Symposium on the Frontiers of Spectsos[25]
copyW B. Miller, S-A. Stiron and D.R. Herschbach, Discussions Faraday Sot. 44 (1967) 108.
599