J. Phys. Chem. 1985,89, 1245-1249 is included in Table 11. The increase in the current efficiency of benzene reduction in the former solution can be related to the lower acidity6 of 2-propanol. The reduction in proton availability results in a repression of the competitive reduction of the alcohol and a decrease in the production of cyclohexane. Sternberg has indicated3 that the reduction of benzene is not dependent on cathode materials (Pt vs. Al) a t 5 mol % ethanol. The reduction products for Pt, Au, and Pb cathodes under identical experimental conditions are shown in Table 111. It can be seen that there is little dependence of the reduction process on the hydrogen overvoltage of the electrode at 21 mol % ethanol. Hence at relatively low ethanol concentrations, the reduction undoubtedly takes place in solution. At 67 mol % ethanol concentration, some benzene reduction was obtained with a lead cathode, whereas only the alcohol was reduced with a platinum cathode. It therefore appears that, at this concentration, some reduction occurs at the cathode, where the higher hydrogen overvoltage of lead represses the reduction of the alcohol. In all of the above reduction experiments, the current applied and time of the reduction process were arbitrarily chosen at 60 mA and 61/2h. The results of increasing the time or of doubling the applied current are given in Table IV. The platinum cathode dimensions were 2 X 8 cm, and the total number of coulombs applied was the same for the first two experiments. The higher percent conversion obtained for the large applied current is consistent with the larger current efficiency. It is not surprising that the total recovered products should be less for the longer experiment, 87.7% vs. 93.9%. The cell is being continuously flushed with a stream of hydrogen gas, and some loss of products would be expected despite the liquid-nitrogen trap. The maximum benzene reduction of 60% was achieved a t 240 mA in 7 h. The electrochemical reduction of benzene and related compounds in aqueous solutions has been reported by Coleman and Wagenknecht.' The authors claim that at optimum conditions the benzene may be converted to l,4-cyclohexadiene at 90% se(6) Hine, J.; Hine, M.J . Am. Chem. SOC.1952, 74, 5266. (7) Coleman, J. P.; Wagenknecht, J. H. J. Electrochem. SOC.1981, 128, 322.
1245
lectivity with nearly quantitative current efficiency. However, the total yield of reduction products obtained in a batch reaction was no better than those reported in this study when a current of 240 mA was applied for 7 h in a cell containing a 21 mol % ethanol-HMPA solution. In the continuous electrolysis process which they describe, a considerable loss in the reduction products occurred, presumably because of the evaporative losses which occurred at the operating temperature of 60 OC.
Conclusions At low ethanol concentrations, the reduction of benzene by solvated electrons appears to take place primarily in solution since it is independent of the hydrogen overvoltage of the electrode used. NMR measurements are consistent with the existence of hydrogen bonding between HMPA and alcohol, and this bonding can be expected to affect the relative rates of benzene and alcohol reduction. At high ethanol concentrations (267 mol % ethanol), no benzene is reduced except when the cathode has a sufficiently high hydrogen overvoltage, e.g., Pb. Only the alcohol is reduced when a platinum cathode is used, and hydrogen is evolved. This dependence upon hydrogen overvoltage indicates that under these conditions some reduction must occur at the cathode surface. The substitution of 2-propanol for ethanol decreases proton availability for the same alcohol concentration. This represses reduction of the alcohol, and consequently greater benzene reduction is observed. Acknowledgment. Acknowledgment is made to the National Science Foundation (Grant DMR 79-23605) for the support of D.P. In addition, we thank the Office of Naval Research (N00014-77-C-0387) for the support of J.F. and K.D. A.W. thanks the GTE Laboratories (Waltham, MA) of the GTE Corporation for partial support during this work. Acknowledgment is also made to Brown University's Materials Research Laboratory program, which is funded through the National Science Foundation. Registry NO. Pt, 7440-06-4; Au, 7440-57-5; Pb, 7439-92-1; HMPA, 680-3 1-9; benzene, 7 1-43-2; cyclohexadiene, 29797-09-9; cyclohexene, 110-83-8; cyclohexane, 110-82-7; ethanol, 64-17-5; 2-propanol, 67-63-0.
Vibrational Excitation due to Atom-Molecule Collision in an Intense Laser Field C. A. S. Lima Instituto de Fhica, Universidade Estadual de Campinas, 13100- Campinas, SP, Brazil
and L. C. M. Miranda* Instituto de Estudos Avancados, Centro TZcnico Aeroespacial, 12200-S.J. Campos, SP, Brazil (Received: July 26, 1984)
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A theory for the nonresonance vibrational excitation of diatomic molecules under impact excitation conditions in an intense laser field is developed. Specific consideration is given to 0 n transitions in He-LiH collisions. It is found for this system that at relatively large collision times direct many-photon processes are dominant. However, with decreasing collision time the compensation for resonance mismatch requires fewer photons to assist the vibrational excitation. The behavior of the different excitation processes for the 0 n transition on the laser field strength is also presented.
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Introduction The development of powerful laser sources in the infrared range has stimulated a great deal of work in the areas of selective excitation of and many-quantum dissociation.F10 In (1) Karlov, N. V.;Petrov, Yu.N.; Prokhorov, A. M.; Stelmarkh, 0. M. JETP Lett. 1970, ZI, 135. (2) Meyer, S.W.; Kwok, M. A.; Gross, R. V. E.; Spencer, D. J. Appl. Phys. Lett. 1970, 17, 516.
0022-3654/85/2089- 1245$01.50/0
the later process, also known as collisionless dissociation of molecules, the original idea was to use a laser frequency w resonant (3) Basov, N. G.; Markin, E. P.; Oraevskii, A. N.; Pankratov, A. V. Sou. Phys.-Dokl. (Engl. Transl.) 1971, 16, 445. (4) Ambartsumyan, R. V.; Letokov, V. S . Appl. Opt. 1972, 11, 354. (5) Jortner, J.; Mukamel, S . In "The World of Quantum Chemistry"; Daudel, R., Pullmann, B., Eds.; Riedel: Dordrecht, The Netherlands, 1974; p 145. (6) Quack, M. J . Chem. Phys. 1978, 69, 1282.
0 1985 American Chemical Society
1246 The Journal of Physical Chemistry, Vol. 89, No. 7, 1985
with the ground-state vibrational frequency wo so that a cascade of stepwise multiquantum processes would eventually lead to the dissociation limit. Apart from suffering from detuning from resonance just after the first few vibrational transitions (due to the anharmonicity), this model also suffers from the fact that to obtain the resonance w = wo in rarefield gases it is usually necessary to vary the laser frequency. This later drawback can, in principle, be accomplished by means of stimulated Raman scattering” such that the new laser frequency becomes w f Aw, where Aw is the appropriate Raman shift. In doing so, however, one usually looses power by a factor of The compensation for the resonant mismatch can alternatively be achieved during collisions’ in a real gas at the expenses of the kinetic energy of the impinging particles. In this paper we investigate the vibrational excitation of diatomic molecules by nonresonant laser radiation, taking into account their collisions with inert-gas atoms.l* For the sake of simplicity we adopt the simple one-dimensional model for the collision of a molecule BC with an inert-gas atom A (collinear collision). The molecule BC is described by a charged Morse oscillator. The inert-gas atom is assumed to be a structureless particle and we neglect the action of the laser field on it. The motion of the inert-gas atom is given a quasiclassical treatment,I3J4 in which approximation the atom is replaced by an effective potential V,(x,t),where x is the deviation from equilibrium of the B-c bond length. We also assume the gas concentration to be such that the time 7 between collisions is shorter than the laser pulse duration rL, i.e., T T but still short enough compared to w-l SO that V&- &t)) = VM(x- 6(t + At)). In other words, for w wo, the particle motion is dominated by the oscillation of the binding potential
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(15) Henneberger, W. C. Phys. Reu. Lett. 1968, 21, 838. (16) Lima, C. A. S.; Miranda, L. C. M. Phys. Reu. A 1981, 23, 3335. (17) Landgraf, T. C.; Leite, J. R.; Almeida, N. S.; Lima, C. A. S.; Miranda, L. C. M. Phys. Lett. A 1982, 92A, 131. (18) Lima, C. A. S.; Miranda, L. C. M. J. Phys. Chem. 1984,88, 3079.
The Journal of Physical Chemistry, Vol. 89, No. 7, 1985
Vibrational Excitation due to Atom-Molecule Collision in the laser field and, consequently, sees a laser-dressed potential. As the laserdressed potential shape, for UT >> 1 , is different from that of the undistorted potential, the energy eigenstates should accordingly become parametrically dependent upon the laser field strength. In what follows we shall restrict ourselves to the discussion of this high-frequency case. The solution to eq 10 for w >> wo can be found by noting that V(x - 6(t)) is a periodic function in time, for any position x. Thus, using eq 6a and expanding V(x - a(?)) in a Fourier series, one getsIs V(X - 6 ( t ) ) = +or
D{1 - 2
+
hwo(W
1
2
S=nD-n; y=2s+1
+m
(-i)'Zy(X)e-iyufe-z y=-m
2D(X) n = 0 , 1 , 2,..., nD; n D = - - -
1241
(-i)"Zy(2X)e-iyufe-2z) (11) "=-m
where Z,(z) is the Bessel function of imaginary argument. In arriving a t eq 1 1 we have introduced the dimensionless variable z = x/ao measuring the position z in units of ao, and the dimensionless parameter A = alao as a measure of the amplitude of the charge oscillation in the laser field a = eA/mcw in units of ao. The parameter X is related to the laser intensity Z by I = cmzaozw4X2/8rez.In the high-frequency limit, where W T >> 1 , the terms eq 1 1 with u # 0 oscillate very rapidly and are therefore vanishing small. Thus, the dominant term in eq 1 1 is u = 0, namely, the time-average laser-dressed potential, hereafter called vdc:
v&= D{l - 2Z0(X)e-Z+ Zo(2X)e-2z)
(12) I.e., for w >> wo, the particle states are actually described by the Schriidinger equation for a particle moving in the time-average laser-dressed potential vdc. We note that, in the absence of the laser field (Le., for X = 0),eq 12 reduces to the field-free Morse potential as given by eq 3. We also note from eq 12 that, in the high-frequency limit, the dressed potential becomes explicitly dependent upon the laser field strength through the parameter A.
This point has been addressed in ref 18 in which it is shown that, as one increases the laser intensity, the time-averaged laser-dressed potential vdc becomes flater and shallower such that the equilibrium internuclear position zo, the dissociation energy D, as well as the vibrational frequency wo = ( 2 D / m ~ ~ become ~)'/~ parametrically dependent upon the laser field strength, namelyI8
where F(-n,2s+ 1 ,5) is the confluent hypergeometric function. Substituting eq 16 into eq 9, and performing the indicated calculations, one obtains the probability amplitude for the 0 .+ n transition due to a collision
wheref,(X) = ( xnlexp(@qx)lxo). Expanding exp(-@$(t)) as a series of Bessel functions, and performing the integration with respect to t with the help of
one gets for
where
(20) with
F(-n,u
+ 1 , ~ , 1 ) (21)
+
Here, 1 = @quo,u = s nD - 1 - 1 , F(-n,o+l,y,l) is the hypergeometric function, and It thus follows from the above discussion that, in the highfrequency case, one may safely replace Hoin eq 10 by its dc component, so that, for w >> ob the eigenstates of the laser-driven Morse oscillator are the solution to
(14) Now, using eq 13 and performing some straightforward calculations, the time-average Hamiltonian H d c of eq 14 can be rewritten as Hdc
P2 =2m
+ D(X){l - 2e-("o) + e-2(P-zd]+ D - D(X)
(15)
so that the exact solutions to eq 14 are readily identified as those of a Morse oscillator centered at zo(X)with dissociation energy D(X) and vibrational frequency wo(X). Le., for w >> wo, the eigenstates and eigenfunctions of the driven Morse oscillator can be written asI9 '
xn = Nne-"25" F(-n,2s
+ 1,5)
- -
Equations 19 and 20 tell us that, in the presence of a laser field, the 0 n transition can now take place with the simultaneous absorption ( u 0) or emission ( u > 0) of IuI photons whose probability amplitude is a+/(X). In the absence of the laser field (Le., X = 0),eq 19 reduces to the usual Landau-Teller expression.I4
Results In this section we apply the above theory for the case of collisions between H e atoms and LiH molecules in the presence of a CO laser (1, = 5.5 pm). For this molecule the values of the physical parameters arezoD = 2.43 eV, wo = 1405.65 cm-I, ro = 1.59 A, and ero = 5.88 D. We assume the collision parameter @ varying from 5 X lo8 to 5 X lo9 cm-' and the velocity range (19) Landau, L. D.; Lifshitz, E. M. 'Quantum Mechanics"; Pergamon Press: Oxford, 1958. (20) Huber,K. P.; Herzberg, G. In 'Constants of Diatomic Molecules"; Van Nostrand Reinhold New York. 1979.
1248 The Journal of Physical Chemistry, Vo1.89, No. 7, 1985
Lima and Miranda TABLE I: Calculated Values of R&.s for the He-LiH Collision with = 5 x 1 P c m - l and Y = 2 x 105cm/s
x 0.1 0.5 1.0 1.5
RE, 7.01 X lo2 1.18 X lo4 2.73 X los 3.46 X lo6
RzZ3 6.08 X lo4 2.33 X lo6 4.08 X lo6 2.41 X lo6
RE3 1.97 7.76 X 10 5.83 X 10 2.57 X 10
TABLE II: Calculated Values of for the He-LiH Collision with @ = 5 x W c m - ' and Y = 2 x W c m / s X
0.1 0.5 1 .o 1.5
Re3 5.19 X 10 1.52 X 10 2.01 x 103 4.53 X lo5
RE3 7.57 x 103 3.50 8.11 x io2 2.04 X los
RzZ3 9.18 x io2 1.01 x 10 1.67 x 103 3.86 X los
I0'
IO
0 . i
~
d05 :
~
!
10
:
:
'
I5
;
:
!
PO
~
Figure 1. Dependence of the ratio R L on the laser field strength parameter X for the He-LiH collision with u = 2 X lo5 cm/s, j3 = 5 X 108 cm-l, and W ~ T ,= 2.64. T~ is the collision time defined by (@I)-'.
considered is between 1 X lo5 and 5 X lo5 cm/s, which corresponds to temperatures between 322 and 8050 K. In terms of these quantities, the parameter A is related to the laser intensity I by I = - cmDw4
Y
R
093
Id
A2
47re2w2
where we have used the well-known relation between the Morse parameter a. and oo,namely, aoz= 2D/mwo2. Before presenting the results of our numerical calculations a word is needed regarding the validity of the present theory. The above results for the excitation probability amplitude are valid provided the collision potential can be treated as a perturbation. This requires that the collision energy p 2 / 2 to be smaller than the vibrational energy hwo(A). Since the vibrational energy of the laser-dressed Morse oscillator decreases with increasing laser field strength, this condition imposes an upper limit value in A, A,, obtained from
hwo(X) = p v 2 / 2 For the specific example of He-LiH collisions with v = 2 X lo5 cm/s considered below this upper limit is roughly A, = 10. To better assess the effects of the intense laser field on the vibrational excitation of the molecule, we consider in what follows the ratio R+/ of the transition probability amplitudes with and without the laser field, namely,
I
IO l
0.8
1.0
A
1.2
1.4
Figure 2. Dpendence of the ratio G3 on the laser field strength parameter X for the He-LiH collision with u = 2 X lo5cm/s, /3 = 5 x lo9 cm-', and w07, = 0.264.
-.
culations of eq 23 for the 0 3 transition due to the He-LiH collision for v = 2 X lo5cm/s and /3 = 5 X lo8 and 5 X lo9cm-I, respectively. In Figures 1 and 2 is depicted the dependence of R L 3 on the laser intensity for the aforementioned values of v and /3. An inspection of Tables I and I1 and Figures 1 and 2 immediately points toward the existence of two distinct regions in the collisional vibrational excitation of a diatomic molecule in a laser field according to whether one has W o 7 , >> 1 or w07,
