J. Phys. Chem. 1984,88, 3079-3082 TABLE VI1 102.
(d In A'/dT)/ K-'
103b/~-1
-1.13
-2.41
103.
103.
102.
(EA)/ K-'
(ED)/
K-1
(9 + E A + ED) /K-l
-1.39
-7.35
-1.22
experiments these functions are linear, but In p,' is also a linear function of T', as is In p, (Figure 6). It is evident that the narrow range of temperature available for the present experiments is not sufficient to define the functional dependence of (3 on temperature. The quantity ( A E ) is the energy transferred in a deactivating collision, as formulated by Tardy and Rabin0~itch.l~ The quantity ( A E T ) ,defined by Troe as the energy transferred per collision, both activating and deactivating, is related to the energy transferred in a deactivating collision only, in the following way.'
( U T=) ( m ) ' / ( ( A E ) + RTI
3079
Values for these three terms have been estimated in the following way. From the present measurements, d4/dT and d In (AE)/dT may be obtained directly. d In ( E ) + / d Tis based on the calculated values of E+ in Table VI. Values of d In ',p --
d4
- 2.80
and
a In p,' d In ( A E )
--
d In
B,',
d In ( E + )
= 1.08
were estimated from the "universal" curves of Tardy and Rabinovitch14 for 4 = 1.25 and E' = 0.5 using the data given in ref 1. The results are summarized in Table VII. The present results show that the variation in the collisional efficiency with temperature is due largely to changes in ( A E ) . Over the temperature range 940-1000 K, ( A E ) has decreased by about 30%. The change in the order with temperature, for constant w, is also significant while the change in E+ with temperature is the least important term.
These energy increments will be similar for strong collisions but not for weak collisions when ( A E ) < RT. At 1000 K ( A E ) = 1.25 kcal mol-' so that (AET) = 50 cal mol-I. This may be compared to the value of 60 cal mol-' calculated earlier from Po. The variation of p,' with temperature was examined with respect to the three principal factors, 4, E+, and ( AE), which determine this dependence. The function may be expressed as follows.' d In 6', -dT d In p,' d In (E') d In p,' d In ( A E ) d In 8,' d 4 -dT dln (AE) dT d4 d T d I n ( E + )
Conclusions The collisional efficiencies of hydrogen and xenon in the unimolecular dissociation of ethane and the corresponding value of (A!?), measured at temperatures in the neighborhood of 1000 K, show that both gases are weak colliders. Both 6, and 6', declined significantly over the relatively small range of temperatures of the present experiments. The decrease in the average increment of energy transferred per deactivating collision, ( a ) , was the most important factor in the decrease in collisional efficiency. The results are in accord with most estimates of the trend in (3, with temperature.
or, briefly, as suggested by Tardy and Rabinovitch' d In 0', -- 4 EA ED dT
Acknowledgment. We thank the Natural Sciences and Engineering Research Council of Canada for financial support of this work. Registry No. Ethane, 74-84-0.
+
+
+
+
Nonresonant Vibrational Excitation of Diatomic Molecules. Laser-Driven Morse Oscillator Model C. A. S . Lima Institute de Fisica, Universidade Estadual de Campinas, 131 00-Campinas, SP, Brazil
and L. C . M. Miranda* Instituto de Estudos Avaqados. Centro TZcnico Aeroespacial, 12200-S. J . dos Campos, SP, Brazil (Received: August 24, 1983)
A discussion is presented for the nonresonance vibrational excitation of a heteronuclear diatomic molecule acted upon by
a laser field. The diatomic molecule is assumed to be represented by a charged Morse oscillator. It is shown that for laser frequenciesgreater than the Morse oscillatorvibrational frequency, the molecular parameters become parametrically dependent upon the laser field such that, when the laser field strength is increased, the dissociation energy as well as the vibrational frequency decrease. A numerical estimate for the laser intensities needed for the observation of the effects discussed in the present paper is also given.
Introduction The rapidly growing field of laser technology has stimulated in recent years the study of the interaction of matter with intense electromagnetic fields. In such intense fields multiphoton transition~'-~ in both atoms and molecules can occur with significant (1) L. V. Keldysh, Sou. Phys.-JETP (Engl. Transl.), 20, 1307 (1965). (2) F. V. Bunkin and I. I. Tugov, Phys. Rev. A , 8, 601 (1973). (3) N. B. Delone, B. A. Zon, V. P. Krainov, and V. A. Khodovoi, Sou. Phys.-Usp. (Bngl. Transl.), 19, 711 (1976).
0022-3654/84/2088-3079$01 S O / O
probability. In particular, the development of intense laser sources in the infrared range has found extensive application in the study of various molecular processes, such as selective excitation of molecule^^-^ and many-quantum diss~ciation.~O-'~ In the later (4) J. E. Bayfield, Phys. Rep., 51, 317 (1977). ( 5 ) V. S. Letokov and A. A. Makarov, Sou. Phys.--Lisp., 24, 366 (1981). (6) N. V. Karlov, Yu.N. Petrov, Am. M. Prokhorov, and 0. M. Stlmarkh, JETP Lett. (Engl. Transl.), 11, 135 (1970). (7) S.W. Mayer, M. A. Kwok, R. V. F. Gross, and D. J. Spencer, Appl. Phys. Lett., 17, 51C (1970).
0 1984 American Chemical Society
3080 The Journal of Physical Chemistry, Vol. 88, No. 14, 1984 process, which is also known as collisionless dissociation of molecules, the original idea was to use a laser resonant with the ground-state vibrational frequency wo so that a cascade of stepwise multiquantum processes would eventually lead to the dissociation limit. This model, however, suffers from the fact that due to the anharmonicity, as well as to the collisions with the other molecules, a detuning from the resonance occurs just after the first few vibrational transitions. This detuning becomes even more critical for the case of polyatomic molecules due to the interaction between the various rotovibrational modes. The following model for the dissociation of polyatomic molecules by resonant laser radiation is nowadays generally accepted. We assume that there are two systems of levels beginning with a series of discrete states and followed by a quasicontinuum. This quasicontinuum appears because of the aforementioned interaction between the several rotovibrational modes. Thus, after the excitation of the first two or three lower discrete states by the resonant laser the molecule then move over to the quasicontinuum where diffusion transitions take place between narrow bands right up to the dissociation limit. Based upon this model, the existing experimental data on the dissociation of some typical polyatomic molecules (e.g., SF, and CF,I) by a resonant C 0 2 laser have been successfully ex~lained.’~-’~ In this paper, we address ourselves to the question of the excitation of diatomic molecules by nonresonant laser radiation. This problem has two adverse features. First, by restricting ourselves to diatomic molecules we rule out the possibility of a quasicontinuum of vibrational states which, as mentioned above, enhances the photodissociation. Secondly, by using nonresonant radiation, the stepwise vibrational excitation becomes inhibited. To examine the problem posed above, we consider, in what follows, the simple one-dimensional model of a charged Morse oscillator driven by a laser field. The laser field is treated as a classical plane electromagnetic wave in the dipole approximation. We also assume the laser frequency w to satisfy the following condition: w f w o
(1)
o’
}
+ - A ( t ) + V ( x ) Ji(x,t)
(2)
where V ( x ) = D(1
-
exp(-x/ao))2
(3)
and A ( t ) = A cos ut is the vector potential for a plane electromagnetic wave of frequency w describing the laser field. In eq 2 and 3, m is the reduced mass of the molecule, e is the effective charge of the oscillator related to the molecule dipole moment =tby ero, where ro is the internuclear equilibrium distance, D is
Lima and Miranda the molecule dissociation energy, and a. is the range parameter of the Morse potential. To find a nonperturbative solution to eq 2, valid for intense fields, we apply to eq 2 a unitary transformation,
Ji
= exp(id(t)p/h) exp(b(t)/h)$
(4)
where d(t)
= -(e/mc)S‘dt’A(t? = -a sin w t a = eA/(mco)
e2 ? ( t ) = -2mc2
s
(5a)
b t ’ A2(t?
We find ificj(x,r) = (lj2/2m
+ V(x-d(t)))$(x,t)
(6)
Equation 6 shows that in the presence of a driving laser the charged particle motion may be alternatively described by the Schrijdinger equation for a particle moving in a time-dependent potential which oscillates with frequency w and amplitude a = eA/(mcw). The potential V(x-G(t)) will be called hereafter the laser-dressed potential. Equation 6 has no exact analytical solutions but can be approximately solved in two different ranges of frequencies w. First, if the period of oscillation of the laser field is much larger than the period of oscillation T of the particle in the potential well ( T wol), namely, if U T T but still short enough compared to so that V(x-G(t)) N V ( x - s ( t + A T ) ) . In other words, for w wo, the particle motion is dominated by the oscillation of the binding potential in the laser field and, consequently, sees a laser-dressed potential. As the laser-dressed potential shape, for U T >> 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 6 for w >> wo can be found by noting that V(x-d(t)) is a periodic function in time, for any position x. Thus, using eq Sa and expanding V(x-G(t)) in a Fourier series, one gets (see Appendix)
-
V(u-G(t)) = +m
D(1- 2
+
(-i)’Zu(X)e-iuwte-n y=--m
+m
1
(-i)’Z1,(2X)e-’”“‘e-2z) (7)
u=-m
(8) N. G. Basov, E. P. Martin, A. N. Oraevskii, and A. V. Pankratov, Sou. Phys.-Dokl. (Engl. Transl.), 16, 445 (1971). (9) R. V. Ambartsumyan and V. S. Letokov, Appl. Opt., 11, 354 (1972). (IO) J. Jortner and S. Mukamel in “The World of Quantum Chemistry”, R. Daudel and B. Pullman, Eds., Riedel, Dordrecht, 1974, p 145. (11) M. Quack, J . Chem. Phys., 69, 1282 (1978). (12) H. S. Kwok, E. Yablonovitich, and N. Bloembergen, Phys. Reu. A ,
where Zu(z) is the Bessel function of imaginary argument. In arriving at eq 7 we have introduced the dimensionless variable z = x / u o measuring the postion x in units of ao, the the dimensionless parameter X = a/uo as a measure of the amplitude of the charge oscillation in the laser field u = eA/(mcw) in units of ao. The parameter A is related to the laser intensity I by Z = cm2a02w4X2/(8~e2). In the high-frequency limit where U T >> 1 the terms in eq 7 with v # 0 oscillate very rapidly and are therefore vanishingly small. Thus, dominant in eq 7 is the v = 0 term,
23, 3094 (1981). (13) P. Kolodner, C. Winterfield, and E. Yablonovitch, Opt. Commun., 20, 119 (1977). (14) V. T. Platonenko, Sou. J. Quantum Electron., 8, 1010 (1978). (15) L. I. Trakhtenberg and G. M. Milikh, Sou. J. Quantum Electron., 12, 1656 (1982).
(16) W. C. Henneberger, Phys. Reu. Lett., 21, 838 (1968). (17) C. A. S. Lima and L. C. M. Miranda, Phys. Rev. A, 23,3335 (1981). (18) T. C. Landgraff, J. R. Leite, C. A. S. Lima, and L. C. M. Miranda, Phys. Lett. A, 92, 131 (1982).
Vibrational Excitation of Diatomic Molecules
The Journal of Physical Chemistry, Vol. 88, No. 14, I984 3081
1 5 ’
I 1.5 1.0 Y
/
1
(rl
05
I 2
I
I
0
I
I
I
I
1
2
3
4
3
7.
Figure 1. Laser-dressed Morse potential V,, (in units of D ) as a function of the internuclear separation z ( z = x / a o ) for several values of the laser field parameter A.
namely, the time-average laser-dressed potential, hereafter called vdc:
= D(1 - 2Zo(X)e-2
+ Z0(2X)e-22]
(8) Le., for w >> wo, the particle states are actually described by the Schrodinger 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 8 reduces to the field-free Morse potential as given by eq 3. We also note from eq 8 that, in the high-frequency limit, the dressed potential becomes explicitly dependent upon the laser field strength through the parameter A. This entails that the molecular parameters such as the dissociation energy, the internuclear equilibrium position zo, and the vibrational frequency wo should accordingly become parametrically dependent upon the laser field. This is seen in Figure 1 in which we plot the laser-dressed Morse potential V& for several values of X. It follows from Figure 1 that, when the laser field strength is increased, the binding potential becomes flatter and shallower, such that zo shifts to high values of z at the same time that the dissociation energy decreases with increasing A. This means that the effect of an intense laser field points in the direction of a weakening of the molecule confinement and stability. These features can also be readily deduced from eq 8 as follows. The equilibrium internuclear distance zo is obtained by requiring that aVdc/az = 0. One gets from eq 8 vdc
e-=O = 10(X)/10(2X)
(9a)
(9b) zo = In V0(29/Io(M For h .- 0, eq 9 gives us zo = 0, whereas for large values of A, using the asymptotic expansion of lo@), one finds that zo scales linearly with A. Substituting eq 9 for zo into eq 8, we can calculate the laserdressed dissociation energy defined as V(z=m) - V(z=zo). One finds D(W = Df(M
f ( X ) = IoZ(V/I0(2X)
(10)
I
I
2
3
A
Figure 2. Dependence of the normalized equilibrium internuclear separation zo, dissociation energyflA), and vibrational frequency g(A), on the laser field parameter A.
Conclusion and Discussion In this paper, using the simpie model of a charged Morse oscillator for a heteronuclear diatomic molecule, we have discussed the vibrational excitation of the molecule by nonresonant laser radiation. It was shown that depending upon the ratio of the laser frequency to the oscillator frequency (Le., w / w o ) the response of our system changes considerably. In the low-frequency regime, where w > wo, when the laser field strength is increased, the equilibrium internuclear separation, the dissociation energy D, as well as the vibrational frequency wo become parametrically dependent upon the laser field strength such that both D and wo decrease with increasing driving field. This entails that, for high-frequency lasers, the molecular excitation occurs via multiphoton transitions between the dressed states. To get an estimate for the laser intensity needed to observe these dressing effects reported here let us consider the case of LiH driven by a CO laser (A, = 5.5 km). For this molecule the values of the physical parameters are19 D = 2.43 eV, wo = 1405.65 cm-’, ro = 1.59 A, p = 5.88 D. It follows from Figure 2 that already for X = 0.6 one should observe a 15% reduction in the dissociation energy, and a 25% shift in the internuclear equilibrium separation. The parameter X is related to the laser intensity I by
Finally, expanding eq 8 in a power series around the equilibrium internuclear position zo one finds for the vibrational frequency w0
wo(W = wog(N g o ) = lf(NI”* (11) In Figure 2, we show the dependence of zo, D, and wo on the laser field strength A. This plot confirms the above conclusions that, when the laser field strength is increased, the molecule becomes highly anharmonic at the same time its dissociation via low-energy processes is greatly enhanced.
where we have used the well-known relation between a. and wo, namely, ao2= 2D/(rnwo2). If we use the values of the physical parameters given above for LiH, it follows from eq 12 that, for (19) K. P. Huber and G. Herzberg in “Constants of Diatomic Molecules”, Van Nostrand-Reinhold, New York, 1979. (20) I. S. Gradshtein and I. M. Ryzhik in “Table of Integrals, Series, and Products”, Academic Press, New York, 1965, p 973.
3082
J. Phys. Chem. 1984, 88, 3082-3085
a C O laser, X = 0.6 corresponds to an intensity I 4 X lOI4 W/cmZ. Even though this value for the laser intensity seems to be quite high, it is certainly within present-day capabilities; it can be achieved with a C O laser pulse of 10 m J of energy and 1-ps duration, focused in a region of roughly 100-pm diameter. For molecules with small dissociation energy it follows from eq 12 that the requirement on the laser intensity becomes more favorable. For instance, for NeAr (D = 0.0062 eV, w,, = 20.9 cm-’, ro = 3.43 A), X = 0.6 corresponds for a H,O laser (AL = 118.6 pm) to an intensity of the order of 2.1 X 10” W/cm2, which requires H 2 0 laser pulse of 10 mJ of energy and 1-ns duration, focused in a region of 100-ym diameter.
Appendix In this Appendix we provide the explicit derivation of the laser-dressed Morse potential as given by eq 7. We begin by noting from eq 3 that the laser-dressed potential can be written as I/(x-a(t)) = ~ ( -12eW/aoe-z + e2W)/aoe-2z) (AI) or, using eq 5a
~ / ( ~ - s ( t )=) ~ ( -12e-ksinwte-z + e-ZXsmwr
-22
e
I
(A2)
Now, from the well-known expansion 20
+-
2
exp(ix cos 0) =
i”J,(x) exp(iv0)
(‘43)
”=-a-
upon making the substitution x = iu and 0 = ?r/2 - wt, one gets +m
exp(-u sin ut) =
2 (-1)”Jv(iu) exp(-ivwt)
(A4)
y=-m
Introducing into eq A4 the definition of the Bessel functions of imaginary argument Iv(u),namely, I,(u) = (-i)’Ju(iu), one finally obtains +m
exp(-u sin wt) =
c (-i)”ZU(u)exp(-ivwt)
(AS)
ye-m
Equation A5 is the expression which used in eq A2 yields the laser-dressed Morse potential expression as given by eq 7 of the text.
Intercalation Kinetics of a Proton into Tisz by the Pressure-Jump Technique Minoru Sasaki, Hiroshi Negishi, Masasi Inoue, and Tatsuya Y asunaga* Faculty of Science, Hiroshima University, Hiroshima 730, Japan (Received: September 7, 1983)
In an aqueous suspension of TiS2(H) prepared by the reduction of Tisz with zinc grains and dilute HC1, where an atomic hydrogen is intercalated into the interlayers of Tis2, double relaxation was observed by using the pressure-jump technique with conductivity detection. The fast relaxation time increases with pH and decreases with particle concentration, while the slow relaxation time is approximately constant. Taking into account the surface potential created by the release of a proton from the interlayers of TiS2(H), the fast and slow relaxations were attributed to protonation-deprotonation of protonated surface sites and proton intercalation-deintercalation in the interlayers of TiS2(H), respectively. The intrinsic values of the protonation and deprotonation rate constants were determined to be 3.0 X lo4 mol-’ dms s-’ and 7.0 s-l, and those of the intercalation and deintercalation rate constants 4.6 and 1.1 s-’ at 25 OC,respectively. The kinetic parameters obtained reveal that the proton intercalated into the interlayers of TiS2(H) acts as a strong solid acid comparable to a mordenite.
Introduction
the pressure-jump technique with conductivity detection.
Transition-metal dichalcogenides, MX,, have a sandwichlike layered structure consisting of a layer of transition-metal atoms M between two layers of chalcogen atoms X which are bonded by a weak van der Waals force, and they are of current interest because their structural and electronic characters are two-dimensional.14 The physical and electrochemical properties of these compounds can be significantly modified by the intercalation of various organic or inorganic guest species. In these studies it is necessary to know how the guest species intercalate into the interlayers and to clarify the elementary intercalation process. However, few kinetic studies have been performed. Recently, we have investigated kinetically the proton intercalation into zirconium phosphates and the Cl- and OH- intercalation into hydrotalcite-like compounds by using a pressure-jump technique, and the detailed intercalation mechanisms have been established.jS6 The purpose of the present investigation is to elucidate kinetically the proton intercalation into the interlayers of Tis2 by using
Experimental Section
~~
~
(1) Wilson, J. A,; Yoffe, A. D. Adu. Phys. 1969, 18, 193. C (Amsterdam) 1980, 99B C , 89. (2) Schdllhorn, R. Physica B ( 3 ) Thompson, A. H. Physica A (Amsterdam) 1980, 100. (4) Whittingham, M. S.; Jacobson, A. J. “Intercalation Chemistry”; Academic Press: New York, 1982; Chapters 7, 9. ( 5 ) Sasaki, M.; Mikami, N.; Ikeda, T.; Hachiya, K.; Yasunaga, T. J. Phys. Chem. 1982, 86,4413. ( 6 ) Sasaki, M.; Mikami, N.; Ikeda, T.; Hachiya, K.; Yasunaga, T. J . Phys. Chem. 1982, 86, 5230.
+
+
0022-3654/84/2088-3082$01.50/0
A titanium disulfide (Tis2) crystal was prepared by a chemical vapor reaction technique with iodine transport. The evacuated quartz ampules (1 2-20 mm in diameter and 150-200 mm long), charged with the elements Ti (99.9%) and S (99.99%) in the stoichiometric ratio, were placed in a homemade two-zone furnace for 1 week; the high temperature TH = 800-900 OC, and the low temperature TL= 700-750 OC. The grown crystals were examined by an X-ray diffractometer to have a 1T-type la ered structure with the lattice parameters a = 3.406 0,001 and c = 5.703 f 0.003 This c value corresponds to the nonstoichiometric Ti,,ojSzaccording to Thompson et a1.,8 indicating the presence of a small amount of interstitial Ti atoms. The grown flakes and crystallites were ground into fine powders (about 1 pm) by a vibrating mixer mill and then reduced to the protonated form H,Ti10jS2with zinc grains and dilute HCl, where an atomic hydrogen (in the nascent state) was intercalated into the interlayers of Tis,. In order to remove contaminants such as the zinc ion in the interlayers of H,Til,osSz, the samples were washed several times with 1 N HCl and with distilled water. The value of x was estimated to be 2.95 X lo-, by using the value of the amount of protonated sites existing in the interlayers obtained from the base
*
8’
A.738
(7) Greenaway, D. L.; Nitsche, R. J . Phys. Chem. Solids 1965,26, 1445. (8) Thompson, A. H.; Symon, C. R.; Gamble, F. R. Muter. Res. Bull. 1975, 10, 915.
0 1984 American Chemical Society
