- Email: [email protected]
Depolarized stimulated gain spectra of liquid CS2 and benzene at room temperature
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November 1991
Depolarized stimulated gain spectra of liquid CS2 and benzene at room temperature J.S. Friedman, MC. Lee ’ and C.Y. She Physics Department,
Colorado State University, Fort Collins. CO 80523, USA
Received 12 July 1991; in final form 21 August 1991
Depolarized Rayleigh-wing spectra of CS2 and benzene have been recorded using stimulated gain spectroscopy (SGS). Each spectrum is fitted to three different phenomenological models. In thecase of CSL,the determined dynamical characteristic parameters are compared to those measured previously by different experimental techniques with good agreement.To adequatelydescribe the SGS spectraof both liquids, an intermediatetime scale contribution is needed in addition to a slow reorientational relaxation and a fast broad underdamped oscillator. Compared to CS2,the slow and intermediate relaxation ratesof benzene are lower, whilethefrequency andrelaxationrateof theunderdamped oscillatorarehigher.
1. Introduction
Depolarized light scattering studies at low frequencies (O-200 cm-l) can provide information on the dynamics of molecular liquids. Spontaneous light scattering (LS) studies have revealed an exponential tail in the wings of the depolarized LS spectra for both monatomic liquids, such as Ar, and anisotropic liquids, such as CS2, suggesting the existence of interaction-induced anisotropies in simple liquids [ l-41. In the case of anisotropic molecular liquids, there is, in addition, a Lorentzian line centered at zero frequency due to reorientational relaxation. Subsequent theoretical studies [ 5-81 have confirmed the existence of the exponential tail, but concluded that the shape of the spectrum, intermediate between the exponential tail and the Lorentzian line, is complex and varies from one liquid to another. Furthermore, it is difficult to distinguish different monotonically decreasing functions in frequency from experimental data. For a liquid with high polarizability, such as CS2, the depolarized LS spectrum may be separated [8,9] into diffusive reorientational anisotropy which dominates the low-frequency dynamics and is characterized by a (Debye) relaxation time of typically 2 ps, and broad interaction-induced contributions which dominate the spectrum at intermediate and higher frequencies. According to the fluctuation-dissipation theorem [ lo], the LS spectrum is proportional to the response funo tion divided by angular frequency in the low-frequency regime. This renders the interaction-induced effects in a LS spectrum much weaker than the reorientational anisotropy and nearly indistinguishable. A stimulated experiment using a pump and probe beam directly monitors the molecular response function, which goes to zero at zero frequency. It can thus display the intermediate frequency structure more readily and reveal the detailed shape of the response function more clearly. There have been four types of depolarized stimulated scattering experiments reported in the literature by different research groups for the study of liquid dynamics. Two time-domain techniques are the transient grating or impulsive stimulated scattering (ISS ) [ 1 I ] and the optical Kerr effect with optical-heterodyne detection (OHD OKE) [ 121, Two frequency-domain techniques are the tunable-laser-induced grating (TLIG) [ 131 and stimulated gain spectroscopy (SGS) [ 141. These stim’ On sabbatical leave from Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan. 0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rightsreserved.
161
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November 1991
ulated techniques probe, respectively, the square of the impulse response function, the impulse response function, the square of the frequency response function, and the imaginary part of the frequency response function. Along with the spontaneous LS spectrum, we express these measured functions below for comparison: ISS intensity
(la)
OHD OKE response R(t) ccG( t ) ,
(lb)
TLIG efficiency
(lc)
?(~)aIG(w)l*,
SGS gain LS spectrum
(IdI Qw)alm[G(w)
I/w,
WI
where G(t) is the impulse response function of the liquid, and its Fourier transform, the frequency response function, is G(w). Unlike ISS and TLIG techniques, which are proportional to the square of the response function, the linear proportionality of OHD OKE and SGS techniques have some advantage in bringing out the relatively weak features in the response function because squaring a strong signal makes it stronger relative to squaring a weak signal. To obtain a stimulated response, fluctuations over a wide range of frequencies must be excited in the liquid. In order to do so the width of the interrogating laser pulse must be narrow enough in the time-domain, and the frequency range of the tunable narrow band laser should be wide enough in a frequency-domain technique. These requirements are now being met by state-of-the-art lasers. Because the detailed theoretical response function is rather complicated and varied, experimentalists have had a long tradition of using a phenomenological response function [ IO,151 with either two or three additive terms to model their experimental data. In 198I, Scarparo et al. [ 141 recorded a stimulated gain spectrum for CS2 up to a shift of 100 cm- ‘, and uncovered a broad underdamped oscillation at 35 cm-’ which was buried under the tail of the corresponding spontaneous spectrum [ 2,4]. A stimulated experiment on CSI in the timedomain was carried out in 1982 by Green and Farrow [ 161, revealing two time constants and associating the shorter relaxation time of 0.28 ps with intermolecular collisions. Recently, the dynamics of liquid CSZhave been studied by several groups. Nelson and co-workers were able to analyze their impulsive stimulated light scattering (ISS) data [ 171 by several models; the model that gave the best fit consists of a Debye relaxation plus an inhomogeneously broadened underdamped oscillator [ 181, Kalpouzos et al. [ 19,201 have fitted their OHD OKE data to a three-term impulse response function [21]: an inhomogeneously broadened underdamped oscillator with an intermediate dephasing time plus two exponentially decaying terms, one with the same intermediate time constant and the other with a diffusive (Debye) relaxation time. They further modified the form of the two exponential terms by imposing inertia in these responses. This model consists of a total of four characteristic dynamical parameters to be determined, the oscillator frequency w,, the inhomogeneous broadening rate (Y,the intermediate dephasing time r, and the Debye relaxation rate r. Barker et al. [ 221 have analyzed their TLIG data in terms of several models, and, based on a stringent statistical test, they favored a model consisting of two exponential terms modified to include the inertial nature of the nuclear response, plus a homogeneously broadened underdamped oscillator. Again, they used four characteristic times: the oscillator frequency, the fast and intermediate dephasing times and the Debye relaxation time, to analyze their frequencydomain
data.
These recently used phenomenological models do not explicitly account for the exponential tail in the frequency response function (or Lorentzian in the impulse response function) that most theories predict [ 5-8 1. However, since even with a semi-log plot it is not possible to ascertain the exact form of a fall-off function over the narrow frequency range associated with a single dynamical mechanism, curve fitting cannot distinguish between different functions, e.g., Gaussian and exponential, in this tail. As a result, the interaction-induced effects have been accounted for by these models in various ways. In the case of CSI, all models seem to fit the experimental data well independent of the attention paid to curve fitting. Recently, McMorrow and Lotshaw [ 231 have suggested that the interaction-induced effects can be more logically represented by a single 162
Volume 186, number2,3
CHEMICAL PHYSICS LETTERS
8 November 1991
intermolecular vibrational degree of freedom consisting of a continuous superposition of overdamped, critically damped and underdamped motion with a dephasing rate comparable to vibration frequencies. They did not, however, present a quantitative fitting of the proposed model to the experimental data. One of the purposes of this paper is to present our recent depolarized SGS spectrum of CS2 and to determine the dynamical parameters of the liquid. Our spectrum is comparable to that of Scarparo [ 141 but with an improved frequency resolution of 0.5 GHz. We also present the SGS spectrum of room-temperature benzene. Except for our preliminary conference report [ 241, the SGS spectrum of benzene has not yet been reported and analyzed. Compared to the existing time-domain data from which the frequency response can be deduced, our measured SGS spectra more closely represent the frequency response function, G(w), of the liquid. This fact has been made clear by a recent paper of McMorrow and Lotshaw [ 25 1. Using pyridine as an example, they pointed out that the distortion caused by a finite pulse width in the measured OHD OKE response can be eliminated only after a deconvolution procedure which requires precise knowledge of the shape of the pulse. Even for the 65 fs pulse they used, a considerable narrowing appeared in the frequency response. Comparison of their pyridine results with our benzene spectra shows that the deconvolution procedure works very well. However, for measured impulse response functions with marginal signal-to-noise, the required deconvolution procedure will introduce errors that would make the time-domain measurement less sensitive to different models. In contrast, the broadening caused by a pump laser of 0.5 GHz linewidth in a measured depolarized SGS spectrum, which typically covers a frequency range of more than 100 cm-‘, is quite negligible. In this paper we analyze our experimental data by three similar dynamical models and compare the results to those of previous reports. The obvious differences between the SGS spectra of CS2and benzene should serve to demonstrate the value of the SGS technique for studying liquid dynamics.
2. Experimental The experimental procedure is similar to those used previously [ 14,26-281 and is shown in fig. 1. A pulsed dye laser serves as a pump source. It is pumped by the second harmonic of a Q-switched Nd:YAG laser at a rate of 10 Hz. The pulsed dye laser is a modified Littman style using two gratings. Light inside the cavity is expanded by a four prism beam expander onto the fixed grating which is at grazing incidence. The reflection from this grating serves as the output beam, and the first diffraction order goes to a second grating, oriented for third or fourth order, which acts as the feedback mirror. We have measured the bandwidth of this laser to be roughly 0.5 GHz, and we can scan it as much as 300 cm-‘. The bandwidth of the pump laser is the limiting factor in this experiment as the probe laser, a cw single mode dye laser, has a bandwidth of less than 2 MHz.
Beam
Splitter
Normallzatian Diode
Fig. 1, Optical layout of the SGS experiment.
163
Volume186,number 2,3
CHEMICALPHYSICSLETTERS
8 November1991
Thus, our spectrometer has an equivalent finesse of 1.8~ 104.The lasers operate near 590 nm. The probe beam is split into 2 parts. The first is simply a normalization beam to eliminate noise due to fluctuations in the probe intensity, and the second is the actual probe. It is polarized perpendicular to the pump polarization and focused loosely into a glass cell containing the sample. The pump and probe lasers cross at a near counter-propagating angle of 179” through the sample, and the left-over pump light is caught by a beam dump. The probe beam is steered through a polarization analyzer and several apertures to eliminate stray scattered pump light that may be on its path, and is then detected by a photodiode. Because the gain/loss signal appears as a gain or loss pulse, a gated integrator and boxcar averager are used to detect the light. Fluctuations in the signal corresponding to pulse-to-pulse fluctuations in the pump laser can be normalized out with a detector that monitors the pump intensity. The frequency scale of the experiment was determined by steering part of the pulsed dye laser through a fixed spacing Fabry-Perot interferometer whose free spectral range of 1.74 cm-’ was determined from the measured spacing. A frequency scale is obtained by simply using the interferometer peaks as markers. Using this frequency scale, we note that as an independent check Raman lines in the measured SGS spectra of chloroform and bromobenzene agree with the known values. The decision of whether to scan both the gain and loss sides of the spectrum or to scan mainly the gain side depended on the width of the spectrum. In CS, the full spectrum is less than 250 cm- ’ wide, well within the tuning range of our pulsed pump source. In contrast, the full speo trum of benzene extends over 300 cm-‘, which is near or over the limit of the tuning range. As the laser is tuned towards the wings of the dye gain curve, a considerable amount of fluorescence is often seen. Before and after each spectrum is taken, the output of the pulsed dye laser is checked and, if necessary, the alignment is adjusted, using an optical multichannel analyzer as a diagnostic tool, to reduce fluorescence in the output to a negligible level.
3. Results and comparisons Many existing dynamical models can be described phenomenologically by an impulse response of the following form:
G(t)= [jAexp( -rt)+iBexp(
-t/r)]
E( w,,t)+Cexp(-t/r’)exp(-~~2t2)sin(w,t),
(2)
where A, B, and Care adjustable constants. The characteristic dynamical parameters include the oscillator frequency w,, the inhomogeneous broadening rate (Y,the homogeneously broadened rate 1/T’, the intermediate dephasing time 7, and the Debye relaxation rate lY The multiplicative function E(o,, t) is set to be [ 1 -exp( - 2wJ) ] if the inertial nature of the nuclear response is imposed. Ruhman et al. [ 181 set E( w,, t) = 1, 1/t’ = 0, and B=O and determined the three characteristic dynamical parameters w,, IY,and r from their 1% CS2data. Kalpouzos et al. [ 201 set E( w,, t) = [ 1-exp( -2w,t) ] and ~=7’, and determined four characteristic dynamical parameters, w,, OL,7= r’, and L Barker et al. [22] also determined four characteristic parameters, w,, l/r’, ‘5,and r, by setting cy=O, and E(w,, t)= [I -exp( -2w,t)]. Since both a and l/r’ are comparable to o,, the accuracy of most experimental data does not justify the inclusion of both dephasing and inhomogeneous broadening rates as independent adjustable parameters for the underdamped oscillator. Furthermore, their inclusion would lead to a convolution integral in the frequency response, G(w), which further complicates the curve fitting procedure. The impulse response we have chosen to represent two of the models we selected may be expressed as G(t)=[fAexp(-ll)+$Bexp(-t/7)]
[l-exp(-2w,t)]+Cexp(-~ct2t2)sin(wOt).
(3)
Here, a factor of f is used with the constants A and B, so that each term contributes to the initial rise of G( 1) in proportion to its respective amplitudes, A, B, or C. By setting B= 0, we have a mode1with three characteristic
dynamic parameters, w,, (Y,and L This mode1 is similar to that of Ruhman et al. [ 181, although they impose 164
CHEMICAL PHYSICS LETTERS
Volume186,number2,3
8 NovemberI991
no inertia requirement by setting E( w,, t ) = 1. By leaving E as an adjustable constant, we have a model of four characteristic dynamic parameters, o,, a!, T, and r This model with four dynamic parameters is similar to that of Kalpouzos et al. [ 201, except that they set t’ = ‘Iand we set l/r’ = 0. The gain spectra for these models, respectively, take the following forms: 1 -!-- - w*+ (r+20,)2 g,(o)=w~ 2 i w2+F
+ ${F[exp(
&(w)=w;
(
- (wyT)i)
&
t $ {;[exp(-
-
)
-exp(-
1 w2t(rt2w0)2
(w;212)-exp(
‘w~~‘2)]}“‘,
)
+w”
(4)
1 1 2 ( w~+(l/7)~-w~t(l/7t2w,)2
- ‘“;JcJ’)D”‘.
)
(5)
As will be seen later, the inclusion of the intermediate relaxation time, r, improves the quality of the fit, especially for liquid benzene. The third model we have chosen is also a model with four dynamic parameters in which we replaced the second tenu in eq. (3) by a term due to (orientationally averaged) collision-induced anisotropy. Bucaro and Litovitz [ 51 pointed out that since interaction-induced effects extend only several atomic radii in liquids, the symmetry and close-packed nature of simple liquids tend to suppress the (first-order) induced dipole effect, and the induced anisotropy in this case is dominated by electronic overlap or higher order effects [ 5 1. Their model calculation gives this form of the LS spectrum: 1o I 2(m- ‘)I7 exp ( - Iw I t), with m = 9 for (orientationally averaged) collision-induced anisotropy. Using this result, the SGS spectrum for our collisional model takes the following form:
H&
&(W)=W ;
- w2t
1 (rt2w,)*
+ 2 {f [exp( - (mi2)2)-exp(
I
t0.873w,r(25/7)~wlwl(4/7)exp(-lwlr)
- (“i2’“)][‘2
.
(6)
Here again, the multiplicative factors are so chosen that the rates of increase for the impulse response, G(t), at t=O due to the three terms are, respectively, Ac+,,Bw,, and Cw,. In this model, the depolarized spectrum consists of three contributions: collision-induced translational anisotropy (the second term) which exists even in an isotropic liquid such as Ar, reorientational relaxation (the first term) and an inhomogeneously broadened underdamped oscillator to account for local oscillations. The second term now contains a functional form that has some theoretical justification, but the high-frequency tail is not just due to the second term as would be expected from previous observations [ 51. We point out that fitting the CS2 data to eq. (6) and forcing the exponential decay rate, l/7, to be greater than m0 also produces a good fit [ 291. Our choice of letting I /T to be less than w,, provides not only a more direct comparison between models depicted by eqs. (5) and (6) but a more reasonable physical picture. If the collision rate is higher than the oscillation frequency, then we believe that evidence of the oscillations would be washed out. This collisiona model is conceptually similar to that used by Scarparo et al. [ 141, although they impose no inertia requirement for the reorientational relaxation (the first term ) , do not have the I w I 4’7 factor in the second term, and use a homogeneously broadened underdamped oscillator for the third term. Although we use the I w I4/7 dependence in eq. (6)) its validity may be questionable. More recent molecular dynamics calculations [ 6,8,9] have shown that higher-order dipoleinduced-dipole effects are not necessary to produce the exponential tail on the one hand, and that different dynamical effects are strongly correlated for liquids with high polarizability on the other. In this sense, the second and third terms of eqs. (5) and (6) together represent the phenomenology of strongly coupled collision165
Volume 186, number 2,3
199I 8 November
CHEMICAL PHYSICS LETTERS
induced dynamics and non-diffusive reorientation as well as cross-correlation between translational and orientational motions which dominate intermediate and high-frequency dynamics. We have fitted our measured SGS spectra of CS2 and benzene to the expressions given in eqs. (4)-( 6). In fig. 2, the measured SGS spectrum of CS2from - 120 to 120 cm-’ is fitted to g3(w), g4( o) and ge(w), and the results are shown, respectively, in figs. 2a, 2b and 2c. The values of the characteristic dynamic parameters so determined are listed in table 1 along with those determined by previous workers. The fit shown in fig. 2a is pretty good, although not as good as those shown in figs. 2b and 2c. The result from the three-parameter model, eq. (4), may be compared with the model of Ruhman et al. [ 181. A fit as good as fig. 2a requires the use of the inertial factor of the first term in eq. (4) [ 291. Another three-parameter model may be obtained by dropping the oscillator term in eq. (6). This gives rise to the Bucaro-Litovitz model [ 5 1. Fitting our CS2
(a)
z
-I1 I .z
3
.
c 3
L”
(0)
_._._..
5l
_.
0
k c
6
PI
“‘ --,..,.... 1
I”
*
I “I)’
-150 -100 -50 Frequency
“I”.
0
I
50
” .‘
100 I50
[cm-‘]
Fig. 2. The SGS spectrum of CS2with curve fits to: (a) eq. (4), three-parameter fit, (b) eq. (5) and (c) eq. (6), four-parameter fits. For CSI the four-parameter tits are only slightly better.
166
i,,, -20
,,, 20
1
,,,,,,,,,,, 60
Frectency
I 40
100 [cm-’
180
1
Fig. 3. The SGS spectrum of benzene with curve fits to: (a) eq. (4), three-parameter tit, (b) eq. (5) and (c) eq. (6), four-parameter fits. In benzene, the four-parameter tits are noticeably better.
8 November 199L
CHEMICAL PHYSICS LETTERS
Volume 186, number 23
Table 1 Comparison of dynamical parameters (in cm-’ ‘) ) in CS2and benzene Ref.
Orientational response
Collisional (intermediate) response
Librational response
B
l/r
c
%
41/r’)
18 28&l 35.4 33*2 33*s 35 34.5
30 33&3 23.5 3lll 3423 20 34.9
36+13 58.2kO.6 52il
45*5 38. I + 0.6 41.7kO.7
A
r
0.259 0.5 0.16 0.33 0.45 0.195
4.98 4.27 k 0.03 3.3 3.48 * 0.04 3.65 + 0.03 3.2 3.99
0.17 0.39 0.13 0.1 0.278
13.3 15.740.6 11.8kO.7 20 12.6
0.741 0.5 0.67 0.28 0.42 0.008 0.527
0.28 0.21 0.18
3.22+ 0.04 f.94i0.03 2.14kO.02
0.18 0.23
11.9kO.3 8.320.1
0.72 0.61 0.59
CS2 Ruhman et al. [25] this work, eq. (4) Kalpouzos et al. [ 17] this work, eq. (5 ) this work, eq. (6) Scarparo et al. [ 1l] b, Barker et al. [ 181 benzene this work, eq. (4) this work, eq. (5) this work, eq. (6)
1
n) I c11-~=6nxIO-*ps-‘~0.189 ps-‘, forexample 3.5 cn-‘co.662 b, Unlike other works listed, At B t C# 1.
ps-‘, corresponding to r=1.51 ps.
data to this three-parameter model yields rc3.46 cm-’ and 1/t= 17.1 cm-‘. Although the value of l/z so determined compares well with the LS value of 19.8 cm-’ [2,5] and the early value of 18.9 cm-’ using a picosecond technique [ 161, the quality of the fit is worse than even fig. 2a [ 29 1. All of these clearly reveal the inadequacy of the original Bucaro-Litovitz model without the oscillator term, and the superiority of the stimulated (nonlinear optical) techniques over conventional LS for extracting detailed information on the intermolecular dynamics in liquids. The dynamic parameters determined from our four-parameter model may be compared to those of Kalpouzos et al. [ 201. The oscillator frequency w,, the intermediate dephasing time T, and the Debye relaxation rate r are comparable. Our inhomogeneous broadening rate (Yis larger, 3 1 cm-’ as compared to their value of 24 cm-‘. This is not surprising, however, because our single broadening rate replaces both homogeneous and inhomogeneous relaxation rates of the underdamped oscillator in their model. The dynamical parameters determined from the collisional model are in qualitative agreement with our four-parameter model, although, due to the difference in the functional shape, the intermediate collision rate is somewhat smaller and the oscillator relaxation rate is somewhat larger for the collisional model. Despite the differences mentioned earlier, the collisional model is conceptually similar to the model used by Scarparo et al. [ 141. Again, the intermediate collision rate is somewhat smaller and oscillator relaxation rate is somewhat larger for the collisional model. From table 1, it is clear that the dynamic parameters determined by Barker et al. [22] are overall closer to our values, especially when the collisional model is used for comparison. In general, due to the ability to adjust relative amplitudes of each term, the experimental spectrum of CS2 is not very sensitive to the models considered. However, fluctuations in three time scales: a slow relaxation, an intermediate relaxation and a fast highly broadened oscillator, are needed to obtain excellent fit to the experimental data. The need for an intermediate time constant becomes very clear when the fittings of the three models to the measured SGS spectrum of benzene are compared. The fit to the three-parameter model, as shown in fig. 3a, is obviously inadequate, while the fits to both the four-parameter model (fig. 3b) and the collisional model (fig. 3c) are very good. The determined dynamical rates for liquid benzene are also listed in table 1. Alms et al. [30], in a depolarized Rayleigh-Brillouin LS experiment, measured the reorientational relaxation rate J’ to be 1.82 cm-‘. Our value of about 2 cm-’ compares favorably with this result. The ISS response has also been measured [ 3 11, but no results were reported. Comments on the differences between the four-parameter 167
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November I99 1
model and the collisional model made for CS2 are also applicable to liquid benzene. The measured spectrum of benzene resembles the pyridine spectrum, computed from time-domain response after the distortion of the short pulse is removed [25]. Comparing the results between CS2 and benzene, one finds that the slow and intermediate rates for benzene are lower, while the frequency and relaxation associated with the underdamped oscillator are higher. These differences may reflect the structural differences between a cigar-shaped molecule and a pancake-shaped molecule.
4. Conclusions The depolarized Rayleigh-wingspectra of liquid CS2and benzene have been recorded using SGS. These spectra have been fitted to three dynamical models, one containing three dynamic parameters and two containing four dynamic parameters. The functional form of the added term to account for the intermediate time-scale dynamics for the latter two models is taken to be exponential in the impulse response in one case and exponential in the frequency response in the other. With the relative amplitudes of the dynamical terms treated as adjustable constants, excellent tits to the measured SGS spectra have been obtained using either of the fourparameter models. The values of the dynamical parameters for CS2determined using these models have been compared to one another as well as compared to the values of previous researchers who determined these values from different types of stimulated light scattering spectroscopies. The four dynamical parameters for liquid benzene are reported for the first time. It is seen that a term with an intermediate relaxation rate in the order of 12 cm-’ is needed to obtain a good fit to the experimental data, especially for the benzene spectrum. General agreement with the parameters determined from the time-domain measurements in CS2 suggeststhat our frequency-domain technique is complimentary. The fact that our raw data spectrum of benzene resembles the pyridine spectrum, computed from the time-domain response after the distortion of the short pulse is removed, suggeststhat our measured SGS spectra mirror the intrinsic liquid dynamics very closely with little or no contamination from the instrument function. The striking differences between the SGS spectra of CS, and benzene establish the feasibility of using the SGS technique for studying liquid dynamics. At this time, other pure organic liquids in the benzene series are being investigated. The fact that the intermediate time is similar for the two liquids with different models should provide impetus for further theoretical study to reveal the mechanisms and functional forms that govern that decay time. Presumably, different molecular dynamics simulations from first principles treat the different degrees of freedom in liquid dynamics as coupled entities [ 8,321 instead of treating them phenomenologically and additively with their relative amplitudes adjusted to the measured spectrum. In this manner, different theoretical models will lead to different simulated SGS spectra that can be compared with the experimental spectra within a simple overall scale factor. Our SGS spectra with high signal-to-noise and negligible instrument effect can be used to distinguish the merits of different theoretical simulations in a detailed comparison. This comparison will help to establish a clearer microscopic dynamical picture of liquids.
Acknowledgement Two of us (JSF and CYS) gratefully acknowledge helpful discussions with Keith Nelson, Dale McMorrow and Branka Ladanyi at different stages of this work.
References [ I ] J.P. McTague, P.A. Fleury and D.B. Dupre, Phys. Rev. 185 (1969) 303.
168
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November 199I
[2] S.L. Shapiro and H.P. Broida, Phys. Rev. 154 (1967) 129. [3] T.I. Cox, M.R. Battaglia and P.A. Madden, Mol. Phys. 38 (1979) 1539. [4] B. Hegemann and J. Jonas, J. Chem. Phys. 82 (1985) 2845. [S] J.A. Bucaro and T.A. Litovitz, J. Chem. Phys. 54 ( 1971) 3846. [6] D. Frenkel and J.P. M&ague, J. Chem. Phys. 72 (1980) 2801. [7]P.A.MaddenandT.I.Cox,Mol.Phys.43 (1981)287. [8] L.C. Geiger and B.M. Iadanyi, J. Chem. Phys. 87 (1987) 191. [9] P.A. Madden and D.J. Tildesley, Mol. Phys. 55 (1985) 969. [lo] A.S. Barker Jr. and R. Loudon, Rev. Mod. Phys. 44 ( 1972) 18. [ 1I ] K.A. Nelson, DR. Lutz, M.D. Fayer and L. Madison, Phys. Rev. B 24 ( 1981) 3261. [ 121E.P. Ippen and C.V. Shank, Appl. Phys. Letters 26 (1985) 93. [ 131R. Trebino, C.E. Barker and A.E. Siegman, IEEE I. Quantum Electron. QE-22 (1986) 1413. [ 141 M.A.F. Scarparo, J.H. Leeand J.J. Song, Opt. Letters6 (1981) 193. [IS] C.Y. She, T.W. Broberg, L.S. Wall and D.F. Edwards, Phys. Rev. B 6 (1972) 1847. [ 161B.I. Green and R.C. Farrow, Chem. Phys. Letters 98 (1983) 273. [ 171S. Ruhman, L.R. Williams, A.G. Joly, B. Koh1erandK.A. Nelson, J. Phys. Chem. 91 (1987) 2237. [ I8 ] S. Ruhman, B. Kohler, A.G. Joly and K.A. Nelson, Chem. Phys. Letters 141 ( 1987) 16. [ 191C. Kalpouzos, W.T. Lotshaw, D. McMorrowand G.A. Kenney-Wallace, J. Phys. Chem. 91 (1987) 2028. [20] C. Kalpouzos, D. McMorrow, W.T. Lotshaw and GA. Kenney-Wallace, Chem. Phys. Letters 150 (1988) 138; I55 (1989) 240. [ 2 I ] D. McMorrow, W.T. Lotshaw and GA. Kenney-Wallace, IEEE J. Quantum Electron. QE-24 ( 1988) 443. [22] C.E. Barker, R. Trebino, A.G. Kostenbauder and A.E. Siegman, J. Chem. Phys. 92 (1990) 4740. [23] D. McMorrowand W.T. Lotshaw, Chem. Phys. Letters 178 (1991) 69. [24] J.S. Friedman, M.C. Lee and C.Y. She, in: International Conference on Quantum Electronics Technical Digest Series 1990, Vol. 8 (Opt. Sot. Am., Washington, 1990) pp. 262-263. [ 251 D. McMorrow and W.T. Lotshaw, Chem. Phys. Letters I74 ( 1991) 85. [26] A.G. Jacobson and Y.R. Shen, Appl. Phys. Letters 34 (1979) 464. [ 271 C.Y. She, G.C. Herring, H. MoosmuIler and S.A. Lee, Phys. Rev. letters 51 ( 1983) 1648. [28]S.Y.Tang,C.Y.SheandS.A.Lee,Opt. Letters 12 (1987) 870. [29] J.S. Friedman, Stimulated Rayleigh-Brillouin gain spectroscopy for the study of low frequency dynamics in simple liquids, Ph.D. Dissertation, Colorado State University (1991) unpublished. [30] G.R. Alms, D.R. Bauer, 1.1.Braumanand R. Pecora, J. Chem. Phys. 58 (1973) 5570. [ 311 S.Ruhman, 8. Kohler, A.G. Joly and K.A. Nelson, IEEE J. Quantum Electron. QE-24 ( 1988) 470. [32] B.M. Ladanyi, J. Chem. Phys. 78 (1983) 2189.
169
CHEMICAL PHYSICS LETTERS
8 November 1991
Depolarized stimulated gain spectra of liquid CS2 and benzene at room temperature J.S. Friedman, MC. Lee ’ and C.Y. She Physics Department,
Colorado State University, Fort Collins. CO 80523, USA
Received 12 July 1991; in final form 21 August 1991
Depolarized Rayleigh-wing spectra of CS2 and benzene have been recorded using stimulated gain spectroscopy (SGS). Each spectrum is fitted to three different phenomenological models. In thecase of CSL,the determined dynamical characteristic parameters are compared to those measured previously by different experimental techniques with good agreement.To adequatelydescribe the SGS spectraof both liquids, an intermediatetime scale contribution is needed in addition to a slow reorientational relaxation and a fast broad underdamped oscillator. Compared to CS2,the slow and intermediate relaxation ratesof benzene are lower, whilethefrequency andrelaxationrateof theunderdamped oscillatorarehigher.
1. Introduction
Depolarized light scattering studies at low frequencies (O-200 cm-l) can provide information on the dynamics of molecular liquids. Spontaneous light scattering (LS) studies have revealed an exponential tail in the wings of the depolarized LS spectra for both monatomic liquids, such as Ar, and anisotropic liquids, such as CS2, suggesting the existence of interaction-induced anisotropies in simple liquids [ l-41. In the case of anisotropic molecular liquids, there is, in addition, a Lorentzian line centered at zero frequency due to reorientational relaxation. Subsequent theoretical studies [ 5-81 have confirmed the existence of the exponential tail, but concluded that the shape of the spectrum, intermediate between the exponential tail and the Lorentzian line, is complex and varies from one liquid to another. Furthermore, it is difficult to distinguish different monotonically decreasing functions in frequency from experimental data. For a liquid with high polarizability, such as CS2, the depolarized LS spectrum may be separated [8,9] into diffusive reorientational anisotropy which dominates the low-frequency dynamics and is characterized by a (Debye) relaxation time of typically 2 ps, and broad interaction-induced contributions which dominate the spectrum at intermediate and higher frequencies. According to the fluctuation-dissipation theorem [ lo], the LS spectrum is proportional to the response funo tion divided by angular frequency in the low-frequency regime. This renders the interaction-induced effects in a LS spectrum much weaker than the reorientational anisotropy and nearly indistinguishable. A stimulated experiment using a pump and probe beam directly monitors the molecular response function, which goes to zero at zero frequency. It can thus display the intermediate frequency structure more readily and reveal the detailed shape of the response function more clearly. There have been four types of depolarized stimulated scattering experiments reported in the literature by different research groups for the study of liquid dynamics. Two time-domain techniques are the transient grating or impulsive stimulated scattering (ISS ) [ 1 I ] and the optical Kerr effect with optical-heterodyne detection (OHD OKE) [ 121, Two frequency-domain techniques are the tunable-laser-induced grating (TLIG) [ 131 and stimulated gain spectroscopy (SGS) [ 141. These stim’ On sabbatical leave from Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan. 0009-2614/91/S 03.50 0 1991 Elsevier Science Publishers B.V. All rightsreserved.
161
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November 1991
ulated techniques probe, respectively, the square of the impulse response function, the impulse response function, the square of the frequency response function, and the imaginary part of the frequency response function. Along with the spontaneous LS spectrum, we express these measured functions below for comparison: ISS intensity
(la)
OHD OKE response R(t) ccG( t ) ,
(lb)
TLIG efficiency
(lc)
?(~)aIG(w)l*,
SGS gain LS spectrum
(IdI Qw)alm[G(w)
I/w,
WI
where G(t) is the impulse response function of the liquid, and its Fourier transform, the frequency response function, is G(w). Unlike ISS and TLIG techniques, which are proportional to the square of the response function, the linear proportionality of OHD OKE and SGS techniques have some advantage in bringing out the relatively weak features in the response function because squaring a strong signal makes it stronger relative to squaring a weak signal. To obtain a stimulated response, fluctuations over a wide range of frequencies must be excited in the liquid. In order to do so the width of the interrogating laser pulse must be narrow enough in the time-domain, and the frequency range of the tunable narrow band laser should be wide enough in a frequency-domain technique. These requirements are now being met by state-of-the-art lasers. Because the detailed theoretical response function is rather complicated and varied, experimentalists have had a long tradition of using a phenomenological response function [ IO,151 with either two or three additive terms to model their experimental data. In 198I, Scarparo et al. [ 141 recorded a stimulated gain spectrum for CS2 up to a shift of 100 cm- ‘, and uncovered a broad underdamped oscillation at 35 cm-’ which was buried under the tail of the corresponding spontaneous spectrum [ 2,4]. A stimulated experiment on CSI in the timedomain was carried out in 1982 by Green and Farrow [ 161, revealing two time constants and associating the shorter relaxation time of 0.28 ps with intermolecular collisions. Recently, the dynamics of liquid CSZhave been studied by several groups. Nelson and co-workers were able to analyze their impulsive stimulated light scattering (ISS) data [ 171 by several models; the model that gave the best fit consists of a Debye relaxation plus an inhomogeneously broadened underdamped oscillator [ 181, Kalpouzos et al. [ 19,201 have fitted their OHD OKE data to a three-term impulse response function [21]: an inhomogeneously broadened underdamped oscillator with an intermediate dephasing time plus two exponentially decaying terms, one with the same intermediate time constant and the other with a diffusive (Debye) relaxation time. They further modified the form of the two exponential terms by imposing inertia in these responses. This model consists of a total of four characteristic dynamical parameters to be determined, the oscillator frequency w,, the inhomogeneous broadening rate (Y,the intermediate dephasing time r, and the Debye relaxation rate r. Barker et al. [ 221 have analyzed their TLIG data in terms of several models, and, based on a stringent statistical test, they favored a model consisting of two exponential terms modified to include the inertial nature of the nuclear response, plus a homogeneously broadened underdamped oscillator. Again, they used four characteristic times: the oscillator frequency, the fast and intermediate dephasing times and the Debye relaxation time, to analyze their frequencydomain
data.
These recently used phenomenological models do not explicitly account for the exponential tail in the frequency response function (or Lorentzian in the impulse response function) that most theories predict [ 5-8 1. However, since even with a semi-log plot it is not possible to ascertain the exact form of a fall-off function over the narrow frequency range associated with a single dynamical mechanism, curve fitting cannot distinguish between different functions, e.g., Gaussian and exponential, in this tail. As a result, the interaction-induced effects have been accounted for by these models in various ways. In the case of CSI, all models seem to fit the experimental data well independent of the attention paid to curve fitting. Recently, McMorrow and Lotshaw [ 231 have suggested that the interaction-induced effects can be more logically represented by a single 162
Volume 186, number2,3
CHEMICAL PHYSICS LETTERS
8 November 1991
intermolecular vibrational degree of freedom consisting of a continuous superposition of overdamped, critically damped and underdamped motion with a dephasing rate comparable to vibration frequencies. They did not, however, present a quantitative fitting of the proposed model to the experimental data. One of the purposes of this paper is to present our recent depolarized SGS spectrum of CS2 and to determine the dynamical parameters of the liquid. Our spectrum is comparable to that of Scarparo [ 141 but with an improved frequency resolution of 0.5 GHz. We also present the SGS spectrum of room-temperature benzene. Except for our preliminary conference report [ 241, the SGS spectrum of benzene has not yet been reported and analyzed. Compared to the existing time-domain data from which the frequency response can be deduced, our measured SGS spectra more closely represent the frequency response function, G(w), of the liquid. This fact has been made clear by a recent paper of McMorrow and Lotshaw [ 25 1. Using pyridine as an example, they pointed out that the distortion caused by a finite pulse width in the measured OHD OKE response can be eliminated only after a deconvolution procedure which requires precise knowledge of the shape of the pulse. Even for the 65 fs pulse they used, a considerable narrowing appeared in the frequency response. Comparison of their pyridine results with our benzene spectra shows that the deconvolution procedure works very well. However, for measured impulse response functions with marginal signal-to-noise, the required deconvolution procedure will introduce errors that would make the time-domain measurement less sensitive to different models. In contrast, the broadening caused by a pump laser of 0.5 GHz linewidth in a measured depolarized SGS spectrum, which typically covers a frequency range of more than 100 cm-‘, is quite negligible. In this paper we analyze our experimental data by three similar dynamical models and compare the results to those of previous reports. The obvious differences between the SGS spectra of CS2and benzene should serve to demonstrate the value of the SGS technique for studying liquid dynamics.
2. Experimental The experimental procedure is similar to those used previously [ 14,26-281 and is shown in fig. 1. A pulsed dye laser serves as a pump source. It is pumped by the second harmonic of a Q-switched Nd:YAG laser at a rate of 10 Hz. The pulsed dye laser is a modified Littman style using two gratings. Light inside the cavity is expanded by a four prism beam expander onto the fixed grating which is at grazing incidence. The reflection from this grating serves as the output beam, and the first diffraction order goes to a second grating, oriented for third or fourth order, which acts as the feedback mirror. We have measured the bandwidth of this laser to be roughly 0.5 GHz, and we can scan it as much as 300 cm-‘. The bandwidth of the pump laser is the limiting factor in this experiment as the probe laser, a cw single mode dye laser, has a bandwidth of less than 2 MHz.
Beam
Splitter
Normallzatian Diode
Fig. 1, Optical layout of the SGS experiment.
163
Volume186,number 2,3
CHEMICALPHYSICSLETTERS
8 November1991
Thus, our spectrometer has an equivalent finesse of 1.8~ 104.The lasers operate near 590 nm. The probe beam is split into 2 parts. The first is simply a normalization beam to eliminate noise due to fluctuations in the probe intensity, and the second is the actual probe. It is polarized perpendicular to the pump polarization and focused loosely into a glass cell containing the sample. The pump and probe lasers cross at a near counter-propagating angle of 179” through the sample, and the left-over pump light is caught by a beam dump. The probe beam is steered through a polarization analyzer and several apertures to eliminate stray scattered pump light that may be on its path, and is then detected by a photodiode. Because the gain/loss signal appears as a gain or loss pulse, a gated integrator and boxcar averager are used to detect the light. Fluctuations in the signal corresponding to pulse-to-pulse fluctuations in the pump laser can be normalized out with a detector that monitors the pump intensity. The frequency scale of the experiment was determined by steering part of the pulsed dye laser through a fixed spacing Fabry-Perot interferometer whose free spectral range of 1.74 cm-’ was determined from the measured spacing. A frequency scale is obtained by simply using the interferometer peaks as markers. Using this frequency scale, we note that as an independent check Raman lines in the measured SGS spectra of chloroform and bromobenzene agree with the known values. The decision of whether to scan both the gain and loss sides of the spectrum or to scan mainly the gain side depended on the width of the spectrum. In CS, the full spectrum is less than 250 cm- ’ wide, well within the tuning range of our pulsed pump source. In contrast, the full speo trum of benzene extends over 300 cm-‘, which is near or over the limit of the tuning range. As the laser is tuned towards the wings of the dye gain curve, a considerable amount of fluorescence is often seen. Before and after each spectrum is taken, the output of the pulsed dye laser is checked and, if necessary, the alignment is adjusted, using an optical multichannel analyzer as a diagnostic tool, to reduce fluorescence in the output to a negligible level.
3. Results and comparisons Many existing dynamical models can be described phenomenologically by an impulse response of the following form:
G(t)= [jAexp( -rt)+iBexp(
-t/r)]
E( w,,t)+Cexp(-t/r’)exp(-~~2t2)sin(w,t),
(2)
where A, B, and Care adjustable constants. The characteristic dynamical parameters include the oscillator frequency w,, the inhomogeneous broadening rate (Y,the homogeneously broadened rate 1/T’, the intermediate dephasing time 7, and the Debye relaxation rate lY The multiplicative function E(o,, t) is set to be [ 1 -exp( - 2wJ) ] if the inertial nature of the nuclear response is imposed. Ruhman et al. [ 181 set E( w,, t) = 1, 1/t’ = 0, and B=O and determined the three characteristic dynamical parameters w,, IY,and r from their 1% CS2data. Kalpouzos et al. [ 201 set E( w,, t) = [ 1-exp( -2w,t) ] and ~=7’, and determined four characteristic dynamical parameters, w,, OL,7= r’, and L Barker et al. [22] also determined four characteristic parameters, w,, l/r’, ‘5,and r, by setting cy=O, and E(w,, t)= [I -exp( -2w,t)]. Since both a and l/r’ are comparable to o,, the accuracy of most experimental data does not justify the inclusion of both dephasing and inhomogeneous broadening rates as independent adjustable parameters for the underdamped oscillator. Furthermore, their inclusion would lead to a convolution integral in the frequency response, G(w), which further complicates the curve fitting procedure. The impulse response we have chosen to represent two of the models we selected may be expressed as G(t)=[fAexp(-ll)+$Bexp(-t/7)]
[l-exp(-2w,t)]+Cexp(-~ct2t2)sin(wOt).
(3)
Here, a factor of f is used with the constants A and B, so that each term contributes to the initial rise of G( 1) in proportion to its respective amplitudes, A, B, or C. By setting B= 0, we have a mode1with three characteristic
dynamic parameters, w,, (Y,and L This mode1 is similar to that of Ruhman et al. [ 181, although they impose 164
CHEMICAL PHYSICS LETTERS
Volume186,number2,3
8 NovemberI991
no inertia requirement by setting E( w,, t ) = 1. By leaving E as an adjustable constant, we have a model of four characteristic dynamic parameters, o,, a!, T, and r This model with four dynamic parameters is similar to that of Kalpouzos et al. [ 201, except that they set t’ = ‘Iand we set l/r’ = 0. The gain spectra for these models, respectively, take the following forms: 1 -!-- - w*+ (r+20,)2 g,(o)=w~ 2 i w2+F
+ ${F[exp(
&(w)=w;
(
- (wyT)i)
&
t $ {;[exp(-
-
)
-exp(-
1 w2t(rt2w0)2
(w;212)-exp(
‘w~~‘2)]}“‘,
)
+w”
(4)
1 1 2 ( w~+(l/7)~-w~t(l/7t2w,)2
- ‘“;JcJ’)D”‘.
)
(5)
As will be seen later, the inclusion of the intermediate relaxation time, r, improves the quality of the fit, especially for liquid benzene. The third model we have chosen is also a model with four dynamic parameters in which we replaced the second tenu in eq. (3) by a term due to (orientationally averaged) collision-induced anisotropy. Bucaro and Litovitz [ 51 pointed out that since interaction-induced effects extend only several atomic radii in liquids, the symmetry and close-packed nature of simple liquids tend to suppress the (first-order) induced dipole effect, and the induced anisotropy in this case is dominated by electronic overlap or higher order effects [ 5 1. Their model calculation gives this form of the LS spectrum: 1o I 2(m- ‘)I7 exp ( - Iw I t), with m = 9 for (orientationally averaged) collision-induced anisotropy. Using this result, the SGS spectrum for our collisional model takes the following form:
H&
&(W)=W ;
- w2t
1 (rt2w,)*
+ 2 {f [exp( - (mi2)2)-exp(
I
t0.873w,r(25/7)~wlwl(4/7)exp(-lwlr)
- (“i2’“)][‘2
.
(6)
Here again, the multiplicative factors are so chosen that the rates of increase for the impulse response, G(t), at t=O due to the three terms are, respectively, Ac+,,Bw,, and Cw,. In this model, the depolarized spectrum consists of three contributions: collision-induced translational anisotropy (the second term) which exists even in an isotropic liquid such as Ar, reorientational relaxation (the first term) and an inhomogeneously broadened underdamped oscillator to account for local oscillations. The second term now contains a functional form that has some theoretical justification, but the high-frequency tail is not just due to the second term as would be expected from previous observations [ 51. We point out that fitting the CS2 data to eq. (6) and forcing the exponential decay rate, l/7, to be greater than m0 also produces a good fit [ 291. Our choice of letting I /T to be less than w,, provides not only a more direct comparison between models depicted by eqs. (5) and (6) but a more reasonable physical picture. If the collision rate is higher than the oscillation frequency, then we believe that evidence of the oscillations would be washed out. This collisiona model is conceptually similar to that used by Scarparo et al. [ 141, although they impose no inertia requirement for the reorientational relaxation (the first term ) , do not have the I w I 4’7 factor in the second term, and use a homogeneously broadened underdamped oscillator for the third term. Although we use the I w I4/7 dependence in eq. (6)) its validity may be questionable. More recent molecular dynamics calculations [ 6,8,9] have shown that higher-order dipoleinduced-dipole effects are not necessary to produce the exponential tail on the one hand, and that different dynamical effects are strongly correlated for liquids with high polarizability on the other. In this sense, the second and third terms of eqs. (5) and (6) together represent the phenomenology of strongly coupled collision165
Volume 186, number 2,3
199I 8 November
CHEMICAL PHYSICS LETTERS
induced dynamics and non-diffusive reorientation as well as cross-correlation between translational and orientational motions which dominate intermediate and high-frequency dynamics. We have fitted our measured SGS spectra of CS2 and benzene to the expressions given in eqs. (4)-( 6). In fig. 2, the measured SGS spectrum of CS2from - 120 to 120 cm-’ is fitted to g3(w), g4( o) and ge(w), and the results are shown, respectively, in figs. 2a, 2b and 2c. The values of the characteristic dynamic parameters so determined are listed in table 1 along with those determined by previous workers. The fit shown in fig. 2a is pretty good, although not as good as those shown in figs. 2b and 2c. The result from the three-parameter model, eq. (4), may be compared with the model of Ruhman et al. [ 181. A fit as good as fig. 2a requires the use of the inertial factor of the first term in eq. (4) [ 291. Another three-parameter model may be obtained by dropping the oscillator term in eq. (6). This gives rise to the Bucaro-Litovitz model [ 5 1. Fitting our CS2
(a)
z
-I1 I .z
3
.
c 3
L”
(0)
_._._..
5l
_.
0
k c
6
PI
“‘ --,..,.... 1
I”
*
I “I)’
-150 -100 -50 Frequency
“I”.
0
I
50
” .‘
100 I50
[cm-‘]
Fig. 2. The SGS spectrum of CS2with curve fits to: (a) eq. (4), three-parameter fit, (b) eq. (5) and (c) eq. (6), four-parameter fits. For CSI the four-parameter tits are only slightly better.
166
i,,, -20
,,, 20
1
,,,,,,,,,,, 60
Frectency
I 40
100 [cm-’
180
1
Fig. 3. The SGS spectrum of benzene with curve fits to: (a) eq. (4), three-parameter tit, (b) eq. (5) and (c) eq. (6), four-parameter fits. In benzene, the four-parameter tits are noticeably better.
8 November 199L
CHEMICAL PHYSICS LETTERS
Volume 186, number 23
Table 1 Comparison of dynamical parameters (in cm-’ ‘) ) in CS2and benzene Ref.
Orientational response
Collisional (intermediate) response
Librational response
B
l/r
c
%
41/r’)
18 28&l 35.4 33*2 33*s 35 34.5
30 33&3 23.5 3lll 3423 20 34.9
36+13 58.2kO.6 52il
45*5 38. I + 0.6 41.7kO.7
A
r
0.259 0.5 0.16 0.33 0.45 0.195
4.98 4.27 k 0.03 3.3 3.48 * 0.04 3.65 + 0.03 3.2 3.99
0.17 0.39 0.13 0.1 0.278
13.3 15.740.6 11.8kO.7 20 12.6
0.741 0.5 0.67 0.28 0.42 0.008 0.527
0.28 0.21 0.18
3.22+ 0.04 f.94i0.03 2.14kO.02
0.18 0.23
11.9kO.3 8.320.1
0.72 0.61 0.59
CS2 Ruhman et al. [25] this work, eq. (4) Kalpouzos et al. [ 17] this work, eq. (5 ) this work, eq. (6) Scarparo et al. [ 1l] b, Barker et al. [ 181 benzene this work, eq. (4) this work, eq. (5) this work, eq. (6)
1
n) I c11-~=6nxIO-*ps-‘~0.189 ps-‘, forexample 3.5 cn-‘co.662 b, Unlike other works listed, At B t C# 1.
ps-‘, corresponding to r=1.51 ps.
data to this three-parameter model yields rc3.46 cm-’ and 1/t= 17.1 cm-‘. Although the value of l/z so determined compares well with the LS value of 19.8 cm-’ [2,5] and the early value of 18.9 cm-’ using a picosecond technique [ 161, the quality of the fit is worse than even fig. 2a [ 29 1. All of these clearly reveal the inadequacy of the original Bucaro-Litovitz model without the oscillator term, and the superiority of the stimulated (nonlinear optical) techniques over conventional LS for extracting detailed information on the intermolecular dynamics in liquids. The dynamic parameters determined from our four-parameter model may be compared to those of Kalpouzos et al. [ 201. The oscillator frequency w,, the intermediate dephasing time T, and the Debye relaxation rate r are comparable. Our inhomogeneous broadening rate (Yis larger, 3 1 cm-’ as compared to their value of 24 cm-‘. This is not surprising, however, because our single broadening rate replaces both homogeneous and inhomogeneous relaxation rates of the underdamped oscillator in their model. The dynamical parameters determined from the collisional model are in qualitative agreement with our four-parameter model, although, due to the difference in the functional shape, the intermediate collision rate is somewhat smaller and the oscillator relaxation rate is somewhat larger for the collisional model. Despite the differences mentioned earlier, the collisional model is conceptually similar to the model used by Scarparo et al. [ 141. Again, the intermediate collision rate is somewhat smaller and oscillator relaxation rate is somewhat larger for the collisional model. From table 1, it is clear that the dynamic parameters determined by Barker et al. [22] are overall closer to our values, especially when the collisional model is used for comparison. In general, due to the ability to adjust relative amplitudes of each term, the experimental spectrum of CS2 is not very sensitive to the models considered. However, fluctuations in three time scales: a slow relaxation, an intermediate relaxation and a fast highly broadened oscillator, are needed to obtain excellent fit to the experimental data. The need for an intermediate time constant becomes very clear when the fittings of the three models to the measured SGS spectrum of benzene are compared. The fit to the three-parameter model, as shown in fig. 3a, is obviously inadequate, while the fits to both the four-parameter model (fig. 3b) and the collisional model (fig. 3c) are very good. The determined dynamical rates for liquid benzene are also listed in table 1. Alms et al. [30], in a depolarized Rayleigh-Brillouin LS experiment, measured the reorientational relaxation rate J’ to be 1.82 cm-‘. Our value of about 2 cm-’ compares favorably with this result. The ISS response has also been measured [ 3 11, but no results were reported. Comments on the differences between the four-parameter 167
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November I99 1
model and the collisional model made for CS2 are also applicable to liquid benzene. The measured spectrum of benzene resembles the pyridine spectrum, computed from time-domain response after the distortion of the short pulse is removed [25]. Comparing the results between CS2 and benzene, one finds that the slow and intermediate rates for benzene are lower, while the frequency and relaxation associated with the underdamped oscillator are higher. These differences may reflect the structural differences between a cigar-shaped molecule and a pancake-shaped molecule.
4. Conclusions The depolarized Rayleigh-wingspectra of liquid CS2and benzene have been recorded using SGS. These spectra have been fitted to three dynamical models, one containing three dynamic parameters and two containing four dynamic parameters. The functional form of the added term to account for the intermediate time-scale dynamics for the latter two models is taken to be exponential in the impulse response in one case and exponential in the frequency response in the other. With the relative amplitudes of the dynamical terms treated as adjustable constants, excellent tits to the measured SGS spectra have been obtained using either of the fourparameter models. The values of the dynamical parameters for CS2determined using these models have been compared to one another as well as compared to the values of previous researchers who determined these values from different types of stimulated light scattering spectroscopies. The four dynamical parameters for liquid benzene are reported for the first time. It is seen that a term with an intermediate relaxation rate in the order of 12 cm-’ is needed to obtain a good fit to the experimental data, especially for the benzene spectrum. General agreement with the parameters determined from the time-domain measurements in CS2 suggeststhat our frequency-domain technique is complimentary. The fact that our raw data spectrum of benzene resembles the pyridine spectrum, computed from the time-domain response after the distortion of the short pulse is removed, suggeststhat our measured SGS spectra mirror the intrinsic liquid dynamics very closely with little or no contamination from the instrument function. The striking differences between the SGS spectra of CS, and benzene establish the feasibility of using the SGS technique for studying liquid dynamics. At this time, other pure organic liquids in the benzene series are being investigated. The fact that the intermediate time is similar for the two liquids with different models should provide impetus for further theoretical study to reveal the mechanisms and functional forms that govern that decay time. Presumably, different molecular dynamics simulations from first principles treat the different degrees of freedom in liquid dynamics as coupled entities [ 8,321 instead of treating them phenomenologically and additively with their relative amplitudes adjusted to the measured spectrum. In this manner, different theoretical models will lead to different simulated SGS spectra that can be compared with the experimental spectra within a simple overall scale factor. Our SGS spectra with high signal-to-noise and negligible instrument effect can be used to distinguish the merits of different theoretical simulations in a detailed comparison. This comparison will help to establish a clearer microscopic dynamical picture of liquids.
Acknowledgement Two of us (JSF and CYS) gratefully acknowledge helpful discussions with Keith Nelson, Dale McMorrow and Branka Ladanyi at different stages of this work.
References [ I ] J.P. McTague, P.A. Fleury and D.B. Dupre, Phys. Rev. 185 (1969) 303.
168
Volume 186, number 2,3
CHEMICAL PHYSICS LETTERS
8 November 199I
[2] S.L. Shapiro and H.P. Broida, Phys. Rev. 154 (1967) 129. [3] T.I. Cox, M.R. Battaglia and P.A. Madden, Mol. Phys. 38 (1979) 1539. [4] B. Hegemann and J. Jonas, J. Chem. Phys. 82 (1985) 2845. [S] J.A. Bucaro and T.A. Litovitz, J. Chem. Phys. 54 ( 1971) 3846. [6] D. Frenkel and J.P. M&ague, J. Chem. Phys. 72 (1980) 2801. [7]P.A.MaddenandT.I.Cox,Mol.Phys.43 (1981)287. [8] L.C. Geiger and B.M. Iadanyi, J. Chem. Phys. 87 (1987) 191. [9] P.A. Madden and D.J. Tildesley, Mol. Phys. 55 (1985) 969. [lo] A.S. Barker Jr. and R. Loudon, Rev. Mod. Phys. 44 ( 1972) 18. [ 1I ] K.A. Nelson, DR. Lutz, M.D. Fayer and L. Madison, Phys. Rev. B 24 ( 1981) 3261. [ 121E.P. Ippen and C.V. Shank, Appl. Phys. Letters 26 (1985) 93. [ 131R. Trebino, C.E. Barker and A.E. Siegman, IEEE I. Quantum Electron. QE-22 (1986) 1413. [ 141 M.A.F. Scarparo, J.H. Leeand J.J. Song, Opt. Letters6 (1981) 193. [IS] C.Y. She, T.W. Broberg, L.S. Wall and D.F. Edwards, Phys. Rev. B 6 (1972) 1847. [ 161B.I. Green and R.C. Farrow, Chem. Phys. Letters 98 (1983) 273. [ 171S. Ruhman, L.R. Williams, A.G. Joly, B. Koh1erandK.A. Nelson, J. Phys. Chem. 91 (1987) 2237. [ I8 ] S. Ruhman, B. Kohler, A.G. Joly and K.A. Nelson, Chem. Phys. Letters 141 ( 1987) 16. [ 191C. Kalpouzos, W.T. Lotshaw, D. McMorrowand G.A. Kenney-Wallace, J. Phys. Chem. 91 (1987) 2028. [20] C. Kalpouzos, D. McMorrow, W.T. Lotshaw and GA. Kenney-Wallace, Chem. Phys. Letters 150 (1988) 138; I55 (1989) 240. [ 2 I ] D. McMorrow, W.T. Lotshaw and GA. Kenney-Wallace, IEEE J. Quantum Electron. QE-24 ( 1988) 443. [22] C.E. Barker, R. Trebino, A.G. Kostenbauder and A.E. Siegman, J. Chem. Phys. 92 (1990) 4740. [23] D. McMorrowand W.T. Lotshaw, Chem. Phys. Letters 178 (1991) 69. [24] J.S. Friedman, M.C. Lee and C.Y. She, in: International Conference on Quantum Electronics Technical Digest Series 1990, Vol. 8 (Opt. Sot. Am., Washington, 1990) pp. 262-263. [ 251 D. McMorrow and W.T. Lotshaw, Chem. Phys. Letters I74 ( 1991) 85. [26] A.G. Jacobson and Y.R. Shen, Appl. Phys. Letters 34 (1979) 464. [ 271 C.Y. She, G.C. Herring, H. MoosmuIler and S.A. Lee, Phys. Rev. letters 51 ( 1983) 1648. [28]S.Y.Tang,C.Y.SheandS.A.Lee,Opt. Letters 12 (1987) 870. [29] J.S. Friedman, Stimulated Rayleigh-Brillouin gain spectroscopy for the study of low frequency dynamics in simple liquids, Ph.D. Dissertation, Colorado State University (1991) unpublished. [30] G.R. Alms, D.R. Bauer, 1.1.Braumanand R. Pecora, J. Chem. Phys. 58 (1973) 5570. [ 311 S.Ruhman, 8. Kohler, A.G. Joly and K.A. Nelson, IEEE J. Quantum Electron. QE-24 ( 1988) 470. [32] B.M. Ladanyi, J. Chem. Phys. 78 (1983) 2189.
169