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Relation between Stokes and anti-Stokes low-frequency light scattering in liquid carbon disulfide
5 January 2001
Chemical Physics Letters 333 (2001) 113±118
www.elsevier.nl/locate/cplett
Relation between Stokes and anti-Stokes low-frequency light scattering in liquid carbon disul®de J. Watanabe *, Y. Watanabe, S. Kinoshita Department of Physics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan Received 24 July 2000; in ®nal form 10 October 2000
Abstract The intensity ratio of Stokes to anti-Stokes light scattering in the low-frequency region (<15 cmÿ1 ) has been investigated within an accuracy of 0.01 for liquid carbon disul®de at 292 K. The spectrum shows a symmetric lineshape about the excitation frequency in the region less than several wavenumber, indicating that the commonly used relation between these scattering intensities based on canonical distribution breaks down in this frequency region. The symmetric lineshape of the spectrum may arise from its dissipative character. Ó 2001 Elsevier Science B.V. All rights reserved.
The intensity ratio between Stokes and antiStokes light scattering usually takes the form n x 1=n x, where n x is the occupation number of phonon modes with an angular frequency x. This relation is derived generally from the time-reversal symmetry between Stokes and anti-Stokes scattering processes [1±3] and is commonly used for the analysis of Raman scattering experiments. Loudon [2] pointed out that the relation breaks down in the systems of ordered magnets or under resonance scattering conditions where the time-reversal symmetry between these processes does not hold. With respect to the timereversal symmetry, another interesting situation is light scattering of a state that cannot be regarded as a stationary state of the system due to the existence of the dissipation process. In such a case,
*
Corresponding author. Fax: +81-6-6850-5365. E-mail address: [email protected] (J. Watanabe).
the argument based on time-reversal symmetry is not applicable since the dissipation process is timeirreversible. Even if the state is not exactly the stationary state of the system, the above-mentioned relation is of course a good approximation when the spectral width due to the dissipation is small enough compared with the frequency of the state as in the case of usual phonon Raman scattering. When the spectral width due to the dissipation is comparable to the frequency of the state, however, the relation should break down. In this case, it may be necessary to reconsider the usual way of analyzing light scattering spectra based on the ¯uctuation±dissipation theorem [3], where Stokes scattering spectra are taken to be proportional to the imaginary part of the frequency response function multiplied by n x 1. In this respect, one of the interesting situations is the low-frequency light-scattering spectrum in liquids. In the low-frequency region of depolarized light-scattering spectra in liquids, a so-called relaxational mode centered at the excitation light
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 3 3 2 - 4
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J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
frequency and structureless broad low-frequency phonon modes are observed [4±12]. The relaxational mode is usually believed to originate from cooperative reorientational motion of liquid molecules. On the other hand, the low-frequency phonon modes are considered to originate from the librational motions of individual liquid molecules around their mean con®guration. The lineshape of the relaxational mode is known to approximately follow a Lorentzian function whose width gives a collective reorientation time. Correspondingly, the time response function observed by time-resolved impulsive stimulated scattering [13,14] or optical Kerr eect spectroscopy [15±17] shows an exponential decay in the long-time region as in the Debye relaxation model. Thus the relaxational mode has a dissipative character. The symmetric lineshape of the Lorentzian does not necessarily contradict the above-mentioned relation of Stokes to anti-Stokes intensity ratio, n x 1=n x, when the spectral width of the relaxational mode is too narrow to experimentally detect the dierence. A question arises whether the relation remains even when its width is large enough to experimentally notice a dierence between n x 1=n x and unity. Shapiro and Broida [4] reported that the intensity ratio between Stokes and anti-Stokes light scattering in carbon disul®de followed n x 1=n x up to x 300 cmÿ1 within an accuracy of several percent. For the relaxational mode, however, it is necessary to measure the intensity ratio within an accuracy less than a percent to clarify whether it follows n x 1=n x or not. So far, to our knowledge, there has been no report on the detailed study of the light scattering intensity ratio for a dissipative state. In the present Letter, we carefully examine the relation between the intensities of Stokes and anti-Stokes spectra measured for liquid carbon disul®de, which shows a strong relaxational mode with a relatively wide width (HWHM ' 3 cmÿ1 at room temperature). The results show that the above relation actually breaks down, and in addition, its lineshape is symmetric about the excitation frequency in a range of several wavenumber but is not a Lorentzian in a whole range. A part of the present Letter has already been reported brie¯y [20].
At ®rst, we brie¯y summarize the theoretical background of the relation above according to the theory of Loudon [2,3]. The dierential cross-section of spontaneous light scattering is generally given by d2 r / x1 x32 S Dx; dX dx2 where S Dx
Z
1
ÿ1
dthay a tieiDxt ;
1
2
with Dx x2 ÿ x1 . x1 and x2 are the frequencies of the excitation and scattered photons, respectively, X is the solid angle and a is the polarizability. Here we consider light scattering under non-resonant excitation and take a refractive index to be independent of photon frequency. We assume that the excitation of the medium responsible for the inelastic light scattering with the scattering vector q is characterized by an amplitude X^ r; t fX^ q; teiqr X^ y q; teÿiqr g, where X^ q; t and X^ y q; t are excitation destruction and creation operators, respectively. We also assume that the modulation of a is expanded to the ®rst order of the amplitude X^ r; t. Then the dierential crosssection for Stokes scattering is given by d2 r Cx1 x32 hX^ qX^ y qix ; dX dx2
3
where C is a constant, x ÿDx and hix shows a power spectrum de®ned by hX^ qX^ y qix R hX^ q;xX^ y q;x0 id x ÿ x0 with X^ q;t X^ q; x exp ÿixt. For the time-reversed process of the Stokes scattering, the dierential cross-section becomes d2 r Cx2 x31 hX^ y ÿqX^ ÿqix : dX dx1
4
Using the ¯uctuation±dissipation theorem, the power spectra for Stokes scattering and its timereversed process are related by n xhX^ qX^ y qix n x 1hX^ y ÿqX^ ÿqix : 5
J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
Eqs. (3)±(5) lead to a general relation x2 x31 n x
d2 r d2 r x1 x32 n x 1 : dX dx2 dX dx1
6
When x2 is very close to x1 , the dierential crosssection for the time-reversed process of the Stokes scattering can be taken to be the same as that of the anti-Stokes scattering, where the angular frequencies of the excitation and scattered photons are x1 and x1 x, respectively. Then S Dx for Stokes (SS ) and anti-Stokes (SAS ) scattering are related, in a good approximation, by SS =SAS n x 1=n x:
7
In the following, we compare experimental results of carbon disul®de with Eq. (7). We employed a cw Ar ion laser as an exciting source for the measurements of low-frequency light scattering spectra of carbon disul®de. The scattered light perpendicularly polarized with respect to the excitation light was introduced to a double monochromator (Jobin Yvon U-1000) with a spectral resolution of 0:6 cmÿ1 , and detected by a photomultiplier (Hamamatsu R464). The temperature of the sample is measured within an accuracy of 0.5 K by a thermocouple in the liquid near the scattering volume. To clarify whether the intensity ratio of Stokes to anti-Stokes scattering takes n x 1=n x or not in the frequency range below 5 cmÿ1 , we need to measure the ratio within the accuracy less than 0.01. Therefore we checked various experimental factors which could distort a spectrum as follows: (1) We carefully made a correction of measured spectra by the wavelengthdependent sensitivity of the monochromator/ detection system. To check the validity of the correction, we measured the spectra at two dier ent excitation wavelengths of 4765 and 4880 A. They coincided with each other within the experimental error and then these spectra were averaged to increase the S/N ratio. (2) The absence of any ¯uorescence component contaminated in the liquid was con®rmed by the agreement of the spectra excited by the two dierent wavelengths. (3) To see the validity of the non-resonant excitation condition assumed in Eq. (7), the absorption spectrum of carbon disul®de was measured using a 245-mmlength cell. We found that the absorption is neg-
115
(4) We simulated the ligible at 4880 or 4765 A. eect of a slit function on measured spectra. For that purpose, we calculated the convolution spectra between a slit function and a Lorentzian (FWHM 6.4 cmÿ1 ) which is distorted by a factor of n x 1=n x about Dx 0. For the slit function, we used a Gaussian (FWHM 0.6 cmÿ1 ) which reproduces well the measured slit function. The dierence in the Stokes to anti-Stokes intensity ratio between the spectra before and after the convolution was less than 10ÿ4 , and thus of negligible order. (5) In the analysis of the spectra, we found that the result was very sensitive to the determination of the origin of the spectra, Dx 0, and that the accuracy required for the determination was about 0.01 cmÿ1 . There are several ways to determine the origin of the spectrum measured by a double monochromator, such as: (a) measuring the spectrum of the laser light, (b) measuring the elastic scattering component from the sample, (c) observing the Stokes and anti-Stokes peaks of a molecular vibrational mode, and (d) comparing the spectrum observed by a double monochromator with that observed by a Fabry± Perot interferometer. We tried these methods and found that they are not applicable to determining the origin within an accuracy of 0.01 cmÿ1 for the following reasons. That is, (a) the reproducibility for the wavenumber of the monochromator is 0.1 cmÿ1 , so that repeated measurement is not applicable in the present case; (b) due to the limitation of the step size in scanning the monochromator, the number of datapoints in the elastic scattering spectrum is not sucient for determining the origin; (c) from Stokes and anti-Stokes lines separated by several hundred cmÿ1 , it is hard to determine the middle point precisely; and (d) the shape of the spectrum obtained by a Fabry± Perot interferometer is very sensitive to the alignment of the interferometer and thus the determination of the origin by comparing the two spectra has uncertainty. We determined Dx 0 for each measured spectrum with an accuracy of 0.01 cmÿ1 , assuming that the spectrum very close to its peak within 1 cmÿ1 is symmetric about Dx 0. This assumption was based on the fact that the time-domain response of the relaxational mode, for example in the optical Kerr
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eect spectroscopy [18,19], is known to show a single exponential decay in the time region longer than several picoseconds. This means that the light-scattering spectrum at least in the region very close to the origin should become a Lorentzian-like symmetric spectrum. In Fig. 1, we show a low-frequency light-scattering spectrum S Dx of carbon disul®de measured at 292 K. In order to see the symmetry of the spectra about Dx 0, we also plot the same spectrum by a broken line in which Dx is converted to ÿDx. A small hump at Dx 0 arises from an elastic scattering component. As seen in the ®gure, S Dx is symmetric about Dx 0 up to jDxj 2 cmÿ1 . With increasing jDxj, the ratio of the Stokes to anti-Stokes scattering intensities increases as is usually expected. In Fig. 2, we show the Stokes to anti-Stokes intensity ratio, SS =SAS , obtained from Fig. 1 as a function of jDxj. The solid line is SS =SAS ejDxj=kB T , which is expected from Eq. (7). The experimental values coincide with those from Eq. (7) within the experimental error in the region of jDxj J 5 cmÿ1 . In the lower frequency region, however, they
Fig. 1. A low-frequency light-scattering spectrum S Dx of carbon disul®de measured at 292 K. Here Dx x2 ÿ x1 ; x1 and x2 are the excitation and scattered photon frequencies, respectively. The broken line is the same spectrum in which Dx is converted to ÿDx.
Fig. 2. The Stokes to anti-Stokes intensity ratio, SS =SAS , obtained from Fig. 1, as a function of jDxj. The solid line represents SS =SAS ejDxj=kB T .
clearly deviate from Eq. (7) and take smaller values. Especially, SS =SAS ' 1 holds in the region of jDxj K 2 cmÿ1 , meaning that the spectrum is symmetric, whereas the solid line almost linearly increases with jDxj. The dierence in SS =SAS between measured value and that from Eq. (7) is only 0.01 at jDxj 2 cmÿ1 . Thus the relation of Eq. (7) expected from the canonical distribution of the system actually breaks down for the light scattering of the relaxational mode in carbon disul®de. In the frequency region of jDxj J 5 cmÿ1 , the experimental values follow Eq. (7), as shown in Fig. 2. The spectrum in this region is considered to originate from the local librational motions of individual liquid molecules. Thus the present result implies that the spectral broadening in this frequency region mainly arises from a frequency distribution of the librational motions, not from the damping of the modes. As is well known, Eq. (7) does not hold in the higher-order light scattering of the phonon modes of angular frequency x. For two-phonon scattering [21], for example, Stokes scattering intensity at angular frequency, 2x is proportional to n x 1n x 2, and the corresponding antiStokes scattering to n xn x ÿ 1. Thus the
J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
intensity ratio SS =SAS should be larger (and much larger in much-higher-order scattering) than that of Eq. (7). Since the value of SS =SAS below 5 cmÿ1 does not take larger values than that of Eq. (7), we consider that there is no contribution from such a higher-order scattering process in this frequency range. We consider that the breakdown of Eq. (7) originates from a dissipative character of the relaxational mode. When it is dissipative, the Stokes scattering and its time-reversed process (or antiStokes scattering and its time-reversed process) cannot simply be related by the time-reversal symmetry like in Eq. (5) since the amplitude X^ q; t should include a damping factor which is timeirreversible. For a study of the dynamics in liquid and glass materials, the frequency response function R x is often evaluated from light-scattering spectra using a relation SS Dx / n x 1ImR x (or SAS Dx / n xImR x). This relation is based on the ¯uctuation±dissipation theorem [3], hX^ qX^ y qix h=p n x 1ImR q;x (or hX^ y qX^ qix h=p n xIm R q; x). Since, for the relaxational mode, the intensity ratio between Stokes and anti-Stokes scattering does not take a value of n x 1=n x, the resultant `response functions' obtained from the Stokes or anti-Stokes scattering do not coincide with each other. Therefore a correction of the measured spectrum by the Bose factor n x 1 (or by n x) does not lead to a correct form of the frequency response function. The symmetric lineshape of the relaxational mode means, in some sense, that the mode behaves like a classical ¯uctuation. Therefore it may be necessary to reconsider the applicability of the quantum-mechanical form of the ¯uctuation±dissipation theorem for the analysis of the light-scattering experiment of liquids in this frequency region. Next we examine the lineshape of the spectrum. It is well known that the spectral shape of the relaxational mode having a much narrower width approximately follows the Lorentzian function. Then a question arises as to in what frequency range the Lorentzian reproduces the spectrum having a wide width. To this end, we consider two possible cases. One is case (1), where the spectrum for jDxj K 2 cmÿ1 consists of only the Lorentzian component (plus elastic scattering). The other is
117
case (2), where the spectrum for jDxj K 2 cmÿ1 consists of the low-frequency phonon modes and the Lorentzian with a smaller width. For case (1), we calculate convolution spectrum between a Gaussian slit function and a Lorentzian function added by a d-function-like component corresponding to elastic scattering at the excitation frequency. In the inset of Fig. 3, we compare the measured spectrum (circles) with a calculated curve (solid line) using a Gaussian (FWHM 0.6 cmÿ1 ), a Lorentzian (FWHM 6.4 cmÿ1 ) and a d-function-like component whose relative intensity 0.003 to the Lorentzian is determined so as to ®t the calculated spectrum. One may consider that the discrepancy shown in jDxj J 2 cmÿ1 simply arises from the low-frequency phonon modes. If that is the case, we can expect that the dierence spectrum DS Dx between the measured and calculated spectra should satisfy the relation of
Fig. 3. Comparison of the spectrum with a Lorentzian function. In the inset, we compare the spectrum at 292 K (circles) with a Lorentzian function (FWHM 6.4 cmÿ1 ) convoluted by a slit function (solid line). We show, by closed diamonds, the intensity ratio of Stokes (DSS ) to anti-Stokes (DSAS ) components for the dierence between the measured and the calculated spectra shown in the inset. The solid line representing Eq. (7) and closed circles are the same with those in Fig. 2.
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J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
Eq. (7). In Fig. 3, we show the intensity ratio between Stokes (DSS ) and anti-Stokes (DSAS ) components for DS Dx by closed diamonds. It is clear that the ratio is larger than that of Eq. (7) (solid line). Especially for jDxj J 5 cmÿ1 , where the total intensity (closed circles) follows Eq. (7), the spectrum cannot be considered as the sum of lowfrequency phonon modes and a relaxational mode of Lorentzian or any other symmetric function, since the existence of the symmetric function would make the low-frequency phonon modes deviate largely from the canonical distribution. For case (2), we must consider a Lorentzian having a negligible contribution for jDxj J 5 cmÿ1 . However, in this case, the phonon modes would make the spectrum unsymmetric for jDxj K 2 cmÿ1 in spite of the symmetric lineshape shown in Fig. 2. Anyway the Lorentzian plus the phonon modes cannot reproduce the measured spectrum. It is concluded that there exists a Lorentzian-like component which is symmetric about the excitation frequency in a limited region of jDxj K 5 cmÿ1 , but it does not have a long tail like a Lorentzian function. From a statistical mechanical viewpoint, a Lorentzian lineshape appears in a Markovian limit where the characteristic time of the mode is much longer than the correlation time of the microscopic motions concerned. In carbon disul®de at room temperature, re¯ecting a relatively wide width of the spectrum, this condition will not be satis®ed. In addition, it has not been fully understood yet how the relaxational mode and its dissipative character appear from the microscopic motions in the liquids. Since the light-scattering intensity and the corresponding frequency response function are strongly temperature-dependent in the frequency less than several tens of wavenumber, we consider that anharmonic couplings between low-frequency phonon modes play an essential role for the cooperative molecular motions in liquid [22]. Summarizing, we have shown that the commonly used relation between Stokes and antiStokes light-scattering intensities breaks down for the relaxational mode having a relatively wide width in carbon disul®de. The spectral shape becomes symmetric about the excitation frequency in
the region less than several wavenumber, and the spectral component does not have a long tail like a Lorentzian function. The symmetric lineshape of the spectrum may result from the dissipative character of the relaxational mode. Acknowledgements This work has been supported in part by a Grant-in-Aid for Scienti®c Research (C-11640315) from the Ministry of Education, Science, Sports and Culture. References [1] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1960. [2] R. Loudon, J. Raman Spectrosc. 7 (1978) 10. [3] W. Hayes, R. Loudon, Scattering of Light by Crystals, Wiley, New York, 1978. [4] S.L. Shapiro, H.P. Broida, Phys. Rev. 154 (1967) 129. [5] J.P. McTague, P.A. Fleury, D.B. DuPre, Phys. Rev. 188 (1969) 303. [6] J.A. Bucaro, T.A. Litovitz, J. Chem. Phys. 54 (1971) 3846. [7] H.D. Dardy, V. Volterra, T.A. Litovitz, J. Chem. Phys. 59 (1973) 4491. [8] G. Enright, B.P. Stoiche, J. Chem. Phys. 60 (1974) 2536. [9] T.I. Cox, M.R. Battaglia, P.A. Madden, Molec. Phys. 38 (1979) 1539. [10] W. Danninger, G. Zundel, Chem. Phys. Lett. 90 (1982) 69. [11] B. Hegemann, J. Jonas, J. Chem. Phys. 82 (1985) 2845. [12] L.C. Geiger, B.M. Ladanyi, J. Chem. Phys. 87 (1987) 191. [13] S. Ruhman, B. Kohler, A.G. Joly, K.A. Nelson, Chem. Phys. Lett. 141 (1987) 16. [14] S. Ruhman, L.R. Williams, A.G. Joly, B. Kohler, K.A. Nelson, J. Phys. Chem. 91 (1987) 2237. [15] B.I. Greene, R.C. Farrow, Chem. Phys. Lett. 98 (1983) 273. [16] C. Kalpouzos, W.T. Lotshaw, D. McMorrow, G.A. Kenney-Wallace, J. Phys. Chem. 91 (1987) 2028. [17] D. McMorrow, W.T. Lotshaw, J. Phys. Chem. 95 (1991) 10395. [18] W.T. Lotshaw, D. McMorrow, N. Thantu, J.S. Melinger, R. Kitchenham, J. Raman Spectrosc. 26 (1995) 571. [19] B.J. Loughnane, A. Scodinu, R.A. Farrer, J.T. Fourkas, J. Chem. Phys. 111 (1999) 2686. [20] J. Watanabe, Y. Watanabe, M. Tango, S. Kinoshita, J. Lumin. 87±89 (2000) 779. [21] A.K. Ganguly, J.L. Birman, Phys. Rev. 162 (1967) 806. [22] S. Kinoshita, Y. Kai, Y. Watanabe, Chem. Phys. Lett. 301 (1999) 183.
Chemical Physics Letters 333 (2001) 113±118
www.elsevier.nl/locate/cplett
Relation between Stokes and anti-Stokes low-frequency light scattering in liquid carbon disul®de J. Watanabe *, Y. Watanabe, S. Kinoshita Department of Physics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan Received 24 July 2000; in ®nal form 10 October 2000
Abstract The intensity ratio of Stokes to anti-Stokes light scattering in the low-frequency region (<15 cmÿ1 ) has been investigated within an accuracy of 0.01 for liquid carbon disul®de at 292 K. The spectrum shows a symmetric lineshape about the excitation frequency in the region less than several wavenumber, indicating that the commonly used relation between these scattering intensities based on canonical distribution breaks down in this frequency region. The symmetric lineshape of the spectrum may arise from its dissipative character. Ó 2001 Elsevier Science B.V. All rights reserved.
The intensity ratio between Stokes and antiStokes light scattering usually takes the form n x 1=n x, where n x is the occupation number of phonon modes with an angular frequency x. This relation is derived generally from the time-reversal symmetry between Stokes and anti-Stokes scattering processes [1±3] and is commonly used for the analysis of Raman scattering experiments. Loudon [2] pointed out that the relation breaks down in the systems of ordered magnets or under resonance scattering conditions where the time-reversal symmetry between these processes does not hold. With respect to the timereversal symmetry, another interesting situation is light scattering of a state that cannot be regarded as a stationary state of the system due to the existence of the dissipation process. In such a case,
*
Corresponding author. Fax: +81-6-6850-5365. E-mail address: [email protected] (J. Watanabe).
the argument based on time-reversal symmetry is not applicable since the dissipation process is timeirreversible. Even if the state is not exactly the stationary state of the system, the above-mentioned relation is of course a good approximation when the spectral width due to the dissipation is small enough compared with the frequency of the state as in the case of usual phonon Raman scattering. When the spectral width due to the dissipation is comparable to the frequency of the state, however, the relation should break down. In this case, it may be necessary to reconsider the usual way of analyzing light scattering spectra based on the ¯uctuation±dissipation theorem [3], where Stokes scattering spectra are taken to be proportional to the imaginary part of the frequency response function multiplied by n x 1. In this respect, one of the interesting situations is the low-frequency light-scattering spectrum in liquids. In the low-frequency region of depolarized light-scattering spectra in liquids, a so-called relaxational mode centered at the excitation light
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 3 3 2 - 4
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J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
frequency and structureless broad low-frequency phonon modes are observed [4±12]. The relaxational mode is usually believed to originate from cooperative reorientational motion of liquid molecules. On the other hand, the low-frequency phonon modes are considered to originate from the librational motions of individual liquid molecules around their mean con®guration. The lineshape of the relaxational mode is known to approximately follow a Lorentzian function whose width gives a collective reorientation time. Correspondingly, the time response function observed by time-resolved impulsive stimulated scattering [13,14] or optical Kerr eect spectroscopy [15±17] shows an exponential decay in the long-time region as in the Debye relaxation model. Thus the relaxational mode has a dissipative character. The symmetric lineshape of the Lorentzian does not necessarily contradict the above-mentioned relation of Stokes to anti-Stokes intensity ratio, n x 1=n x, when the spectral width of the relaxational mode is too narrow to experimentally detect the dierence. A question arises whether the relation remains even when its width is large enough to experimentally notice a dierence between n x 1=n x and unity. Shapiro and Broida [4] reported that the intensity ratio between Stokes and anti-Stokes light scattering in carbon disul®de followed n x 1=n x up to x 300 cmÿ1 within an accuracy of several percent. For the relaxational mode, however, it is necessary to measure the intensity ratio within an accuracy less than a percent to clarify whether it follows n x 1=n x or not. So far, to our knowledge, there has been no report on the detailed study of the light scattering intensity ratio for a dissipative state. In the present Letter, we carefully examine the relation between the intensities of Stokes and anti-Stokes spectra measured for liquid carbon disul®de, which shows a strong relaxational mode with a relatively wide width (HWHM ' 3 cmÿ1 at room temperature). The results show that the above relation actually breaks down, and in addition, its lineshape is symmetric about the excitation frequency in a range of several wavenumber but is not a Lorentzian in a whole range. A part of the present Letter has already been reported brie¯y [20].
At ®rst, we brie¯y summarize the theoretical background of the relation above according to the theory of Loudon [2,3]. The dierential cross-section of spontaneous light scattering is generally given by d2 r / x1 x32 S Dx; dX dx2 where S Dx
Z
1
ÿ1
dthay a tieiDxt ;
1
2
with Dx x2 ÿ x1 . x1 and x2 are the frequencies of the excitation and scattered photons, respectively, X is the solid angle and a is the polarizability. Here we consider light scattering under non-resonant excitation and take a refractive index to be independent of photon frequency. We assume that the excitation of the medium responsible for the inelastic light scattering with the scattering vector q is characterized by an amplitude X^ r; t fX^ q; teiqr X^ y q; teÿiqr g, where X^ q; t and X^ y q; t are excitation destruction and creation operators, respectively. We also assume that the modulation of a is expanded to the ®rst order of the amplitude X^ r; t. Then the dierential crosssection for Stokes scattering is given by d2 r Cx1 x32 hX^ qX^ y qix ; dX dx2
3
where C is a constant, x ÿDx and hix shows a power spectrum de®ned by hX^ qX^ y qix R hX^ q;xX^ y q;x0 id x ÿ x0 with X^ q;t X^ q; x exp ÿixt. For the time-reversed process of the Stokes scattering, the dierential cross-section becomes d2 r Cx2 x31 hX^ y ÿqX^ ÿqix : dX dx1
4
Using the ¯uctuation±dissipation theorem, the power spectra for Stokes scattering and its timereversed process are related by n xhX^ qX^ y qix n x 1hX^ y ÿqX^ ÿqix : 5
J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
Eqs. (3)±(5) lead to a general relation x2 x31 n x
d2 r d2 r x1 x32 n x 1 : dX dx2 dX dx1
6
When x2 is very close to x1 , the dierential crosssection for the time-reversed process of the Stokes scattering can be taken to be the same as that of the anti-Stokes scattering, where the angular frequencies of the excitation and scattered photons are x1 and x1 x, respectively. Then S Dx for Stokes (SS ) and anti-Stokes (SAS ) scattering are related, in a good approximation, by SS =SAS n x 1=n x:
7
In the following, we compare experimental results of carbon disul®de with Eq. (7). We employed a cw Ar ion laser as an exciting source for the measurements of low-frequency light scattering spectra of carbon disul®de. The scattered light perpendicularly polarized with respect to the excitation light was introduced to a double monochromator (Jobin Yvon U-1000) with a spectral resolution of 0:6 cmÿ1 , and detected by a photomultiplier (Hamamatsu R464). The temperature of the sample is measured within an accuracy of 0.5 K by a thermocouple in the liquid near the scattering volume. To clarify whether the intensity ratio of Stokes to anti-Stokes scattering takes n x 1=n x or not in the frequency range below 5 cmÿ1 , we need to measure the ratio within the accuracy less than 0.01. Therefore we checked various experimental factors which could distort a spectrum as follows: (1) We carefully made a correction of measured spectra by the wavelengthdependent sensitivity of the monochromator/ detection system. To check the validity of the correction, we measured the spectra at two dier ent excitation wavelengths of 4765 and 4880 A. They coincided with each other within the experimental error and then these spectra were averaged to increase the S/N ratio. (2) The absence of any ¯uorescence component contaminated in the liquid was con®rmed by the agreement of the spectra excited by the two dierent wavelengths. (3) To see the validity of the non-resonant excitation condition assumed in Eq. (7), the absorption spectrum of carbon disul®de was measured using a 245-mmlength cell. We found that the absorption is neg-
115
(4) We simulated the ligible at 4880 or 4765 A. eect of a slit function on measured spectra. For that purpose, we calculated the convolution spectra between a slit function and a Lorentzian (FWHM 6.4 cmÿ1 ) which is distorted by a factor of n x 1=n x about Dx 0. For the slit function, we used a Gaussian (FWHM 0.6 cmÿ1 ) which reproduces well the measured slit function. The dierence in the Stokes to anti-Stokes intensity ratio between the spectra before and after the convolution was less than 10ÿ4 , and thus of negligible order. (5) In the analysis of the spectra, we found that the result was very sensitive to the determination of the origin of the spectra, Dx 0, and that the accuracy required for the determination was about 0.01 cmÿ1 . There are several ways to determine the origin of the spectrum measured by a double monochromator, such as: (a) measuring the spectrum of the laser light, (b) measuring the elastic scattering component from the sample, (c) observing the Stokes and anti-Stokes peaks of a molecular vibrational mode, and (d) comparing the spectrum observed by a double monochromator with that observed by a Fabry± Perot interferometer. We tried these methods and found that they are not applicable to determining the origin within an accuracy of 0.01 cmÿ1 for the following reasons. That is, (a) the reproducibility for the wavenumber of the monochromator is 0.1 cmÿ1 , so that repeated measurement is not applicable in the present case; (b) due to the limitation of the step size in scanning the monochromator, the number of datapoints in the elastic scattering spectrum is not sucient for determining the origin; (c) from Stokes and anti-Stokes lines separated by several hundred cmÿ1 , it is hard to determine the middle point precisely; and (d) the shape of the spectrum obtained by a Fabry± Perot interferometer is very sensitive to the alignment of the interferometer and thus the determination of the origin by comparing the two spectra has uncertainty. We determined Dx 0 for each measured spectrum with an accuracy of 0.01 cmÿ1 , assuming that the spectrum very close to its peak within 1 cmÿ1 is symmetric about Dx 0. This assumption was based on the fact that the time-domain response of the relaxational mode, for example in the optical Kerr
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eect spectroscopy [18,19], is known to show a single exponential decay in the time region longer than several picoseconds. This means that the light-scattering spectrum at least in the region very close to the origin should become a Lorentzian-like symmetric spectrum. In Fig. 1, we show a low-frequency light-scattering spectrum S Dx of carbon disul®de measured at 292 K. In order to see the symmetry of the spectra about Dx 0, we also plot the same spectrum by a broken line in which Dx is converted to ÿDx. A small hump at Dx 0 arises from an elastic scattering component. As seen in the ®gure, S Dx is symmetric about Dx 0 up to jDxj 2 cmÿ1 . With increasing jDxj, the ratio of the Stokes to anti-Stokes scattering intensities increases as is usually expected. In Fig. 2, we show the Stokes to anti-Stokes intensity ratio, SS =SAS , obtained from Fig. 1 as a function of jDxj. The solid line is SS =SAS ejDxj=kB T , which is expected from Eq. (7). The experimental values coincide with those from Eq. (7) within the experimental error in the region of jDxj J 5 cmÿ1 . In the lower frequency region, however, they
Fig. 1. A low-frequency light-scattering spectrum S Dx of carbon disul®de measured at 292 K. Here Dx x2 ÿ x1 ; x1 and x2 are the excitation and scattered photon frequencies, respectively. The broken line is the same spectrum in which Dx is converted to ÿDx.
Fig. 2. The Stokes to anti-Stokes intensity ratio, SS =SAS , obtained from Fig. 1, as a function of jDxj. The solid line represents SS =SAS ejDxj=kB T .
clearly deviate from Eq. (7) and take smaller values. Especially, SS =SAS ' 1 holds in the region of jDxj K 2 cmÿ1 , meaning that the spectrum is symmetric, whereas the solid line almost linearly increases with jDxj. The dierence in SS =SAS between measured value and that from Eq. (7) is only 0.01 at jDxj 2 cmÿ1 . Thus the relation of Eq. (7) expected from the canonical distribution of the system actually breaks down for the light scattering of the relaxational mode in carbon disul®de. In the frequency region of jDxj J 5 cmÿ1 , the experimental values follow Eq. (7), as shown in Fig. 2. The spectrum in this region is considered to originate from the local librational motions of individual liquid molecules. Thus the present result implies that the spectral broadening in this frequency region mainly arises from a frequency distribution of the librational motions, not from the damping of the modes. As is well known, Eq. (7) does not hold in the higher-order light scattering of the phonon modes of angular frequency x. For two-phonon scattering [21], for example, Stokes scattering intensity at angular frequency, 2x is proportional to n x 1n x 2, and the corresponding antiStokes scattering to n xn x ÿ 1. Thus the
J. Watanabe et al. / Chemical Physics Letters 333 (2001) 113±118
intensity ratio SS =SAS should be larger (and much larger in much-higher-order scattering) than that of Eq. (7). Since the value of SS =SAS below 5 cmÿ1 does not take larger values than that of Eq. (7), we consider that there is no contribution from such a higher-order scattering process in this frequency range. We consider that the breakdown of Eq. (7) originates from a dissipative character of the relaxational mode. When it is dissipative, the Stokes scattering and its time-reversed process (or antiStokes scattering and its time-reversed process) cannot simply be related by the time-reversal symmetry like in Eq. (5) since the amplitude X^ q; t should include a damping factor which is timeirreversible. For a study of the dynamics in liquid and glass materials, the frequency response function R x is often evaluated from light-scattering spectra using a relation SS Dx / n x 1ImR x (or SAS Dx / n xImR x). This relation is based on the ¯uctuation±dissipation theorem [3], hX^ qX^ y qix h=p n x 1ImR q;x (or hX^ y qX^ qix h=p n xIm R q; x). Since, for the relaxational mode, the intensity ratio between Stokes and anti-Stokes scattering does not take a value of n x 1=n x, the resultant `response functions' obtained from the Stokes or anti-Stokes scattering do not coincide with each other. Therefore a correction of the measured spectrum by the Bose factor n x 1 (or by n x) does not lead to a correct form of the frequency response function. The symmetric lineshape of the relaxational mode means, in some sense, that the mode behaves like a classical ¯uctuation. Therefore it may be necessary to reconsider the applicability of the quantum-mechanical form of the ¯uctuation±dissipation theorem for the analysis of the light-scattering experiment of liquids in this frequency region. Next we examine the lineshape of the spectrum. It is well known that the spectral shape of the relaxational mode having a much narrower width approximately follows the Lorentzian function. Then a question arises as to in what frequency range the Lorentzian reproduces the spectrum having a wide width. To this end, we consider two possible cases. One is case (1), where the spectrum for jDxj K 2 cmÿ1 consists of only the Lorentzian component (plus elastic scattering). The other is
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case (2), where the spectrum for jDxj K 2 cmÿ1 consists of the low-frequency phonon modes and the Lorentzian with a smaller width. For case (1), we calculate convolution spectrum between a Gaussian slit function and a Lorentzian function added by a d-function-like component corresponding to elastic scattering at the excitation frequency. In the inset of Fig. 3, we compare the measured spectrum (circles) with a calculated curve (solid line) using a Gaussian (FWHM 0.6 cmÿ1 ), a Lorentzian (FWHM 6.4 cmÿ1 ) and a d-function-like component whose relative intensity 0.003 to the Lorentzian is determined so as to ®t the calculated spectrum. One may consider that the discrepancy shown in jDxj J 2 cmÿ1 simply arises from the low-frequency phonon modes. If that is the case, we can expect that the dierence spectrum DS Dx between the measured and calculated spectra should satisfy the relation of
Fig. 3. Comparison of the spectrum with a Lorentzian function. In the inset, we compare the spectrum at 292 K (circles) with a Lorentzian function (FWHM 6.4 cmÿ1 ) convoluted by a slit function (solid line). We show, by closed diamonds, the intensity ratio of Stokes (DSS ) to anti-Stokes (DSAS ) components for the dierence between the measured and the calculated spectra shown in the inset. The solid line representing Eq. (7) and closed circles are the same with those in Fig. 2.
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Eq. (7). In Fig. 3, we show the intensity ratio between Stokes (DSS ) and anti-Stokes (DSAS ) components for DS Dx by closed diamonds. It is clear that the ratio is larger than that of Eq. (7) (solid line). Especially for jDxj J 5 cmÿ1 , where the total intensity (closed circles) follows Eq. (7), the spectrum cannot be considered as the sum of lowfrequency phonon modes and a relaxational mode of Lorentzian or any other symmetric function, since the existence of the symmetric function would make the low-frequency phonon modes deviate largely from the canonical distribution. For case (2), we must consider a Lorentzian having a negligible contribution for jDxj J 5 cmÿ1 . However, in this case, the phonon modes would make the spectrum unsymmetric for jDxj K 2 cmÿ1 in spite of the symmetric lineshape shown in Fig. 2. Anyway the Lorentzian plus the phonon modes cannot reproduce the measured spectrum. It is concluded that there exists a Lorentzian-like component which is symmetric about the excitation frequency in a limited region of jDxj K 5 cmÿ1 , but it does not have a long tail like a Lorentzian function. From a statistical mechanical viewpoint, a Lorentzian lineshape appears in a Markovian limit where the characteristic time of the mode is much longer than the correlation time of the microscopic motions concerned. In carbon disul®de at room temperature, re¯ecting a relatively wide width of the spectrum, this condition will not be satis®ed. In addition, it has not been fully understood yet how the relaxational mode and its dissipative character appear from the microscopic motions in the liquids. Since the light-scattering intensity and the corresponding frequency response function are strongly temperature-dependent in the frequency less than several tens of wavenumber, we consider that anharmonic couplings between low-frequency phonon modes play an essential role for the cooperative molecular motions in liquid [22]. Summarizing, we have shown that the commonly used relation between Stokes and antiStokes light-scattering intensities breaks down for the relaxational mode having a relatively wide width in carbon disul®de. The spectral shape becomes symmetric about the excitation frequency in
the region less than several wavenumber, and the spectral component does not have a long tail like a Lorentzian function. The symmetric lineshape of the spectrum may result from the dissipative character of the relaxational mode. Acknowledgements This work has been supported in part by a Grant-in-Aid for Scienti®c Research (C-11640315) from the Ministry of Education, Science, Sports and Culture. References [1] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1960. [2] R. Loudon, J. Raman Spectrosc. 7 (1978) 10. [3] W. Hayes, R. Loudon, Scattering of Light by Crystals, Wiley, New York, 1978. [4] S.L. Shapiro, H.P. Broida, Phys. Rev. 154 (1967) 129. [5] J.P. McTague, P.A. Fleury, D.B. DuPre, Phys. Rev. 188 (1969) 303. [6] J.A. Bucaro, T.A. Litovitz, J. Chem. Phys. 54 (1971) 3846. [7] H.D. Dardy, V. Volterra, T.A. Litovitz, J. Chem. Phys. 59 (1973) 4491. [8] G. Enright, B.P. Stoiche, J. Chem. Phys. 60 (1974) 2536. [9] T.I. Cox, M.R. Battaglia, P.A. Madden, Molec. Phys. 38 (1979) 1539. [10] W. Danninger, G. Zundel, Chem. Phys. Lett. 90 (1982) 69. [11] B. Hegemann, J. Jonas, J. Chem. Phys. 82 (1985) 2845. [12] L.C. Geiger, B.M. Ladanyi, J. Chem. Phys. 87 (1987) 191. [13] S. Ruhman, B. Kohler, A.G. Joly, K.A. Nelson, Chem. Phys. Lett. 141 (1987) 16. [14] S. Ruhman, L.R. Williams, A.G. Joly, B. Kohler, K.A. Nelson, J. Phys. Chem. 91 (1987) 2237. [15] B.I. Greene, R.C. Farrow, Chem. Phys. Lett. 98 (1983) 273. [16] C. Kalpouzos, W.T. Lotshaw, D. McMorrow, G.A. Kenney-Wallace, J. Phys. Chem. 91 (1987) 2028. [17] D. McMorrow, W.T. Lotshaw, J. Phys. Chem. 95 (1991) 10395. [18] W.T. Lotshaw, D. McMorrow, N. Thantu, J.S. Melinger, R. Kitchenham, J. Raman Spectrosc. 26 (1995) 571. [19] B.J. Loughnane, A. Scodinu, R.A. Farrer, J.T. Fourkas, J. Chem. Phys. 111 (1999) 2686. [20] J. Watanabe, Y. Watanabe, M. Tango, S. Kinoshita, J. Lumin. 87±89 (2000) 779. [21] A.K. Ganguly, J.L. Birman, Phys. Rev. 162 (1967) 806. [22] S. Kinoshita, Y. Kai, Y. Watanabe, Chem. Phys. Lett. 301 (1999) 183.