J . Phys. Chem. 1985,89,2240-2253
2240
Thus, some degree of disparity in chemical structure between the components seems to be necessary to invoke the occurrence of fictitious critical points. On the other hand, their existence is obviously not limited to systems consisting of a single solvent
and a polymer mixture, chosen in this letter for illustration. They should be found just as often, e.g., in solutions of a polymer in a mixed solvent, and in ternary and higher low molecular weight systems.
FEATURE ARTICLE Studies of Vibrational Relaxation In Low-Temperature Molecular Crystals Using Coherent Raman Spectroscopy S. Velsko and
R. M. Hochstrasser*
Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania I9104 (Received: October 23, 1984;In Final Form: March 5, 1985)
We review recent studies of vibrational phase and population relaxation in molecular crystals at liquid helium temperatures. Since coherent anti-Stokes Raman scattering is becoming the method of choice for very high-resolution Raman spectroscopy in those systems, important aspects of both time and frequency-resolved CARS measurements are discussed. The contributions to the residual low-temperaturevibron line width are systematicallyexplored, drawing on experimental observations in crystalline N2, H2, benzene, and naphthalene for examples. These observations are brought into relation with current theories of exciton dynamics in condensed molecular systems. An important conclusion from these studies is that vibrational population relaxation rates show a high degree of mode selectivity, which has an impact on theories of chemical reaction dynamics.
1. Introduction
The study of the relaxation dynamics of the internal vibrational modes of molecular crystals bridges two major areas of chemical physics. It fits naturally into the context of research on exciton transport in solids and a t the same time provides information on the flow of vibrational energy in complex molecular systems which has a bearing on issues in chemical dynamics. The natural modes of vibration of a crystal are collective, with each internal vibrational state of the molecules in the crystal forming a band of elementary excitations called vibrons. These are the Frenkel excitons formed from molecular vibrations.’ In a macroscopic crystalline array pulsed optical excitation can create vibron wavepackets which, in principle, can transport energy through the crystal as they propagate. In a real crystal two basic processes degrade the wavepacket: It can lose its coherence through dephasing interactions or the energy of the wavepacket can be lost to other vibrational modes. These same processes affect the propagation of thermal or acoustic energy in solidsZand the transport of electronic or magnetic excitations in weakly interacting crystal^.^ All of the issues which have been raised concerning those processes are, a priori, appropriate here too. This includes questions about the effects of various types of disorder, the role of impurities as scattering centers or traps, and the nature of the coupling which leads to energy loss through the excitation of other degrees of freedom of the crystal. To a large extent, vibrational energy is chemical energy and the flow of vibrational energy in molecules or molecular aggregates can control the rates and pathways of chemical reactions. An important problem which has emerged in recent years is to develop
simple mechanical intuitions to explain systematically the kinetic pathways for vibrational relaxation in isolated molecules, in molecular complexes, and in condensed phases.4” Molecular crystals are well-defined model systems in which to study this question. The study of vibron dynamics in molecular crystals is a relatively recent subject, and the tools used are primarily laser spectroscopies. In this review we mainly describe results which have been obtained by using coherent Raman spectroscopy to probe the dynamics of Raman-active bands in a number of simple systems. The paper is organized as follows: In section I1 we discuss the methodology of CARS in the time and frequency domains, including the important analysis of artifacts and considerations from crystal vibrational spectroscopy and crystal optics. Sections I11 and IV are devoted to exploring the various contributions to vibrational coherence decay in crystals, illustrated by the results of CARS measurements in several model systems. 11. Coherent Raman Scattering in Molecular Crystals A . The CARS Process. The Raman scattering process has proved an invaluable tool for studying the vibrational spectroscopy of molecular crystals. Ordinary spontaneous Raman spectroscopy has provided a great deal of the basic knowledge about vibrational states of solids.’ It has been known for many years that one can learn much about vibrational dynamics from line-shape analysis, and this has been exploited in a number of liquid and high-temperature solid-state However, the utility of spontaneous (4) E. L. Sibert 111, W. P. Reinhardt, and J. T. Hynes, J . Chem. Phys., 81, 1115 (1984).
( I ) G. W. Robinson, Annu. Rev. Phys. Chem., 21, 429 (1970).
(2) ‘Phonon Scattering in Condensed Matter”, H. J. Harris, Ed.,Plenum, New York, 1980. (3) “Spectroscopy and Excitation Dynamics of Condensed Molecular Systems”, V. M. Agronovich and R. M. Hochstrasser, Eds., North Holland, 1983.
0022-3654/85/2089-2240$01 S O / O
(5) T.A. Stephanson and S. A. Rice, J . Chem. Phys., 81, 1083 (1984). ( 6 ) A. Fendt, S. Fischer, and W. Kaiser, Chem. Phys. Lett., 82, 350 (1981). (7) W. Hayes and R. Loudon, ‘Scattering of Light by Crystals”, WileyInterscience, New York, 1978. (8) S. Bratos in ‘Vibrational Spectroscopy of Molecular Liquids and Solids”, S. Bratos and R. M. Pick, Eds., Plenum, New York, 1980.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2241
Feature Article AVAIL A B L E S P E C T R A L RES0 L U T IO N OF R A M A N SPECTROSCOPIES
I
I
I
I
1
I
CARS (Coherent Anti-Stokes Raman Scattering)
I
'CONVENTION AC *,,RAMAN FABRY PEROT
RAMAN
CARS
Figure 2. Schematic representation of the CARS process. In frequency domain CARS, w1 and wl'usually correspond to the same laser beam and the arrows refer to driving fields which overlap in time. w2 is scanned, and the intensity of the scattered light at w2 is measured. In time domain CARS, the w1 and w2 arrows represent time coincident pulses, and w,' refers to the time delayed probe pulse. The lower figure shows the wavevector matching condition which must be satisfied by the four waves.
others21-26have begun using this technique to learn about line shapes and relaxation times of vibrons in molecular crystals at liquid helium temperatures. The physical principle behind the CARS technique is illustrated in Figure 2. A pair of laser pulses incident on the sample are tuned so that the difference in their central frequencies matches a vibrational frequency in the solid. This vibration is driven by the stimulated Raman process, and modulates the polarizability of the crystal. This modulated polarizability interacts with the higher frequency incident field to produce an electric polarization oscillating at the anti-Stokes frequen~y.~' We may formalize this description starting with the fact that the signal in CARS is generated by the third-order polarization induced in the sample:28
In this expression, E, is the a t h component of the total incident electric field. There are two ways that the third-order response can be used to yield information about vibrational dynamics. In the time-resolved CARS technique the incident electric field is the sum of three laser pulses. The first two are time coincident and tuned so that their central frequencies and aZ match a vibrational resonance. This excites a coherently oscillating vibrational amplitude which modulates the crystal polarizability. The transient change in the polarizability can be written (9) P. N. Prasad, Mol. Cryst. Liq. Cryst., 58, 39 (1980). (IO) R. Quillon, P. Ranson, and S. Califano, Chem. Phys., 86, 115 (1984). (11) P. Esherick and A. Owyoung, "Advances in Infrared and Raman Spectrascopy", Vol. 9, R. J. H. Clark and R. E. Hester, Eds., Heyden and Son, London, 1982. (12) R. L. Fork, B. I. Greene, and C. V. Shank, Appl. Phys. Lett., 38,671 (198 1). (13) I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, and D. White, J . Chem. Phys., 71, 153 (1979). (14) I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, and D. White Chem. Phys. Lett., 71, 405 (1980). (15) R. M. Hochstrasser, G. R. Meredith, and H. P . Trommsdorff, J. Chem. Phys., 73, 1009 (1980). (16) P. L. Decola, R. M. Hochstrasser, and H. P. Trommsdorff, Chem. Phys. Lett., 72, 1 (1980). (17) F. Ho, W. S.Tsay, J. Trout, and R. M. Hochstrasser. Chem. Phvs. Lett., 83, 5 (1981). (18) F. Ho, W. S. Tsay, J. Trout, S. Velsko, and R. M. Hochstrasser, Chem. Phys. Lett., 83, 5 (1981). (19) S. Velsko, J. Trout, and R. M. Hochstrasser,J. Chem. Phys., 79,2114 (1983). (20) J. Trout, S. Velsko, R. Bozio, P. L. Decola, and R. M. Hochstrasser,
J . Chem. Phys., in press.
The response function consists of two parts: (3)
R,, arises from terms in x ( ~for ) which no resonance conditions (21) B. H. Hesp and D. A. Wiersma, Chem. Phys. Lett., 75,423 (1980). (22) K. Duppen, B. M. Hesp, and D. A. Wiersma, Chem. Phys. Lett., 79, 399 (1981). (23) D. D. Dlott, C. L. Schosser,and E. L. Chronister, Chem. Phys. Lett., 90, 386 (1982). (24) E. I. Chronister and D. D. Dlott, J. Chem. Phys., 79, 5286 (1983). ( 2 5 ) C. L. Schosser and D. D. Dlott, J. Chem. Phys., 80, 1369 (1984). (26) C. L. Schosser and D. D. Dlott, J . Chem. Phys., 80, 1394 (1984). (27) A. Laubereau and W. Kaiser, Rev. Mod. Phys., SO, 607 (1978). (28) P. W. Butcher, "Nonlinear Optical Phenomena", Engineering Bulletin 200, Ohio State University Press, Columbus, 1965.
2242 The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
Velsko and Hochstrasser
are satisfied. If the "other" resonances are far away, the nonresonant response becomes a 6 function.
R",(t) = 2TX"I 6(t)
(4)
It is straightforward to show that, under the conditions in which a single Raman resonance condition is satisfied, the resonant part of the response function reduces to the form: Rrss(t) = ( c u , , q y ( t ) )
+
(~,j3"uy(t))
(5)
This form is consistent with the ideas of linear response theory, and it is clear that to extract meaningful information about the response function R(t) the duration of the pulses must be short compared with the response time. At some time delay 7 after the excitation pulses have left the sample a probe pulse El' interacts with the excited sample. The field El'(t) beats with Acu(t) to produce a polarization oscillating with w, + fi3)(t,7) = Aa(t) EI'(t - T)
(6)
One may then measure the intensity of the signal field generated by p 3 )as a function of the time delay between El' and E I E 2 . (7)
In effect, the probe pulse samples the free induction decay of the vibrational coherence introduced by the pump pair.27 In practice, the light pulses for time domain CARS have often been generated by synchronously pumping dye lasers with a The pulse powers are bemode-locked ion or YAG laser.17-21*23 tween 100 W at repetition rates of 100 MHz and 100 kW at 500 Hz. In this case, time delays of up to 10 ns are conveniently obtained with optical delay lines. For measuring much longer coherence decay times electronically synchronized N2laser pumped dye lasers have been used.13 These have pulse powers of ca. 20 kW and repetition rates of 10 Hz. The second, complimentary method of probing vibrational dynamics is frequency domain CARS. This is the "ordinary" CARS which has been reviewed recently by Levinson and Song29 and other^.^^,^^ In the context of the above discussion, frequency domain CARS with two laser beams is described by letting El' = El and making both El and E2 long in time and spectrally narrow about central frequencies P, and 02. Then, if we denote the slowly varying envelopes of the fields by z,(r)and Z,(t): To the extent that R,(t) contains a 6 function (eq 4), the Fourier transform of the nonresonant response is the constant xnr.Aa(t) oscillates at the frequency P, - a2and has a duration equal to that of the laser pulses. The amplitude of Aa(t) is determined by the Fourier transform of R(t) at the frequency P,- a2. The signal field is produced by the beating of El(t) and Aa(t), and also has the duration of the light pulses. One measures the intensity of the field produced as a function of the detuning 0, - 02. (Usually, is fixed and a2is scanned.)
Thus, both the frequency-resolved CARS spectrum SFR and the time-resolved CARS decay profile S , are functionals of the same system response. In this paper, we will refer interchangeably to either the coherence decay time or the line width-these two quantities are, in fact, reciprocally related and thus carry the same information. It should be noted that eq 2-9 are derived rigorously in a straightforward fashion from eq 1 by using the approximation (29) M.D. Levinson and J. J. Song, "Coherent Raman Spectroscopy" in "Topics in Current Physics", Vol. 21, Feld and Letohkov, Eds., Springer. Verlag, Berlin, 1980. (30) J. Valentini, "Coherent Antistokes Raman Spectroscopy" in 'Spectroscopic Techniques", Vol. 4, Academic Press, New York, 1983. (3 I ) G. L. Eesley, 'Coherent Raman Spectroscopy", Pergamon Press, Oxford, 1981.
0 50 IO0 I50 TIME DELAY IN PICOSECONDS ( T ) Figure 3. Time-resolved CARS decay profile for the u, A, (991-cm-') transition in crystalline benzene of natural isotopic composition at 1.6 K. (a) is the measured signal and a best-fit curve obtained by using eq 11 and an exponential response function with 7 - 40 ps. (b) is an instrument -50
function obtained from liquid benzenes. (Note the log scale.) that a single Raman resonance condition is satisfied by the choice of fields. Spontaneous Raman scattering also measures the Fourier transform of R(t).
I,&)
= S_,oe'l'c, R,up,(t)t, dt
(10)
Here, {e,) are polarization components of the incident and scattered light. Because of the inelastic and incoherent nature of spontaneous Raman scattering only the resonant part of the third-order susceptibility is probed. Thus, with this exception, all three techniques give the same information about the material response. However, as we have mentioned, there may be vast experimental advantages to CARS. For example, etalon-narrowed N2or YAG pumped dye lasers provide a CARS spectral resolution of ca. 0.03 cm-1.20 Coherence decay times of 10 ns and longer have been measured in crystals of hydrogen at 4 K,13implying an equivalent frequency resolution of 3 X cm-'. In our own laboratory, the dynamic range of experiments at these resolutions is typically greater than three orders of m a g n i t ~ d e . High ~ ~ , ~dynamic ~ range is crucial if line shapes or decay profiles are to be analyzed accurately although in CARS this is achieved at the expense of a somewhat more complex dependence of the signal on the spectral or temporal properties of the laser fields. The interpretation of CARS data has two aspects. First, the response function must be extracted from the measured decay profile or line shape by taking into account possible artifacts and finite time or frequency resolution. Only then can one go on to the analysis of the contributions to the measured response and a comparison to theory. An important step, intermediate to these two is understanding which vibrational resonance is being probed and its relationship to the underlying vibrational band structure of the crystal. In the next two parts of this section we discuss aspects of fitting CARS data to response functions and some relevant points about crystal vibrational spectroscopy. In sections 111 and IV we turn to the analysis of contributions to the line width of Raman-active vibrations in very low-temperature molecular crystals. B. The Structure of CARS Signals. A CARS line shape or decay profile must be fit to a model response function to extract detailed material information from experiment. A successful fit must take into account the structure of the signal, i.e., distortions due to the finite spectral or temporal width of the laser pulses used in the experiment. Because of the nonlinear dependence of the signal on the fields, this is not always straightforward. In addition, proper account must be taken of the statistical properties of the laser pulses-their deviation from transform limited behavior. In Figure 3 we show a typical time domain CARS decay profile (a) (in this case, for the uI A, vibron of crystalline benzene at 1.6 K) and an instrument function (b) obtained in liquid benzene where the dephasing time is much shorter than the pulse width. For times much greater than the duration of the instrument
The Journal of Physical Chemistry, Vol. 89, No. 11, I985
Feature Article function the decay looks exponential, but the overall decay does not fit well to the convolution of a single-exponential decay with instrument function (b). A straightforward convolution of an exponential decay with the measured instrument function produces a curve which severely undershoots the decay profile around t = 0 and has its maximum shifted too far toward positive times. The distortion of the signal for times near T = 0 is termed a “coherence artifact” and is present in all time-resolved CARS data. It arises because the probe pulse E,’(?)(eq 6) can generate population when it is in time coincidence with E2 and the “pump” pulse El can probe this excitation, as well as vice versa. These two processes satisfy the same phasematching condition, regardless of the incident beam configuration, if the polarization of the E l and E,’ beams are the same. This effect was first described by Ippen and Shank in regard to a different experiment (anisotropic ab~orption).’~ The first quantitative formal analysis of this effect in time domain CARS was reported in ref 16. An important feature of this analysis is the observation that a considerable simplification of the structure of the formal expression which describes the CARS decay profile S T R ( t ) can result when the laser pulses are not transform limited. If the pulses were transform limited, one could derive E ( t ) directly from autocorrelation measurements and calculate S T R ( T ) explicitly through eq 2-7. However, unless great care is taken to narrow the bandwidth of a synchronously pumped dye laser, and in controlling the pumping power, pulses can deviate from transform limit by as much as a factor of three or four. In our system, the auto- and cross-correlation measurements and spectral widths of the dye laser pulses all conform well to a Gaussian noise burst r n ~ d e l . ~ ’ Under . ~ ~ this general assumption, the time domain CARS signal takes the form’*
STR(T) = S_:IR(.
- t?I2
dt’+
Q2(7)
(11)
in general Qland Q2 are third-order correlation functions of the three laser pulses and are asymmetric, resembling the crosscorrelation function of E , and E2 on the rising side and the autocorrelation of E , on the other. When the two-pulse cross- and autocorrelation functions are not too different
QIa Q2
(12)
Qlis essentially equal to the CARS instrument function that can be measured either by probing the instantaneous nonresonant background response far from a vibrational resonance or by resonant excitation of a vibration whose relaxation time is much shorter than the pulse duration. Figure 3 shows that a fit using eq 1 1 and 12 with R(t) = Aedr2 does fit the data well over the full time range. It has been incorrectly assumed in the past that the artifact Q2(t)arises solely from the nonresonant background contribution to x(’). This does contribute to the coherence artifact, but in benzene, for example, this contribution is at least 30 times smaller than the actual observed artifact. Q2(t)is generally a resonant contribution to the signal intrinsic to the experiment itself. Of course, if the resonance is weak, or XNR is particularly large as in the case of naphthalene, where there is a low-lying two-photon resonance, it may be the dominant contribution to Q2(t).23It is somewhat puzzling that the data of Wiersma on the 766-cm-, mode of naphthalene do not show clearly the coherence artifact while that of Dlott on the same mode and apparently under the same conditions does.23q21 The data of Wiersma on the lower energy librons of naphthalene also show no obvious difference between the decay profiles of the A, and B, levels even though the excitation conditions for the B, state preclude the presence of the coherence artifact.22 Other types of artifacts have been noted in experiments utilizing high-power laser pulses. These appear as oscillation^^^ or dis(32) E. P. Ippen and C. V. Shank in “Ultrashort Light Pulses”, S. L. Shapiro, Ed., Springer-Verlag, Berlin, 1977. (33) D. B. McDonald, J. L. Rossel, and G . R. Fleming, IEEE J . Quunt. Electron, QEt7, 1344 (1981). (34) D. P. Millar and A. H. Zewail, Chem. Phys., 72, 381 (1982).
-
n
2243
41
m
3
v
ti 0 a
0
111111111111
989 990 991 992 993 994 ( W , -W,)/hc (cm’l) Figure 4. Frequency domain CARS spectrum of v, A, at 1.6 K. These
data were fit to a Lorentzian response function and the line-width parameter so obtained (y = 0.067 crn-l) agrees within experimental error with the decay time obtained from the data of Figure 3. tortions22of the rising edge of the CARS signal. While these effects have not been definitively analyzed, it has been suggested in at least one case that stimulated Brillouin scattering may play a role.22 Finally, a distinction must be drawn between the time-resolved CARS experiment described above and the transient-stimulated Raman scattering experiments of, for example, Kaiser2’ and Harris.3s In those experiments, the Stokes beam is spontaneously generated, the w1 laser has a high depletion3s and the signal can develop a high degree of k selectivity. This so-called phase-selective experiment can distinguish between inhomogeneous and homogeneous broadening but has not yet been applied to low-temperature molecular crystals. As we shall see in section I11 the ability to unambiguously test hypotheses about the functional form of R ( t ) is of the utmost importance. Hence, it is necessary to have a thorough understanding of the factors contributing to STR(t). Lineshape distortions in frequency domain CARS arise because of the finite spectral width of the pulses. The spectral width of nitrogen laser pumped dye lasers with intracavity etalons is about 0.03 cm-I, while the pulse duration is about 5 ns. Thus, the dye laser pulses most likely contain substructure whose correlation time is at least ten times shorter than the pulse duration. The spectral bandshape is approximately Gaussian, and for lack of more precise information we have assumed a Gaussian noise burst model for these pulses too. With this hypothesis the CARS spectrum can be written:
Pi(@)= X,dT e‘”’(Ej(t) Ej(t
+
T))
The functions rjare the power spectra of the laser fields. Figure 4 shows the CARS line shapes for the same resonance as Figure 3, under the same experimental conditions. We have fit this data using a Lorentzian form for the response function: R ( w ) = (w
A
- w,)
+ iy
+ XNR
We have assumed that xNRis real. The laser power spectra were assumed to be Gaussian functims with widths of 0.03 cm-I. The best fit using eq 13 and 14 gave y = 0.067 cm-I. The parameter (35) S. M. George, A. L. Hannis, M. Beng, and C. B. Harris, J . Chem. Phys., 80, 83 (1984).
Velsko and Hochstrasser
2244 The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 y is related to the decay time
7
of Figure 3 by 1
TABLE I: Model Molecular Crystals
?=Zz
H2
molecules/ unit cell
crystal structure
1 4 4 2
hcp’ cubic orthorhombic monoclinic
symmetry
The numerical values we have quoted for y and T (Figure 3) satisfy this relationship to within experimental uncertainty. The effect of omitting the laser spectral width (I’,(o) a 6(w - ~ 3 ) ~in) eq 13 leaves the fit unchanged in the wings but causes a severe overshoot near the peak. Thus, when the laser spectral width is comparable with the width of the resonance, it is absolute necessary to use the convolved form (13) or other appropriate models to obtain an unambiguous fit across the whole frequency range. This is especially true in the region of the interference minimum, although this is not seen in Figure 4. A simpler approximation to (13) was suggested in ref 16, apparently on intuitive grounds. For Gaussian power spectra, this corresponds to taking only the second term in eq 13. By numerical simulation, we have found that this leads to very little qualitative difference in the line shape, although the relationship of the parameters in the fit to the laser spectral parameters is modified. A more detailed discussion of laser line-width effects in CARS can be found in ref 36-38. The excellent agreement between the results of the measurements in Figures 3 and 4 and similar measurements in naphthalene is important as it strengthens our confidence that no large systematic errors are present in these experiments-at least in the time and frequency regime where the two methods overlap. We have stated that the methods are complimentary in that time domain and frequency domain measurements with currently available laser systems are convenient in disparate time or frequency regimes. The two methods can also be complementary in the regions where they do overlap. Theories for R(t) or R(o, - 0 2 )use different asymptotic approximations. For example, the Markovian approximation of relaxation theories ignores short-time behavior.3g It has long been recognized that non-Markovian dynamical effects might be seen in the spectral wings of a line shape. Details of the asymptotic time dependence of the coherence decay are buried in the line center of the spectrum. C . CARS and Crystal Spectroscopy. The interactions among molecules in organic or van der Waals crystals are quite weak and, as a consequence, the crystal-induced splitting of vibrational levels both by reduction of site symmetry and by dynamical interactions between the molecules can be very small. In a crystal with n molecules per unit cell, each nondegenerate molecular vibration is split into n branches or subbands. The k = 0 state of each subband is the optically accessible Davydov component and transforms as an irreducible representation of the crystal factor group.’,@ Measurements of such splittings give valuable information on the intermolecular vibrational interactions and, thus, the vibrational dispersion relation. From the dispersion relation the structure of the entire band may be deduced. For Ramanactive modes factor group splittings are often only fractions of a wavenumber.20 Therefore, CARS spectroscopy, with its capability of high resolution at high dynamic range, promises to be a valuable tool for the vibrational spectroscopy of molecular crystals. In this section we will review some details about the correspondence between crystal orientation, laser polarization, and the identification of Davydov components, pointing out with specific examples the relationship of these ideas to CARS. As we shall see in section 111, knowledge of the band structure, and in particular the position of the optically excitable components within the band, is important for interpreting the line widths of those components. In this respect, the ideas presented here form the link between real experimental systems and the theoretical models which will be introduced in section 111.
L is a local field correction factor. Note that the fourth-rank tensor components of the third-order susceptibility for CARS are conand transform as the structed as symmetrized products of (al;k)) identity representation. Therefore, in the parametric coherent Raman scattering process to excite a single Raman active mode one chooses polarizations correspondin to the elements of x(j) which come from the product of the ai:) for that mode. In general, to excite a single Davydov component the polarizations of the incident fields must be chosen so that the material responds through Raman tensor components corresponding to a single irreducible representation of the crystal factor group. Under these conditions for an n-fold degenerate molecular vibration one can observe up to n-site split states in the vicinity of the resonance of interest. A table relating the symmetry species of Raman scattering tensors in the factor groups appropriate to benzene and naphthalene can be found in ref 19. One should note that the title of this table is somewhat misleading since the actual symmetry species of all the third-order susceptibility components are A,; the symmetry species in the table relate to the vibration of interest excitable by the interaction of light with the given tensor element. In biaxial crystals, angle-tuned phase matching for a single resonance is most easily accomplished with parallel polarized beams propagating in a principal plane and using the dispersion of the corresponding principal In this configuration, the problems of beam walk-off, etc. are minimized, but one cannot
(36) M. A. Yuratich, Mol. Phys., 38, 625 (1979). (37) R. E.Teets, Opt. Lett., 9, 226 (1984). (38) L. A. Rahn, R. L. Farrow, and R. P. Lucht, Opr. Lett., 9,223 (1984). (39) B. J. Berne and K. Pecora, “Dynamic Light Scattering”, Wiley, New York, 1976. (40) R. M. Hochstrasser, “Molecular Aspects of Symmetry”, Benjamin, New York, 1966. Chapter 10.
(41) (42) (43) (44) (45) (1968). (46)
LY-N~ benzene naphthalene a Orientationally
7“6 D2 h
C2h
disordered.
Table I gives the basic crystal structural parameters for four model crystals which have been studied by CARS, and to which the results in this review pertain. Much of the basic (low-resolution) vibrational spectroscopy of these systems has been worked out by using ordinary infrared and Raman We should mention that the exciton splittings for infrared-active vibrations are much larger than for Raman-active ones, and the spectroscopy (though not the dynamics) of several IR active bands of benzene and naphthalene are well u n d e r s t o ~ d . ~ ~ ~ ~ ~ A Raman-active mode GR is excited through polarizability derivatives
Under symmetry operations these derivatives transform like ( Q R ) and form a second rank tensor basis for the irreducible representations of the crystal factor group. The thud-order polarization, which generates the CARS signal, can be expressed for monochromatic fields in the laboratory frame by15
Greek letters refer to the laboratory axis system while (ijkl)refer to the crystal. The (4 are direction cosines. Under the conditions that the local field tensor is approximately isotropic, the third-order susceptibility can be related to the crystal Raman tensor elements by
Q
V. Soots,E. J. Allin, and H. L. Welsh, Can. J . Phys., 43, 1985 (1965). M. M. Thieky and D. Fabre, Mol. Phys. 32, 257 (1976). A. R. Gee and G.W.Robinson, J. Chem. Phys. 46, 4847 (1967). D. M. Hanson and A. R. Gee, J. Chem. Phys., 51, 5052 (1969). E. R. Bernstein and G.W. Robinson, J . Chem. Phys., 49, 4962 E. R. Bernstein, J . Chem. Phys., 50, 4842 (1969).
The Journal of Physical Chemistry, Vol. 89, No. 11, I985 2245
Feature Article I
1
5.0
5
3.0
3
;; 2.0 W
-50
0 50 IO0 I50 TIME DELAY IN PICOSECONDS (TI Figure 5. Time domain CARS profile showing coherent beating between the A, and B2, Y, vibrons. Insert shows the orientation of the benzene crystal and laser polarizations used in this experiment.
generally probe all possible vibrations. In addition, this configuration may not drive the strongest susceptibility component for a given vibration. Mixed polarization conditions are generally useful only if beam walk-off is not too large an effect, compared with the size of the beam overlap volume. The most accurate experimental method of obtaining the relative positions of the Davydov components is to measure the splitting between the transitions observed simultaneously in a single spectrum. To observe two or more Davydov components in the same spectrum, the incident electric fields must have polarization components which drive Raman tensors for more than one irreducible representation of the crystal factor group. This often requires the simultaneous excitation of diagonal and off-diagonal Raman tensors and, hence, an effective mixed polarization condition. In the time domain, this is equivalent to the situation in which one simultaneously excites and probes the coherence decay of two or more Davydov components (or even more than one vibrational level, if the exciting pulse is short enough), and forms the basis for Raman beating spectroscopy which we have recently applied to the benzene 991-cm-' mode.lg (Raman beating has been seen in several other e ~ p e r i m e n t s , 2 ~but * ~apparently ~*~~ arises from other sources.) Because of the optical anisotropy of the (biaxial) crystal, there are strict requirements on crystal orientation to allow phase matching of both signals when a single polarization condition is used for the laser beams. In general, if one chooses a configuration in which the CARS beams propagate in a direction perpendicular to a principal axis of the dielectric tensor with their polarizations lying in the plane perpendicular to that axis, one can have projections of polarization along two principal axes. Because the light beams travel as single "extraordinary" rays in this configuration, simultaneous phase matching is possible by angle tuning using the dispersion of the "extraordinary index" for polarization vectors in the chosen principal plane. In other configurations, birefringence will influence phase matching of the signal arising from the mixed polarization tensor c ~ m p o n e n t . ' ~ Figure 5 shows the time-resolved CARS signal from the 991cm-I band of benzene for a mixed polarization condition (6- 2) in which both the Raman tensors transforming as A, and B2 contribute to the coherent polarization. Thus, both the A, and the B2, Davydov components are excited and the third-order polarization (eq 6) shows the coherent beating of these two levels. The level separation was determined to be 0.64 cm-'.19 By a combination of ordinary Raman data and the CARS data of Figure 5 we have deduced the density of states of the v 1 band using the dispersion re1ati0n.l~ Figure 6 shows this, along with the positions of the Davydov components. We have also observed factor group splittings of several other modes using mixed polarization frequency domain CARS, including a pair of lines from the 1603-cm-' mode separated by 0.3 cm-1.20 Except in cases where extremely high resolution is re(47) K.Duppen, D. Weitekamp, and D. A. Wiersma, J . Chem. Phys., 79, 5835 (1983).
WAVENUMBER SHIFT FROM BAND MEAN
Figure 6. Density of vibron states calculated for the Y, 991-cm-' band of benzene, with the positions of the k = 0 level of each of the four branches. Note that A, lies at the bottom edge of the band.
quired, it is certainly more convenient to extract the spectrum of factor group components by spontaneous Raman scattering since the experimental orientation and polarization requirements are much simpler to achieve. Nonetheless, we believe that CARS will prove to be a valuable tool for elucidating vibrational interactions in molecular crystals. Finally, it is to be noted that orientational uncertainty can be a source of systematic error in line-shape analysis if unresolved vibrational transitions are present. For example, we have seen an apparent broadening of about 0.01 cm-l (Le., -20%) of a factor group component of vg, when the crystal orientation is changed.20 Small distortions of the line shape are to be expected if there are small deviations from perfect orientation. This result again dramatizes the importance of knowing the locations of the k = 0 states before making dynamical interpretations of the line shapes. 111. Coherence Decay in Molecular Crystals
A . Overview. The study of vibrational line widths in lowtemperature molecular crystals by high-resolution CARS is only a few years old, but a clear picture of certain aspects of coherence decay in these systems is beginning to emerge. As the temperature of a crystal is lowered, vibrational line widths decrease until (around 10 K) it becomes close to a temperature-independent limit.25*48The residual ( T = 0) line width is due to one or more of the following causes: (a) dephasing due to strain or defect scattering; (b) dephasing due to impurity scattering; (c) energy transfer to impurities; (d) energy transfer to other modes of the host material. From a time domain point of view, the first two processes represent mechanisms by which initially well-defined phase relations among the excited oscillators in the wavepacket are randomized. Dephasing can be pictured as the evolution of a relatively smooth Fourier-limited initial excitation wavepacket into a "noise burst" structure characterized by a correlation length inversely proportional to the width of the distribution of k states into which the wavepacket has evolved. The last two items represent mechanisms by which vibrational amplitude "drains away" evenly at all excited states of the crystal, preserving all initially present phase relationships. This kind of decay is accompanied by the spontaneous emission of phonons and corresponds to a loss of vibrational population from the initially excited mode. Hence, (c) and (d) are denoted "T1"processes by analogy with the terminology used in magnetic resonance. If one examines the line shape of a particular vibrational transition of one of the species (A) of a binary mixed crystal, as a function of the concentration of the other species (B), it will vary in a complex and subtle way. In the extreme case of a pure A crystal, from which all chemical and isotopic impurities have been removed, only strain-induced dephasing and lifetime broadening due to spontaneous emission of phonons contribute to the vibrational line width. In section IIIB, we will present experimental and theoretical evidence which indicates that vibron line shapes are "motionally narrowed" in neat c r y ~ t a l s . ' ~Thus, J~ in simple solids such as a - N 2 the Raman line widths of funda(48) L.A. Hess and P. S . Prasad, J . Chem. Phys., 78,626 (1983).
2246 The Journal of Physical Chemistry, Vol. 89, No. 11, 1985
’1
!
( a ) Y , PURE
A
6
(b)
x 51
:50%
Velsko and Hochstrasser
MIXED
( c ) u I :3 % MIXED
0 IO 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 TIME ( n s )
989
9 9 0 991
992 993 9 9 4 (u~-u h c~ (cm-1) )/
Figure 7. Dependence of the benzene Y, A, line width on concentration of perdeuteriobenzene. (a) Isotopically pure crystal with fit to a Lorentzian line shape, y = 0.043 cm-l. For clarity, a small satellite peak corresponding to the Bzs Davydov component has been omitted as indicated by the dashed line. (b) 50% C6&/C6D6. The solid line represents the best fit to the measured points obtained by using a convolution of Gaussian and Lorentzian line shapes. For this fit, y = 0.13 cm-’ and u = 0.25 cm-l. (c) Spectrum of dilute (3%)protiobenzene in perdeuterio host. The line shape is fit to a convolution with y = 0.073 cm-I and u = 0.133 cm-I. The high-energy side of the peak displays the interference with the nonresonant background of the host which is characteristic of
CARS spectra of dilute species. mental modes are far narrower than the width of the distribution of site energies caused by strain disorder. In more complex crystals of polyatomic molecules like benzene or naphthalene this effect implies that lifetime broadening can dominate the observed line width. Therefore, in isotopically pure molecular crystals a t low temperatures, line widths can be used to test hypotheses concerning mode dependence of population decay rates. As dilute impurities B are added (or for neat crystals with natural abundances of isotopic impurities) line-broadening mechanisms (b) and (c) become operative. Simple theoretical models indicate that dephasing due to scattering is a slow process, except possibly in the “resonant regime” where the impurity levels lie close to the host band.49 In organic crystals trapping appears to dominate the impurity influence on the line shape for certain mode^.'^,^^ Energy transfer in this regime is “coherent” in the sense that the trap degrees of freedom are driven by the delocalized collective oscillations of the host. In the intermediate concentration regime (C, C,) a number of processes become interwoven: Impurity scattering may make an appreciable contribution to dephasing. Separation of A molecules undermines the motional narrowing effect, allowing strains to contribute an inhomogeneous character to the line width. Population decay also changes character, becoming an ”incoherent” process, as will be discussed in section IV. All of these effects are not necessarily separable and additive. Finally, for dilute A molecules in B host, the line shape takes on a form typical of impurity spectra.50 If the guest level is sufficiently far from nearby host bands, the line shape will have a strong inhomogeneous component which does reflect the distribution of site energies due to crystal strain fields. The homogeneous component of the line reflects population decay, both to the surrounding host material and to lower modes of the guest molecules. If these two processes are approximately independent the line shape will be a convolution of an inhomogeneous
-
(49) S. Velsko and R. M. Hochstrasser, J . Chem. Phys., in press. (50) A. S. Barker, Jr., and A. J. Sievers, Reu. Mod. Phys., 47, 52 (1975).
Figure 8. Coherence decay of the A vibron of a - N 2 at 1.6 K. The solid line is a fit to the function Ad‘/‘)’’’with T = 17 ns. The dashed curves indicate the Fourier transform of the spectrum predicted by approximate frequency domain theories (after ref 13).
(Gaussian) and homogeneous (Lorentzian) functions. As an example, Figure 7 shows the line shape of the v, A, mode in crystalline benzene for a neat C6H6crystal, for a 50% mixed C&/C$6 Crystal, and for dilute (3%) C6H6 in C6D6 host. Note both the nonmonotonic dependence of overall line width on concentration and the qualitative change in the line shapes. One expects different modes to exhibit different behavior depending on where the k = 0 state lies within the band, and on how close it lies to the levels of the isotopic diluent. Dlott has studied the concentration dependence of vibrational coherence decay times in mixtures of protio- and perdeuterionaphthalene, and has observed a several distinct behaviors for different modes.24 It is clear that to interpret these dependences it is necessary to know in detail the vibrational levels of the host and guest molecules and their band structures, as well as the intrinsic strain-induced site energy distribution. Vibrational energy relaxation and energy transfer to impurities are the most “chemically” interesting processes insofar as understanding these modes of decay sheds light on broader questions about vibrational relaxation in molecular systems. There are now a few cases where coherence decay can be unambiguously assigned to these mechanisms. However, the arguments used to eliminate pure dephasing are indirect and caution must be exercised in applying them. In this respect an important experimental challenge is to make systematic studies of vibrational relaxation in crystals using direct T, measurements via transient spontaneous anti-Stokes scattering or its variant^.^' Even when it can be established that population relaxation dominates the line width one must distinguish between (c) and (d) to make comparisons with theoretical models. We have discovered that there is often a surprising sensitivity of the line width to small amounts of naturally occurring isotopic impurities, and there are cases where the origin of these effects is far from clear.” Of course, from the point of view of exciton transport theory, dephasing processes are of interest in themselves, because of their potential for affecting the dynamics of population decay processes, especially trapping. From the discussion above, it should be evident that the dependences of dephasing and population relaxation on impurity concentration are necessarily complex and interrelated. While line-shape studies cannot always distinguish between dephasing and population decay it is convenient to discuss these issues separately. Therefore, in the rest of this section we will consider in turn the effects of strain inhomogeneity (weak disorder) and the presence of impurities on vibron dephasing. In section IV we will review ideas about energy decay in mixed crystals and the evidence gathered about mode selectivity in the self-relaxation of pure crystals.
The Journal of Physical Chemistry, Vol. 89, No. 11, 1985 2247
Feature Article
B. Weakly Disordered Crystals. Figure 8 shows the coherence decay of the 2328-cm-l vibron of a-NZat 4.2 K. This experiment followed the protocol of eq 2-7.14 The decay did not depend on temperature (between 1.3 and 4.2 K), the excitation intensity, or the addition of up to 10% 15N2impurities. The decay was sensitive to the mode of crystal preparation and these observations imply that scattering due to the strains and defects of the Nz crystal is the source of the low-temperature coherence loss. The exact nature of the strain disorder in the nitrogen crystal is not understood. Abram and H o c h ~ t r a s s e r ’ ~have . ~ ~suggested that the strain field arises from extended dislocations produced during the freezing process. The spectrum of such disorder must be characterized by both a strength and a correlation length. Simple theoretical models have been proposed to account for dephasing due to this kind of disorder. Klaftner and Jortners2 derived an approximate line-shape theory and applied it to the absorption spectrum of triplet excitons. Because of its general nature, this theory ought to be applicable to vibrons as well. Abram and Hochstrasser have treated directly the time evolution of ~ o h e r e n c e .Both ~ ~ of these theories use a simple tight binding Hamiltonian to which a site diagonal perturbation is added to describe the effect of the crystal strain field on the site energies. That is H=Ho+ V where
and
The pure crystal exciton (vibron) band is characterized by a width W. The site energy perturbations are taken to be Gaussian random variables with a correlation length equal to the lattice spacing, and mean squared value u: ( A i A j ) = dijuZ
(20)
These theories predict that (1) when u > W the band is inhomogeneously broadened, (2) when u
