J. Phys. Chem. 1992, 96, 9660-9666
9660
motion involves this type of conformational adjustment. The high-frequency modulation of the electric field gradient by these librations induces relaxation through a Raman-type indirect process. At temperatures above 200 K, the modulation of the EFG by the macrocyclic rotation/conformational adjustment dominates and 23Na. the TIrelaxation of both Acknowledgment. This research was supported by NSF Grants DMR 87-14751, DMR 90-17292, INT. 86-19636, and B.P. (VRU). We would also like to express our gratitude to the Michigan State University Center for Fundamental Materials Research for some of the instrumentation used in this work. We thank Professor James Yesinowski for helpful discussion.
References and Notes (1) For reviews and references see: Dye, J. L.; Jackson, J. E.; Cauliez, P. In Proceedings of the Fifth International Kyoto Conference on New Aspects of Organic Chemistry, in press. Dye, J. L. Science 1990, 247, 663. Dye, J. L.; DeBacker, M. G.Annu. Rev. Phys. Chem. 1987,38,271. Dye, J. L. Prog. Inorg. Chem. 1984,32, 327. (2) Doeuff, S.; Tsai, K.-L.; Dye, J. L. Inorg. Chem. 1991, 30, 849. (3) Buchanan, G. W.; Kirby, R. A. Tetrahedron Lett. 1987, 28, 4783. (4) Buchanan, G.W.; Morat, C.; Ratcliffe, C. 1.; Ripmeester, J. A. J . Chem. SOC.,Chem. Commun. 1989, 1306. (5) Ratcliffe, C. 1.; Ripmeester, J. A.; Buchanan, G. W.; Denike, J. K. J . Am. Chem. SOC.1992, 114, 3294. (6) IUPAC name: 18-crown-6; 1,4,7,10,13,16-hexaoxacyclooctadecane, abbreviation 18C6. (7) Fyfe, C. A. Solid State NMR for Chemists; C.F.C. Press: Ontario, 1983.
(8) IUPAC name: 15-crown-5; 1,4,7,10,13-pentaoxacyclopentadecane, abbreviation 15C5. (9) Kim, J. Ph.D. Dissertation, Michigan State University, 1989. (10) Dye, J. L. J . Phys. Chem. 1984,88, 3842. (1 1) McMills, L. E. Ph.D. Dissertation, Michigan State University, 1989. (12) Dawes, S.B.; Ward, D. L.; Fussa-Rydel, 0.;Huang, R. H.; Dye, J. L. Inorg. Chem. 1989, 28, 2132. (13) Huang, R.; Dye, J. L. Unpublished results. (14) Ellaboudy, A.; Tinkham, M. L.; VanEck, B.; Dye, J. L.; Smith, P. B. J . Phys. Chem. 1984,88, 3852. (15) Ellaboudy, A.; Dye, J. L. J . Magn. Reson. 1986, 66, 491. (16) Dye, J. L.; Andrews, C. W.; Ceraso, J. M. J . Phys. Chem. 1975,79, 3076. (17) Edwards, P. P.; Ellaboudy, A.; Holton, D. M.; Pyper, N. C. Annu. Rep. NMR Spectrosc. 1988, 20, 3 15. (18) Dawes, S. B.; Ellaboudy, A. S.; Dye, J. L. J . Am. Chem. Soc. 1987, 109, 3508. (19) Ellaboudy, A.; Dye, J. L. J . Am. Chem. SOC.1983, 105, 6490. (20) Abragam, A. Principles of Nuclear Magnetism; Oxford University Press: London, 1986. (21) Waugh, J. S.; Fedin, E. I. Sou. Phys. Solid State 1963, 4, 1633. (22) Hubbard, P. S. J. Chem. Phys. 1970,53, 985. (23) Bobel, D.; Muller-Warmuth, W.; Olyschlager, H. J . Magn. Reson. 1979, 36, 371. (24) Woessner, D. E. J. Chem. Phys. 1962,37, 647. (25) Farrar, T. C.; Baker, E. D. Pulse and Fourier Transform NMR; Academic Press: New York, 1971; Chapter 4. (26) Cohen, M. H.; Reif, F. Solid State Phys. 1957, 5, 321. (27) Chihara, H.; Nakamura, N. Adv. Nucl. Quadrupole Reson. 1980,4. (28) Gutowsky, H . S.; Pake, G. E. J . Chem. Phys. 1950, 18, 162. (29) Andrew, E. R.; Eada, R. G.Proc. R. Soc. (London)1953, A216,405. (30) Das, T. P. J . Chem. Phys. 1957, 27, 763. (31) IUPAC (1990) recommends the use of 'natride" rather than "sodide". The latter term is used here because of a long-standing precedent in our work.
A Picosecond Raman Study of Vibrational Relaxatlon in Naphthalene at High Pressure and Low Temperature Robert A. Crowell and Eric L.Chronister* Department of Chemistry, University of California, Riverside, California 92521 (Received: July 9, 1992)
Pressure-dependent vibrational dephasing in crystalline naphthalene is observed by picosecond coherent Raman measurements in a diamond anvil cell at low temperature. The results are analyzed in terms of both pressure-induced changes in the phonon density of states and pressure-induced anharmonicity in the intermolecular potential. We observe that both of these mechanisms can be the origin of pressure-induced vibrational relaxation. Furthermore, the relative importance of anharmonic versus density of states effects is found to be vibrational mode specific. When pressure-induced relaxation is controlled by anharmonic effects, we discuss the use of such measurements as a probe of the intermolecular potential.
Introduction High-pressure spectroscopic studies have proven to be a valuable tool for investigating intermolecular interactions in condensed UPressure-tuning"of electronic and vibrational energy levels has proven to be a unique tool for determining electronic structure and intermolecular interactions, respectively? Because molecular crystals are relatively compressible, modest pressures (i.e., 110 kbar) can strongly affect molecular interactions and lead to signiiicant increases in the frequencies of the intermolecular vibrational modesas Since the effect of pressure on the frequency of internal modes is much weaker, pressure is also an effective way to differentiate bonding strengths in solids.6 In this study, we utilize time-resolved coherent Raman spectroscopic measurements at high pressure to probe the intermolecularinteractions that control solid-state vibrational relaxation. Raman spectroscopy under high-pressure conditions can be used to probe intermolecular interactions in a variety of molecular solids.' In a high-pressure molecular solid, the increased intermolecular interactions can increase the often hard to resolve Davydov splitting of the internal modes, as well as cause spectral expansion of the lattice mode^.^,^ In addition, high-pressure
Raman line-shape studies have been a valuable probe of the origin of the spectral broadening in liquids.'*I2 However, there have been very few pressure-dependent measurements of vibrational relaxation in molecular solids. Recent time-resolved coherent Raman experiments have measured pressure-induced vibrational dephasing in solid benzenei3J4and in liquid nitromethanel5 at room temperature. However, such measurements are complicated by thermally induced dephasing and relaxation processes that obscure the origin of these effects.I3J6In a pure strain-free molecular crystal at low temperature, the only dephasing mechanism is spontaneous vibrational relaxation. Furthermore, the number of high-pressure, low-temperature studies of vibrational dynamics has been limited due to the narrow vibrational Raman linewidths at low temperature (Le., $0.1 C ~ - ~ ) . I ' - * ~ We have combined time-resolved coherent spectroscopy and high-pressure techniques to investigate solid-state vibrational dynamics. Although phenomenologicalsolid-state intermolecular potential energy functions can be used to model static observable quantities such as crystal structure and dispersion curves,zoa quantitative understanding of solid-state dynamics is still missing.
0022-365419212096-9660%03.00/00 1992 American Chemical Society
Vibrational Relaxation in Naphthalene
The Journal of Physical Chemistry, Vol. 96, NO. 24, 1992 9661
In this study, pressure-dependent measurements of solid-state vibrational relaxation rates are used as a unique probe of the anharmonic interactions that control solid-state relaxation. Furthermore, “pressure-tuned dynamics” is used to determine the relative importance of density of states and anharmonic coupling effects. We present high-pressure picosecond Coherent anti-Stokes Raman scattering (psCARS) results for crystalline naphthalene at low temperature. Although the information obtained by the psCARS technique is formally equivalent to the study of spontaneous Raman line ~hapes,2’-~~ the psCARS technique has distinct advantages for high-resolution studies on small sample volumes. These experiments have been performed on strain-free crystals grown at high pressure, since the narrow Raman lines at low temperature are particularly sensitive to sample inhomogeneities and crystal strain. Pressure-dependent infrared,25Raman,26and neutron diffract i ~ studies n ~ ~ of pressure-induced shifts in the frequencies of the intramolecular and lattice phonon modes of crystallinenaphthalene provide the information needed to calculate pressure-induced changes in the vibrational density of states (DOS).In addition, previous time-resolved Raman measurements on crystalline naphthalene at low temperature and ambient p r e s ~ u r e ~ ’ have ,~~,*~ identified several long-lived vibrational modes with lifetimes exceeding 50 ps. This study focuses upon the three longest-lived Raman active modes of crystalline naphthalene, v5 (1385 cm-l), us (766 cm-I), and u, (51 1 cm-l) with lifetimes of 92,62, and 140 ps, r e s p e c t i ~ e l y . ~The ~ . ~corresponding ~*~~ low-temperatureambient pressure line widths for these vibron modes range from 0.038 to 0.085 cm-I. The narrow line widths and corresponding long dephasing times for these Raman modes provides a wide dynamic range with which to investigate pressure-induced changes in vibrational dephasing.
Experimental Section The picosecond CARS experiments were performed using two cavity-dumped dye lasers synchronously pumped by the second harmonic of a mode-locked, Q-switched, Nd3+:YAGlaser, which has been described previ0us1y.l~~~~ The tunable dye laser was cavity dumped at a rate of 800 Hz, producing pulses with energies of 5-10 pJ, pulse widths of 30-40 ps, and a bandwidth of 1 cm-’. The excitation and probe pulses were all focused through a 20-cm lens into a miniature Merrill-Bassett diamond anvil celPo in thermal contact with the cold tip of a closed-cycle helium refrigerator (9-300 K) or immersed in pumped liquid helium (1.1 K). The CARS signal was dispersed with a monochromator, detected with a cooled photomultiplier, and subsequently signal averaged and digitized. The psCARS technique involves the coherent excitation of a Raman-active vibration by a pair of laser pulses whose frequency separation corresponds to the vibrational frequency. Following this excitation a third laser pulse of variable delay is used to induce coherent anti8tokes emission. The pump pulse, the probe pulse, and the detected anti-Stokes emission were all polarized parallel to each other. Signal due to the nonresonant thud-order nonlinear susceptibility of the diamonds was eliminated by minimizing the interaction length of the three pulses in the sample, and care was taken not to exceed the laser damage threshold for the diamonds of 3 G W / C ~ ~The . ~tightly ~ , ~ focused ~ lasers, the nearly linear optical geometry, and the directionally coherent emission are well matched to the small aperture size and restricted geometry of the diamond anvil cell. A miniature Merill-Bassett diamond anvil cel130was utilized for growing naphthalene crystals over a 0-12.3 kbar pressure range. Naphthalene was purchased from Aldrich Chemical Co. and purified by zone refining (-200 passes). Strain-free single crystals at high pressure were obtained by allowing two or three separate naphthalene crystals to grow from the melt simultaneously within the pressurized diamond anvil cell.”-29 Pure single crystals were grown on the order of 200 pm in dimension. Using this protocol, crystal strain upon cooling of the samples can be relieved by relative motion of the different crystals.33 To facilitate the growth of strain-free crystals, we annealed them near the melting
point at different pressures. Of the naphthalene crystals produced in this way, only samples with sharp phonon lines in their Raman spectra34 were further investigated with psCARS. Pressure calibration was obtained by the frequency shift of the Rl Ruby fluorescence lineP5which has been well established over a pressure range of tens of kilobars and down to liquid helium temperaPressureinduced phonon frequency shifts are also used as a secondary measure of the sample pressure. In addition, the P vs T equilibrium phase diagram for neat samples has been used to obtain a variety of crystallization pressures at different temperatures and yields a third measure of the sample pressure.
Pressure-Induced Vibrational Dephasing in Solids In this section we outline the framework in which we describe the origin of the effect of pressure on vibrational relaxation. Specifically, pressure-dependent vibrational dephasing is evaluated in terms of density of states and anharmonic effects. In the following sections we also discuss the importance of pressure-ind u d vibrational dephasing as a unique tool with which to probe the intermolecular interactions that control vibrational dephasing. The interactions between neutral molecules in condensed phases can be classified as repulsive, dispersive, polarization, and elect r o ~ t a t i c . Since ~ ~ the repulsive and dispersive interactions are difficult to express analytically and ab initio quantum mechanical calculations for molecular crystals are complex, many lattice dynamics calculations on van der Waals crystals use phenomenological atom-atom potential energy functions to represent the repulsive and dispersive interactions. In addition, the intermolecular potential for molecular crystals is often calculated as a sum of intermolecular atomatom interactions. The Buckingham “6-exp” potential, 9, = A, exp(-B,lri,) - Ci,rij4,is a widely used phenomenological form for the van der Waals atom-atom interactions in aromatic hydrocarbons such as naphthalene.39v40For a three-dimensional lattice of particles, the potential energy per particle is the sum of all atomatom interactions, a = 1 / 2 ~ i z j ~ i , , Such potential energy functions have been useful for calculating structural and spectral changes for molecular crystals under pre~sure.~’ Although simple two-body potential energy functions of this form have been successfully used for calculating the pressure-induced structural and spectral changes for molecular crystals of aromatic hydrocarbons such as naphthalene:’ they often fail for crystals of small molecules or molecules with strong dip o l e ~ .Due ~ ~ to the harmonic nature of molecular crystals, a perturbative anharmonic analysis has proven useful for calculating vibrational lifetimes in molecular crystals utilizing phenomenological potential energy functions.38 Furthermore, since the intermolecular potential is rather harmonic, a series expansion of the potential energy function in terms of normal coordinates is used to determine the most important anharmonic decay processes.
The lowest order anharmonic term in the intermolecular potential energy function is the cubic term which couples three normal coordinates qi, qj, and qk. This coupling provides a mechanism by which an excited vibrational mode qi fissions into a pair of excited vibrations qj and qk. At low temperature the resulting “golden rule” relaxation rate for the mode qi is given by38,43-46
where ( @P(3)ijk) is the anharmonic matrix element which couples the three vibrational modes, and where the sum of b(wi-w,-ok). b(k,-k,-kk) determines the density of accepting vibrational states
9662 The Journal of Physical Chemistry, Vol. 96, No. 24, 1992
Crowell and Chronister
that conserve both energy and crystal momentum. Thus, pressure-induced changes in the relaxation rate are directly related to the anharmonicity in the intermolecular potential and the density of vibrational states. The cubic coupling matrix elements - can be characterized by an average coupling strength, I(@(3))i12, between the initial state and all two-phonon combination states into which this state can relax by a cubic fission p r o c e s ~ . ~In~ addition, *~~ the optically excited state has nearly zero crystal momentum (i.e. ki 0), resulting in
-
which can be written in the compact form
0
where p2&) is the sum of the number of two-phonon combination states with a total energy hi.However, since cubic relaxation typically occurs by fission of an excited vibration into a lower energy intramolecularvibration (v), and a low-frequency phonon mode (p), pzp(wi) represents a two-phonon density of states (2PDos)consisting of the sum of all vibration phonon combination states with a combined energy of h q z 8
+
250
500 750 FREQUENCYjcm'
ldoo
1250
1%
Figure 1. Intramolecular vibrons (the discrete lines between 250 and 1500 cm-I), the lattice phonon density of states (the curve below 250 cm-I), and the two-phonon density of states (ZPDOS), ~ ~ ~ (calculated w ) , using eq 6, arc shown for crystalline naphthalene at atmospheric prcssure. The molecular vibrons are given a bandwidth of 1 cm-I, and we have reduced the height of these peaks by a factor of 8 for display purposes. The three intramolecular vibrations investigated with psCARS are u5 (1385 cm-I), us (766 cm-I), and ug (511 cm-I), as indicated by bolder lines.
2PDOS distribution, as seen in Figure 1. Thus, the relatively small 2PDOS for relaxation for these vibrons appears to be responsible for the rather long lifetimes of these modesOz8Furthermore, DOS where the sum is over all lower frequency vibrations and ~~(q-0,) effects can be expected to dominate pressure-induced vibrational relaxation for such modes since a pressure-induced change will is the single phonon DOS at the difference in energy between the most likely result in an increase in the 2PDOS at these frequencies. initially excited vibration wiand a lower frequency vibration wV. 2. PressureInduced Anharmonic Coupling. Using eqs 2 and Furthermore, the sum over crystal momentum states, k, for the 3, cubic vibrational relaxation rates can be calculated given a phonons is implicit in the phonon DOS since the intramolecular knowledge of the potential energy function and the normal covibrons are nearly dispersionless, which enables the momentum ordinates for the vibrational modes of the ~rystal.'~However, = -knbrm to be satisfied for every decay conservation relation anharmonic lattice dynamics calculations of vibrational relaxation pathway from one vi ron branch to another. rates in molecular crystals have met with only moderate success. Vibrational dephasing in a perfect crystal at low temperature The extension of theoretical and experimental studies to presis determined strictly by vibrational relaxation. Thus, a pressureinduced dynamics in molecular crystals is motivated by the sure-induced change in the low-temperature homogeneous deneed for more parameters with which to investigate relaxation phasing rate (or line width) of a strain-free single crystal can be processes.14 This connection between anharmonicity in the indue to either anharmonic effects or due to changes in the vibratermolecular potential and the rate of vibrational relaxation is a tional DOS, according to eq 5. Furthermore, if pressure-induced motivation for our efforts to experimentally correlate vibrational vibrational relaxation is due to increased anharmonicity in the dynamics with the solid-state potential energy function. The intermolecular potential, then such results, in conjunction with magnitude of pressure-induced anharmonicity for the v I mode of eq 2, provide a unique probe with which to evaluate phenomebenzene has recently been calculated using an intermolecular nological potential energy functions. potential derived from the atom-atom potentials of ref 48. The 1. PreawffLndUced chrnges in vibrational Dedties of states. results of this calculation indicate that the square of the average A determination of pressure-induced changes in the 2PDOS is cubic coupling matrix element increases by only a factor of 2 over straightforward given the available pressure-dependent spectral the pressure range of 0-40 kbar.14 However, the calculated viand volume compression data. The competition between spectral brational lifetimes were 2-3 times longer than the measured room expansion of the lattice phonons and volume compression of the temperature vibrational dephasing times over most of this pressure crystal yields an average density of phonon states per volume at range.13J4 This discrepancy is most likely due to thermal dephasing a given frequency that is relatively constant with pressure. and higher order relaxation effects that can be dominant at these However, the pressure-induced phonon spectral expansion creates higher temperatures, but which were not accounted for in the higher frequency phonon modes that can open new decay pathways c a l c ~ l a t i o n . In ~ ~ addition, at higher temperatures, thermally even though the average phonon DOS is relatively pressure inactivated relaxation processes, such as phonon absorption and dependent. Furthermore, the phonon DOS has considerable stimulated phonon emission, require that the pressure-dependent structure and pressure-induced shifts of this distribution can density of ZPDOS be thermally weighted, further complicating modulate the 2PDOS at a given vibrational frequency. the comparison of theory with experiment. The current lack of The pressure effect on the 2PDOS in threefold; pressure can experimental data has limited the progress in this area and we modulate pZp(o,)due to phonon frequency shifts; it can drahope that density-dependent data on several vibrons in low-temmatically increase pZp(wi)at particular frequencies by opening perature naphthalene will spawn more detailed calculations of new relaxation channels; and it can cause an overall increase due pressure-dependent vibrational relaxation. Nevertheless, there to the volume compression of the solid. The effect of pressure are several general conclusions that can be made concerning on the crystal volume and geometry is straightforward to obpressureinduced anharmonicity in the intermolecular For an applied pressure P, the equilibrium nearesta crystal grown at high pressure will have smaller nearest-neighbor neighbor separation is obtained from the intermolecular potential distances, resulting in increased vibrational anharmonicity due by solving P = -(a@/av) . A calculation of the 2PDOS at to interactions with the repulsive van der Waals potential. ambient pressure yields a wi";le range of values for the naphthalene Qualitatively, a monotonic increase in vibrational relaxation with vibrons, as shown in Figure 1. It is interesting that the frequencies applied pressure is to be expected since the anharmonic repulsive of two of the three longest-lived vibrons of naphthalene, v5 (1 385 part of the potential will tend to dominate at higher pressures. cm-I) and v8 (760 cm-I), correspond to relative minima in the PdUi)
CPdwi-Uv) V
2-
(6)
Vibrational Relaxation in Naphthalene A similar correlation is seen in liquids where repulsive forces lead to line width increases, while attractive forces tend to narrow the line widthss0 This qualitative analysis indicates that higher pressures should cause an increase in the anharmonic vibrational coupling matrix elements; however, the magnitude of these increased anharmonic effects requires detailed calculations.'4 The need for experimental measurements of anharmonic contributions to the intermolecular potential energy function is highlighted by the fact that the intermolecular potential is relatively insensitive to the details of the van der Waals atom-atom potentials from which it is typically constructed. For example, two very different sets of atom-atom van der Waals interact i o n ~ can ~ ~do, ~an~equally good job of reproducing the crystal structure, intermolecular phonon frequencies, and even some of the anharmonic Gruneisen parameters for n a ~ h t h a l e n e .Given ~~ a potential energy function48 and normal coordinates5* for naphthalene, it is possible to calculate the anharmonic coupling matrix elements and from these obtain vibrational relaxation In this report we present pressure-dependent vibrational relaxation data for three different vibrons in crystalline naphthalene at low temperature? A comparison of the pressure-dependent vibrational relaxation rates for these three vibrons with anharmonic calculations will provide a clear test of both the fidelity of phenomenological intermolecular potential energy functions and lattice dynamical calculations. 3. Inhomogeneous Dephasing. An analysis of the magnitude of inhomogeneous vibrational effects in high-pressure molecular crystals has recently been discussed.53 A solid-state pressure change typically causes broadening in the vibrational Raman spectrum due to sample inhomogeneities. Although pressuremediating fluids can be utilized for studies at elevated temperatures, this is not the case for cryogenic studies. Therefore, in this study strain-free naphthalene crystals were produced by growing crystals from the high-pressure melt by slow cooling. Once a crystal was grown, the pressure was not changed except for the unavoidable -5% change upon cooling to cryogenic temperatures. The small pressure change upon cooling and the ability of the multiple crystals within the cell to relieve strain yielded strain-free crystals, and the exponential psCARS decays observed for these samples at low temperature are attributed to vibrational relaxati~n.~~ ReSdtS
1. Calculation of p&) versus Pressure. The calculation of the 2PDOS at a frequency w, is performed according to eq 6. The ZPDOS, pzp(w), is calculated by summing all lower energy molecular vibrations scaled by the density of lattice phonons at an energy equal to the energy difference between the intramolecular vibrons involved. In Figure 1 we show the results of such a calculation for a naphthalene crystal at ambient pressure. Figure 1 also identifies the phonon DOS" and the intramolecular vibrons used to generate the 2PDOS. Our calculation of the pressure dependence of the 2PDOS is obtained by recalculating pzp(w) with the pressure-dependent lattice phonon DOS. The pressure-dependent phonon DOS utilized in the calculation of pzp(w) was obtained by parametrizing this function in terms of the known pressure-dependent phonon frequency shiftsSSobtained from pressure-dependent neutron diffraction,%Raman,s5 and IRZ5spectra, as well as known phonon Gruneisen parameterss7for naphthalene. Over the pressure range of this study the intramoIecular vibrational frequencies typically shifted no more than a few wavenumbers and these shifts were neglected in our calculation of the pressure induced changes in pzp(w). The low-frequency region 1250 cm-' includes a lowfrequency molecular out-of-plane butterfly mode of naphthalene that has a relatively large bandwidth, due to its large amplitude. Since the frequency of this mode is quite low and its dispersion is rather large, this vibron is treated as a lattice phonon in eq 6. The 2PDOS was calculated for all pressures at which psCARS results w m obtained, and at all pressures the lattice phonon region (excluding the butterfly mode) was renormalized to 12states/unit cell.
The Journal of Physical Chemistry, Vol. 96, No. 24, 1992 9663 10Naphthalene vs 766cni'
T = 15K
3
$lo-
rii
z
g- lo22
