Homogeneous vibrational dynamics and inhomogeneous ... · The dynamics cover temperature regions from the low tem-perature glass, through the glass transition and supercooled liquid,

Homogeneous vibrational dynamics and inhomogeneous broadeningin glass-forming liquids: Infrared photon echo experiments from roomtemperature to 10 K

A. Tokmakoffa) and M. D. FayerDepartment of Chemistry, Stanford University, Stanford, California 94305

~Received 23 December 1994; accepted 9 May 1995!

A study of the temperature dependence of the homogeneous linewidth and inhomogeneousbroadening of a high-frequency vibrational transition of a polyatomic molecule in three molecularglass-forming liquids is presented. Picosecond infrared photon echo and pump–probe experimentswere used to examine the dynamics that give rise to the vibrational line shape. The homogeneousvibrational linewidth of the asymmetric CO stretch of tungsten hexacarbonyl~;1980 cm21! wasmeasured in 2-methylpentane, 2-methyltetrahydrofuran, and dibutylphthalate from 300 K, throughthe supercooled liquids and glass transitions, to 10 K. The temperature dependences of thehomogeneous linewidths in the three glasses are all well described by aT2 power law. Theabsorption linewidths for all glasses are seen to be massively inhomogeneously broadened at lowtemperature. In the room temperature liquids, while the vibrational line in 2-methylpentane ishomogeneously broadened, the line in dibutylphthalate is still extensively inhomogeneouslybroadened. The contributions of vibrational pure dephasing, orientational diffusion, and populationlifetime to the homogeneous line shape are examined in detail in the 2-methylpentane solvent. Thecomplete temperature dependence of each of the contributions is determined. For this system, thevibrational line varies from inhomogeneously broadened in the glass and low temperature liquid tohomogeneously broadened in the room temperature liquid. The homogeneous linewidth isdominated by the vibrational lifetime at low temperatures and by pure dephasing in the liquid. Theorientational relaxation contribution to the line is significant at some temperatures but neverdominant. Restricted orientational relaxation at temperatures below;120 K causes thehomogeneous line shape to deviate from Lorentzian, while at higher temperatures the line shape isLorentzian. ©1995 American Institute of Physics.

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I. INTRODUCTION

Vibrational line shapes in condensed phases containdetails of the interactions of a normal mode with its enviroment. These interactions include the important microscodynamics, intermolecular couplings, and time scales ofvent evolution that modulate the energy of a transition,addition to essentially static structural perturbations. Anfrared absorption spectrum or Raman spectrum gifrequency-domain information on the ensemble-averagedteractions that couple to the states involved intransition.1–3 Line shape analysis of vibrational transitionhas long been recognized as a powerful tool for extracinformation on molecular dynamics in condensed phase4,5

The difficulty with determining the microscopic dynamicfrom a spectrum arises because linear spectroscopic tniques have no method for separating the various contrtions to the vibrational line shape. The IR absorption or Rman line shape represents a convolution of the varidynamic and static contributions to the observed line shaIn some cases, polarized Raman spectra can be used torate orientational and vibrational dynamics from the lishape, yet as with all linear spectroscopies, contributifrom inhomogeneous broadening cannot be eliminated.6

In order to completely understand a vibrational li

a!Present address: Physik Department E11, Technische Universita¨t Munchen,D-85748 Garching, Germany.

2810 J. Chem. Phys. 103 (8), 22 August 1995 0021-9606/9

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shape, a series of experiments are required to characteeach of its static and dynamic components. These expments can be effectively accomplished in the time domawhere well-defined techniques exist for measuring the vaous quantities. Nonlinear vibrational spectroscopy canused to eliminate static inhomogeneous broadening fromand Raman line shapes.6 Techniques such as the infrarephoton echo7–9 and Raman echo10–13 can determine the ho-mogeneous vibrational line shape which contains the imptant microscopic dynamics when this line shape is maskedinhomogeneous broadening. Yet nonlinear time-domain teniques alone cannot separate the various dynamic contrtions to the homogeneous vibrational line shape. Traditiotime-domain experiments, such as transient absorptpump–probe experiments, are needed to observe populadynamics such as the vibrational lifetime and orientationrelaxation. To further characterize the long time-scale dnamics of spectral diffusion within the line, additional experiments, such as stimulated photon echoes14 or transienthole burning,15 are required.

In this paper we present a detailed study of the tempeture dependence of the infrared line shape of a higfrequency vibrational transition of a solute in three molecuglass-forming liquids between 10 and 300 K using picoseond infrared photon echo and pump–probe experimenThese experiments are the first to follow the evolution witemperature of the dynamics that comprise the vibratio

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2811A. Tokmakoff and M. D. Fayer: Infrared photon echo in glass-forming liquids

line shape in amorphous condensed phases and to quathe degree of inhomogeneous broadening for these systeThe dynamics cover temperature regions from the low teperature glass, through the glass transition and supercoliquid, to the room temperature liquid. The results aresignificance for the understanding of various vibrational dnamic phenomena in condensed phases, and have relevto the study of fast dynamics in the passage through the gtransition. These results also form a basis for compariswith a number of theories of dephasing in liquids,4,16–18andin glasses.19,20

Temperature-dependent studies of condensed mattebrational dynamics are of importance for understandingcoupling of vibrational modes with the external degreesfreedom of the solvent. The population and density of staof low frequency modes of a solvent and the couplingthese modes to internal molecular vibrations dictate thebrational dynamics. Despite their importance in a wide vaety of fields of chemistry, biology, and physics, relativelittle is known about the temperature-dependent dynamicsmolecular vibrations in polyatomic liquids and glasses. Athough a great deal of work has been done using line shanalysis of vibrational transitions, there have only been a fstudies that unambiguously eliminated inhomogeneobroadening and examined the temperature dependence ohomogeneous dynamics. Raman echo measurementsdephasing of C–H stretching modes in ethanol liquid aglass13 have demonstrated that the line is homogeneoubroadened at all temperatures between 10 and 300 K. Itbeen proposed that in this system fast intramolecular vibtional energy redistribution determines the linewidth. Mesurements of line narrowed vibrational line shapes in glasand crystals have been performed with persistent infrahole burning.21–24While hole burning and Raman echoes avibrational line narrowing techniques, in general they canndistinguish among the contributions to the homogeneousbrational line shape from lifetime, rotation, and pure dephing. In addition, hole burning cannot discriminate betwehomogeneous dephasing and additional line broadening ctributions from longer time scale spectral diffusion.14 Manyof these problems are being overcome with time-resolvresonant infrared experiments. Temperature-dependent psecond infrared studies of discrete vibrations in liquids aglasses have observed homogeneous vibrational dephawith photon echoes7–9 and vibrational population lifetimeswith pump–probe experiments7,15,25,26and picosecond tran-sient hole burning.27

Below we present the temperature-dependent vibratiodynamics of the triply degenerateT1u CO stretching mode oftungsten hexacarbonyl@W~CO!6# in the molecular glass-forming liquids 2-methyltetrahydrofuran ~2-MTHF!,2-methylpentane~2-MP!, and dibutylphthalate~DBP!. Twoaspects of the vibrational line shape in these systemsdiscussed in detail. Initially, we compare the behavior of thomogeneous linewidths in the glasses and the transitiothe room temperature liquids for the three glasses, usingIR photon echo experiments. The temperature dependencthe vibrational dephasing and the degree of inhomogenas the liquids approach room temperature are described.

J. Chem. Phys., Vol. 103,

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temperature dependence of the homogeneous vibrationlinewidth in the three glassy solvents isT2 within experimen-tal error, but the behavior is distinct in each of the liquidsWhile in 2-MP the vibrational line is homogeneously broad-ened at room temperature, the line in DBP is massively inhomogeneously broadened in the room temperature liquid

Following the comparison of the temperature-dependendephasing in the three solvents, one of the systems, W~CO!6in 2-MP, is analyzed in greater detail. The contributions tothe vibrational line shape from different dynamic processeare delineated by combining the results of photon echo mesurements of the homogeneous line shape8 with pump–probemeasurements of the lifetime and reorientational dynamics.15

This combination of measurements allows the decompositioof the total homogeneous vibrational line shape into the individual components of pure-dephasing (T2* ), population re-laxation ~T1!, and orientational relaxation. The results demonstrate that each of these can contribute significantly, butvarying degrees at different temperatures.

W~CO!6 was chosen as a well-understood vibrationachromophore to use as a dilute probe of intermolecular sovent interactions while discriminating against solute–soluteffects. It is soluble in a wide variety of solvents, and it isstable under normal experimental conditions. The CO stretcof W~CO!6 has a particularly strong transition dipole mo-ment in the infrared, allowing very dilute solutions to beprobed and strong photon echo signals to be observed. Tabsorption linewidth for this transition is very narrow, per-mitting dynamics to be characterized with picosecond pulseAlso, several aspects of the dynamics of this probe molecuhave previously been characterized in a number of liquidand glasses.7,8,9,15,25,28,29The degenerate nature of the transi-tion studied makes the analysis of the orientational relaxatiomore involved, yet ultimately contributes more informationon the temperature-dependent dynamics.

The paper is divided as follows. In Sec. II, the dipolecorrelation function formalism that relates the microscopicdynamics of the vibrational dipole to the infrared line shapeand the infrared photon echo experiment is discussed. Thmaterial is presented because the influence of orientationrelaxation and restricted orientational relaxation on the photon echo observable has not been discussed previously. Tresults are given in the text while the full treatments arereserved for the appendices. The experimental procedurand results are described in Secs. III and IV. The discussiois divided into two sections. Section V A is a comparison othe temperature dependences of vibrational dephasing in tthree glasses. Section V B is a detailed discussion of thliquid 2-MP, in which the temperature dependence of all dynamic contributions to the vibrational line shape are delineated. A summary and concluding remarks are given in SeVI.

II. THEORETICAL BACKGROUND

A. Correlation function description of the vibrationalline shape

For a dilute solution of a vibrational chromophore, thehomogeneous vibrational linewidth,G, generally has contri-

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2812 A. Tokmakoff and M. D. Fayer: Infrared photon echo in glass-forming liquids

butions from the rate of population relaxation~lifetime!,1/T1, the rate of pure dephasing, 1/T2* , and the rate of ori-entational relaxation,Gor . T1 processes are caused by thanharmonic coupling of the vibrational mode to the bawhich includes other vibrational modes of the solute andsolvent and the low frequency continuum of intermolecusolvent modes or phonons.17,25,30,31Population relaxation isthe only dynamic process that can contribute to the vibtional linewidth in the limit thatT→0 K.31,32Pure dephasingdescribes the adiabatic modulation of the vibrational enelevels of a transition caused by thermal fluctuations ofenvironment.18,33Measurement of this quantity provides detailed insight into the fast dynamics of the system. Althougenerally equated with physical rotation of the dipole, orietational relaxation is defined as any process that causesloss of angular correlation of an ensemble of dipoles. ForT1u mode of W~CO!6 studied here, orientational relaxatiooccurs through the time evolution of the coefficients of tthree states in the superposition state created by the inexcitation of the triply degenerateT1u mode. Both puredephasing and orientational relaxation will vanish in thlimit that T→0 K, since they are thermally induced processes.

The infrared absorption line shape is related to themicroscopic dynamics through the Fourier transform of ttwo-time transition dipole correlation function1,2,3,33

I ~v!5~2p!21E2`

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The infrared absorption spectrum gives dynamic information ^P1(m(t)m* (0))&, wherePl is the Legendre polynomialof order l . A similar equation is valid for Raman or lighscattering experiments where the time correlation functionthe molecular polarizability is considered; these experimegive information in the form ofP2.

34

If the vibrational and rotational motions are decouplethen the dipole correlation function can be factored

^m~ t !–m* ~0!&5^m~ t !m* ~0!&^m~ t !–m~0!&. ~2!

The first term in Eq.~2! describes only the time dependencof the nuclear motions and is often written as35

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HereD describes the variations in the transition energiesthe ensemble of vibration transition dipoles and includes ainhomogeneous broadening.14 The second term in Eq.~2! isthe ensemble averaged correlation function for unit vectalong the dipole direction and describes the orientational mtion of the dipole.

In the Markovian limit, in which these quantities ardescribed by independent exponentially relaxing dipole crelation functions, the total dipole correlation function dscribed by Eq.~2! decays exponentially at a rate of 1/T2, thedephasing time. Equation~1! gives a Lorentzian line shapeand contributions to the full line width at half-maximum~FWHM! are additive

G51/pT251/pT2*11/2pT11Gor . ~4!

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The orientational contribution to the infrared line shape duto isotropic rotational diffusion of the dipole is determinedby ^P1(m(t)–m(0))& and, as shown in Appendix A, is givenby

Gor52Dor /p, ~5!

whereDor is the orientational diffusion constant. This formarises from the general relation of the time-dependent decof ^Pl(m(t)–m(0))& to the expansion coefficients of theGreen’s function solution to the orientational diffusionequation36,37

clm~ t !5e2 l ~ l11!Dort. ~6!

Equation~4! allows the contribution of pure dephasing to thevibrational line shape to be determined from a knowledgethe homogeneous linewidth, the vibrational lifetime, and thorientational diffusion constant.

B. The infrared photon echo experiment

The infrared photon echo7–9 is a time-domain nonlineartechnique that allows inhomogeneity to be removed from thvibrational line shape. It is the infrared vibrational equivalenof photon echoes conducted on electronic states with visibpulses and spin echoes in magnetic resonance. Two shortpulses, tuned to the molecular vibration of interest, arcrossed in the sample. The first pulse creates a coherent sthat begins to dephase due to both its interactions with tdynamic environment and the static inhomogeneity. A seconpulse, delayed by timet, initiates rephasing of the inhomo-geneous contributions to the vibrational transition; howevehomogeneous dephasing is irreversible. This rephasingsults in a macroscopic polarization that is observed asecho pulse in a unique direction at time 2t. The integratedintensity of the echo pulse is measured as a function oft, andits decay time witht is proportional to the homogeneousdephasing time of the vibrational transition. The rephasingthe inhomogeneous broadening eliminates its contributionthe signal.

The infrared absorption line shape, including any inhomogeneity, is governed by a two-time correlation functionThe two-pulse photon echo, and other line narrowing tecniques such as the stimulated photon echo and hole burnieliminate the inhomogeneous contribution to the line shapand are described by four-time correlation functions6 of theform

C5^m* ~ t31t21t1!–m~ t21t1!–m~ t1!–m* ~0!&, ~7!

wheret1, t2, andt3 refer to three consecutive time intervalsWe have written this form38 of the correlation function fordirect comparison to Eq.~1!. Just as the quantum mechanicacorrelation function in Eq.~1! can also be written in terms ofthe anticommutator$m(t),m~0!%/2&, the correlation functionin Eq. ~7! is normally written as a trace over three nestecommutators of a dipole operator with the density matriusing the Heisenberg representation.39 This form yields fourcorrelation functions~and their complex conjugate! in theform of Eq.~7! that contribute to the observed signal, two owhich describe photon echo experiments.40,41The two-pulsephoton echo is a third-order nonlinear experiment that us

, No. 8, 22 August 1995

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three input fieldsEi resonant with a vibrational transition togenerate an output echo signal described byC(t3 ,t2 ,t1). Thefirst field interaction is during the first pulse, and the secoand third interactions are with the second pulse. For a puseparationt, the experiment is described by correlation funtions of the form^m* ~2t!–m~t!–m~t!–m* ~0!&.

As with the two-time correlation function, the assumtion that vibrational and rotational degrees of freedom aindependent allows the four-time correlation functionC tobe separated asC5CVCOR. The vibrational contribution tothe four-time correlation functionCV is given by6,39,42

CV~ t3 ,t2 ,t1!5umu4K expF i Et11t2

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Unlike Eq. ~3!, Eq. ~8! allows for time reversal processes tremove inhomogeneity. The contribution of orientational rlaxation to the correlation functionCOR can be written as

COR~ t3 ,t2 ,t1!5^m~ t31t21t1!–m~ t21t1!

–m~ t1!–m~0!&, ~9!

which is the ensemble averaged four-time correlation funtion for unit vectors along the transition dipole directioCOR can be treated as a classical probability average wthe orientational relaxation is diffusive.

The description of the third-order nonlinear polarizatiothat governs infrared photon echo experiments in termsthe dynamics of lifetime, pure dephasing, and orientatiodiffusion is discussed in Appendix A. For the case that trelaxation processes that contribute to the line shapeseparable, the photon echo signal with delta function puldecays exponentially as

I ~t!/I ~0!5exp@24t~1/T2*11/2T112DOR!# ~10a!

5exp@24t/T2#. ~10b!

The signal decays at a rate four times faster than the decathe homogeneous dipole correlation function. The decaythe photon echo in this limit can be written as proportionalu^m~t!–m* ~0!&u4, and thus gives the homogeneous vibrationline shape through Eq.~1!. It is important to note that due tothe separability of the rotational and vibrational degreesfreedom and the Markovian form of the correlation functiodecay, the four-time correlation function can be writtenthe product of two-time correlation functions.

III. EXPERIMENTAL PROCEDURES

Vibrational photon echoes were performed with infrarepulses at approximately 5mm generated by the Stanfordsuperconducting-accelerator-pumped free electron la~FEL!. The FEL generates Gaussian pulses that are transflimited with pulse length~bandwidth! variable from about0.7 to 2.0 ps. The wavelength is tunable in 0.01mm incre-ments; active frequency stabilization allows waveleng


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drifts to be limited to,0.01%, or,0.2 cm21. The pulselength and spectrum are monitored continuously with an atocorrelator and grating monochromator.

Acquisition of data requires manipulation of the unusuFEL pulse train shown in Fig. 1~a!. The FEL emits a 2 msmacropulse at a 10 Hz repetition rate. Each macropulse csists of the ps micropulses at a repetition rate of 11.8 M~84 ns!. The micropulse energy at the input to experimenoptics is;0.5 mJ, and the corresponding energy in the fuFEL beam is 120 mW. In vibrational experiments, virtualall power absorbed by the sample is deposited as heatavoid sample heating problems, micropulses are selectedof the macropulse at a reduced frequency.

The experimental apparatus is shown in Fig. 1~b!. Theinfrared beam enters the experimental area roughly comated with a 16 mm diam. Beam pointing stability is montored with two position sensitive detectors. A 1:63telescope~L1 & L2 ! reduces the beam size. At the focus of the tescope is a Ge acousto-optic modulator~AOM! for pulse se-lection, within a 1:1 cylindrical telescope using CaF2 lenses.Micropulses are selected out of each macropulse at a reption rate of 50 kHz by the AOM single pulse selector. Thcylindrical telescope makes the AOM rise time less than tinterpulse separation. This pulse selection yields an effecexperimental repetition rate of 1 kHz, and an average pow,0.5 mW. A ZnSe beam splitter allows 1% of the IR beambe directed into a HgCdTe reference detector. The referedetector was used for shot intensity windowing; all data fropulses with intensities outside of a 10% window were dcarded.

The two pulses for photon echo or pump–probe expements were obtained with a 10% ZnSe beam splitter. T10% beam~first pulse in echo sequence and probe pulse! issent through a computer-controlled stepper motor delay li

FIG. 1. ~a! FEL pulse train structure. The macropulse repetition rate10 Hz, with a 2 msenvelope. The micropulse repetition rate is 12 MHwithin the macropulse.~b! FEL experimental apparatus. Pulse selectionaccomplished with AOM1, while AOM2 is used for chopping of pulseAOM, acousto-optic modulator; A/D, analog-to-digital converter; BS, ZnSbeam splitter; CL, cylindrical lens; D, detector; DL, optical delay line; Ggated integrator; L, lens; PC, personal computer; PO, pick-off; PR, off-aparabolic reflector; Q, position sensitive detector; S, sample.

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The remaining portion~second echo pulse or pump pulse! ischopped at 25 kHz by a second Ge AOM. A HeNe beammade collinear with each IR beam for alignment purposThe two pulses were focused in the sample to 220mm diam-eter using an off-axis parabolic reflector, for achromaticcusing of the IR and HeNe. The beams and echo signal wrecollimated with a second parabolic reflector, and focuinto a HgCdTe signal detector with a third parabolic reflectBy selecting the desired beam with an iris between the sond and third parabolic reflectors, either the photon echopump–probe signal could be observed. The photon echonal and an intensity reference signal were sampled bygated integrators and digitized for collection by compute

Careful studies of power dependence and repetitiondependence of the data were performed. It was determthat there were no heating or other unwanted effects wdata were taken with pulse energies of;15 nJ for the firstpulse,;80 nJ for the second pulse, and the effective reption rate of 1 kHz~50 kHz during each macropulse!.

Vibrational photon echo and pump–probe data wtaken on the triply degenerateT1u asymmetric CO stretchingmode of W~CO!6. Solutions of W~CO!6 in the glass-formingliquids were made to give a peak optical density of 0.8 wthin path length. Concentrations varied from 0.6 mM with200mm path length for 2-MP, to 4 mM with a 400mm pathlength for DBP.~The difference arises because of the mubroader inhomogeneous line found in DBP, which is dcussed below.! These solutions correspond to mole fractioof <1024, and are dilute enough to eliminate Fo¨rster reso-nant energy transfer.28 2-MP ~.99.9%!, DBP ~.99%!, andW~CO!6 ~99%! were purchased and used without further prification. 2-MTHF ~99%! was distilled in order to removeperoxide inhibitors added by the vendor. The temperaturthe sample was controlled to60.2 K using a closed-cycledHe refrigerator. After equilibration, the temperature gradiacross the sample never exceeded60.2 K.

The temperature dependences of the vibrational abstion spectra were characterized by infrared absorption stra. Fourier transform infrared spectra of the 2-MTHF a2-MP samples were taken with 0.25 cm21 resolution usingthe same closed cycled helium refrigerator. Spectra onDBP solution were taken with 2 cm21 resolution using avariable temperature liquid nitrogen cryostat.

IV. RESULTS

The results of temperature-dependent photon echoperiments on theT1u mode of W~CO!6 in the three glass-forming liquids are shown in Fig. 2. Examples of the actuecho decay data have been shown elsewhere.7–9 The homo-geneous linewidth,G51/pT2, is shown, derived from fitsassuming that the photon echo signal decays exponentialEq. ~10b!. Data on 2-MTHF were taken from 10 up to 120 Kat which point the decay rate exceeded the instrumentsponse. This data, although very similar, supersedes anlier report on this liquid.7 Echo data in the other liquids werobservable up to room temperature. A preliminary desction of the data for 2-MP has been given previously.8 Thedata in DBP, taken with 0.7 ps pulses, has been discu


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recently up to 150 K.9 At temperatures above 150 K, rapidamping of additional decay processes from higher vibtional levels gave rise to the large error bars in the Fig. 2

The temperature dependence of the homogeneouswidth increases monotonically in the glass. Above the gltransition temperature, the linewidth in 2-MTHF~Tg586 K!and 2-MP~Tg580 K! rises rapidly in a manner that appeathermally activated. In 2-MP, the rapid increase in dephasrate slows above 130 K and becomes temperature indedent. In DBP~Tg5169 K!, the linewidth appears to decreaswith temperature above 200 K although the error barslarge enough that it is possible that the temperature depdence is essentially flat.

Pump–probe measurements of the population dynamin 2-MP and 2-MTHF taken with parallel pump and probpolarizations are shown in Fig. 3. A detailed analysis ofpopulation dynamics and their temperature dependencesthe 2-MP data are given in Ref. 15. The interpretationidentical for 2-MTHF. The data at all temperatures are bie

FIG. 2. Temperature dependence of the homogeneous linewidths of theT1uCO stretching mode of W~CO!6 in 2-MTHF, 2-MP, and DBP, determinedfrom infrared photon echo experiments using Eq.~10b!, i.e., the data decaysexponentially at four times the homogeneous dephasing rate. Arrows mthe glass transition temperature. Note the different temperature andwidth scales.

, No. 8, 22 August 1995

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ponential. The decay times of the two decay componentsgiven in Fig. 3. The long decay component is due to vibrtional population relaxation~lifetime!, and decays with anexponential decay time ofT1. Both liquids have a counter-intuitive temperature dependence, with the rate of relaxatdecreasing as the temperature increases. This phenomhas recently been observed in crystallizing and glass-formliquids.15,25 The fast component in the pump–probe datadue to orientational relaxation of the chromophore, andcays exponentially at a rate of~6Dor11/T1!. With magicangle probing to eliminate orientational relaxation effecthe fast component vanished from the decay, leaving onlylong component. The amplitudes of the decay componeare consistent with complete orientational relaxation abo;120 K and restricted relaxation at lower temperatures. Tpump–probe data on 2-MTHF supersede the prelimindata reported on this system.7

The absorption linewidths for theT1u mode of W~CO!6in the three glass-forming liquids are shown in Fig. 4. Tabsorption linewidth in DBP is 26 cm21 and is temperatureindependent. In 2-MTHF, the linewidth varies somewhfrom 16 cm21 at room temperature to 19 cm21 at 10 K. Thespectrum in 2-MP shows the most change with temperatuIn the glass, the absorption line is;10 cm21 wide. Abovethe glass transition, the line narrows rapidly to a minimulinewidth of 3.2 cm21 at 250 K and broadens slightly ahigher temperatures. The absorption linewidths for all so

FIG. 3. Temperature-dependent population relaxation dynamics of theT1uof W~CO!6 in ~a! 2-MP and~b! 2-MTHF. Each set of data shows the twdecay components of the biexponential data from parallel-polarized pumprobe experiments. A detailed analysis of the dynamics and their tempture dependences for 2-MP are given in Ref. 15. The interpretation is idtical for 2-MTHF. The solid squares represent the population lifetimeT1.The open circles are a fast component that is due to orientational relaxaof the chromophore.


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tions are wider than the homogeneous linewidths measurwith the echo experiments at all temperatures with the eception of 2-MP above 200 K. As will be discussed belowproper interpretation of the echo decays above 200 K demonstrates that the absorption line in 2-MP is homogeneousbroadened.

The temperature dependences of the homogeneousbrational linewidths in the three glasses are compared usinreduced variable plot in Fig. 5. Although the absolute linewidths for each system may vary, this is a reflection of thstrength of coupling of the transition dipole to the bath, ancan be removed by normalization to the linewidth at thglass transition temperature. Likewise, using a reduced teperatureT/Tg allows thermodynamic variables that contrib-ute to determining the glass transition to be normalized. Sunormalization allows comparison of the functional form othe temperature dependences, independent of differencesTg and coupling strengths. Figure 5 shows that the tempeture dependences of the homogeneous linewidths are idecal in the three glasses and are well described by a power lof the form

G~T!5G01ATa. ~11!

The offset at 0 K,G0, represents the linewidth due to the lowtemperature vibrational lifetime. A fit to Eq.~11! for all tem-

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FIG. 4. Temperature dependence of the infrared absorption linewidth of tT1uCO stretching mode of W~CO!6 in ~a! DBP, ~b! 2-MTHF, and~c! 2-MP.

FIG. 5. Comparison of the homogeneous infrared linewidth in three organglasses. The homogeneous linewidth normalized to the linewidth at the glatransition temperature to remove the coupling strength is plotted againstreduced temperature. The temperature dependence follows a power law wexponenta52.160.2.

, No. 8, 22 August 1995


TABLE I. Fit parameters for all glasses to Eq.~11!.

Power law fit toTg1/pT2

Power law fit toTg1/pT221/2pT1

LiquidTg~K!

log~A!~GHz! a

T2~0 K!~ps!

log~A!~GHz! a

2-MP 79.5 23.260.4 2.060.3 12462 22.560.3 2.260.22-MTHF 86 23.260.4 2.360.3 2662 22.960.3 1.960.2DBP 169 23.360.3 2.060.1 3362 ••• •••

All liquids 0.826 2.160.1 ~0.246 0.02! ••• •••~Fig. 5! 0.03 3G(Tg)

sd

B

.salleth

i-ureseenof

s

ve anten

theva-a-b-n 10

u-iffu-

tureea-

ree

atureThesesthemaybe

s

le

peratures below the glass transition is shown in Fig. 5, ayields an exponent ofa52.160.2 and G0/G(Tg) 50.2460.02. At temperatures above;1.2Tg , the linewidths of thethree liquids diverge from one another. All of the liquidwere individually fit to Eq.~11!. The results are summarizein Table I, and demonstrate that the temperature dependeis well described by a power law of the formT2.

Because of its high glass transition temperature, Dallows the largest temperature range over which to obsethe power law. The data are presented in two ways in FigThe solid circles are the data and the line through them ifit to Eq. ~11!. To show more clearly the power law temperture dependence, the open circles are the data with thetemperature linewidth,G0 @corresponding to a vibrationalifetime,T1~0 K!533 ps# subtracted out. The line through thdata isT2. It can be seen that the power law describesdata essentially perfectly over a change of linewidth of;500from 10 to 200 K. This temperature dependence is similarthat observed for infrared vibrational transitions in inorganglasses with infrared hole burning.22

FIG. 6. Homogeneous linewidth of theT1u mode of W~CO!6 in DBP be-tween 10 and 200 K. The homogeneous linewidth is shown in solid circand the corresponding data with the low temperature lifetime ofT1~0 K!533 ps removed are shown with open circles. The open circle data fipower law ofa52.060.1.


nd

nce

Prve6.a-ow

e

toic

V. DISCUSSION

A. Comparison of vibrational dephasing in 2-MP, 2-MTHF, and DBP

1. Temperature dependence of dephasing in theglasses

The dynamics of optical dephasing in glasses19 are gen-erally interpreted within the model of two-level systems, intially proposed to describe their anomalous low temperatheat capacities.43,44 In this model, pure dephasing dynamicare described in terms of phonon-induced tunneling betwtwo structural potential wells. Above the Debye frequencythe glass, a power law ofT2 is theoretically predicted fordephasing due to two-phonon~Raman! scattering processebetween potential wells.19 Huber has pointed out that inglasses, this temperature dependence is expected aboeffective Debye temperature, which can often be two totimes lower than the true Debye temperature.45 The a'2temperature dependence is almost universally seen forhigh temperature dephasing of electronic transitions in ariety of glasses.19 The temperature dependencies of vibrtional linewidths using infrared hole burning have been oserved to have similar temperature dependencies betweeand 60 K. These includea52.3 for N–D stretches in mixedammonium salts,23 a52.2 and 2.0 for S–H and CO2 defectsin chalcogenide glasses,22 and a52.0 for CN2 defects inCsCl crystals.24 The same temperature dependence,a52.08,is seen for ReO4

2 impurity vibrations in alkali halidecrystals.20 Hole burning measures a linewidth with contribtions from both homogeneous broadening and spectral dsion.

Figure 5 demonstrates that the underlying temperadependence of the vibrational homogeneous dephasing msured with the photon echoes is, in fact, identical in the thglasses studied here, and is described by a power lawT2. Theresults presented here are the first to examine the temperdependence of vibrational dephasing in organic glasses.observation of theT2 dependence suggests that these glasare in the high temperature limit above 10 K and thattemperature dependence of vibrational pure dephasingbe universal in all high temperature glasses. This wouldconsistent with dephasing caused by Raman~two quantum!phonon scattering.

Fits to Eq. ~11! assume that the vibrational lifetime i

s,

t a

, No. 8, 22 August 1995

m

b

du

e

o

psai

u

r

thesn

es

uuteTetthe

e

erlov

if

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yg

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ingy,onal


constant with temperature. However,T1 can vary by;50%over the temperature range. This variation ofT1 with tem-perature is relatively small compared to the power law teperature dependence. Equation~11! was fit to the 2-MP and2-MTHF data after using the lifetime data of Fig. 3 to sutract out the temperature-dependent lifetime contributionthe linewidth according to Eq.~10!. The parameters obtainefrom these fits, given in Table I, are slightly different bremain within the error bars of the original fit.

2. Temperature dependence above the glass transition

The temperature dependences of the rate of dephasin2-MP and 2-MTHF~Fig. 2! both show a gradual increaswith temperature until shortly after the glass transition,which point a rapid increase in dephasing is observed. Ccidentally, the glass transition temperatures for 2-M~Tg580 K! and 2-MTHF~Tg586 K! are very similar. Thus,the rapid increase in dephasing can be explained by twosible phenomena: either by the emergence of additionalvent dynamics above the glass transition, or by thermactivated dephasing through coupling to a low frequencytramolecular mode of W~CO!6. If one of the low frequencymodes of W~CO!6 is quadratically coupled to theT1u mode,thermal population of this low frequency mode would resin thermally activated dephasing of theT1u mode. For thismechanism, the break in the functional form of the tempeture dependence would be independent ofTg . The tempera-ture dependence of theT1u vibrational linewidth in DBPshows no dramatic change near 90 K, demonstratingsuch a mechanism is not the cause of the rapid increasdephasing rates near 90 K in the other glasses. Rather,the mechanism is dependent on the solvent, the dephasidictated by the particular temperature-dependent dynamof the solvent.

At temperatures above the glass transition, the normized linewidths in the three liquids diverge from each othindicating the emergence of distinct relaxation processethe various liquids. However, the temperature dependenobserved in the glasses do not deviate from each otherT/Tg.1.15. In 2-MTHF and 2-MP, the rate of homogeneodephasing increases rapidly above the glass transitionperature, in a manner that appears thermally activated.temperature dependences for the two liquids are in fact vsimilar. Apart from coupling strength they are identical un;130 K. At this temperature, the rate of increase ofdephasing in 2-MP slows as it approaches its high tempture behavior. The temperature dependence observedDBP also deviates from the universal behavior observedthe glassy state forT/Tg.1.15, i.e., 30 degrees above thglass transition temperature.

The observation of a temperature for crossover betwshort time- and long time-scale structural relaxation pcesses in supercooled liquids is predicted by mode-couptheory to occur slightly above the glass transititemperature.46 This crossover temperature has been obserin many organic liquids to beTc51.18Tg .

47 Tc is regardedas the temperature below which relaxation processes bcate into two branches,a relaxations andb relaxations.aprocesses are longer time-scale structural evolution of a


-

-to

t

g in

atin-P

os-ol-llyn-

lt

a-

atin

inceg isics

al-r,incesntilsm-heryilera-forin

eno-ingned

ur-

su-

percooled liquid which appear to freeze out on the relevaexperimental time scale at the glass transition.b processesinvolve high frequency, more or less localized, moleculmotions. Thea relaxation processes are described by tranport processes that can be related to the viscosity, trantional and rotational diffusion, and dielectric relaxation.brelaxation processes are interpreted in terms of microscomolecular fluctuations that exist at temperatures abovebelow the glass transition.47 If b processes are coupled to thvibrational transition, they can contribute to the homogneous vibrational dephasing. Although a change occurs intemperature dependence of the vibrational dephasing at;Tcabove the glass transition, this may be a coincidence,related to the bifurcation point. The onset ofa relaxationshave been reported near the glass transition using neuscattering48 and light scattering49 experiments. The infraredphoton echo is sensitive to fast dephasing dynamics~perhapsb relaxations! while it should be insensitive to slower structural processes, such asa relaxations nearTc . However,there is clearly a change in the dephasing dynamics inrange of temperatures aroundTc . It is possible that the merg-ing of the two relaxation branches atTc modifies the cou-pling to the intramolecular vibration of the high frequencpart of the fluctuation spectrum. The additional couplinmight contribute substantially to the dephasing.

In contrast to the solvents 2-MP and 2-MTHF, thereno rapid increase in the homogeneous linewidth aboveglass transition temperature in DBP. This observation shothat the temperature dependence is sensitive to the individdynamics of the solvent. The differences in the dephasdata in the three solvents aboveTg suggest that the rates odephasing cannot be related to the solvent viscosity, ethough the changes in the rates of dephasing are clearlyto the onset of additional relaxation processes aboveTg . Theadditional processes may be new or involve dynamics twere too slow to cause dephasing belowTg . A fragilityplot50 of the viscosities of the three liquids51–53 shows theysuperimpose. The temperature dependence of the viscosand thus the ability to assume structural configurations olonger time scales, is virtually identical for the three liquidThus differences in the temperature dependence ofdephasing in the three solvents at temperatures greaterT/Tg51.15 cannot be ascribed to differences in the solvefragilities.

In 2-MTHF and in 2-MP below 150 K, the echo decayyield a homogeneous linewidth that is much narrower ththe width of the absorption spectrum. These results demstrate that the vibrational lines of these system are inhomgeneously broadened in the glass and supercooled liquidshown below, in 2-MP the W~CO!6 T1u line becomes homo-geneously broadened at room temperature. However, in Dthe line is clearly inhomogeneous at all temperatures. Thomogeneous linewidth at 300 K is;1 cm21, while theabsorption spectrum linewidth is 26 cm21. This is the firstconclusive evidence for intrinsic inhomogeneous broadenof a vibrational line in a room temperature liquid. Previouslmeasurements of room temperature homogeneous vibratidephasing of the C–H stretch of neat acetonitrile,10 the C–Hstretch of 1,1-d2-ethanol,

13 and CwN stretch of neat

, No. 8, 22 August 1995

tua

s.

o

el

a

o-ea-

usly

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ta-

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benzonitrile12 using Raman echo experiments have demined that these vibrational bands are homogeneobroadened. Inhomogeneous broadening at room temperhas been observed with Raman echo experiments onC–H stretch in CH3I/CDCl3 mixtures.

11 The source of inho-mogeneity in this case is due to concentration fluctuationthe two types of molecules within the first solvation shell

B. Total analysis of contributions to thehomogeneous line shape in 2-MP

1. Transition to a homogeneously broadened line

The measured homogeneous vibrational linewidthtained from infrared photon echo measurements~Fig. 2! as-sumed that the echo decays as Eq.~10b!. This equation isvalid in the inhomogeneous limit, when the width of thinhomogeneous distribution of homogeneous linesD far ex-ceeds the homogeneous linewidthG. The inhomogeneouslimit implies that a separation of time scales exists betwthe fast fluctuations of homogeneous dephasing andtime-scale inhomogeneous structural evolution.

Above 150 K, the homogeneous linewidth begins to aproach the measured absorption linewidth, as illustratedFig. 7. Under these conditions, the time scale for the rephing of the echo pulse is shortened by the polarization dedue to homogeneous dephasing. This causes the echrephase at times betweent and 2t. Thus, the echo signadecays at a slower rate than the 4/T2 given by Eq.~10b!.

54,55

In the limit that the absorption line is homogeneously broened, a free induction decay~FID! will be observed along theecho phase matching direction. If the homogeneous lineLorentzian, then an exponential decay with a decay consof 2/T2 will be observed.55

FIG. 7. Temperature dependence of the homogeneous and absorptionwidths for theT1u mode of W~CO!6 in 2-MP. The homogeneous linewidth~solid circles! are taken from Fig. 2 and the absorption linewidths~dia-monds! are those taken from Fig. 3~c!. The ‘‘echo’’ data at 250 and 300 Kare actually free induction decays that match the observed absorption~open circles!, showing the spectra are homogeneously broadened. At 20~dotted circle!, the homogeneous and inhomogeneous contributions toabsorption line are of equal magnitude. This point is only an interpolabetween the 160 K and 250 K points. Below 200 K, the homogenelinewidth is much narrower than the inhomogeneous width.


er-slyturethe

of

b-

e

enong

p-inas-cayo tol

d-

is atant

The data above 150 K reflect this transition to a homgeneously broadened line. If near room temperature the msured echo data are actually FIDs from a homogeneobroadened line, the decay constants 2/T2 yield a linewidthgiven by the open circles in Fig. 7. The FID linewidthmatch the measured absorption linewidths exactly, demstrating that the line is homogeneously broadened for teperatures>250 K. Notice that the absorption linewidthwhich narrows for temperatures up to 200 K actually broaens slightly for higher temperatures, in a manner that pcisely follows the echo data. Without experimentally timresolving the rephasing of the echo pulse, it is not possibldetermine from an echo experiment alone if the observaba true echo decay, a FID, or an intermediate case. Howea comparison to the absorption spectrum provides a tSince treating the point at 160 K as a FID still results inlinewidth that is narrower than the absorption line, this pocorresponds to a true photon echo. The point at 200 K rresents the transition between a homogeneous and inhogeneous absorption line shape, and is merely added ainterpolation.

2. Orientational relaxation and restricted orientationalrelaxation

Pump–probe studies of the population dynamics ofT1u mode of W~CO!6 in 2-MP have observed dynamics thcontribute to the infrared absorption line shape.15 In additionto pure dephasing, population relaxation and orientatiorelaxation of the initially excited dipole contribute to thhomogeneous linewidth. The contributions to the homoneous vibrational linewidth measured with the photon ecare given by Eqs.~4! and ~5!. Using previous pump–probemeasurements of the orientational diffusion constantDor andthe population relaxation timeT1,

15 the contribution due topure dephasing can be determined from the homogenelinewidth.

Although the viscosity of 2-MP changes over 14 ordeof magnitude between 300 and 80 K, orientational relaxatslows by less than one order of magnitude over this rang15

The orientational relaxation is distinctly nonhydrodynamThis is because the orientational relaxation is not duephysical rotation of the molecule, but rather an evolutionthe nature of the initially excited superposition of the tripdegenerateT1u modes.

15 The initial excitation creates a superposition state of the three basis states taken to be ametric CO stretches along the three molecular axes. Theplitudes, or coefficients, of a given basis state in the inisuperposition is given by the projection of theE-field ontothe direction of the basis state. Fluctuations of the envirment cause time evolution of these coefficients, resultingorientational randomization of the initially excited ensembof dipoles. This reorientation is well described as isotroBrownian orientational diffusion of the dipole.15 Pump–probe experiments allow the determination of the orientional diffusion constant in Eq.~5!.

Although this describes the high temperature~T.125 K!behavior of the 2-MP system, at lower temperatures,orientational relaxation does not occur. Rather, the oriention is restricted and can be interpreted as diffusion withi

line-

line0 Ktheionous

, No. 8, 22 August 1995

f

e

exp

eetv

e

-

I

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iot

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hroi

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eous


cone of semiangleu0.56–58 Although the interpretation o

pump–probe data within this model is well understood,15 nosuch theory exists for the analysis of photon echo expments. The influence of restricted orientational relaxationthe echo data is not dramatic at most temperatures. Hownear the glass transition, echo decays are distinctly nonenential, leading to non-Lorentzian homogeneous line shaIn Appendix B, we discuss the details of a treatment ofstricted orientational relaxation for the photon echo expment. As demonstrated in the following section, this nonponentiality does not invalidate the discussion ofprevious sections in terms of exponential fits since the detions from Lorentzian line shapes are small.

As with the treatment of complete orientational relaation, we consider the case where the orientational andbrational degrees of freedom are decoupled, so that thpole correlation function can be written as Eq.~2!. In thelimit of small cone angles~u0,60°!, the two-time orientational correlation function decays as

COR~ t i !5~S1!21~12~S1!

2!cn111

~ t i !, ~12!

whereS15~11cosu0!/2. The exponential termcnnmmis defined

as in Eq.~6!, but for the noninteger factorsnnm. The factors

nnm are functions ofu0, and are discussed in Appendix B.the limit of complete orientational relaxation,S150 andnnm→ l , and the expression for complete orientational reation is obtained. Equation~12! shows the primary feature orestricted orientational relaxation: the correlation functdecays to a nonzero value related to the degree of restrici.e., the cone angle. For the two-time restricted orientatiocorrelation function, the long time value~t5`! is given by^m&25S1

2 which is the constant offset in Eq.~12!. For thefour-time correlation function, the long time offset^m&45S1

4.If, as with the complete orientational relaxation, t

four-time correlation function can be described as the puct of two two-time correlation functions, then the echo snal ~see Appendix B! is given by

I ~t!/I ~0!5uCVCORu25exp~24t~1/T2*11/2T1!!

3@~S1!21~12~S1!

2!cn111

~ t i !#4. ~13!

In the limit that the cone angle grows to 180°,S150 andn11→l51, and the signal for complete orientational relaxatin Eq. ~10a! is recovered. Likewise for the case whenorientational relaxation occurs at all,u050°, and the exponential decay of the signal is due only to the vibratioterms. For intermediate cases, it is observed that the efferestricted orientational relaxation on the echo signal decarelaxation to an offset of~S1!,

8 the result predicted for thsquare of the four-time correlation function. This high ordependence demonstrates the sensitivity of the echonique to restricted orientational relaxation. The offset~S1!

8

allows the cone angle to be determined, and the cone adeterminesnn

m for the fit of the orientational diffusion constant.

From Eq. ~13! it is clear that when the dynamicsrestricted orientational relaxation are faster than those


ri-onver,po-es.re-ri-x-heia-

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population relaxation and pure dephasing, the echo decallows the orientational contribution to be separated from tother dynamics influencing the echo decay. Once the orietational dynamics have decayed, the residual decay is donly to the vibrational terms. A fit to the tail of the decaydetermines (1/T2* 1 1/2T1) independently from the orien-tational dynamics. This is particularly useful in the smacone angle limit. As the cone angle shrinks, the orientationdecay to (S1)

2 becomes faster. This rapid orientational decay, followed by the exponential relaxation due to the vibrtional term, is observed as a clear division of time scalesthe decay.

An example of a fit of Eq.~13! to echo data at 85 K isshown in Fig. 8~solid line!. Also shown is an exponential fit~dashed line! and the residuals of the fits. The data are clearnonexponential, with an initial contribution due both to thvibrational and orientational dynamics, followed by a tathat will decay at a rate determined by the vibrational corrlation function. The quality of the fit obtained with Eq.~13!is good for all temperatures. At low temperatures, it fits thfast but small amplitude restricted orientational decay component, which is observed as a small deviation from expnentiality. At intermediate temperatures, given the goosignal-to-noise ratio, the deviations from exponential decaare pronounced, as in Fig. 8. At higher temperatures, whthe cone angle approaches complete orientational relaxatobtaining accurate values of the orientational parametsolely from the echo decay becomes difficult, since the raof pure dephasing far exceeds the orientational contributio

The functional form of Eq.~13! is correct, i.e., the re-stricted orientational relaxation will cause a decay to a nozero value. It can be used to extract the time dependencethe orientational and vibrational contributions to homogneous dephasing. However, there are clear indications thais not quantitative. A quantitative relationship would perm

FIG. 8. Photon echo data, fits, and residuals for theT1u mode of W~CO!6 in2-MP at 85 K. To demonstrate that the data are not exponential, two fitsshown: a single exponential fit~dashed line! and a fit to the model includingrestricted orientational relaxation~solid line! given by Eq.~13!. The residu-als for the exponential~top! and Eq.~13! ~bottom! fits are given to allowcomparison. The nonexponential echo decay means that the homogenline shape is not Lorentzian.

, No. 8, 22 August 1995

t

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the cone angle and the orientational diffusion constant toobtained from the decay constant and magnitude of the decaused by restricted orientational relaxation. Analyzingecho data with Eq.~13! does not yield the cone angles andiffusion constants obtained for this system from the analyof the pump–probe experiments.15 In the glass, the coneangles obtained from the echo data are consistently smby ;50% than the cone angles obtained from pump–prdata. The diffusion constants obtained from the echo dataequal within error bars to those derived from pump–prodata at low temperatures, but are anomalously high betw70 and 100 K.

Since the theoretical analysis used to analyze the pumprobe experiments is well defined,15 the lack of agreemensuggests uncertainty in the more complex problem oftheoretical description of the influence of restricted orientional relaxation on the echo signal. The separability offour-time correlation function is based on assuming a Mkovian process. In general, higher-order correlation functican be expressed in their two-time equivalents if the prability function for a given time interval is independentthe history of previous processes.59 This is not the case forestricted orientational relaxation. The true form needs todetermined using the full four-time correlation function. Fcomplete orientational relaxation, which occurs at the higtemperatures, the decay is exponential, and the four-timerelation function can be factored. The derivation of the ecdecay for complete orientational relaxation is given in Apendix A and for restricted orientational relaxation is givin Appendix B.

3. Contributions to the homogeneous vibrational lineshape

By combining the photon echo and pump–probe dataof the dynamics that contribute to the homogeneousshape are known. At higher temperatures, pump–probeperiments demonstrate that there is full orientational relation and can be used to measure the orientational diffuconstant.15 Full orientational relaxation results in an expnential decay of the correlation function, and thus a Lorezian contribution to the line shape. Therefore, all contribtions to the echo decay are exponential, andhomogeneous line shape is Lorentzian with width givenEqs.~4! and~5!. The homogeneous infrared line shape in tabsence of inhomogeneous broadening is given by Eq.~1!.

As described in Appendix B, the two-time correlatiofunction for restricted orientational relaxation has been givby Wang and Pecora.57 The small cone angle result is Eq~12!. The photon echo data at long times give 1/T2*1 1/2T1, and pump–probe experiments giveT1, Dor , andu0.

15 Since these are the dynamic parameters that contrito the line shape, the homogeneous line shape can be dmined from Eq.~1!, where

^m~ t !m* ~0!&5exp~2t~1/T2*11/2T1!!

3@~S1!21~12~S1!

2!cn111

~ t !# ~14!


becayhedsis

llerbearebeeen

p–

theta-hear-nsb-f

beorercor-hop-n

allineex-x-ion-nt-u-thebyhe

nen.

uteeter-

in the small cone angle limit. For data below 125 K, Eq.~14!applies,15 and the homogeneous line shape is given by thsum of two Lorentzians.

As seen from Eq.~14!, one of the Lorentzian contribu-tions to the line shape is due purely to the vibrational relaxation terms (1/T2* 1 1/2T1), and the other is a convolutionof the orientational and vibrational terms. Combining thephoton echo and pump–probe results allows the individuadynamic contributions to the total linewidth to be quantifiedAt temperatures.125 K, the line is Lorentzian with contri-butions to the linewidth~FWHM! described by Eq.~4!. Us-ing theT1 andDor measured with the pump–probe experi-ment, the contribution of pure dephasing to the line can bobtained. At lower temperatures, the vibrational line is nonLorentzian, but all dynamics contribute to the FWHM line-width based on their amplitudes. Using the linewidth fromline shapes obtained from Eq.~14!, T1 from the pump–probeexperiments, andT2* obtained from the echo data, the orien-tational contribution can be determined by difference.

In Fig. 9, the various contributions to the homogeneoulinewidths are given. Although the manner of analysis of thdynamics changes at 125 K, the temperature dependenceeach of the contributions is continuous. The total linewidthobtained directly from the dynamic variablesT1, T2* , Dor ,

FIG. 9. Log–log plot of the dynamic contributions to the homogeneouvibrational linewidth of theT1u mode of W~CO!6 in 2-MP. The totallinewidth—squares; lifetime contribution—diamonds; pure dephasingcontribution—solid circles; orientational contribution—open circles. Thelifetime dominates the homogeneous linewidth at the lowest temperaturand is only mildly temperature dependent. At high temperatures, purdephasing dominates the homogeneous linewidth. Orientational relaxatinever dominates but makes significant contributions to the linewidth at intermediate temperatures. The total homogeneous linewidths were detmined directly from the dynamic variablesT1, T2* , Dor , andu0, and areidentical to the data in Fig. 7 within experimental error. Lifetime contribu-tions are taken from Fig. 3. The orientational contribution, shown with aT2.2

power law line, is continuous over all temperatures. The line through thpure dephasing data is a fit to Eq.~16!. Error bars on the total linewidth,lifetime, and pure dephasing are approximately the size of the symbolsmost temperatures. The orientational error bars vary from the size of thsymbols at high temperature to650% at 30 K.

, No. 8, 22 August 1995

h

eaeiol

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a

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re

reth

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and u0, match the numbers from an exponential fit to thecho data in Fig. 2 almost exactly. There is only a sligdeviation ~,10%! for the temperatures between 80 and 9K. The single exponential fit, although inadequate for detmining the line shape in regions of restricted orientationrelaxation provides a very good number for the FWHM linwidth. This conclusion provides the basis for the discussof the temperature dependence of the homogeneousewidths of the other liquids given in Secs. IV and V A.

Figure 9 displays the decomposition of the temperatudependence of the homogeneous vibrational linewidth iits three dynamic components. At low temperatures, the litime is the dominant contribution. At high temperatures, pudephasing is the dominant contribution. Only at intermeditemperatures does orientational relaxation make a substacontribution. The linewidth contribution from lifetime broadening remains significant until;50 K. The mild temperaturedependence ofT1 shows that it is adequate to subtractconstant,G0, for the lifetime contribution in Eq.~11!. Above50 K, the contributions to the linewidth from pure dephasiand orientational relaxation dominate. By 100 K, the pudephasing is a substantially larger contribution than eiththe lifetime or the orientational relaxation, and the ovewhelmingly dominant component of the homogeneous linwidth above;150 K.

At low temperature, where the contributions from pudephasing and orientational relaxation are negligible,contribution to the linewidth from the lifetime~from pump–probe data! and the linewidth determined from the decaythe echo are equal within error. This is the expected ltemperature limit for the homogeneous vibrational linewidwhere processes caused by thermal fluctuations disappand only lifetime broadening is possible.31,32

From low temperature to slightly aboveTg , both puredephasing and orientational relaxation have power law teperature dependences. The earlier discussion of the cbined dynamics in 2-MP and the other two solvents showa T2 temperature dependence when the lifetime contributwas taken out. Here, the contributions from each mechanare shown to follow the sameT2 power law behavior in theglass. Thus the total temperature dependence, excludingT1,is T2, as shown in Figs. 5 and 6. In addition, the power laobserved for the orientational relaxation in the glass is oserved to continue into the liquid. In Fig. 9, a power lawof a52.260.2 is shown. The orientational dynamics are idependent of the glass transition and distinctly nonhydronamic. The orientational relaxation does not have the teperature dependence of the viscosity. This is consistent wthe results of pump–probe experiments.15

In Fig. 9 it is seen that there is a rapid increase in pudephasing above the glass transition temperature. Thisplies the onset of an additional mechanism operating onappropriate time scale to cause pure dephasing and linkethe glass-to-liquid transition. The onset of dynamic procesnear the glass transition is often described with a VogeTammann–Fulcher equation50,60,61

t5t0 exp~B/~T2T0!!. ~15!

This equation describes a process characterized by a timt,


et5r-l-nin-

reto-etetial

geer--

e

fwhar,

-m-dnm

-t-y--

ith

em-hetoes–

,

with a temperature-dependent activation barrier that diverat a temperature,T0, below the nominal glass transition temperature@h(Tg)51013 poise#. T0 can be linked thermody-namically to an ‘‘ideal’’ glass transition temperature thwould be measured with an ergodic observable.50 This equa-tion describes the temperature dependence of the viscosi2-MP well, and givesT0559 K.

If the Vogel–Tammann–Fulcher~VTF! equation appliesto pure dephasing near and above the glass transition,the full temperature dependence is given by the sum oflow temperature power law and a VTF term:

G* ~T!5A1Ta1A2 exp~2B/~T2T0* !!. ~16!

A fit to Eq. ~16! for all temperatures below 300 K is showas the line through the pure dephasing data in Fig. 9. Thdescribes the entire temperature dependence exceedwell, but yields a reference temperature ofT0* 5 80 K. Thisreference temperature matches the laboratory glass transtemperatureTg exactly not the ideal glass transition tempertureT0. We can thus infer that the onset of the dynamics tcause the rapid increase in homogeneous dephasing in 2is closely linked with the onset of structural processes nthe laboratory glass transition temperature. This may bmanifestation of the short time scale of the measurementa reflection of the nonergodicity of the system.

The temperature dependence given by Eq.~16! does notdescribe the decrease in the pure dephasing linewidthserved for the 300 K point. This may be explained by mtional narrowing of the line,35 which is consistent with theobserved transition to a homogeneously broadened abstion line at 250 K.

4. Relationship to theories of pure dephasing inliquids

The data presented here can be compared with the pcations of a number of theories for the temperature depdence of vibrational dephasing in liquids.4,16,17,33A numberof the theories for vibrational pure dephasing in liquids abased on vibrational–translational coupling and collisioinduced frequency perturbations and yield results thatrelated to viscosity. Lynden-Bell relates vibrational dephing to translational diffusion within the liquid potential anobtains a temperature dependence 1/T2* } rh/T.62 The iso-lated binary collision ~IBC! model of Fischer andLaubereau63 relates the linewidth to dephasing collisioprobability yielding 1/T2* } hT/r. The hydrodynamicmodel of Oxtoby64 obtains results similar to the IBC mode1/T2* } hT. In these expressions, the temperature depdence of the viscosity,h, is the dominant factor.

None of these theories can describe the data in this stAlthough the viscosity in liquid 2-MP increases by approxmately 14 orders of magnitude between 80 and 300 K,rate of pure dephasing changes by less than 2 orders of mnitude. The problem may be that the viscosity does not foan adequate approximation for the collision frequency,4 orthat an explicit mechanism for dephasing in a particular stem may be necessary to explain the temperature dedence.

No. 8, 22 August 1995

l

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e

fr

i

r

,ui

r

A

t

m

nalonde-reiq-

y-de-ra-ne

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hefi-line-sing

tran-umon.m--

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Schweizer and Chandler have calculated the vibratiodephasing due to fast interactions with the repulsive walthe potential based on an Enskog collision time.18 Thistheory predicts a temperature dependence of 1T2*} rT3/2g(s), whereg~s!, the contact value of the radial distribution function for a spherical solvent with diameters,may be mildly temperature dependent. This temperaturependence comes much closer to describing the pure deping of W~CO!6 in 2-MP between 130 and 275 K, but does nreflect the rapid increase in the viscous, supercooled liqbetween 80 and 130 K.

A description of vibrational dephasing above the glatransition that is valid for viscous fluids is clearly necessaSuch a theory might benefit from some of the approacheamorphous media that are used for the theories of high tperature dephasing in glasses. The data presented heredemonstrated the existence of continuousT2 behaviorthrough the glass transition with the rapid onset of an adtional or modified dephasing process above the glass tration at a temperature.

VI. CONCLUDING REMARKS

The infrared photon echo and pump–probe experimepresented above have allowed a complete characterizatiothe temperature dependence of the infrared vibrationalmogeneous line shape of theT1u CO stretching mode oW~CO!6 in three molecular glass forming liquids. This wohas quantified the degree of inhomogeneous broadeningdynamic contributions to the homogeneous vibrational lshape, and the temperature dependences of these quanResults are followed from the nonequilibrium low tempeture glass, through the glass transition and supercooleduid, to the equilibrium room temperature liquid.

To completely characterize the infrared line shapecombination of experimental methods is necessary to eldate each of the dynamic and static contributions to the lPicosecond infrared photon echo experiments were usemeasure the homogeneous line shape and remove inhomneity. Infrared pump–probe experiments were used to msure the vibrational lifetime and orientational relaxation. Ttotal vibrational line shape was determined by absorptspectroscopy. Together, these experiments yield a compcharacterization of the dynamics that make up the homoneous line, and the extent of inhomogeneity of the infraabsorption line.

Several key finding of this study are worth stating.low temperatures, the homogeneous linewidth is determiby the vibrational lifetime. From 10 K to slightly above thglass transition, aT2 temperature dependence for the vibrtional pure dephasing is observed in all three solvents, incating dephasing through a two-phonon Raman scatteprocess. In the glass and supercooled liquid, orientatiorelaxation is incomplete, leading to nonexponential echocays and non-Lorentzian homogeneous line shapes. Hever, the orientational relaxation contribution to the total hmogeneous line shape is never dominant, and the deviafrom Lorentzian are not great. In the liquid, above;125 K,orientational relaxation is complete and, therefore, the ho


nalof

-

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di-nsi-

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geneous line shape is Lorentzian. The role of orientatiorelaxation and the effect of restricted orientational relaxation photon echo observables has been examined in sometail theoretically in the Appendices. While the dynamics auniversal in the three glasses studied, they differ in the luids for temperatures above;1.2Tg . The homogeneousdephasing in the liquids are independent of their hydrodnamic characteristics, having a much milder temperaturependence than the viscosity. At room temperature, the vibtional line is homogeneous in the solvent 2-methylpentabut substantially inhomogeneous in dibutylphthalate.

The homogeneous line shape in the 2-methylpentanevent was decomposed into its three components, pdephasing, orientational relaxation, and vibrational lifetimover the entire temperature range of 10–300 K. The vibtional lifetime has a very mild temperature dependence adominates the linewidth at low temperature. The orientional relaxation contribution has aT2 temperature depen-dence over the entire range with no discontinuity near tglass transition temperature. Its contribution, while signicant in some temperature ranges, never dominates thewidth. The temperature dependence of the pure dephacontribution to the linewidth is described by aT2 power lawin the glass but becomes much steeper above the glasssition. Its temperature dependence can be fit well to the sof the power law and a Vogel–Tammann–Fulcher equatiHowever, the Vogel–Tammann–Fulcher divergence teperature isTg rather than the ‘‘ideal’’ glass transition temperature 20 K belowTg .

While the study presented here gives detailed resufrom three solvents, it examines only one mode of one mecule. Clearly the time-domain methods that have been eployed here to obtain a complete determination of ttemperature-dependent vibrational dynamics need to beplied to other systems. Vibrational photon echoes combinwith pump–probe experiments will allow accurate decompsition of vibrational line shapes that are convolutions of mutiple dynamic and static contributions acting on widely varable time scales.

ACKNOWLEDGMENTS

The authors gratefully acknowledge David Zimdars, DRandy Urdahl, Dr. Bernd Sauter, Rick Francis, and Dr. Afred Kwok who contributed to the acquisition of the dadiscussed in this paper. The authors thank Professor ASchwettman and Professor Todd Smith of the DepartmenPhysics at Stanford University and their groups for the oportunity to use the Stanford Free Electron Laser. We athank Dr. Camilla Ferrante for many helpful discussionThis work was supported by the National Science Foundtion ~DMR93-22504!, the Office of Naval Research~N00014-92-J-1227!, the Medical Free Electron Laser Program~N00014-91-C-0170!, and the Air Force Office of Sci-entific Research~F49620-94-1-0141!.

, No. 8, 22 August 1995

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APPENDIX A: CONTRIBUTION OF ORIENTATIONALRELAXATION TO THE THIRD-ORDERPOLARIZABILITY

Here we describe the effect of orientational relaxationthe stimulated photon echo response function for an inhomgeneously broadened line comprised of motionally narrowhomogeneous lines, i.e., the Bloch limit. The photon echothe limiting case of the stimulated echo in which the secoand third pulses are simultaneous. Generally, the third-ormaterial response39,65 is written for isotropic media, or unpo-larized light. A proper treatment of orientational anisotropin time-domain third-order~four-wave mixing! experimentsrequires a tensorial nonlinear response function formalisThis formalism has been presented by Choet al.,66 but onlyderived for pump–probe experiments.

The density matrix description of resonant third-ordnonlinear experiments in terms of four-time correlation funtions has been given by others,6,39–42 and is not discussedhere. The experimental four-time correlation functions agiven by the third-order response functionR, and are relatedto the signal through the third-order polarizationP39,40,41,66

Ph~k,t !5~2 i 3!E0

`

dt3E0

`

dt2E0

`

dt1Rhabg~ t1 ,t2 ,t3!

3exp~ i ~v11v21v3!t31 i ~v11v2!t2

1 iv1t1!E3a~ t2t3!E2b~ t2t32t2!

3E1g~ t2t32t22t1!. ~A1!

Here Rhabg(t1 ,t2 ,t3) is the third-order tensorial nonlinearesponse function. This equation describes the signal geated at timet with wave vectork and polarization compo-nenth, due to three input pulsesE1, E2, andE3 with polar-ization componentsg, b, anda. In Eq. ~A1!, t1 and t2 referto the time intervals between interactions with the fieldsE1,E2, andE3; t3 is the time interval betweenE3 and the ob-servation timet.39

To greatly simplify this problem, it is assumed that thorientational motion is separable from the conventional istropic response, i.e., the vibrational and orientational motioare not coupled in the system. Further, the orientationalhavior of the excited coherent state is assumed to be simto that of the ground state. These assumptions~discussedbelow! allow us to write

Rhabg~ t1 ,t2 ,t3!5ROR~ t1 ,t2 ,t3!RV~ t1 ,t2 ,t3!, ~A2!

whereROR is the orientational contribution to the responsfunction, andRV is the conventional isotropic vibrational response function.ROR contains only the time-dependent orentational information and the dependence on the input fipolarizations,a, b, and g, while the remaining dipole-coupling terms are contained inRV .

To describe the contribution of orientational diffusion tthe third-order response function in the Bloch limit, we cosider the case of three delta function pulses incident alothex laboratory axis, with parallel linearz axis polarizations.The generalization to any polarization is straightforward. Wconsider an ensemble of spherical rotors with nondegenedipole transitions with orientationsV. We write V to


no-disder

y

.

r-

e

er-

-se-lar

ld

-g

eate

refer to the spherical coordinates~u,f!, with*dV5*0

2pdf*0p sinudu. The classical orientational diffu-

sion of such a dipole is well known.36,37Orientational diffu-sion is described byG(V,tuV0), the probability evolutionGreen’s function that satisfies the orientational diffusioequation, which gives the probability that ifm has directionV0 at time t50, then it will have directionV at time t.G(V,tuV0) is a well-known function that is given by anexpansion in spherical harmonics67

G~V i ,t i uV0!5(l50

`

(m52 l

l

clm~ t i !@Yl

m~V0!#*Ylm~V i !,

~A3!

where the expansion coefficientsc1m are given by Eq.~6!.

Evaluation of probability functions for linear spectroscopieare greatly simplified by the orthogonality of spherical harmonics. Such correlation functions can generally be epressed in the form ofPl(m(t)•m(0))&, wherePl is the l thorder Legendre polynomial~m50!, and can be expressed interms of the expansion coefficients@Eq. ~A6!#. In particular,it can be shown that67

^P1~m~ t !•m~0!!&5c1~ t !, ~A4a!

^P2~m~ t !•m~0!!&5c2~ t !. ~A4b!

The first equality is of importance in dielectric relaxationmeasurements and infrared line shapes, while the secondused for Raman scattering, light scattering, pump–probe eperiments, and fluorescence depolarization.3,34

For the stimulated echo, the orientational contribution tthe third-order polarizability is given by a four-time jointprobability function66

ROR~ t3 ,t2 ,t1!5E dV3E dV2E dV1E dV0~m3• eh!

3G~V3 ,t3uV2!~m2• ea!G~V2 ,t2uV1!

3~m1• eb!G~V1 ,t1uV0!~m0• eg!P~V0!,

~A5!

where P~V0! is the initial dipole distribution and (mi•ej )represents the interaction of the field with the distribution odipoles at time t i . For an initially isotropic sampleP~V0!51/4p. Here we consider all incident pulses polarizedparallel along thez axis, so that

~m i• e j !5cosu i5P1~cosu i !. ~A6!

Substitution of Eqs.~A3! and ~A6! into Eq. ~A5!, with par-allel detection polarization, yields

ROR~ t1 ,t2 ,t3!5 19c1~ t1!@11 4

5c2~ t2!#c1~ t3!. ~A7!

This result shows that the polarization of the two pulse photon echo experiment decays with time exponentially aexp@22Dort#. Equation~A7! also reproduces all polarizationdependence of all parallel polarized four-wave mixing experiments. For pump–probe and transient grating expements, t15t350, and the well-known expression for theparallel-probed orientational signal is obtained. The orienttional component of the signal decays exponentially aexp@26Dort# to a constant offset of 0.56. Notice that thefour-time correlation function given by Eq.~A7! can be writ-

No. 8, 22 August 1995

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ten, using Eq.~A4!, as the product of two-time correlatiofunctions. The two-pulse photon echo signal~t250! gives theorientational dynamics in terms of^P1(m(t)•m(0))&, just asrepresented in the infrared line shape.

The vibrational contribution to the response functionRV

has been evaluated for a number of models,6,42,68and is notdiscussed here. For the dephasing and relaxation of thebrational modes studied here, the fast modulation~Bloch!limit applies, and the vibrational response function has bgiven as11

RV~ t1 ,t2 ,t3!5exp@2~ t11t3!/T2* #

3exp@2~ t112t21t3!/2T1#

3exp@2D2~ t32t1!2/2#. ~A8!

Here any dipole operator terms that reflect the amplitudethe signal have been suppressed. The excitation fieldsthe inhomogeneous line, which is described by a Gausdistribution of homogeneous packets,G~v!, with a standarddeviation in time ofD. Notice that in the Markovian limit,wheret15t3 , the four-time vibrational response function cabe factored into the product of three two-time correlatifunctions. Thus, the entire four-time correlation functionthe Bloch limit can be expressed as two-time correlatfunctions.38

For delta-function pulses,Ei(t2t8)5d(t2t8), the po-larization is given by the response function. For the twpulse photon echo,t15t35t, t250, andD50, and the signalis calculated from

I ~2k22k1 ,t!}uP~2k22k1 ,t!u2. ~A9!

For ROR andRV given by Eqs.~A7! and ~A8!, respectively,the echo signal is given by Eq.~10!. For an echo signadecay due only to rotational diffusion, the echo signal decexponentially at rate of eight times the orientational diffusiconstant. This is in contrast to the anisotropy decaypump–probe or fluorescence depolarization experimewhich decay as 6Dor . Notice that for this Markovian limit, inwhich the correlation functions decay exponentially, the snal is proportional tou^m~t!–m* ~0!&u4.

The derivation of the expressions used here for the efof orientational relaxation is based on the assumption this uncorrelated with the vibrational dynamics of the dipoThis assumption is often invoked to simplify the descriptiof rotational motion in complex systems.34,38,66This assump-tion may be considered if a separation of time scales exfor the dynamics of vibrational dephasing and orientatiorelaxation. The rapid fluctuations of the bath allow its intaction with the vibrational coherence to be considered aMarkovian process in the weak coupling limit. Further, tinfluence of these fast bath fluctuations on the motion ofdipole allow the orientational motion to be treated as Browian diffusion.

These generalizations are different for the current cawhen we consider that the orientational motion of the itially excited dipole is not due to the physical rotation of tmolecule. In fact, the orientational relaxation occurs throuthe evolution of the initially excited superposition state of tT1u mode.

15 This process is independent of the solvent v


vi-

en

ofpanian

nnnn

-

ysnints,

g-

ctt it.n

stsalr-a

ehen-

e,i-ehe-

cosity, and occurs due to thermal fluctuations in the envirment that cause the coefficients of the basis states to evoHowever, the observation that reorientational motionlinked to spectral diffusion in the line15 allows us to decouplethese dynamics from the pure dephasing. Spectral diffusobserved for temperatures below 200 K, is the evolutionthe homogeneous transition frequencies in an inhomoneous line on a time scale longer then that of the homoneous pure dephasing.

A further assumption in the description of the rolerotational diffusion on the echo experiment is that the rotional behavior of the evolving excited and ground vibrtional states are similar. This allows the rotational characistics to be treated as uniform, and as classical diffusion. Tassumption would most likely not be valid for electronspectroscopy, when the differing electronic configuratioand change of size or shape of the electronic cloud codramatically effect the rotational characteristics. For the cof ground electronic state vibrational spectroscopystretching modes, this is not unreasonable. Vibrational etation leads to an increased root-mean-squared displaceof the oscillating dipole, but this should not alter the rottional characteristics greatly.

In this treatment of orientational relaxation of the vibrtional line shape, all expressions have been written fowell-defined dipole transition direction. It has been assumthat the results derived above can be applied to the degeateT1u mode. The basis for this assumption is the influenof orientational relaxation on the pump–probe experimeFor parallel excitation, the pump–probe data show a zetime orientational amplitude of 0.44, which yields a polariztion anisotropy ofr ~0!50.4, as predicted for the isotropireorientation of a nondegenerate system. The applicabilitthe results to a degenerate system needs to be confirmedfull theoretical calculation, which is very complex.

APPENDIX B: RESTRICTED ORIENTATIONALRELAXATION

This section describes the influence of restricted orientional relaxation on the orientational correlation functioCOR, and its effect on the two pulse photon echo experimeInstead of diffusion on the surface of a sphere, restricorientational diffusion is often modeled as Brownian diffsion of the dipole within a cone of semiangleu0 centeredabout thez axis~0°<u0<180°!. In this discussion, we followthe treatment of two-time correlation functions for restrictorientation relaxation within a cone by Wang and Pecora57

As shown previously, the description of the infrared linshape requires a knowledge ofCOR5P1^m(t)•m(0)&. Thetreatment of restricted orientational relaxation fP2^m(t)•m(0)& has been given in a number ostudies.56,58,69,70

In general, the two-time dipole correlation function fothe orientational relaxation within a cone will decay tononzero value, (Sl)

2, which gives a measure for the extentorientational relaxation.69 The factorSl , the generalized or-der parameter, satisfies the inequality 0<Sl<1, whereSl50describes unrestricted reorientation, whileSl51 is no orien-

, No. 8, 22 August 1995

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.


tational motion. The value of (Sl)2 can be determined from

the relationship69

limt→`

^A~0!B~ t !&5^A&^B&. ~B1!

For the two-time dipole correlation functionP1^m(t i)•m(0)&, the decay should be to

F ~2p~12x0!!21E dVP1~x0!G25~~11x0!/2!25~S1!2,

~B2!

wherex0[cosu0. The factor@2p~12x0!#21 normalizes the

expression to ensure a uniform distribution within the conet5`. The functional form of the decay of the correlatiofunction toSl

2 cannot be written in closed form, but can bapproximated by a number of methods.57,58,71In the limit ofsmall cone angles, the decay is well approximated byexponential-with-offset of the form

COR~ t !5S121~12S1

2!exp~2kDort !, ~B3!

wherek is an proportionality constant that relates the effetive decay time to the orientational diffusion constant. In tlimit of complete orientational relaxation,S150 andk5l ( l11), and Eq.~6! is recovered.

The probability evolution functionG(V,tuV0) for thediffusion within a cone is given by an expansion in sphericharmonics of noninteger degreenn

m

G~V i ,t i V0!5~2p~12x0!!21(n51

`

(m52`

`

cnnmm

~ t i !

3@Ynnmm

~V0!#*Ynnmm

~V i !, ~B4!

where the expansion coefficients are given by

cnnmm

~ t i !5exp~2nnm~nn

m11!Dort i !. ~B5!

The parameternnm is a function of the cone angle, yet cann

be obtained in closed form as such. The calculation ofnnm

from the diffusion equation and its relation to the cone anghas been outlined elsewhere.57

Wang and Pecora have calculated the two-time dipcorrelation function m(t)•m~0!& for first and second degreespherical harmonics. These are given by an infinite sumexponentials of the form57

^m~ t !•m~0!&} (n51

`

Dnmcn

nmm

~ t !, ~B6!

where the cone angle determines the coefficientsDnm and the

factorsnnm. Dipole correlation functions can be approximate

by truncating the sum in Eq.~B6!. For first degree sphericaharmonics, the dipole correlation function is well approxmated for any cone angle by57

P1^m~ t i !•m~0!&5D101D2

0 exp~2n20~n2

011!Dort i !

1D11 exp~2n1

1~n1111!Dort i !. ~B7!

For small cone angles~u0,60°!, D20'0 and the two-time

dipole correlation function is observed to decay exponetially to a constant valueD1

0. In this limit, comparison to Eq.


at

an

c-e

al

t

le

le

of

d

i-

n-

~B3! shows that (S1)25D1

0 and the exponential decay isgiven by cn

111(t i). In this limit, the parametern1

1 is well ap-

proximated byn11~n1

111!524/7u02. In the limit of complete

orientational relaxation~u05180°!, the parametersn11 andn2

0

converge towardl51 andD1050, so that Eq.~6! is recovered.

For purposes of fitting in this work, calculated values ofnnm

as a function ofu0 were fit to empirical functions.72

The four-time correlation function for restricted orientational relaxation will likewise decay to a constant offsegiven by

^P1~x0!&45~~11x0!/2!45~S1!

4. ~B8!

The four-time correlation function for two-pulse photon echexperiments has been shown in Appendix A to be writtenthe Markovian limit as a product of two-time correlationfunctions. Thus, the four-time orientational correlation function for the photon echo experiment would be written as thproduct of two-time correlation functions

COR~ t1 ,t3!5P1^m~ t1!•m~0!&P1^m~ t11t3!•m~ t1!&.~B9!

Within this description, the two-pulse echo signal in thesmall cone angle limit is described by Eq.~13!. Notice Eq.~13! shows the correct limits by decaying toS1

4. In the limitof complete orientational relaxation,S150 andn1

15l51, andthe result given by Eq.~10! is obtained. For completely re-stricted orientational relaxation,S151, and orientational dif-fusion does not contribute to the signal. In each of theslimits, the Lorentzian line shape is obtained. The limitationof this descriptions are pointed out in Sec. V B 2.

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tt.

n,

s

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1 in the limitof small cone angles:n1

1~n1111!524/7u0

2. An empirical relationship thatgivesn1

1 within 1% for 50°,u0,150° isn11 5 101.676u0

20.8334, whereu0is in degrees. Foru0,170°, n0

2 5 102.496u021.122.

No. 8, 22 August 1995

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Homogeneous vibrational dynamics and inhomogeneous ... · The dynamics cover temperature regions from the low tem-perature glass, through the glass transition and supercooled liquid,