J. Phys. Chem. 1987,91, 2237-2240
2237
Nonrelaxational Inertial Motlon In CS2 Liquid Observed by Femtosecond Time-Resolved Impulsive Stimulateii Scattering* S. Ruhman,+ Leah R. Williams,*Alan G. Joly, Bern Kohler, and Keith A. Nelson* Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: February 20, 1987)
Femtosecond time-resolved impulsive stimulated scattering data from CS2liquid are shown which clearly indicate an inertial component of the short-time motion which cannot be described in terms of Debye relaxational dynamics. This is interpreted in terms of the vibrational character of intermolecular motion in the liquid. A discussion of time-domain and frequency-domain light-scattering techniques is given to illustrate the comparative difficulty of characterizing this type of motion by conventional methods.
Introduction The dynamics of orientational and intermolecular motion in molecular liquids has long been an area of intense theoretical and experimental study.’ Much of the experimental work has involved light-scattering spectroscopy,14 which in principle can provide most of the dynamical information of primary interest. Depolarized quasielastic light-scattering spectra of simple fluids often show an approximately Lorentzian central peak “riding” on top of another, broader feature, with both features centered at zero frequency. Both features are attributed primarily to orientational and “collisional-induced” fluctuations in the fluid. The narrower line width gives in some cases a collective molecular reorientation time which is associated with entropy-driven return to the isotropic state following a fluctuation-induced partial alignment of molecules. The broader line width has been interpreted in some cases in terms of a faster “local” relaxation time which is generally associated with an individual molecule coming into local equilibrium with its immediate neighbors. This kind of “interaction-induced” local motion leads to subpicosecond relaxRecent subpicosecond timeation times in simple resolved optical Kerr effect experiments on CS2clearly resolved two lifetimeslo which were compatible with light-scattering (LS) results.’ Although a *two-parameter” model can in many cases yield reasonable fits to LS spectra of simple molecular liquids, it is clear that the underlying dynamics are usually more complicated. More realistic models have been discussed at great length.’ Perhaps the simplest of these recognizes that the intermolecular interactions in the fluid may produce local potential minima with finite restoring forces against intermolecular libration and translation. “Local” motion must have nonrelaxational character, i.e. underdamped or overdamped vibrational character12-21described by at least two parameters, a natural undamped frequency, wo (coming from the configuration-averaged interholecular potential), and a damping rate, y. Here we present femtosecond time-resolved “impulsive” stimulated scattering (ISS) data on CS2liquid which clearly indicate nonreiaxational short-time motion. Some of these results were presented and discussed earlier in preliminary report^.^^-^^ The dynamical information obtained through these ISS experiments can also be obtained by optical Kerr effect experiments, results of which are also shown. The data are fit very roughly by a simple “local overdamped oscillator” model. W e also provide some background on the ISS experiment. We show that although the information content of time-domain ISS data is in principle ideritical with that of frequency-domain LS data, in some cases nonrelaxational dynamics may in practice be more easily resolved in the time domain than in the frequency d ~ m a i n . * ’ * ~ ~ - ~ ’
’
Weizmann Postdoctoral Fellow. *AT&T Bell Laboratories Ph.D. Scholar. * Presidential Young Investigator Awardee. #Presentedin part at the International h e r Science Topical Group of the APS, Seattle, Oct 23, 1986.
0022-3654/87/2091-2237$01.50/0
I SExperiment The impulsive stimulated scattering experiment is illustrated schematically in Figure 1. Two ultrashort “excitation” laser pulses derived f r o p the same-laser, with central frequency and wave vectors (wL,kl)and (y,k,), are overlapped spatially and temporally inside the sample. If the pulse duration, TL, is sufficiently short, then the intersecting pulses can exert a spatially periodic, temporally impulsiveforce on a LS-active mode of the sample through stimulated s ~ a t t e r i n g . 2 ~This 9 ~ ~“impulse“ ~ driving force produces in the medium a coherent, spatially periodic, standing-wave re(1) Recent reviews are given by Patterson, G. D.; Carroll, P. J. J . Phys. Chem. 1985,89, 1344. Madden, P.; Kivelson, D. Adv. Chem. Phys. 1984,16, 467. Steele, W. A. Ado. Chem. Phys. 1976, 34, 1. (2) Keyes, T.; Kivelson, D. J. Chem. Phys. 1972, 56, 1057. (3) Madden, P. A. Mol. Phys. 1978, 36, 365. (4) Frenkel, D.; McTague, J. P. J . Chem. Phys. 1980, 72, 2801. (5) Ladanyi, B. M.; J. Chem. Phys. 1983, 78, 2189. (6) Fleury, P. A.; Worlock, J. M.; Carter, H. L. Phys. Reu. Lett. 1973, 30, 591. (7) Madden, P. A. In Ultrafast Phenomena IV, Auston, D. H., Eisenthal, K. B., Ed.; Springer-Verlag: Berlin, 1984; p 244. (8) Kenney-Wallace, G. A. In Applications of Picosecond Spectroscopy to Chemistry Eisenthal, K. B., Ed.; Reidel: New York, 1984; p 139. (9) Greene, B. I.; Farrow, R. C. J. Chem. Phys. 1982, 77,4779. Halbout, J.-M.; Lang, C. L. Appl. Phys. Lett. 1982, 40, 765. (10) Greene,B. I.; Farrow, R. C. Chem. Phys. Left. 1983, 98, 273. (11) Greene, B. I.; Fleury, P. A.; Carter, Jr., H. L.; Farrow, R. C. Phys. Rev. A 1984, 29, 271. (12) Cox, T. I.; Battaglia, M. R.; Madden, P. A. Mol. Phys. 1979, 38, 1539. (13) Rahman, A. J . Chem. Phys. 1966, 45, 2585. (14) Ben-Reuven, A.; Zamir, E. J. Chem. Phys. 1971, 55,475. (15) Kushick, J. N. J . Chem. Phys. 1977, 67, 2068. (16) An, S. C.; Fishman, L.; Litovitz, T. A.; Montrose, C. J.; Posch, H. A. J. Chem. Phys. 1979, 70, 4626. (17) Guillot, B.; Bratos, S.; Birnbaum, G. Phys. Rev.A 1980, 22, 2230. (18) Lynden-Bell, R. J.; Steele, W. A. J. Phys. Chem. 1984, 88, 6514. Kabadi, V. N.; Steele, W. A. J. Phys. Chem. 1985, 89, 1467. (19) Madden, P. A.; Cox, T. I. Mol. Phys. 1981,43, 287. Madden, P. A,; Tildesley, D.J. Mol. Phys. 1985, 55, 969. (20) Steinhauser. 0. Chem. Phvs. Lett. 1981. 82. 153. (21j Lynden-Bell, R. M.; H u t c h s o n , D. J. C.; Doyle, M. J. Mol. Phys. 19 . - , -307. . -~-,6-SR. (22) Williams, L. R.; Ruhman, S.;Joly, A. G.; Kohler, B.; Nelson, K. A.
Invited oral presentation at APS Topical Meeting on International Laser Science (ILS), Seattle, WA, Oct 1986. (23) Williams, L. R.; Ruhman, S.; Joly, A. G.; Kohler, B.; Nelson, K. A. In Advances in Laser Science-II (Proceeding of 1986 ILS Conference); A I P New York, in press. (24) Initial results obtained by L. R. Williams and B. I. Greene, although not conclusive, showed the qualitative features of interest and were presented tentatively by K. A. Nelson at the Quasielastic Light Scattering Spectroscopy -11 (QELSS-11) Conference, Worcester, MA, June 1986 (Plenary Lecture). (25) Fanar, M.R.; Williams, L. R.; Yan, Y.-X.; Cheng, L.-T.; Nelson, K. A. in Ultrafast Phenomena V, Fleming, G. R., Siegman, A. E., Ed.; Springer-Verlag: Berlin, 1986; p 532. (26) Yan, Y.-X.; Nelson, K. A. J . Chem. Phys., submitted. (27) Yan, Y.-X.; Cheng, L.-T.; Nelson, K. A. Adu. IR Raman Spectrosc., in press. (28) Robinson, M. M.; Yan, Y.-X.; Gamble, Jr., E. B.; Williams, L. R.; Meth, J. S.; Nelson, K. A. Chem. Phys. Lett. 1984, 212, 491. De Silvestri, S.; Fujimoto, J. G.; Ippen, E. P.; Gamble, Jr., E. B.; Williams, L. R.; Nelson, K. A. Chem. Phys. Lett. 1985,116, 146. Farrar, M. R.; Cheng, L.-T.; Yan, Y.-X.; Nelson, K. A. IEEE J. Quantum Electron. 1986, QE-22, 1453.
0 1987 American Chemical Society
2238 The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
ISS
Letters
SIMULATED D A T A
PULSE SEQUENCE
DIFFRACTED PROBE \, PULSE \\,
WEAKLY DAMPED:
WEAKLY OAMPED:
1 INDUCED STANDING WAVE
SAMPLE c _
0
1
2
0
2
4
6
8
1
0
wqt/2n-
W/WO--
J
\
OVEROAMPED: Y=1.5wo
OVEROAMPED: y=1. 5WO
DELAY ED PROBE PULSE 1
0
Figure 1. Schematic diagram of the impulsive stimulated scattering experiment. The ultrashort, crossed excitation pulses "impulsively" excite a standing-wave material response which is monitored by coherent scattering of variably delayed probe pulses.
sponse whose time dependence is given directly by the impulse response function, G ( t ) ,of the LS-active mode.25-27 Th_e tiqedependent motion, Q(@,t) G ( t ) 6(q' f qo) where qo = k , - k,, modulates the dielectric constant, e ( g , t ) a Q(&t), where a = ( & / c ~ Qis) the ~ differential polarizability. (A full tensor treatment of ISS has been p r e ~ e n t e d . ~ ~The . ~ motion ~ ~ ~ ~ is ) monitored by coherent scattering (see Figure 1) to yield signal of the form
-
Z(&t)
-
a
lG"(G,0l2
(1)
where C", the dielectric response function, is given by G"(ij,t) = C,(a*)2Gu(&t)for uncoupled LS-active modes labeled by a. Simulated ISS data from underdamped and overdamped vibrational modes and relaxational modes is shown on the right-hand side of Figure 2. ISS data from underdamped vibrational modes (Figure 2a) shows damped oscillations. ISS data from overdamped vibrational modes (Figure 2b) shows a gradual rise after t = 0, followed by monotonic, nonoscillatory decay. The dynamical parameters, wo and 7,can be extracted readily from data like that in Figure 2, a and b. ISS data from relaxational modes (Figure 2b, dashed curve) show an instantaneous rise followed by exponential decay whose time dependence yields the dynamical parameter l?. Figure 2b shows that relaxational and overdamped vibrational responses are clearly distinguishable in ISS data, if the time resolution is sufficient to resolve the initial rise of the overdamped response (Le., laser pulse duration T~ < y2-'). In frequency-domain quasielastic light scattering, the spectrum can be described by30 Z(q',w) a
kBT
Im w
[G"(q',w)]
2
0.0
w/Y-
EXCITATION PULSES
(2)
where k B is the Boltzmann constant and Gff(?j,,w), the Fourier transform of C=(G,t),is related to thermal fluctuations of the mode through the fluctuation-dissipation theorem. Equations 1 and 2 show that, in principle, the information content of ISS and LS data is identical. However, the extraction of dynamical information from the time- and frequency-domain data differs considerably in practice. The left-hand side of Figure 2 shows simulated LS spectra for underdamped and overdamped vibrational modes and relaxational modes with the same parameters used for the ISS simulations. Figure 2a shows the familiar result for weakly damped modes, namely a well-defined Lorentzian peak in the LS spectrum whose frequency shift and width yield u,,and y. Figure (29) Nelson, K. A. J. Appl. Phys. 1982, 53, 6060. (30) Berne, B. J.; Pecora, R. Dynamic Light Scattering, New York, 1976.
2. 5
5. 0
rt-
Figure 2. Simulated light-scattering spectra (left-hand side, Stokes side of spectra only) and ISS data (right-hand side) from vibrational and relaxational modes. (a) Underdamped vibrational mode described by G(t) e-' sin w,t, where w, = (woz - Y * ) ' / ~ and y = w0/50. (b) Overdamped vibrational mode (solid curve) described by G ( t ) e?'' e-'*', where yZ,,= y f (y2 - wo2)1/z and y = 1.5 wo; relaxational mode (dashed curve) described by G ( t ) e-rr,with relaxation rate r (=2y/ woz) chosen to yield a similar LS spectrum. The LS spectra are essentially indistinguishable, even though the material dynamics are very different. The ISS data can be distinguished and analyzed accurately, given sufficient time resolution.
-
-
-
2b shows that overdamped modes give rise to central peaks in
LS spectra (solid curve) which are nearly identical in form with the central peaks due to relaxational modes (dashed curue). Thus it is extremely difficult to characterize overdamped modes accurately from LS spectra. It is even difficult to make the qualitative distinction between overdamped and relaxational responses, even for modes which are barely overdamped. This is especially true in quasielastic LS spectra of liquids, which usually contain several overlapping contributions. The main point of this discussion with respect to the ISS data to be shown is that nonrelaxational short-time dynamics may be apparent in time-domain data but not in frequency-domaindata, even when the two are equivalent in principle in terms of information content. More complete comparison between ISS and LS data has been
Experimental Section The laser system used was similar to one which has been described in detail earlier.31 Briefly, a Nd:YAG laser is cw mode-locked, and its frequency-doubled output synchronously pumps a dye laser (rhodamine 6G gain dye) with an antiresonant ring which contains a saturable absorber dye jet (DODCI absorber dye). The output of the dye laser is a stream of 65-fs, 615-nm pulses. A second Nd:YAG laser regeneratively amplifies a pulse from the first, and its synchronized output (a 1.2-mJ, 100-ps, 1.06-rm pulse) is frequency-doubled and used to longitudinally pump a homebuilt three-stage amplifier chain. The amplified output after grating-pair compression is a 65-fs, 615-nm, 6-p.J pulse with a 500-Hz repetition rate. For the CS2experiments, the amplified pulse was filtered to avoid intensity-dependent effects and split three ways to yield 75-nJ excitation pulses and a 30-nJ probe pulse. These were polarized vertically (V) or horizontally (H) relative to the scattering plane, focussed to 180-pm spot sizes, and crossed (with a 5 O angle between excitation pulses) inside a 2-mm cuvette filled with CS2 (31) Sizer, T.; Kaka, J. D.; Duling, I. N.; Gabel, C. W.; Mourou, G . A. IEEE J . Quantum Electron. 1983,QE-19, 506.
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2239
Letters CS2 V-H DATA
AUTOCORR.
-
>
4J .ri
ffl
C 0)
4J
C H
-0
-0.15
0.0
0.15
Time
0.30
0.45
0.60
0.
(ps)
Figure 3. V-H ISS data from CS2liquid. The signal after t = 0 continues to rise, indicating the nonrelaxational character of short-time motion in the fluid. After reaching a maximum, the signal decreases rapidly and then more slowly. An approximate fit (dashed curve) assuming overdamped vibrational short-time dynamics and relaxational longtime dynamics is shown. The light curve is an autocorrelation of the amplified pulses.
liquid. The variable delay of the probe pulse was controlled by a 1-pm stepping-motor delay line. The energy of each amplified pulse was measured, and pulses whose energies fell outside a 0.5% range were rejected by an on-line computer which stored and averaged the signal from the remaining pulses. For V-H ISS experiments (analogous to V-H LS), the excitation pulses were V and H polarized, the probe pulse was V polarized, and the coherently scattered signal was H polarized. For V-V ISS experiments, all three pulses and the signal were V-polarized. For optical Kerr effect (OKE) experiments, a single V-polarized excitation pulse and a nearly collinear probe pulse, linearly polarized at +45O, were used. The intensity of probe light transmitted through the sample and a film polarizer oriented at -45O was measured.
Results and Discussion V-H ISS data from CS2liquid is shown in Figure 3. Centered near t = 0 is a ”spike” which is due to the nearly instantaneous electronic response of the CS2 to the nonresonant light fields. This feature appears only when the excitation and probe pulses are all inside the sample simultaneously. We have defined t = 0 by replacing the CS2 with H20 liquid and recording ISS data, which showed only a symmetric “spike”. The peak of the H 2 0 signal defines t = 0. The H 2 0 data was nearly identical with the autocorrelation trace shown in Figure 3 (light curve). The most interesting feature in the data is the rise in signal which is apparent between about 75 and 175 fs after t = 0. It is clear that the t = 0 “spike” due to the electronic response is declining while a slower Tesponse is growing in. At the low laser intensities used, this feature is extremely reproducible and shows little if any intensity dependence. After 175 fs, the ISS signal begins to decay rapidly. However, the rapid decay does not continue to zero but rather to a low level which then decays much more slowly. This is in accord with optical Kerr effect data reported which show a rapid decay followed by a slower decay. All of the features in the data except the t = 0 spike can be attributed to intermolecular and orientational motions of CS2 molecules, in accord with earlier interpretations of frequencydomain and time-domain light-scattering experiments on CS2.7-io,19-21 In the ISS (or OKE) experiment, the excitation pulses exert impulsive torques on the molecules through the single-particle molecular polarizability. This interaction of the polarized light field with the induced single-molecule polarization results in a net orientational alignment of the sample. The excitation pulses also exert impulsive forces on nearest-neighbor
-
20
0 0
0 20
0 40
0.60
0.80
1 00
Time i n P i c o s e c o n d s Figure 4. V-V ISS data from CS2liquid. The data are similar in form to that in Figure 3.
molecule pairs, triplets, etc. (i.e., intermolecular or collision-induced stimulated scattering), causing intermolecular librational and translational motions. Molecular positions and orientations (on which ISS signal depends) do not change instantaneously, however, and so the signal (other than the t = 0 spike) rises gradually after t = 0. The gradual rise in ISS signal reflects the nonrelaxational, inertial nature of intermolecular andlor orientational motion in thefluid. In CS2the time scales for local intermolecular motion and collective orientational motion are reasonably well separated.7312919The slow decay, whose lifetime, I?-’ = 1.4 ps, was characterized accurately through ISS experiments run out to 6-ps delay, gives a collective orientational relaxation time.I0 The short-time dynamics, previously characterized as relaxational:*10 reflect intermolecular motions which are essentially vibrational in character. The rapid decay of the short-time ISS signal probably reflects very rapid dissipation of energy in these modes (“homogeneous dephasing” due to T , processes) as well as the wide range of local vibrational frequencies arising from many different intermolecular configurations (“inhomogeneous dephasing”).18 An approximate fit to the data, intended only as a semiquantitative guide to its interpretation,is shown in Figure 3. The fit was generated by assuming an overdamped oscillator response at short times (“local” intermolecular motion) and a relaxational response at long times (“collective” orientational relaxation). From eq 1, the functional form used to generate the fit is Z ( t ) = [ A ( e r l t- eY2‘)
+ Be-rt]2
(3)
with y1 = 8.8 X 10l2 s-l, y2 = 9.8 X 10l2 s-l, r = 0.73 X 10l2 s-l, A = 44, and B = 1.0 (arbitrary units). Equation 3 was convoluted with the temporal profile of the probe pulse, which was approximated as a Gaussian with 65 fs fwhm, to produce the dashed curve in Figure 3. Similar data were also obtained from V-V ISS and optical Kerr effect experiments, as shown in Figures 4 and 5. Most importantly, the gradual rise after t = 0 still appears, as well as the fast and slower decays. A more complete discussion of these results will be given elsewhere. We note here that ISS time-resolved observations of coherent acoustic and optic phonon^^^-^^ and molecular vibrationd2 have been carried out and, in general, the material dynamics observed in V-V ISS,V-H ISS,and OKE experiments differ. In CS2,since the single-molecule polarizability and many of the interaction-induced polarizabilities arising from different local motions are V-V or V-H LS-active, all three experiments reveal similar features with similar intensities. Although the nonrelaxational aspect of the short-time data is unmistakable and reproducible, we emphasize the preliminary nature of the fit in Figure 3 and the qualitative nature of its interpretation here for several reasons. First, the electronic contribution to signal a t t N 0 was not subtracted in any way, (32) Ruhman, S.; Joly, A. G.; Nelson, K. A. J . Chem. Phys., in
press.
2240
J . Phys. Chem. 1987, 91, 2240-2242
C 0 r(
(0
m d
E 0)
C
m
L
t-
Time in Picoseconds Figure 5. Optical Kerr effect (OKE) data from CS2 liquid. The data are similar in form to those in Figures 4 and 5 . Note that the OKE experiment in this case is similar to ISS in that the same type of “impulse” force is exerted by the excitation pulse. The OKE experiment is essentially a forward-scattering ISS experiment.
but rather the first part of the data was ignored during fitting. A more fundamental limitation on our interpretation of the data is the simplicity of the “local oscillator” model used. The wide range of local vibrational frequencies (Le., inhomogeneous dephasing) has not been accounted for.18 Moreover, the more slowly relaxing orientational motion may also show “nonrelaxational” short-time behavior since an impulsive torque should produce a delayed, not instantaneous, orientational response. More systematic experimental work currently under way should suggest the proper theoretical description. These reservations notwithstanding, we close by discussing the important implications of these results. They support basic notions about short-time dynamics in molecular fluids which have been discussed at length but which are difficult to observe directly by
conventional experimental methods. Time-domain methods with sufficient resolution provide the most direct means for observation of these short-time nonrelaxational dynamics. We have observed similar effects in benzene33 and expect similar results in many liquids. Systematic examination of a variety of pure and mixed fluid systems under different experimental conditions (sample composition, temperature and pressure, laser power, etc.) should further clarify the nature of the short-time motion and may yield configuration-averaged intermolecular potentials, dissipation rates, and additional information fundamental to understanding the forces felt by molecules in a fluid. Determination of absolute electronic and nuclear polarizabilities should also be possible by measurement of absolute diffraction intensities.28 We note finally that the “vibrational” nature of intermolecular motion in liquids may play a major role in electronic excited-state relaxation (e.g. Stokes-shifting dynamics) in fluids and in some liquid-state chemical reactions whose rates are influenced by the solvent.34
Acknowledgment. We acknowledge the participation of Dr. B. I. Greene in the initial stages of this work, and many helpful discussions with Y.-X. Yan and Profs. D. Kivelson and I. Oppenheim. We thank T. Dougherty for assistance with data analysis and presentation. This work was supported in part by NSF Grant No. DMR-8306701 and by NSF and private contributors to a Presidential Young Investigator Award. Contributions from Monsanto, AT&T Bell Labs, Chevron, Perkin-Elmer, and DuPont are gratefully acknowledged. We thank G. A. Kenney-Wallace for inviting us to present our results at the APS Topical Group Meeting22and for her interest in our results at that time. Note Added in ProoJ We have learned that G. A. KenneyWallace et al. have recently reported observations similar to ours in CS2 and other (33) Ruhman, S.;Kohler, B.; Nelson, K. A. to be published. (34) van der Zwan, G.; Hynes, J. T. J . Chem. Phys. 1985, 89, 4181. Hynes, J. T. Annu. Reo. Phys. Chem. 1985, 36, 573. (35) Kalpowzos, C.; Lotshaw, W. T.; McMorrow, D.; Kenney-Wallace,G. A. J . Phys. Chem. 1987, 91, 2028.
Effect of Correlated Proton Jumps on the Zero Field NMR Spectrum of Solid p-Toluic Acid T. P. Jarvie, A. M. Thayer,+J. M. Millar,* and A. Pines* Department of Chemistry, University of California and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720 (Received: February 12, 1987)
Zero field NMR spectra of polycrystalline p-toluic acid are compared to computer simulations assuming static, uncorrelated, and correlated motions for the protons in the hydrogen-bonded dimers. Correlated jumps seem the most reasonable, in agreement with previous work on single crystals.
Introduction Small amplitude motions in condensed matter are amenable to study by N M R through their effects 011 observed chemical shifts and dipolar or quadrupolar couplings. The high-field powder patterns of conventional N M R are generally not very sensitive to these motions. Zero field NMR,’ however, should be ideal for the study of subtle motions in polycrystalline or disordered mat Current address: ATBT Bell Laboratories, Murray Hill, N J 07974.
*Current address: Department of Chemistry, Yale University, New Haven, CT 06511.
0022-3654/87/2091-2240$01.50/0
terials since the resulting asymmetry produces line splittings of the dipole spectrum.2 An important example of interesting small (1) (a) Zax, D. B.; Bielecki, A.; Zilm, K. W.; Pines, A.; Weitekamp, D. P. J. Chem. Phys. 1986,83,4817. (b) Thayer, A. M.; Pines, A. Ace. Chem. Res., in press. ( c ) Pines, A. Lectures on Pulsed NMR,Proceedings of the 100th Fermi School on Physics, Maraviglia, B., Ed.; in press. (2) (a) Millar, J. M.; Thayer, A. M.; Zax, D. B.; Pines, A. J. Am. Chem. SOC.1986, 108, 5113. (b) Jonsen, P.; Luzar, M.; Pines, A.; Mehring, M. J . Chem. Phys. 1986,85,4873. (c) Hennel, J. W.; Birczynski, A.; Sagnowski, S. F.; Stachurowa, M. Z . Phys. E 1985, 60, 49. (d) Meier, P.;Kothe, G.; Jonsen, P.; Trecoske, M.; Pines, A., to be published. (e) Luzar, M.;Thayer, A. M.; Pines, A. submitted to Mol. Phys.
0 1987 American Chemical Society
