J. Phys. Chem. 1996, 100, 7981-7988
7981
Velocity Dependence of Quasi-Resonant Vibrotational Transfer in Li2*-Rare Gas Collisions Thomas P. Scott,† Neil Smith,‡ Peter D. Magill,§ and David E. Pritchard Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02138
Brian Stewart* Department of Physics, Wesleyan UniVersity, Middletown, Connecticut 06459 ReceiVed: NoVember 15, 1995; In Final Form: February 29, 1996X
We report measurements of velocity-dependent cross sections for vibrotationally inelastic scattering in the 1 + system Li2 A1Σ+ u (Vi,ji) + X f Li2 A Σu (Vf,jf) + X, with Vi ) 9, ji ) 42, and X ) Xe, Ar, and Ne. The measurements range over a factor of 30 in energy. Quantum levels were chosen to elucidate the quasiresonant vibrotational transfer process studied previously. A reduction in collision velocity results in both an increase in total vibrationally inelastic cross section and an enhancement of the quasi-resonant effect, with final rotational state distributions as narrow as 2.5p (fwhm) observed. The largest cross section for ∆V < 0 is given by the formula ∆jpeak ) -4∆V and possesses a roughly 1/Vrel velocity dependence. For ∆V ) +1, energy thresholds shift the peak cross section at low velocity to the nearly energy resonant value of ∆jpeak ) -6. The similarity of the final state distributions for different target gases observed in previously measured rate constants does not hold in the velocity-dependent data; this results in part from the appearance of lowvelocity dynamical thresholds for exoergic cross sections in the Ar and Ne systems. We compare the experimental results for X ) Ne with cross sections calculated from quasi-classical trajectories on an ab initio potential energy surface; for jf g 46, agreement is quantitative, while for jf < 46, the calculation overestimates the cross sections.
I. Introduction We have measured the velocity dependence of absolute levelto-level cross sections for vibrotationally inelastic collisions in the system 1 + Li2 A1Σ+ u (Vi,ji) + X f Li2 A Σu (Vf,jf) + X
where X is Ne, Ar, or Xe. The cross sections involve numerous final rotational states along with changes in vibrational quantum number of (1 and, for Xe, -2 and -3. The relative kinetic energy of the collisions ranges over a factor of more than 30. This paper deepens our exploration of vibrotationally inelastic collisions in this system, in which we have focused on a phenomenon we term rotation-induced quasi-resonant V T R transfer.1-4 In previous papers,2,4 we presented thermally averaged rate constants for collisions in this system for a variety of initial vibrotational states with Vi ) 4-9 and ji ) 14-76. Target gases He, Ne, Ar, and Xe were employed. The properties of rotation-induced quasi-resonant V T R transfer are described in detail in these previous papers; the phenomenon is characterized by the following features:2 (1) Increased initial rotation results in an increase in the vibrationally inelastic rate constant and an increase in specificity for jf. (2) For ji ) 42, distributions of final rotational levels are sharply peaked around a jf value that corresponds to about 70% interconversion of vibrational and rotational energy, prompting the designation “quasiresonant.” At ji ) 64, the process is resonant and specificity is * To whom correspondence should be addressed. † Current address: Concurrent Computer Corporation, Westford, MA 01886. ‡ Current address: San Diego Laboratories, Eastman Kodak Company, San Diego, CA 92121. § Current address: AT&T Bell Laboratories, Holmdel, NJ 07733. X Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(95)03365-X CCC: $12.00
at a maximum. (3) The shapes of the final state distributions are identical to within experimental uncertainty for the targets Ne, Ar, and Xe for ji ) 42 and ∆V ) jf - ji ) -1. However, the distribution is broader for He, leading us to suspect that the quasi-resonant peaking of the final state distribution is dependent on collision velocity. The classical dynamical origin of quasi-resonant V T R transfer has been discussed previously;3 we review its essential characteristics here for the convenience of the reader. At elevated rotational angular momentum, the molecule rotates a number of times during a collision. The interaction potential achieves a maximum near the point of atom-molecule collinearity, with the result that the collision becomes resolved into a series of subcollisions (“collisionettes”). Each of these collisionettes is quite sudden for a steeply repulsive interaction potential owing to the substantial molecular anisotropy and because the potential is further modulated by the molecular vibration. The peak in the potential occurs between the moment of atom-molecule collinearity and the time at which the molecular vibration achieves its outer turning point. Impulses that increase j thus tend to decrease V, and Vice Versa. At a resonance, i.e., a value of j for which the rotational and vibrational frequencies are commensurate, successive collisionettes add in phase, and the phenomenon is most pronounced; a 4:1 resonance occurs in this system at ji ) 64. However, the strong correlation between ∆V and ∆j persists away from the resonant value of ji, resulting in the quasi-resonant behavior seen, for example, at ji ) 42. As the collision velocity is lowered, each collision breaks up into more collisionettes whose cumulative effect is thus enhanced. In order to probe the velocity dependence of this phenomenon, we have employed the velocity selection by Doppler shift (VSDS) technique. This technique has been used previously to measure the velocity dependence of purely rotationally © 1996 American Chemical Society
7982 J. Phys. Chem., Vol. 100, No. 19, 1996 inelastic cross sections in Na2*-Xe5,6 and in Li2 colliding with Ne, Ar, and Xe.7,8 It makes possible wide ranges of selection for both ji and relative velocity Vrel; it thus turns out to be ideally suited for the study of rotationally induced quasi-resonant V T R transfer, because this phenomenon is most conspicuous at high ji and low Vrel, as is revealed in the present study. Supersonic molecular beam techniques, in contrast, are intrinsically limited to Vi ) 0 and low ji because of relaxation in the expansion and are generally characterized by higher velocity as well. In section II we review the VSDS technique and describe changes in it since the previous work. In section III we present the data as well as results of quasi-classical trajectory calculations on an ab initio potential energy surface for Li2*Ne, and in section IV we discuss the velocity dependence of V T R transfer. In section V we summarize the behavior and make comparisons with other work. II. Experiment The velocity selection by Doppler shift (VSDS) technique has been described extensively in its application to the measurement of rotationally inelastic cross sections.5,7 The Doppler shift is used in conjunction with the laser-induced fluorescence technique to prepare Li2 molecules in a specific vibrotational level in the A 1Σ+ u electronic state with a specific velocity component along the laser beam axis, VL. Collisions involving these level-selected electronically excited Li2 molecules and inert gas atoms contained in the same cell transfer population from the laser populated vibrotational level (which yields parent line fluorescence) to nearby vibrotational levels (which emit satellite line fluorescence). Magic-angle detection is employed to overcome the differential sensitivity of the monochromator to different fluorescence polarizations. The variation of the ratio of satellite line intensity to parent line intensity with target gas density is used to determine the rate constant for the singlecollision process that populates the upper state of each satellite line, resulting in a rate constant as a function of VL. The laser selected velocity VL determines the relative velocity distribution for collisions in the excited state. Proven numerical deconvolution techniques5,7 are used to obtain rate constants as a function of relative velocity k(Vrel) from the experimental rate constants k(VL). The VSDS technique, dependent only on relative fluorescence intensity ratios, determines only the relative dependence of the rate constant on collision velocity. Data taken to determine velocity-averaged rate constants4 are used both to provide absolute scaling of the cross sections and to correct the VSDS data for multiple-collision effects and reduction of the radiative lifetime due to quenching collisions. The techniques used in the acquisition and analysis of data were very similar to those used in earlier rotationally inelastic measurements in this system.7 In this section we describe only changes made in those techniques since the earlier work. These changes were motivated by problems with laser light rejection and with our Voigt fits to the parent line intensity dependence on laser frequency. In addition, we must depend to a greater extent on the velocity-averaged data to ensure that the spectral lines selected for VSDS are free from fluorescence interference, since the lack of known vibrationally inelastic scaling relationships makes the prediction of interference more difficult than in the previous rotationally inelastic studies.9 Because many of the measured rate constants are quite independent of velocity, we have striven to increase the signalto-noise ratio in our data so that our deconvolution is more sensitive to small deviations from the nearly 1/Vrel cross section dependence. We count photons over a longer period of time, as long as 10 s per detuning, and run the oven at a slightly
Scott et al. higher temperature, 607 °C, to increase the density of Li2 molecules and hence our signal. The use of greater laser powers has necessitated improvements in our Voigt fitting procedure. The parent line fluorescence intensity as a function of detuning is fit to a Voigt profile both to determine the Doppler width (and hence calibrate the laser selected velocity with laser frequency) and to determine the Lorentzian homogeneous width used to correct our data for the fraction of molecules in the cell excited off-resonance.7 When laser power was increased above the approximately 30 mW used previously, the optimum Voigt profile for data with detunings greater than 3 GHz was found to be systematically high at line center by about 10%. We have added a correction factor to our Voigt fit10 that accounts for velocity-dependent changes in the Lorentzian width, an effect enhanced by increased laser power (see also ref 11). Only a 10% variation in Lorentzian width is needed to account for the deviation of the experimental line shape from a true Voigt profile. This correction factor is used only for the Li2*-Xe data, the only data taken at the higher laser power of approximately 60 mW. Problems with laser light rejection have prompted some changes in our experimental procedure and data analysis. We employ a longitudinal heat pipe configuration that gives high signal rates but permits stray laser light to enter our double monochromator. This laser light adds a constant background signal to the parent intensity versus laser detuning data, causing a systematic error in the Lorentzian width determined from Voigt fits to the parent line shape. We now include the parent background in our fits to the parent line shape and have extended the frequency range of laser scans from (4.5 to (5.5 GHz so that the fits can better distinguish the background from the Voigt profile’s Lorentzian wings. The laser scattering background is not a problem for the satellite channel because, unlike the parent channel, the satellite is viewed through both stages of the double monochromator. The laser light was not a problem for the earlier rotationally inelastic VSDS experiment7 because the fluorescence band studied, the 9,7 band, was far enough in frequency from the 9,1 band used for laser excitation that the laser rejection of the first stage of our monochromator was sufficient. We perform the vibrationally inelastic experiment closer in frequency to the excitation frequency to take advantage of bands located there that have large Franck-Condon factors. The effect of the addition of two new fit parameters to the Voigt fits, parent intensity background and Lorentzian width velocity dependence, is to add some uncertainty to the value of some of the cross sections in the largest fifth of their velocity range. Only cross sections that differ significantly from a 1/Vrel dependence are affected. It is in this range and for these cross sections that the Lorentzian tail correction is significant and susceptible to uncertainties in the Lorentzian width increased by the two additional fitted unknowns. We estimate this uncertainty to be about 15% for the worst case of Li2*-Xe collisions, where the effects of large laser power were discovered for the ∆V ) -1 data and where exceptionally large parent line backgrounds were unavoidable (as large as 2 × 104 counts s-1 under a 3 × 106 counts s-1 line) for ∆V ) +1 data. For other systems studied, we kept the laser power at ≈30 mW and were successful in avoiding parent lines with large backgrounds. III. Results In this section we present the velocity-dependent absolute cross sections. Relative cross sections measured by the VSDS technique were normalized by the absolute velocity-averaged rate constants of ref 4. The error bars given in Figure 1 are typical of all the data and reflect only the errors of deconvolution
Li2*-Rare Gas Collisions
J. Phys. Chem., Vol. 100, No. 19, 1996 7983 TABLE 1: Energy Thresholds for Endoergic Vibrotationally Inelastic Li*2-X Collisions ∆V
jf
∆E/cm-1
-1
48 50 54 36 38 40 42 48
13.2 96.2 270.8 15.9 79.4 145.9 215.4 441.5
+1
Figure 1. Velocity dependence of the absolute vibrotationally inelastic cross sections for Li2*-Xe with Vi ) 9, ji ) 42, and ∆V ) -1. The data have been divided into two graphs (a) and (b) for clarity. Representative error bars are given; they are typical of the experimental uncertainties in all the cross sections reported in this work and do not include systematic errors, which we estimate to be on the order of 10% or less of the values of the cross sections. The velocity resolution is discussed in the text. Arrows indicate the locations of thresholds for endoergic processes (see Table 1).
and not the normalization error, which is generally smaller. Details of the uncertainties for all the data are given in ref 10. The VSDS method results in cross sections as a function of velocity in sp thermal units, where 1 thermal unit is given by sp ) xkToven/mLi2 ) 7.23 × 104 cm s-1. Velocity resolution is estimated to be better than 0.35, 0.45, and 0.60 sp respectively for Xe, Ar, and Ne.7 For Ne, the resolution is nearly independent of relative velocity, while for Xe, resolution improves with increasing relative velocity.7 The measurements range in velocity from 0.7 to 4.0 sp, a range of 5.06 × 104 cm s-1 e Vrel e 2.89 × 105 cm s-1. This corresponds to a range of collision energies 135 cm-1 e Erel e 4400 cm-1 in the case of X ) Xe. Inelastic channels were selected primarily to determine the velocity dependence of the quasi-resonant V T R energy transfer phenomenon. All our data are for Vi ) 9, ji ) 42 in the A 1 + Σu state of Li2*, the high initial j being essential to observe the phenomenon. We have measured the velocity dependence
of the cross section for five or more values of jf in the neighborhood of the quasi-resonant peak for ∆V ) -1 in Li2*Ne, Ar, and Xe and for ∆V ) +1 in Li2*-Xe. For Li2*-Xe, we have measured the velocity dependence of the peak cross section for ∆V ) -2 and -3 as well. A. Xe Collisions for ∆W < 0. Data for ∆V ) -1 collisions with Xe are presented in Figure 1. The level-to-level cross sections are large, approaching 8 Å2 for jf ) 46 at the lowest velocity measured. With the exception of jf ) 50 and 54, which have obvious energy thresholds, all these cross sections decrease with increasing relative velocity. Even inelastic channels with low velocity thresholds, jf ) 50 and 48, decrease for Vrel > 105 cm s-1. This behavior, characteristic of quasi-resonant V T R transfer, stands in marked contrast to the behavior ordinarily observed at thermal velocities, in which the vibrationally inelastic cross section rises with increasing relative velocity. Energy thresholds for the cross sections presented in this paper are tabulated in Table 1 and are indicated with arrows above the velocity scale of the figures. The jf ) 46 cross section is the largest cross section throughout the range of Vrel measured and is also the peak of the jf distribution of the thermally averaged rate constant data. At high velocity, however, it falls to the value of the jf ) 48 cross section to within error. The velocity dependence is approximately 1/Vrel. At low velocity, a number of cross sections increase strongly. The nearly energy resonant jf ) 48 and to a smaller extent jf ) 44 and 42 have roughly flat velocity dependences above about 105 cm s-1; below this point, the cross sections show sharp increases for diminishing velocity. This rise is great enough to suggest that σjf)48 may possibly overtake the peak σjf)46 for velocities lower than those resolvable by our experiment but above threshold (1.6 × 104 cm s-1). This divergent behavior as Vrel f 0 is not observed in Li2-Ne and Ar collisions (see section III.C). The only cross section lacking an energetic threshold that does not show enhancement with decreasing velocity is that of jf ) 22. For that channel, the cross section at low velocity flattens out. This behavior may be due to the rotational suppression that has been proposed to explain threshold-like behavior in large negative ∆j collisions in studies of rotational transfer in this system.7,12 At low velocities the time-integrated torque necessary for large changes in angular momentum is averaged away by the molecular rotation. Data for ∆V ) -2 collisions, shown in Figure 2, indicate that the peak position, as in ∆V ) -1, does not change with relative velocity. Three points, jf ) 48, 50, and 52, were selected from the velocity averaged rate constant peak for VSDS analysis. The velocity dependence of these three values of jf is very similar to that of ∆V ) -1, and the position of the peak is stationary at jf ) 50 for all velocities measured. The peak cross sections for ∆V ) -3, -2, and -1 are all similar, as shown in Figure 3. All show a 1/Vrel dependence. The ∆V ) -1 channel remains the largest inelastic channel for
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Figure 2. Velocity dependence of the absolute vibrotationally inelastic cross sections for Li2*-Xe with Vi ) 9, ji ) 42, and ∆V ) -2.
Figure 3. Velocity dependence of the peak inelastic cross sections for Li2*-Xe with Vi ) 9, ji ) 42, and ∆V ) -1, -2, and -3.
all measured velocities. The rule describing the position of the peak, jfpeak ) ji - 4∆V, observed in velocity averaged rate constants,1,4 holds at all velocities for negative ∆V. This rule corresponds to roughly 70% interconversion of vibrational and rotational energy, regardless of collision velocity or ∆V. B. Xe Collisions with ∆W ) +1. Data for ∆V ) +1 collisions are shown in Figure 4. The cross sections are somewhat smaller than for ∆V ) -1, about 5 Å2 at the lowest velocity measured. Since the quasi-resonant cross section is endoergic for ∆V > 0, the low-velocity dependence of many of the cross sections (jf g 38) is dominated by energy thresholds. For this reason, and in contrast to the data for ∆V < 0, the most probable inelastic channel varies with velocity. Channels not constrained by thresholds have cross sections that increase with decreasing Vrel. The largest thermally averaged rate constant for velocities near the thermal average of 1.2 × 105 cm s-1 is jf ) 38. It is exceeded, however, by the nearly energy resonant jf ) 36 (∆E ) 15.9 cm-1) cross section at low velocities and is only slightly larger than the cross sections for jf ) 34, 36, and 40 at high velocity. The very strong low velocity increase of channels without thresholds seen in the ∆V ) -1 data is also present for ∆V )
Figure 4. Velocity dependence of the absolute vibrotationally inelastic cross sections for Li*2-Xe with Vi ) 9, ji ) 42, and ∆V ) +1. The data have been divided into two graphs (a) and (b) for clarity.
+1. The cross sections for jf ) 36, 34, and 30 are roughly flat above 105 cm s-1, but increase rapidly with decreasing velocity. The jf ) 20 cross section does not show this behavior, presumably because of the adiabatic suppression referred to in the previous subsection. C. Ar and Ne Collisions. Data for Ar and Ne with ∆V ) -1, depicted in Figures 5 and 6, show velocity dependences that distinguish them from the Xe data. The Ne data have been summarized in a previous publication.2 Peak cross sections are still largest at the lowest velocity (about 5 and 3.5 Å2 at jf ) 46 for Ar and Ne, respectively), but the universal enhancement of exoergic cross sections near the quasi-resonant peak found in Xe at low velocity is not present. Instead, threshold-like suppressions are evident for all but the peak cross section, resulting in very narrow jf distributions at low Vrel. The effect is especially apparent for jf ) 42 and 40. For both Ar and Ne, as in Xe, the largest cross section is for jf ) 46 over the entire range of velocities measured. Also as in Xe, it is approached by the cross section for jf ) 44 and 48 at high velocities. The sharp rise of the jf ) 48 cross section observed in collisions with Xe at low velocities is not so evident in Ar and is nonexistent in Ne.
Li2*-Rare Gas Collisions
Figure 5. Velocity dependence of the absolute vibrotationally inelastic cross sections for Li2*-Ar with Vi ) 9, ji ) 42, and ∆V ) -1.
Figure 6. Velocity dependence of the absolute vibrotationally inelastic cross sections for Li2*-Ne with Vi ) 9, ji ) 42, and ∆V ) -1. Representative uncertainties are shown.
Dynamical thresholds with behavior similar to these have been observed previously in exoergic cross sections of rotationally inelastic processes with high ji and large negative ∆j.7 In that work, the rapid initial rotation of the Li2 molecules was suggested to be responsible, at low collision velocities, for averaging the short-range anisotropic torques responsible for the large ∆j changes. There the observed decrease in rotational suppression in going from Ne to Xe was explained by noting that the attractive well in the Li2*-Xe potential limited the minimum velocity with which atoms hit the respulsive core. We have calculated cross sections from quasi-classical trajectories13 run on an ab initio Li2 A 1Σ+ u - Ne potential surface;14 details of the method have been given previously.15,16 The calculated cross sections for ∆V ) -1 are shown in Figure 7. Agreement with the experimental results is quantitative for jf g 46 within the experimental velocity resolution of approximately 4 × 104 cm s-1; this is a remarkable success for the first quantitative comparison of experimental and calculated absolute velocity-dependent vibrotationally inelastic cross sections. For jf ) 42 and 44, the calculated cross sections are somewhat larger than the experimental cross sections, particularly in the case of jf ) 44. It appears that whereas the
J. Phys. Chem., Vol. 100, No. 19, 1996 7985
Figure 7. Velocity dependence of the absolute vibrotationally inelastic cross sections for Li2*-Ne with Vi ) 9, ji ) 42, and ∆V ) -1, calculated from quasi-classical trajectories on the ab initio potential energy surface of Alexander and Werner. Statistical uncertainties, where larger than the symbols, are shown.
calculation captures the absolute size and shape of cross sections with energetic thresholds, it is not as successful at reproducing exoergic cross sections with dynamical thresholds. Since the width of the jf distribution is about 2.5p, only slightly larger than the quasi-classical bin width, it is not surprising that classical trajectories binned using the standard histogram method with a bin width of 2p should broaden the distribution somewhat, but it is not clear why this should occur only for cross sections on the exoergic side of the peak. Another possible explanation lies in the fact that the distribution of atommolecule orientations in the experiment is not isotropic. The laser excites molecules with j in the plane normal to the plane defined by the laser direction and the polarization. This results in a distribution of angular momenta with respect to the collision velocity that is anisotropic in a way that changes with detuning.17,18 The effect is smallest for light collision partners such as Ne, and while it merits further consideration, it cannot explain excellent agreement for some cross sections at all relative velocities and relatively poor agreement for others. It is possible that classical mechanics cannot correctly reproduce the dynamical thresholds that result from the adiabatic reduction of these cross sections at low velocity. The cross sections calculated from classical trajectories may be compared with quantum mechanical cross sections obtained using the same potential surface.19 The authors of this study obtained results for Vi ) 9, ji ) 42, and Erel ) 123.8 cm-1 using the coupled-states approximation. As in the experimental results, their largest cross section at this energy is for jf ) 46, although at =0.9 Å2 it is more than 3 times smaller than the experimental and classical trajectory values. The next largest cross section is for jf ) 48 with the value =0.7 Å2, about half the experimental value. The relative sizes of the cross sections more nearly match the experiment than those from quasiclassical trajectories, but their absolute magnitudes are systematically small. It is known that the coupled-states approximation is inadequate to model rotationally inelastic cross sections in this system,14 although it is the shape and not the magnitude of the velocity-dependent cross section that it fails to reproduce. The discrepancies between the experiment, classical mechanical, and quantum mechanical results merit further study.
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Figure 9. Rotationally summed absolute vibrotationally inelastic cross sections for Li2*-Xe with Vi ) 9, ji ) 42, and ∆V ) (1. The quasiresonant enhancement of the peak cross sections at low Vrel dominates the overall vibrational transfer process at elevated ji.
Figure 8. Absolute vibrotationally inelastic cross sections for Li2*Xe with Vi ) 9, ji ) 42, and (a) ∆V ) -1, (b) ∆V ) +1, shown for four representative collision velocities as a function of jf. The symbols indicate the following velocities: O, Vrel ) 5.1 × 104 cm s-1; 0, Vrel ) 7.2 × 104 cm s-1; 4, Vrel ) 1.4 × 105 cm s-1; 3, Vrel ) 2.2 × 105 cm s-1. The quasi-resonant narrowing of the distribution at low velocities is apparent.
IV. Quasi-Resonant Enhancement at Low Velocity The data presented in section III reveal a dramatic increase in the vibrationally inelastic cross section as the relative velocity decreases. We now examine the data in greater detail and show that, in addition, the width of the quasi-resonant peak decreases with decreasing velocity. This effect is most pronounced in the cases of Ne and Ar due to the dynamical thresholds for cross sections adjacent to the peak that were noted in section III. A. Xe Collisions. The data for ∆V ) -1 collisions with Xe are presented in Figure 8a as a function of jf for four representative collision velocities. Once again, the dramatic growth of the quasi-resonant cross sections with decreasing velocity is obvious. This results in a decrease in the full width at half-maximum (fwhm) of the jf distribution; below 1.5 × 105 cm s-1, it is about 6p. The data for ∆V ) +1, shown in Figure 8b, are complicated by the presence of velocity thresholds. The quasi-resonant peaks in the jf distributions of the thermally averaged rate constants4 correspond to 70% V T R energy transfer; thus, the peak cross section for ∆V > 0 possesses an energy (velocity) threshold (as do all cross sections with jf > jfpeak; see Table 1). These thresholds result in a shift of the peak of the ∆V ) +1 distribution from nearly rotationally elastic at the highest measured velocity of 2.9 × 105 cm s-1, through the quasi-
resonant value of jf ) 38 at Vrel = 1.5 × 105 cm s-1, to jf ) 36 at the lowest velocities resolved. The width of the jf distribution at low velocity is narrower than that for ∆V ) -1 as a result, falling from about 15p at the highest measured velocity to only 4p at the lowest. Because the peak jf changes with collision velocity, the jf distribution of the thermally averaged rate constants for ∆V ) +1 is broader than the data of Figure 8b, evaluated at the average thermal velocity, would indicate (cf. Figure 2b of ref 4). Total vibrationally inelastic cross sections as a function of velocity appear in Figure 9. They were obtained by summing our data from jf ) 36 to 54 for ∆V ) -1 and from jf ) 30 to 48 for ∆V ) +1, with cross sections not measured estimated by linear interpolation. The range of jf included in the sums apparently suffices to determine σ∆V; extension of the data by linear extrapolation increases σ∆v by no more than one error bar at any velocity. The velocity dependence of σ∆V)-1 is roughly 1/Vrel, while for ∆V ) +1 it is nearly constant above 1.5 × 105 cm s-1; the two cross sections become identical to within error at Vrel ) 1.5 × 105 cm s-1, where the collision energy of 4420 cm-1 is much larger than the vibrational energy spacing of 255 cm-1. B. Ar and Ne Collisions. Comparison of the Ar and Ne data (Figures 10 and 11) with those for Xe (Figure 8a) reveals a qualitatively similar velocity dependence. At low Vrel, the quasi-resonant peak dominates the cross section. The peak cross section has an approximate 1/Vrel dependence for all three target gases. However, the almost exact similarity of the final state distributions found in ref 4 for the thermally averaged data does not hold for these more detailed measurements. On going from Xe to Ne, the width of the quasi-resonant peak decreases from 6p to about 2.5p at Vrel ) 5 × 104 cm s-1. This dramatic narrowing of the already narrow jf distribution results from the suppression of cross sections adjacent to the peak discussed in section III.C. Rotationally summed cross sections for Ar and Ne have a velocity dependence quite different from that for Xe. Because the decreasing width counters the rise of the peak cross section at low velocity, k∆V diminishes at the lowest velocities for these targets. While the range of data is not sufficient to determine k∆V accurately, it appears to possess a maximum in the range (0.7-1.5) × 105 cm s-1.
Li2*-Rare Gas Collisions
Figure 10. Absolute vibrotationally inelastic cross sections for Li2*Ar with Vi ) 9, ji ) 42, and ∆V ) -1, shown for four representative collision velocities as a function of jf. The symbols indicate the following velocities: O, Vrel ) 5.1 × 104 cm s-1; 0, Vrel ) 7.2 × 104 cm s-1; 4, Vrel ) 1.4 × 105 cm s-1; 3, Vrel ) 2.2 × 105 cm s-1. Dynamical thresholds at low collision velocity make the distribution narrower than in the xenon case.
Figure 11. Absolute vibrotationally inelastic cross sections for Li2*Ne with Vi ) 9, ji ) 42, and ∆V ) -1, shown for four representative collision velocities as a function of jf. The symbols indicate the following velocities: O, Vrel ) 5.1 × 104 cm s-1; 0, Vrel ) 7.2 × 104 cm s-1; 4, Vrel ) 1.4 × 105 cm s-1; 3, Vrel ) 2.2 × 105 cm s-1. Dynamical thresholds at low collision velocity are even more pronounced than in the argon case, resulting in a distribution with a fwhm of 2.5p at Vrel ) 5 × 104 cm s-1.
V. Summary and Conclusion We summarize here the results of this study in order to emphasize the unique features of the velocity dependence of the quasi-resonant V T R process and to facilitate comparison with other work. We also make comparisons with previous results on rotationally inelastic collisions in this system. 1. In marked contrast with vibrational transfer at low ji, the quasi-resonant peak in the cross sections increases dramatically as velocity decreases. This holds for all ∆V and all target gasses studied, as well as for the rotationally summed cross section for ∆V ) -1 collisions with Xe; all these cross sections have an approximate 1/Vrel velocity dependence. The increase is restricted to jf values near the peak; as shown in Figures 1 and 8, only three jf values contribute substantially to the vibrationally inelastic cross section at low velocities in Xe collisions. In Ar and especially Ne collisions, dynamical thresholds in cross sections for off-peak values of jf make the restriction even more dramatic. This behavior is in sharp contrast to that of previously
J. Phys. Chem., Vol. 100, No. 19, 1996 7987 measured rotationally inelastic cross sections in this system.7 In the rotationally inelastic case, cross sections for ji ) 42 and all jf were observed to decrease with decreasing velocity. In addition, whereas the quasi-resonant enhancement in the vibrotationally inelastic case results in increased cross section with increased ji, the rotationally inelastic cross section decreases as ji increases. 2. The fwhm of the jf distribution decreases as velocity decreases. This effect is most pronounced in collisions with Ar and Ne, for the last of which the fwhm is only 2.5p at the lowest collision velocity of 5 × 104 cm s-1 (corresponding to a kinetic energy of 88.4 cm-1). Although in the case of rotationally inelastic collisions, the determination of a width for the final state distribution is not as meaningful, comparison of cross section sizes for the ∆j ) (2 process with those for ∆j ) (4 indicates that no increase in final state specificity occurs as Vrel is lowered. 3. In contrast to the thermally averaged results of ref 4, the final state distribution and its velocity dependence depend on the target gas. This is due primarily to the suppression of cross sections adjacent to the peak that occurs at low velocity in Ar and Ne collisions and possibly also to the slightly different energy ranges of the measurements for the three target gases. 4. For ∆V ) +1, the position of the peak in the jf distribution changes with velocity, owing to energy thresholds that move the peak to smaller jf as velocity is lowered. 5. There may be an underlying nonresonant process that is responsible for changes in vibrational level at high velocity. This may be inferred from the broad, flat final state distributions apparent at Vrel g 1.5 × 105 cm s-1 in Figures 8, 10, and 11 and from the flattening out at large velocity of the velocity dependence of the rotationally summed cross sections shown in Figure 9. 6. Quasi-classical trajectory calculations on the AlexanderWerner ab initio potential reproduce Li2*-Ne collisions quantitatively for ∆V ) -1 and jf g 46. The discrepancies at lower jf suggest that the classical calculation may incorrectly capture the onset of adiabatic reduction in the cross section at low velocity. The present work constitutes the first detailed comparison between experimental and computed absolute cross sections for a vibrotationally inelastic process using an ab initio potential surface. Previous coupled-states calculations at a single energy19 substantially underestimate the cross section. A more detailed comparison between classical and quantal calculation and experiment is called for. Comparison with Other Work. As far as we know, this study is unprecedented in measuring the velocity dependence of the absolute vibrotationally inelastic cross section with both initial and final vibrational and rotational quantum levels specified. Moreover, there are few theoretical calculations at this level of detail that in addition include the high ji crucial to the quasi-resonant phenomenon. In the best studied case, H2He, it is well established that high ji enhances the vibrotationally inelastic cross section and increases its specificity. There have been several studies20-23 that have employed quantum mechanical methods on accurate potential surfaces to obtain energydependent cross sections at low to moderately high ji. The narrow jf distribution obtained for 6 e ji e 10 is found to become even narrower as the collision energy is decreased from 2 to 1 eV,21 a result similar to ours, although this effect does not persist at still lower velocity.22 However, the cross section is found to increase monotonically with increasing energy and is small, ranging from a maximum of 0.02 Å2 for the 1,10 f 0,12 process at 2 eV to less than 10-4 Å2 for all vibrotationally inelastic processes at 1.2 eV.
7988 J. Phys. Chem., Vol. 100, No. 19, 1996 One H2-He study stands out for the range of ji and Vrel employed,24 using quasi-classical trajectories to determine cross sections for ji as high as 24 and 0 e Erel e 4.3 eV. The cross sections determined for low ji and Erel < 1 eV agree remarkably well with the coupled-states results of ref 22. At ji ) 24, narrowing of the jf distribution at low energy is observed; in addition, the cross section falls with increasing energy above the dissociation threshold, which occurs at a collision energy of 0.26 eV for ji ) 24. Our experiments, in contrast, were carried out well below the dissociation threshold of Li2. Quasi-classical trajectory calculations on other systems also exhibit both similarities and differences when compared with our results. Thompson has studied vibrotational transfer in HFAr at energies ranging from 0.2 to 1.0 eV25 and in HCl-Ar at energies ranging from 0.3 to 0.8 eV.26 In the former system, he found strong enhancement of V T R transfer at elevated ji, but only a weak enhancement in the ∆V ) -1 cross section for Vi ) 4, ji ) 20 with declining velocity. In HCl-Ar, the systematic increase in the vibrationally inelastic cross section with decreasing velocity is also only weakly observable, as is the narrowing of the jf distribution. Our earlier publication detailing the dynamics of quasiresonant V T R transfer3 has stimulated numerous treatments of the phenomenon.19,27-29 Two of these publications dealt with the velocity dependence in detail. The work of Alexander et al.19 has been discussed in section III; in addition, Maricq28 has carried out coupled-states calculations on a breathingellipsoid model potential2 for Vi ) 5, ji ) 64. While we have no velocity-dependent experimental data for these initial quantum numbers, the results are consistent with the present results for X ) Ne, revealing a dramatic enhancement of the peak cross section with decreasing velocity and a suppression of off-peak cross sections, resulting in very sharp jf distributions for the lowest energy of the calculation (200 cm-1). In summary, we have thoroughly documented the velocity dependence of vibrotationally inelastic collisions in Li2* at high rotational angular momentum, providing additional information about the conditions necessary for the quasi-resonant V T R process. Quasi-classical trajectories on an ab initio potential energy surface for Li2-Ne agree with most of the measured cross sections to within experimental uncertainty; this constitutes as unprecedented level of agreement between measurements and calculations of absolute vibrationally inelastic cross sections. However, agreement is not quantitative for two measured channels. An extension of the coupled-states calculations of Alexander et al.19 to other collision velocities would help
Scott et al. establish the degree to which quasi-classical dynamics is suited to calculating cross sections for processes with narrow final state distributions. A fully close-coupled calculation, involving as it would thousands of states, remains out of reach. Acknowledgment. We acknowledge the National Science Foundation for the support of this research. We are indebted to Professor James L. Kinsey for many enjoyable and enlightening discussions during the time this work was being carried out. References and Notes (1) Saenger, K. L.; Smith, N.; Dexheimer, S. L.; Engelke, C.; Pritchard, D. E. J. Chem. Phys. 1983, 79, 4076. (2) Stewart, B.; Magill, P. D.; Scott, T. P.; Derouard, J.; Pritchard, D. E. Phys. ReV. Lett. 1988, 60, 282. (3) Magill, P. D.; Stewart, B.; Smith, N.; Pritchard, D. E. Phys. ReV. Lett. 1988, 60, 1943. (4) Magill, P. D.; Scott, T. P.; Smith, N.; Pritchard, D. E. J. Chem. Phys. 1989, 90, 7195. (5) Smith, N.; Brunner, T. A.; Pritchard, D. E. J. Chem. Phys. 1981, 74, 467. (6) Smith, N.; Brunner, T. A.; Karp, A. W.; Pritchard, D. E. Phys. ReV. Lett. 1979, 43, 693. (7) Smith, N.; Scott, T. P.; Pritchard, D. E. J. Chem. Phys. 1984, 81, 1229. (8) Smith, N.; Scott, T. P.; Pritchard, D. E. Chem. Phys. Lett. 1982, 90, 461. (9) We have recently found that an extremely simple scaling relationship captures the jf dependence of vibrotationally inelastic rate constants at high ji (Stewart, B. Chem. Phys. Lett. 1995, 245, 643). (10) Scott, T. P. Ph.D. thesis, M.I.T., unpublished, 1985. (11) Fell, C. P.; McCaffery, A. J.; Ticktin, A. J. Chem. Phys. 1989, 90, 852. (12) Smith, N. J. Chem. Phys. 1984, 81, 5625. (13) Smith, N. J. Chem. Phys. 1986, 85, 1987. (14) Alexander, M. H.; Werner, H.-J. J. Chem. Phys. 1991, 95, 6524. (15) Gao, Y.; Stewart, B. J. Chem. Phys. 1995, 103, 860. (16) Gao, Y.; Gorgone, P. S.; Davis, S.; McCall, E. K.; Stewart, B. J. Chem. Phys. 1996, 104, 1415. (17) Monchick, L. J. Chem. Phys. 1981, 75, 3377. (18) Bain, A. J.; McCaffery, A. J. J. Chem. Phys. 1984, 80, 5883. (19) Alexander, M. H.; Berning, A.; Degli-Esposti, A.; Jo¨rg, A.; Kliesch, A.; Werner, H.-J. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 1253. (20) Rabitz, H.; Zarur, G. J. Chem. Phys. 1974, 61, 5076. (21) Alexander, M. H. J. Chem. Phys. 1974, 61, 5167. (22) Alexander, M. H. J. Chem. Phys. 1977, 66, 4608. (23) Flower, D. R.; Kirkpatrick, D. J. J. Phys. B 1982, 15, 1701. (24) Dove, J. E.; Raynor, S.; Teitelbaum, H. Chem. Phys. 1980, 50, 175. (25) Thompson, D. L. J. Chem. Phys. 1982, 76, 5947. (26) Thompson, D. L. J. Phys. Chem. 1982, 86, 630. (27) Hoving, W. J.; Parson, R. Chem. Phys. Lett. 1989, 158, 222. (28) Maricq, M. M. Phys. ReV. A 1989, 39, 3710. (29) Miklavc, A.; Markovic´, N.; Nyman, G.; Harb, V.; Nordholm, S. J. Chem. Phys. 1992, 97, 3348.
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