J. Phys. Chem. 1995, 99, 1101-1114
1101
State-Resolved Cross Sections and Collision-Induced Alignment from Counterpropagating Beam Scattering of NH3 He
+
Henning Meyer Institut f i r Anorganische, Analytische und Physikalische Chemie, Universitat Bern, Freiestrasse 3, CH3000 Bern 9, Switzerland Received: September 22, 1994; In Final Form: November 21, 1994@
The rotational excitation of NH3 in collisions with He is investigated at a collision energy of 140 meV using the method of counterpropagating pulsed molecular beam scattering in combination with resonance-enhanced multiphoton ionization detection. Initial state preparation is achieved through the adiabatic expansion of N H 3 seeded in Ne. The scattered intensity exhibits a strong dependence on the direction of the laser polarization relative to the molecular beam axis. From the polarization dependence, degeneracy-averaged state-resolved integral and differential cross sections as well as their quadrupole moments could be determined for many final states in both modifications. The degeneracy-averaged cross sections are in reasonable agreement with the results of an earlier calculation based on a semiempirical potential surface. Final states, corresponding to large energy transfer, show strong negative collision-induced alignment for backward scattering. For states representing small energy transfer, we find a smaller degree of alignment which changes from positive to negative values with increasing scattering angle. This general behavior is consistent with a strong propensity for transitions with IAml , are labeled by the quantum number j for the rotational angular momentum and k for its component onto the symmetry axis of NH3. In addition, each level is characterized by a symmetry index E , indicating a symmetric or antisymmetric linear combination of symmetric top rotor functions. Levels, which differ only in the symmetry index, are probed in different vibronic bands of the B state due to symmetry restrictions. Furthermore, NH3 molecules can be distinguished according to the resultant total nuclear spin of the hydrogen atoms giving rise to the independent modifications of orthoand para-NH3. A summary of the relevant beam data is given in Table 1. Efficient rotational cooling for NH3 is achieved by expanding a mixture of 5% NH3 in Ne at a stagnation pressure of 2.0 bar. Details of the experimental procedures for the beam characterization have been given before.32 The He target beam is
+
''
Counterpropagating Beam Scattering of NH3
+ He
J. Phys. Chem., Vol. 99, No. 4, 1995 1103 Target
NH3 in He
too+>
1Bbar
752~s
loo+>
1
550
600
I
I
650
700
I
750
I
000
Delay TBI ps
Figure 2. Time profile of the He target beam. Traces of NH3 are detected in the scattering center as a function of the relative beam delay. At large delays the beginning of the target beam pulse is probed. TABLE 1: Characterization of the Molecular Beam Pulses primary beam target beam 5% NH3 in Ne He stagnation pressurebar 2.0 1.6 0.5 0.5 nozzle diameter/” nozzle to skimmer distance/mm 40 35 890 f 50 2024 f 85 average velocity/m s-’ velocity spread (fwhm)/% ‘15 6 relative pulse delay/ps 745 105 pulse width (fwhm)/ps 35 42 rotational temperatureK 13 collision enerev/meV 140 population/%.
loo+ > llO+>
rest Ill+> Ill-’ rest
80 19 = loo+>, was probed. The time profile of the He target beam pulse, detected in the scattering region, is displayed in Figure 2. The beginning of the pulse, which is probed at large delays, is represented very well by a Gaussian distribution with a half width of 30 ,us. At larger delays, the NH3 intensity forms a long tail. These two different regimes of the pulse can also be distinguished when local velocity distributions are probed through ion TOF spectra using the following field parameters: accelerating field, 3.7 V/cm; mirror field, 18.5 Vkm; MCP voltage, 1680 V. Examples are taken at the delays marked in Figure 2 and displayed in Figure 3. For comparison, also the TOF distribution for the central part of the primary beam pulse, detected with identical parameters for the ion optics but with greatly decreased laser
13500
14000 14500 TOF I ns
15000
Figure 3. Ion TOF distributions for various delays of the target beam. In the lower part the TOF distribution for the primary beam is superimposed. NH3 molecules from the primary beam reach the detector 1.3 ps before NH3 traces from the target beam source. The relative delay corresponds to a velocity difference of 2900 d s . intensity, is superimposed in the lower part of Figure 3. At the field strength of 3.7 Vkm, peaks from both beams are separated by nearly 1300 ns. From the half width (fwhm) of these distributions and their relative spacing, a velocity spread of 6% for the center of the He pulse can be deduced. In the tail of this pulse, we find NH3 molecules with a significantly broader distribution. The half width increases from 52 ns to about 80 ns. In addition, the peak shifts toward smaller flight times, indicating a decrease of the local velocity by 7%. When the TOF distributions are fitted with a Gaussian distribution (dashed curves in Figure 3), we find deviations of the local velocity distribution corresponding to a fast tail in the center and a slow tail in the end of this pulse. For the NH3 primary beam, we find an upper limit for the velocity spread of about 18% which is still influenced by broadening due to Coulomb repulsion. On weaker transitions, probing either 111 or 111- > , a half width corresponding to a velocity spread of 15% could be detected.
+>
3. Results and Discussion I. Integral Cross Sections. To measure integral and differential cross sections, the relative delay of the primary beam is fixed in such a way that the laser probes the maximum of this pulse in the scattering region. The delay for the target beam is determined by optimizing the scattering signal. In addition, this delay should be as close as possible to the delay at which the maximum of the target pulse is probed. In this way contributions to the scattering signal from the last part of the target beam pulse can be neglected. All scattering experiments have been performed with the delay TB = 742 ps. At this delay we find a depletion of the primary beam of (1 1 & 3)%, as can be seen in Figure 4. No influence of the laser polarization could be detected. The dependence of the scattered signal for the state Vke> = 143- > on the target beam delay is given in Figure 5. At the largest delays probed, the target beam is still far from the scattering center and only the thermal population of the 143-> state is probed. Superimposed onto this background is the time profile of the target beam recorded through traces of NH3 (dotted curve in Figure 5). Decreasing the delay further, both signals show the same onset and a very similar slope. The deviations at smaller delays are caused by the larger velocity range of the scattered particles. From the kinematic or Newton
Meyer
1104 J . Phys. Chem., Vol. 99,No. 4, 1995
I
Figure 6. Newton diagram for the counterpropagating beam scattering of NH? He: The solid circle represents elastic scattering, and the dotted circle, collisions with 50% energy transfer. The (in-plane) directions of the KA and GA axes are indicated for e,, = 50".
+
13200
13700
14200 14700 TOF I ns
15201
Figure 4. TOF distributions of the primary beam with the target beam turned on and off probed with the laser beam vertically (V) or horizontally (H) polarized with respect to the molecular beam axis. For clarity the spectra recorded with horizontal polarization are shifted by 1 pus toward larger flight times. The difference spectra in the lower part represent a depletion of the primary beam of 11 =k 3%. 20 I
I
NH3 - He 140 meV
I
I
t
3xp. d
0 3041 0
30450
30490
Frequency I cm-1
Figure 7. Frequency dependence of the scattered intensity with the laser horizontally polarized: (a) = simulated spectrum Sf;(b) = experimental spectrum Sf; (c) = simulated spectrum Ti; (d) = experimental spectrum Ti. The region of 0- and P-branches of the B(2) vibronic band is probed. 500
600 700 Delay TB I ps
000
Figure 5. Dependence of the scattered flux in like> = 143-> as a function of the target beam delay. Superimposed, as the dotted line, is the time profile of the target beam. diagram in Figure 6, we see that particles scattered into the backward direction are characterized by very small velocities in the laboratory (LAB) frame. These molecules reach the laser probe volume only some time after the maximum of the target beam has passed the same probe volume. Again, the delay chosen for the scattering experiments is marked by an arrow in Figure 5. To determine state-resolved integral cross sections, the scattered intensity is measured as a function of the laser frequency and the direction p of the laser polarization with respect to the molecular beam axis. Spectra are recorded with horizontal (labeled H, p = 0") and vertical (labeled V, p = 90°) polarization. Typical examples, probing NH3 through the B(2) and B(3) band with the laser horizontally polarized, are displayed in Figures 7 and 8. Experimentally, a base line subtraction mode (BSM) is employed to determine simultaneously the population change, Anf due to scattering and the
thermal population nfthem of a specific final state. In this mode, the target beam source is operated at 5 Hz, corresponding to half the frequency of the primary beam source. The signal from the detector is divided and recorded with two boxcar averagers. One detects the 5 Hz component T O ) of the signal while the other is operated in the normal averaging mode, detecting a signal Stcp):
/3 = angle of polarization Here we have introduced the factor Df,which describes the reduction of the thermal population due to scattering from the target beam. As expected from the small depletion of the primary beam, the positive scattering signals are typically 2 orders of magnitude smaller than the population of the initial states. Therefore, for various final states, we find population changes which are comparable to the thermal population in the beam. Under these circumstances, it is important to take into
+ He
Counterpropagating Beam Scattering of NH3
J. Phys. Chem., Vol. 99, No. 4, 1995 1105 TABLE 2: Degeneracy-Averaged Population Changes for Different Final State*
NH3 - He
/
like>
Sf
140 meV
21 22 31 32 41 *42 44 51 52 54
668.0 703.0 232.1 208.2 84.3 78.5 194.8 60.2 46.0 101.3 72.0 21.5 31.5 37.4 57.9 18.5 25.0
p-NH3 119.0 117.0 72.5 47.6 24.7 24.2 68.9 14.0 13.3 56.2 17.2 7.5 15.1 15.1 26.2 11.0 14.0
604.0 294.4 453.6 590.4 142.7 136.4 417.1 68.7 66.4 85.2 41.4 44.8 35.5 18.0 84.4
100.1 78.6 38.5 202.7 52.8 35.2 286.0 41.6 12.2 37.3 14.3 9.6 12.5 9.8 64.8
55
860
30900 30940 Frequency / cm-1
3098C
Figure 8. Frequency dependence of the scattered intensity with the laser polarized horizontally. Labels as in Figure 7. The region of 0and P-branches of the B(3) vibronic band is probed. Marked lines are due to either I5NH3 = 15) or the 2’1-hot band transition in I4NH3 (mlon= 14).34 Since only very small densities of these two species are found in the beam, they do not contribute to the scattered flux and, consequently,related resonances only appear in the experimental S type spectrum labeled b. account the depletion Df of the thermal population due to the target beam. For simplicity, we assume Df to be independent of the final state and identical to the value found for the ground states of both modifications: Df = 0.89 & 0.03. In addition, Df should not depend on the direction of the laser polarization. These assumptions are confirmed experimentally for final states representing a large energy transfer. Since the corresponding TOF spectra (see below) are dominated by backward scattering, the depletion of the thermal population is not obscured by the scattering signal and found to be in agreement with the assumptions given above. On the other hand, as was shown previously, some rotational lines exhibit a strong polarization dependence for the scattered intensity.33 This effect is especially pronounced when the excitation is accompanied by large energy transfer. Due to the cylindrical symmetry of the scattering setup, the polarization dependence in a two-photon absorption is caused only by the nonvanishing state multipole moments A t ’ and A t ’ as defined in ref 35. Although, in principle, these moments could be determined from a detailed polarization study, we only derive degeneracy-averaged ICSs from these spectra. This allows us to perform frequency scans with a relatively small number of laser shots averaged, e.g. 30 or 100 shots, while influences due to long term drifts in laser power, molecular beam densities, etc. are minimized. Reliable quadrupole moments are derived from ion TOF spectra which represent averages over up to 4000 laser shots. If we neglect the contribution due to the moment A:), the monopole moment IC0) is calculated from a weighted average of intensities recorded with the laser polarized horizontally and ~ e r t i c a l l y : ~ ~
to’= ‘3(ZH
+ 21,)
and
Tfl2
I
Z = S, T as defined in eq 1 (2)
The quantities, Sf(O) and TfcO),are determined in a least square fit procedure while the related population changes and thermal populations are calculated using the inverse of eq 1. Results are given in Table 2. The error limits are estimated to be f10%
61 62 *64 *65 *74 75
Anr
ndthermall
298.5 298.4 162.5 112.8 55.9 54.4 151.6 33.1 30.2 117.4 40.4 16.5 39.0 32.6 55.9 22.9 29.2
549.0 586.0 159.7 160.6 59.6 54.3 125.9 46.2 32.1 45.1 54.8 14.0 23.8 22.4 31.7 7.5 11.0
255.6 180.9 122.7 448.0 115.6 81.5 586.5 86.1 30.3 79.9 31.6 23.0 27.5 20.6 131.8
503.9 215.8 415.1 387.7 89.9 101.2 131.0 27.2 54.2 47.9 27.1 35.2 23.0 8.2 19.6
O-NH3
20+ 30+ 33+ 33*40+ 43+ 4350+
53+ 5360+ *63+ *63*73*76+
a A depletion of Df= 0.89 was used for the determination of Anf. The polarization dependence for final states marked with an asterisk
could be determined from TOF spectra. while population changes for states with jf= 2 have an estimated uncertainty of f 1 5 % due to the larger thermal population. Probing states of the para modification in both vibronic bands with different symmetry, no significant differences in the population changes could be detected. Therefore, we omit the symmetry quantum number E for the final states of p-NH3 in Table 2. With the achieved initial state preparation, we find the following relationship between population changes and integral cross sections: o-NH,:
Anf = C(08 .00+ ,,
+ 0.19alO+,,)
p-NH,:
Anf = C(0.570,,+,~
+ 0.43a,1-4f)
(3)
and
While population changes for ortho-NH3 are mainly determined by the fully state-resolved cross section om++, parity-averaged cross sections are determined for para-NH3. For pNH3 a maximum energy transfer of 45% of the collision energy is observed. The overall decrease of the ICSs with increasing energy transfer is well described by an exponential energy gap law37with a = 9.2 f 1.8, as can be seen in Figure 9:
(4) If we interpret the ratio of E,, and a as a temperature, we find that the ensemble of scattered NH3 molecules has been heated
Meyer
1106 J. Phys. Chem., Vol. 99, No. 4, 1995 1o2
33
43
O
D
0
0
+
NH3 - He 140 meV 1130 cm-1
G $
1
'5 10 -$
. -+ c? -- . 7
.-
o
c
10
100
100 200 300 400 Energy transfer / cm-1
500
Figure 9. Degeneracy-averaged ICSs as a function of the energy transfer. Several final states are marked. The solid line represents the exponential energy gap law with a = 9.0 1.8. For clarity the ICSs for p-NH3 have been multiplied by 0.1. The dash-dot-dot line connects the ICSs for the excitation of CS-forbidden transitions.
+
to a temperature of nearly 180 K in a single collision. Deviations from the scaling law indicate a propensity for collisions with Ak = f 3 . Large cross sections are found for the final states 144>, 154>, 164>, and 174'. Especially the ICs for 154> is significantly larger than other cross sections leading to slightly smaller or larger energy transfer. For the largest energy transfers, we observe also a strong propensity for the levels 165> and 175>. Excitation to these levels cannot be mediated by the potential direct, since only states with Ak = 1 3 n are coupled. A similar picture arises for the ortho modification. Here we also observe a strong decrease of the ICSs with increasing energy transfer which can be represented by a scaling law similar to the one found for p-NH3. A strong preference for transitions with Ak = 3 is observed. Of the probed final states, we observe the largest cross section for the excitation of the 143-> state. Similar to the ICs for 1 5 4 f > , this transition is characterized by Aj = 4 and Ak = 3. Thus, the excitation will be mediated by the potential expansion terms V43 or a combination of the dominant terms Vlo and V33. For final j values larger than 5 , we find the propensity for Ak = 3 transitions considerably relaxed. The cross section for the excitation of the state 176+> is very large and deviates considerably from the scaling law. In addition, for levels with k t 0, two different symmetry states within the ortho modification can be probed experimentally. So far only pairs of final states with k = 3 have been probed. A clear preference for the states with negative symmetry number is found. This propensity is easily understood in terms of the centrifugal sudden (CS) a p p r o ~ i m a t i o n . ' ~ Within ,'~ this approximation the centrifugal coupling of different magnetic sublevels in the body-fixed frame is neglected, resulting in a selection rule cf = (-)kf for transitions starting from the ground state of O - N H ~ With . ~ ~ increasing energy transfer this preference diminishes, leading to very similar ICSs for the 163' level. This is not surprising, since the CS approximation is especially valid for transitions resulting in small energy transfer. l 2 A number of theoretical calculation^^^^^.^^ of inelastic cross sections for NH3 He have been performed while ICSs have been measured recently by Schleipen and ter Meulen in a crossed beam experiment.1° The results of these studies are listed for comparison with the present data in Tables 3 and 4.
+
TABLE 3: Comparison of Degeneracy-Averaged Integral Cross Sections Z(O)Given in AZfor p-NH3 He" ref 10 this work ref 39 ref 8 140 meV 98meV 54meV Ilk> 65meV 1.76 3.91 1.77 2.67 21 1.56 1.76 0.57 1.37 22 1.29 1.16 1.24 0.96 31 1.17 0.72 0.86 1.24 32 0.32 0.50 41 0.31 0.31 0.35 0.69 42 0.50 0.45 1.46 0.89 44 1.50 1.58 0.19 0.17 51 0.18 52 0.06 0.69 0.61 54 0.80 0.24 0.42 0.33 55 0.09 0.05 61 0.23 0.01 62 0.19 0.02 64 0.33 65 0.28 0.13 0.00 74 0.17 75 0.03 ICSs of this work have been scaled in such a way that the cross section for 143-2 coincides with the experimental result of ref 10.
TABLE 4: Comparison of Degeneracy-Averaged Integral Cross Sections Z(O)Given in A2 for o-NH1 He" ref 39 ref 8 ref 10 this work jlkc> 65meV 98meV 54meV 140 meV 4.53 1.51 6.20 5.16 20+ 1.07 1.29 1.37 1.03 30+ 1.30 0.72 0.04 0.00 33+ 2.69 6.68 4.11 2.64 330.53 0.81 0.50 0.59 40+ 0.62 0.48 0.04 0.00 43+ 3.46 3.46 3.16 4.48 430.51 0.38 50+ 0.18 53+ 0.00 0.47 0.22 530.19 60+ 0.06 0.18 63+ 0.00 0.22 630.02 0.17 0.01 731.10 0.62 76+
+
ICSs of this work have been scaled in such a way that the cross section for 143-2 coincides with the experimental result of ref 10. Since in our work no absolute ICSs are determined, we scale our results in such a way that the cross section for the 143-> excitation coincides with the result of ref 10. Although the collision energies vary by more than a factor of 2 , we find satisfactory agreement for all studies. Comparing our data with the experimental results of Schleipen et al., we find a substantial smoothing of the data due to the increased collision energy. This effect is especially pronounced for p-NH3 and can also be noticed for the results of the CS calculation at 98 meV in ref 8. Furthermore, with increasing collision energy the scattered flux is distributed over many more channels. Simultaneously, the Ak = 3 propensity is reduced for states which represent intermediate and large energy transfer. On the other hand, final rotational levels with large values of k are favored. Especially the cross section for the excitation of the 176+> level, a CSallowed transition, becomes clearly dominant at higher collision energies. 11. Differential Cross Sections. DCSs are determined from TOF spectra recorded for different laser frequencies and directions of polarization using the field parameters given in the experimental section. Due to the small radius of the Newton sphere in velocity space, the off-axis component of the final velocity vector of the scattered particles is restricted to values less than 600 d s . Therefore, during the total flight time of
Counterpropagating Beam Scattering of NH3
+ He
J. Phys. Chem., Vol. 99, No. 4, 1995 1107
the ions of about 14 ps, a maximum off-axis distance of 8.5 mm can be traversed. This ensures that all trajectories are detected by the 25 mm diameter microchannel plate detector. A different baseline subtraction mode is applied in order to distinguish between the scattered particles and the thermal population. Now both sources are operated at a repetition rate of 10 Hz, and pairs of TOF spectra are recorded consecutively: one with both beams properly overlapped and one with the target beam fired 1 ms after the laser pulse. For each single TOF spectrum, 400 shots are accumulated and the resulting speck” is transferred to a PC for further processing. A typical difference spectrum is obtained from the average of 10 pairs of TOF spectra. By storing both types of TOF spectra, the depletion of the thermal population due to the target beam can be taken into account. In total, the accumulation of a difference TOF spectrum, representing a state-resolved polarization dependent differential cross section, takes less than 15 min. In order to determine the first two nonvanishing multipole moments of the cross section, difference spectra are measured with the laser beam vertically and horizontally polarized with respect to the molecular beam axis. The procedure to extract differential cross sections in the center-of-mass frame has been described in detail p r e v i ~ u s l y . ~ ~ In addition to the scattering kinematics, the distribution functions for each molecular beam pulse, like intensity, velocity, and time profile, are also incorporated into a Monte Carlo simulation of an apparatus function G. This function depends parametrically on the relative delays of all three pulses and the shape of the detection volume. Since the laser beam is focused with a 300 mm lens, the detection volume is approximated by a cylinder with a radius of 250 p m and a length of 10 mm. In principle, the apparatus function should describe the correlation between the center-of-mass (c.m.) scattering angle 0 and the ion TOF t. To decouple the influences of the scattering geometry and kinematics from the ion imaging optics, an apparatus function G(cos 0,vk, vfz) is calculated which describes the correlation between scattering angle (measured through cos O), the onaxis (vfz), and one off-axis (vk) velocity component. The cylindrical symmetry for the ion detection is only preserved when the off-axis velocity component of the ion can be neglected against the on-axis component at the end of the accelerating field. Since in our experiment this is only approximately true, a knowledge of the dependence on one offaxis velocity component of the neutral is necessary to describe correctly the ion trajectories. Therefore, in a second step, the function G(cos O,vk,vfz)is transformed into +e final correlation function COS 0 , t ) by applying an operator P,which describes the electric fields used for the ion TOF analysis. This operator projects a point in three-dimensional velocity space (representing the initial conditions for a specific ion trajectory) onto the onedimensional ion TOF space. A similar procedure can be applied for the detection of species whose Dopppler profile cannot be neglected. Having defined the apparatus function, a measured TOF spectrum can be related to the differential cross section in the following way: F(t) = J[hG(cos O,v,,vfz)](cos @,t) o(O)d cos 0
=
COS 0 , t )a(@) d cos O
(5)
With the help of eq 5 , known c.m. velocity distributions or DCSs are transformed easily into TOF spectra for comparison with experimental data. While this forward convolution procedure is unique, assumptions have to be made when DCSs are to be extracted from the experimental TOF data. To avoid the necessity of assuming a specific analytical form of the DCS,
Figure 10. Difference TOF spectra for various final states of p-NH3. The type of transition is indicated as follows: Aj = 0, Q;Aj = -1, P Ak = +1, r. For these lines no polarization dependence is found within the experimental uncertainties.
we assume the cross sections to be constant over specific angle intervals. The constants are determined in a least squares fitting procedure. In practice, we find that the number of intervals is restricted by the achieved angular resolution as well as the signal-to-noiseratio. If the number is chosen too large, a highly oscillating cross section results, which is physically not meaningful. For the NH3-He system, typically five to eight intervals are used. The determined step functions are then approximated by a linear or quadratic angular dependence which usually gives a slightly better fit to the experimental TOF spectra. From these considerations, we derive an angular resolution of A(cos 0)= 0.3. The uncertainty for the DCSs is determined by the experimental error of the TOF spectra and the uncertainty introduced by the model for the apparatus function. Delay studies for the NH3-Ar system demonstrated that the error introduced by the model can be estimated to be &15%. Since this error, which affects all DCSs in the same way, clearly exceeds the uncertainty for the TOF spectra, we assume an upper limit of &15% for the error of the DCSs. Typical normalized TOF difference spectra are shown in Figures 10-13. NH3 molecules scattered into the forward direction arrive after 13.7 ps at the detector while NH3 scattered into the backward direction arrives 0.5 ps later. Careful inspection also reveals the narrowing of the TOF distribution caused by the contraction of the Newton sphere with increasing energy transfer. With increasing energy transfer, we see a clear shift of intensity toward flight times corresponding to backward scattering. At the largest energy transfers, e.g. 173-> or 174- > , no intensity for forward scattering is observed. In these spectra the depletion of the thermal population caused by the target beam is found to be identical to the depletion of the prepared initial states within the experimental uncertainty: Df = 0.89 & 0.03. While the spectra recorded on P- and Q-branch lines exhibit no strong polarization dependence, we observe a pronounced polarization effect for 0-branch lines. Using eq 5 , polarization dependent DCSs are fitted, which in turn are used to derive the degeneracy-averaged DCSs according to eq 2. Since the angular dependence of the cross section is determined in a fit to the normalized difference TOF spectra, relative DCSs for different final states are derived simply by multiplying the fitted DCS with the population change determined in the previous section. The resulting DCSs, which are
Meyer
1108 J. Phys. Chem., Vol. 99, No. 4, 1995 oNH3-He
b
I
.3 c x
. .-
$ 5 23 v)
a c
+ c -
0 13400
13000 14200 TOF I n s
14600
Figure 11. Difference TOF spectra for various final states of 0-NH3. The type of transition is indicated as in Figure 10. displayed in Figures 14 and 15, can be grouped into three categories. Cross sections for transitions involving small energy transfer are characterized by a maximum at cos 0 = 1 and an exponential decrease toward larger scattering angles. Excitation to states representing intermediate energy transfer, e.g. the levels 132>, 142>, and 154>, results in a maximum which shifts to larger angles with increasing energy transfer, a behavior characteristic of a rotational rainbow.40 And finally, those states corresponding to large energy transfer show a threshold behavior and a maximum excitation probability for backward scattering. Comparing the angular dependence with results from the CS calculation in ref 8 at a collision energy of 98 meV, we find good qualitative agreement. For the calculation, a semiempirical potential model based on SCF calculations with damped long range dispersion coefficients has been used. The damping function was adjusted in order to reproduce experimental energy loss spectra. Apart from the diffraction oscillations in the small angle regime, which cannot be resolved with our experimental angular resolution, we find a very similar angular dependence for many DCSs. For 0-NH3, the calculation predicts the 133-> cross section to be dominant at small angles, even larger than the 120+> cross section. In addition, a shoulder is observed at intermediate angles. For large angle scattering, the 143->
is the dominant channel, which is also found in our experiment. For the )76+> cross section, the calculation predicts a threshold behavior and the cross section approaches the one for the 133-> excitation at the largest angles. In our experiment, we find the 176+> cross section for backward scattering to be even larger than the ones for the 133-> and 120+>, which might be due to the higher collision energy. For p-NH3, each initial state, bkc> = I l l f l > , has three magnetic sublevels, and theoretical cross sections have to be averaged over the initial magnetic quantum number. Nevertheless, the CS calculations for individual projections Q of the initial angular momentum predict a similar ordering of the DCSs as observed experimentally. At small angles, the 121’ DCS is dominant. In agreement with experiment, the DCS for 131> decreases at small angles while the 132’ shows a threshold behavior. The difference in their experimentally determined counterparts is less pronounced, which again might be caused by the difference in collision energy. Also the DCSs for the excitation of 154’ and 165> exhibit a clear threshold behavior. As pointed out in the previous section, parity pairs of final states in the ortho modification yield very different ICSs with a clear propensity for states with k = 3 and E = -1. The question arises if these cross sections show also a different angular dependence. So far TOF spectra have been recorded for the pairs with Vk> = 133> and 163’. While the 133> states have been probed using P-branch lines, which show no polarization dependence, 163> states are probed on lines of the 0-branch. As can be seen in Figure 16, a strong polarization dependence is observed. The resulting degeneracy-averaged DCSs are displayed in Figure 17. As expected from results for the ICSs, we find a DCS for the 133-> excitation which is considerably larger than the DCS for the CS-forbidden transition to 133+>. On the other hand the angular dependence for both DCSs is very similar over the whole range. This holds true also for the 1631> pair, which represents an energy transfer of about 33%. For the 163-> DCS, careful inspection reveals a slightly steeper rise for small scattering angles, which is associated with the shoulder in the TOF spectrum. Both cross sections have nearly the same value for large angle scattering. This also holds true for the pair of DCSs for the 133> level when we compare normalized cross sections. More intensity is found at small scattering angles for the CS-allowed transition while at intermediate angles a steeper slope is observed for the CS-forbidden state. This can be understood in terms of the centrifugal coupling, which is neglected in the CS approximapNH3- He 140 meV horizontal
140 meV vertical
-h 132->
0
13400
13000
14200
TOF I ns
14600
-13400
13800
132-> 14200
14600
TOF I ns
Figure 12. Difference TOF spectra for various final states for p-NH3 with the laser polarized horizontally and vertically. Spectra for a specific final state are displayed with the correct relative intensity. All states are probed on lines of the 0-branch.
Counterpropagating Beam Scattering of NH3
+ He
J. Phys. Chem., Vol. 99, No. 4, 1995 1109 I
ONH3140 meV He
vertical
30
10 0
a 140+>
13400
13800 14200 TOF / ns
14600
Figure 13. Difference TOF wectra for various final states for 0-NH3 with the laser polarized horizontally and vertically. All states are probed on lines of the 0-branch.
lo3,
L-54 140 meV
l o-1.0 o
0.0
0.5
1.0
cos e
cos e
Figure 14. Degeneracy-averaged differential cross sections for different final states Ofp-NH3. The relative magnitude of the DCSs is determined by the overall population change given in Table 2, while the angular dependence is extracted from difference TOF spectra. tion. The small differences can be attributed to a direct mechanism which changes simultaneously the projection s2 as well as the magnitude of the rotational angular momentum. Since this direct mechanism will be most effective for large impact parameters, we expect it to influence mainly collisions resulting in forward scattering. On the other hand, the close similarity of the angular dependence suggests that the dominant S2 coupling takes place already at large distances with nearly no rotational excitation. In a second step, the final rotational excitation is then caused by the anisotropy of the repulsive wall probed at smaller internuclear distances. III. Collision-InducedAlignment. The procedure to extract degeneracy-averaged cross sections and alignment parameters has been described p r e v i o ~ s l y .The ~ ~ polarization dependence of the detected scattering intensity can be described in terms of different state multipole moments T o ( Q ) . ~ ~ ~ ~ ’
+
-0.5
+
Z ~ , k j , ; P= ) C(To(o)p(0) To(2’~(2’@)T0’4)p(4’@)) (6) The To@)values are directly proportional to the corresponding moments Ao(Q) defined by Green and Zare.35
Figure 15. Degeneracy-averaged differential cross sections for different final states of o-NH3. The relative magnitude of the DCSs is determined by the overall population change given in Table 2, while the angular dependence is extracted from difference TOF spectra. For the 2-photon transition in NH3, the functions p(Q)are products of a Legendre polynomial of order Q, an angular momentum factor, and a line strength factor:
The factor G@)describes the depolarization due tb unresolved fine or hyperfine structure. For the excited state, no depolarization is expected, since for the rotational levels of the B state a natural line width corresponding to a lifetime of only 6 ps is observed e ~ p e r i m e n t a l l y . ~For ~ the electronic ground state, depolarization will result from the hyperfine interaction. Since the time interval between a collision and the detection of the product will be on average several microseconds, any interaction of the nuclear spins and the rotational angular momentum vector resulting in splittings of more than a few hundred kilohertz will
1110 J. Phys. Chem., Vol. 99, No. 4, 1995
n
oNH3- He 140 meV
I 1
I
I
13300
Meyer
14300 TOF I ns
TABLE 5: Depolarization Factors G@) Due to Different Hyperfine Interactions for Various Rotational Angular Momenta jia
I
15300
Figure 16. TOF difference spectra for the excitation of the symmetry pair jike> = 163&> recorded with the laser polarized horizontally (H) and vertically (V).
2 3 4 5 6 7 8 9 10
0.47 0.70 0.81 0.87 0.91 0.93 0.94 0.96 0.97
0.37 0.61 0.75 0.83 0.87 0.91 0.93 0.94 0.95
0.20 0.38 0.55 0.67 0.75 0.81 0.85 0.88 0.90
2 3 4 5 6 7 8 9
0.23 0.25 0.46 0.6 1 0.71 0.78 0.82 0.86 0.88
0.08 0.16 0.35 0.51 0.63 0.7 1 0.77 0.81 0.84
0.04 0.06 0.14 0.25 0.38 0.48 0.57 0.64 0.70
10
“ I n the first column only the interaction due to the nuclear quadrupole moment of 14Nis taken into account. For the other columns the magnetic hyperfine interaction for p-NH? and o-NH3 is included assuming that the relevant time scale of the experiment is much longer than the resulting precessional period.
interval between collision and ionization. In this case the overall d e p o l a r i z a t i ~ nis~ ~given . ~ by
E CS forbidden ___---_-_----------
loo -1.0
-0.5
0.0
0.5
1.0
cos e
Figure 17. Degeneracy-averaged differential cross sections for the excitation of the symmetry pairs like> = 1331’ and 163f> of o-NH?. Dashed lines represent the CS-forbidden transitions. cause a noticeable depolarization of the alignment. For NH3, the largest splitting is caused by the interaction of the nuclear quadrupole moment which couples the nuclear spin IN of the N atom to the rotational angular momentum vector j to form the resultant F1. A value of -4.084 MHz is reported for the coupling constant eqQ(14N).42 The depolarization can be expressed quantitatively in terms of a depolarization factor G(Q)(j).43,44While the splitting depends on the quantum number k, this dependence disappears if an average over time intervals which are long compared to the precessional period has to be taken. Values of G(Q)for Q = 2 and Q = 4 are given in the first column of Table 5 . Only for small j values, the depolarization exceeds the experimental uncertainties. Additional effects might be caused by the magnetic hyperfine interaction due to the H atoms. In this case the total nuclear spin ZH of the H atoms (IH = l/2 for p-NH3, ZH = 3/2 for o-NH3) will couple with F1 to form the total angular momentum F. Typical . ~ ~upper splittings are on the order of less than 100 ~ C H Z An limit due to this mechanism can be derived when the precessional period is assumed to be short compared to the time
The effective G factors for o-NH3 (IH = 3/z) and p-NH3 (IH = l/2) are given in Table 5 as well. The largest depolarization effects occur for small j , values and for the hexadecupole moment To(4). Since eq 8 gives only an upper limit for the depolarization, which might not be reached experimentally due to the small hyperfine splitting, the reported values for the quadrupole moments have not been corrected for any depolarization. For the extraction of alignment parameters, the 1 type doubling,34which causes a coupling of final states with kf = k, - 1 and kf = k, 1, has been neglected in eq 7. Due to this coupling, both k dependent 3 - j symbols in eq 7 are weighted with an expansion coefficient and contribute only to the overall line strength. Since no coupling of states, which differ in M , occurs, the general structure of eq 7 will not be affected. The functions P(Q)@)are proportional to the square of the bodyfixed matrix element p l of~the two-photon transition operator between rotationless vibronic states. Therefore this factor can be moved into the overall proportionality constant. In principle, the multipole moments can be determined on rotational lines of all branches. In practice, it is found that lines of the O-branch are especially suited to determine the first alignment moment On the other hand, lines of the P-branch show only little modulation due to the quadrupole moment, which makes them more suitable to the determination of the next higher moment A0(4). If we neglect the contribution due to the hexadecupole moment, the quadrupole moment can be determined from the intensities IH and IV in the following way:36
+
Counterpropagating Beam Scattering of NH3
+ He
J. Phys. Chem., Vol. 99, No. 4, 1995 1111 TABLE 6: Degeneracy-AveragedInte ral Cross Sections
(9) Numerical values for the constants t,which depend on vectorcoupling coefficients of the involved angular momenta, and the proportionality constant CgJ2)have been tabulated in ref 36. In the case of a significant contribution due to a A0(4)moment, eq 9 is still valid approximately if the intensities are determined from a line of the 0-branch (Aj = -2) or the S-branch (Aj = +2). Experimentally, TOF difference spectra were recorded with the laser polarized vertically or horizontally with respect to the molecular beam axis. Due to the complicated and congested spectrum, final states could be detected unambiguously only on one or two different transitions. Nevertheless, 0-branch lines have been probed for a variety of final states in both modifications. Examples are given in Figures 12 and 13. The TOF spectra show an increased intensity at flight times corresponding to small angle scattering when the laser is polarized vertically. This behavior is especially pronounced for final states corresponding to small energy transfer. For large angle scattering, we find a strong increase in scattered intensity when the laser is polarized horizontally. At the largest observed energy transfers, the intensity increases by more than a factor of 2. The derived differential quadrupole moments, Ao(~)(cos 0) = CgJ2) (cos are shown in Figure 18. The uncertainty in is estimated to be better than k0.12. This value is exceeded at small scattering angles for those states which show only intensity for backward scattering, namely the states 173- >, /76+> and 165->, 174->. Therefore, the differential alignment is shown only for those angles where the error does not exceed the stated limit. All probed final states exhibit negative alignment for backward scattering which decreases with increasing scattering angle. At small angles, we find positive alignment. The zero point is found to be close to cos 0 = 0.45 & 0.15. In addition, the degree of alignment at large scattering angles increases with increasing energy transfer. While this trend is not that obvious for forward scattering, it is observed also for the integral quadrupole moments listed in Table 6 and displayed in Figure 19. Here we find a nearly linear dependence of the observed integral alignment parameter A#) on the involved energy transfer. To explore the origin of the observed alignment effects, we compare the experimental data with the predictions of the kinematic and geometric apse model developed by Khare, Kouri,
e),
Po) = Ao(O)and Quadrupole Moments )!(Z = A,,(*)for
Different Final States Derived from TOF Spectra Recorded on 0-Branch Lines ~
bkc >
Iv
IH
31 32 42 64 65 74
164.6 121.4 59.3 43.5 15.0 33.2
p-NHs 161.4 108.6 51.9 27.1 46.4 17.8
40+ 63+ 637376-t
137.1 29.8 36.4 28.3 188.9
104.8 19.5 23.0 16.7 103.2
PO)
PI
162.5 12.8 54.4 32.6 55.9 22.9
-0.01 1 -0.066 -0.089 -0.369 -0.375 -0.510
15.6 23.0 27.5 20.6 131.8
-0.182 -0.331 -0.359 -0.427 -0.492
0-NH3
and H~ffmann.~’In this work, they could show how a clear propensity for conserving the magnetic quantum number m results when a geometric apse (GA) frame is used as the relevant quantization axis. The direction of this axis is given by the difference of the unit vectors of the initial and final relative velocity vector. This model has its origin in the classical hard sphere-hard ovaloid scattering, from which the conservation of J, along the kinematic apse (KA) direction is derived. The latter direction is defined as the direction of the difference vector of the initial and final relative velocity vectors. While the KA axis is dependent on the final state, the GA axis is not. Both axes are shown in the kinematic diagram displayed in Figure 6 while the calculation and averaging of the moments are described in the appendix. Typical results for o-NH3 are presented in Figure 20. The solid line represents the result for the GA model, which is independent of the inelastic transition. To demonstrate the possibility to extract reliable quadrupole moments A&*) from 0-branch lines using eq 9, we calculated apparent quadrupole moments using the right hand side of eq 9 and including the next higher moment AO(4). The results for the states 133’ and 1731 are presented in Figure 20 as dashed curves. The deviations are well within the experimental uncertainty. Also shown is the prediction within the KA model for these two states (dash-dotted lines). As expected, deviations from the GA model occur for large energy transfer and small angle scattering. Since, within both models, the moments are multiplied by a Legendre polynomial of the same order, we expect a vanishing quadrupole moment at an angle pg = /3k =
oNH 3 - He 140 meV
c8
v
“8
-1.o -1.0
-0.5
0.0
0.5
1.0
-1 .o
-1.0
-0.5
0.0
0.5
1.0
e COS e Figure 18. Differential quadrupole moments for different final states of p-NH3 (left panel) and 0-NH3 (right panel) as a function of cos 8. The data are not corrected for depolarization effects due to unresolved fine or hyperfine structure. COS
1112 J. Phys. Chem., Vol. 99, No. 4,1995
Meyer
NH3 - He N O
a E
0.0
E
Ea,
0
n 2 -0.25 m3 0U
!2
cn
a, +
= -0.50 -0.75 0
100 200 300 400 Energy transfer / cm-1
500
Figure 19. Integral quadrupole moments A0Iz' for different final states as a function of energy transfer: dashed line,p-NH3; dotted line, o-NH3. The data are not corrected for depolarization effects due to unresolved fine or hyperfine structure.
oNH3 - He 140 meV
0.5
-w u 0
;N
a
0.0
-0.5
1 ' ,
-1.OVl 1 -1.0 -0.5
' ' ' ' ' 0.0
COS
e
0.5
1
1.0
Figure 20. Comparison of differential quadrupole moments AoIZ)in the LAB frame derived from the GA and KA model for two different final states of o-NH?. For further explanations see the text. 125.3'. This corresponds to cos 0 = 0.33 for the GA model, while, in the KA model, it will be shifted toward larger values depending on the exact energy transfer. Within this model there exists a maximum energy transfer of 33% in order to find a zero point which will be located at cos 0 = 0.81. This range of zero points is in good agreement with experiment. Also the calculated quadrupole moments are in qualitative agreement with the experimental findings over a wide range of scattering angles. At very small angles the experimental data show positive alignment in agreement with the GA model while the KA model predicts strong negative alignment. Quantitatively, the variation of alignment with energy transfer is not predicted correctly by both models. For transitions leading to large energy transfer, good quantitative agreement is observed when a maximum depolarization due to the magnetic hyperfine interaction is assumed. As can be seen in Table 5, this correction is smaller for p-NH3. On the other hand, the repulsive models predict less alignment for states of this modification because of the
degeneracy of the initial state. For states involving small energy transfer, we observe significantly less alignment than predicted by the models, even when the maximum depolarization is assumed. Intuitively, one might think that these cross sections will be influenced more strongly by the attractive interaction, and deviations from the purely impulsive models are expected. Therefore, these findings confirm the theoretical results of Davis.31 In this context it is interesting to compare the differential alignment of the 163> symmetry pair in Figure 18. Within the error limits, we find a very similar angular dependence for the pair of final states 163- > and 163+> except for the small angle scattering. Although the error bars are quite large, we find for the CS-forbidden transition a small positive alignment while, for the CS-allowed transition, we find a value much closer to 0.5, the value predicted by the GA model. This might manifest the difference in the excitation mechanism, since only one requires the coupling of different S2 states in the body-fixed frame.
4. Concluding Remarks The rotational excitation of NH3 in collisions with He has been studied at a collision energy of 140 meV in a counterpropagating beam experiment. For the first time, a collision system is characterized by the monopole and quadrupole moment of state-resolved integral and differential cross sections. For both types of cross sections, the monopole moment is identical to the degeneracy-averaged cross section while the quadrupole moment describes the collision-induced alignment. The measured integral cross sections follow roughly an exponential energy gap law while the general angular dependence of the different DSCs is in good agreement with results of a CS calculation based on the semiempirical potential surface of ref 8. ICSs for transitions with Ak = 3 and 6 deviate strongly from the energy gap law, confirming the importance of the anisotropy in the azimuthal angle CP. In addition, at larger energy transfers, the selection rules concerning the excitation of different parity states implied by the CS approximation are relaxed considerably. No pronounced differences in the angular behavior of DCSs for the excitation to symmetry pairs is observed, indicating a two-step mechanism for the change in !2 and the rotational excitation via the change in j or k. Quadrupole moments of the ICSs as well as the DCSs show a strong correlation with energy transfer. Differential moments change from negative values for backward scattering to positive values at small angle scattering. Due to this reversal of the alignment, a smaller integral alignment is observed. While the angular dependence for the differential moments is in qualitative agreement with the purely repulsive geometric apse model, the variation with final state is not reproduced. Theoretical scattering calculations should help to clarify how the alignment is influenced by the PES. Acknowledgment. The author would like to thank Prof. S. Leutwyler for continuous interest and support. The financial support of this research by the Schweizerische Nationalfonds and the Hochschulstiftung der Universitat Bern is gratefully acknowledged. Appendix: Kinematic and Geometric Apse Models In the kinematic apse model the relevant quantization axis is defined by the direction of linear momentum transfer. As can be seen in Figure 6, this direction is inclined by an angle ,& against the direction of the initial relative velocity vector, which coincides with the symmetry axis of the counterpropagating
Counterpropagating Beam Scattering of NH3
+ He
J. Phys. Chem., Vol. 99, No. 4, 1995 1113
molecular beam setup. The two coordinate systems are thus related by consecutive rotations through Euler angles P k and Ipk. Strict conservation of the magnetic quantum number in the kinematic apse frame results in the population of those magnetic sublevels which have been populated already in the initial state. Since in our experiment efficient rotational cooling is achieved in the adiabatic expansion, only states withji = 0 and ji = 1 are populated. In addition, polarization studies of the 111+> state of p-NH3 give no indication of alignment. Therefore the population of the different magnetic sublevels of a final state is given by
Under these conditions the initial and final velocity vectors, Zi and Zf, in the c.m. system are given by
-ui = ui(l,O)
and
Zf = bui(cos 0,sin 0) with
Here AE and E denote the energy transfer and the c.m. collision energy, respectively. The change in linear momentum, corresponding to the scattering angle 0 is directed along the difference vector AZ = Zf - E,, which determines cos Pk:
W: = dm, for 0-NH3
bcos 0- 1 cos p k =
and
(A2) Ji12+1-2~cos0
The related state multipole moments in the apse frame are given according to ref 43 by
In the geometric apse model the quantization axis is assumed to be directed along the difference vector formed from the unit vectors of the initial and final velocity. Therefore, letting D = 1. we find for the GA model cospg=-
which, for 0-NH3, reduces to
From the properties of the 3 - j symbol it is evident that only moments with even Q can be found. This holds also true for the excitation of para-NH3 where we find
(1_1
'1 OQ)
+ +
Here for j j Q = odd the terms with Im(= 1 cancel each other, while the term with m = 0 vanishes. To determine the corresponding moments of the M distribution in the laboratory or collision frame (CF), we have to transform these moments into a coordinate frame which is rotated by the Euler angles P k and q J k with respect to the apse frame:
After averaging over the azimuthal angle q k , we find the relevant moments :
Since for the Legendre polynomials we have P Q ( & ~= ) 1, we see immediately that, for P k = 0" and 180°, the kinematic apse (KA) frame coincides with the laboratory frame in our experiment. The angle P k is determined for each scattering angle from the kinematic or Newton diagram. Due to the cylindrical symmetry with respect to the laboratory fixed quantization axis, we can determine P k from a two-dimensional velocity diagram.
J T1 - cos y 0
From eq A1 it is evident that the quadrupole moment in the LAB frame will vanish for P k , g = 125.3'. While in the GA model this corresponds to cos 8 = '13, the condition can only be fulfilled within the KA model for energy transfers less than 33%. At this energy transfer, we find cos 6 = (2/3)1'2. References and Notes (1) Ho, P. T. P.; Townes, C. H. Annu. Rev. Astron. Astrophys. 1983, 21, 239. (2) Aroui, H.; Broquier, M.; Chevalier, M.; Picard-Bersellini, A.; Billing, G. D. Mol. Phys. 1991, 74, 897. Broquier, M.; Picard-Bersellini, A,; Hall, J. Chem. Phys. Lett. 1987, 136, 531. (3) Oka, T. Adv. At. Mol. Phys. 1973, 9, 127. (4) Klaasen, D. B. M.; Reynders, J. M. H.; ter Meulen, J. J.; Dymanus, A. J. Chem. Phys. 1982, 77, 4972. Klaasen, D. B. M.; ter Meulen, J. J.; Dymanus, A. J. Cliem. Phys. 1983, 78, 767. (5) Schwartz, P. R. Astrophys. J. 1979, 229, 45. (6) Bickes, R. W.; Duquette, G.; van den Meijdenberg, C. J. N.; Rulis, A. M.; Scoles, G.; Smith, K. M. J. Phys. 1975, B8, 3034. (7) Slankas, J. T.; Keil, M.; Kuppermann, A. J. Chem. Phys. 1979, 70, 1482. (8) Meyer, H.; Buck, U.; Schinke, R.; Diercksen, G. H. F. J. Chem. Phys. 1986, 84, 4976. 19) Seelemann. T.: Andresen. P.: Schleiuen. J.: Bever. B.: ter Meulen, J. J. 'Chem. Phys. 1988, 126, 27. (10) Schleiuen. J.: ter Meulen, J. I. Chem. Phys. 1991, 156, 479. (11) Greenis. J. Chem. Phys. 1976,64,3463;-1979,70,816; 1980, 73, 2740. (12) Pack, R. T. J. Chem. Phys. 1974, 60, 633. (13) McGuire, P.; Kouri, D. J. J. Chem. Phys. 1974, 60, 2488. (14) Ahlrichs, R.; Penco, R.; Scoles, G. Chem. Phys. 1979, 19, 119. (15) Schleipen, J.; ter Meulen, J. J.; van der Sanden, G. C. M.; Wormer, P. E. S.; van der Avoird, A. Chem. Phys. 1991, 163, 161. (16) Bergmann, K. In Atomic and Molecular Beam Methods; Scoles, G., Ed.; Oxford University Press: London, 1988; Vol. I. (17) Simpson, W. R.; Om-Ewing, A. J.; Zare, R. N. Chem. Phys. Lett. 1993, 212, 163. Shafer, N. E.; Xu,H.; Tuckett, R. P.; Springer, M.; Zare, R. N. J. Phvs. Chem. 1994, 98, 3369. (18) Brouard, M.; Duxon, S. P.; Enriquez, P. A,; Simons, J. P. J. Chem. Phys. 1992, 97, 7414. (19) Johnston, G. W.; Satyapal, _ _ S.; Bersohn, R.; Katz, B. J . Chem. Phys. 1990, 92, 206. (20) Buntine, M. A.; Baldwin, D. P.; Zare, R. N.; Chandler, D. W. J. Chem. Phys. 1991, 94, 4672. (21) Jons, S. D.; Shirley, J. E.; Vonk, M. T.; Giese, C. F.; Gentry, W. R. J. Chem. Phys. 1992, 97, 7831. (22) Schnieder, L.; Seekamp-Rahn, K.; Liedeker, F.; Steuwe, H.; Welge, K. H. Faraday Discuss. Chem. SOC. 1991, 91, 259. (23) Bontuyan, L. S.; Suits, A. G.; Houston, P. L.; Whitaker, B. J. J. Phys. Chem. 1993, 97, 6342. Suits, A. G.; Bontuyan, L. S.; Houston, P. L.; Whitaker, B. J. J. Chem. Phys. 1992, 96, 8618.
1114 J. Phys. Chem., Vol. 99, No. 4, 1995 (24) Case, D. A,; McClelland, G. M.; Herschbach, D. R. Mol. Phys. 1978, 35, 541.
(25) Hall, G. E.; Houston, P. L. Annu. Rev. Phys. Chem. 1989,40, 375. (26) Docker, M. P.; Hodgson, A,; Simons, J. P. In Molecular photodissociation dynamics; Ashfold, M. N. R., Baggot, J. E., Eds.; Royal Society of Chemistry: London, 1987. (27) Khare, V.; Kouri, D. J.; Hoffmann, D. K. J. Chem. Phys. 1981, 74, 2275; 1981, 74, 2656; 1982, 76, 4493. (281 Mattheus. A,: Fischer. A,: Zieeler. G.: Gottwald, E.; Bergmann, K. Phys. Rev. Lett. 1986, 56, 712. 129) Treffers. M. A,: Korving, - J. J. Chem. Phys. 1986, 85, 5076; 1986, 85, 5085. (30) McCafferty, A. J.; Proctor, M. J.; Whitaker, B. J. Annu. Rev. Phys. Chem. 1986, 37, 223. (31) Davis, S. L. Chem. Phys. 1985, 95, 411. (32) Meyer, H. J . Chem. Phys. 1994, 101, 6686; 1994, 101, 6697. (33) Meyer, H. Chem. Phys. Lett. 1994, 230, 519. (34) Ashfold, M. N. R.; Dixon, R. N.; Stickl;and, R. J.; Western, C. M. Chem. Phys. Lett. 1987, 138, 201. Ashfold, M. N. R.; Dixon, R. N.; Little, N.; Stickland, R. J.; Western, C. M. J. Chem. Phys. 1988, 89, 1754 I
Meyer (35) Green, C. H.; Zare, R. N. J. Chem. Phys. 1983, 78, 6741. (36) Meyer, H. Chem. Phys. Lett. 1994, 230, 510. (37) Steinfeld, J. I.; Ruttenberg, P.; Millot, G.; Fanjoux, G.; Lavoret, B. J . Phys. Chem. 1991, 95, 9638 and references therein. (38) Alexander, M. H.; Davis, S. L. J. Chem. Phys. 1983, 79, 227. (39) Billing, G.; Poulsen, L. L.; Diercksen, G. H. F. Chem. Phys. 1985, 98, 397. ICSs have been taken from ref 10. (40) Schinke, R.; Bowman, J. M. In Molecular Collision Dynamics; Bowman, J. M., Ed.; Springer Verlag: Berlin, 1983. (41) Kummel, A. C.; Sitz, G. 0.;Zare, R. N. J . Chem. Phys. 1986, 85, 6874. (42) Townes, C. H.; Schawlow, A. L. Microwave Spectroscopy; Dover Publications: New York, 1975. (43) Blum, K. Density Matrix Theory and Applications; Plenum Press: New York, 1981. (44) Guest, J. A,; O’Halloran, M. A,; Zare, R. N. Chem. Phys. Lett. 1984, 103, 261.
JP942558X
