Fully State-Resolved Differential Cross Sections for the Inelastic Scattering of the Open-Shell NO Molecule by Ar H. Kohguchi, T. Suzuki, M. H. Alexander

The observed images are not symmetric about the relative velocity vector v_rel(the collision axis). This apparent asymmetry originates from laser detection of products in a pulsed crossed molecular beam experiment. The number of scattered products is proportional to the temporal and spatial overlap of the two molecular beams. Products are scattered in space according to their velocities in the laboratory frame (v_LAB) that is the vector sum (v_LAB = u + v_cm) of the center-of-mass velocity (u) and the scattering velocity in the center-of-mass frame (v_cm).

The flux of scattered particles into a given center-of-mass scattering angle (_cm), which forms a cone around v_rel, is independent of the azimuthal angle because of the cylindrical symmetry of the collision. However, the magnitude of v_LAB varies with azimuthal angle on this cone because of the differing relative angle between u and v_cm. Since a laser beam ionizes scattering products in a volume fixed in the laboratory frame, particles with fast v_LAB escape from the laser volume immediately after generation, while particles with slow v_LAB accumulate in the laser volume during the molecular beam pulse. This results in a detection efficiency which is dependent on v_LAB. It is for this reason that the left side (slow v_LAB) of the images always exhibits stronger intensity than the right side (fast v_LAB) (1).

Image analysis

DCSs were extracted from the observed images after correcting for the v_LAB-dependent detection efficiency (2, 3). Ion-imaging provides a two-dimensional (2D) projection of a three-dimensional (3D) scattering velocity distribution. The 2D image, I(v_x, v_y), projected along the z axis perpendicular to the collision plane is expressed by;

(1)

I (v_x, v_y) = dt dR' [n_I(t,R) f (R') d / d (_cm)]

where, n_I(t, R), n_II(t, R), and f(R') are, respectively, the densities of beams I and II, and the spatial ionization efficiency of the laser beam (4). Here d/d (_cm) is the DCS. Also R and R' are the position vectors of the scattering event and ionization, respectively. They are related by R' = R + t × vLAB, where t is the time difference between the scattering event and the laser pulse. In Eq. (1), symbolically denotes an operator for the 2D projection of a 3D object.

The experimental parameters, n_I(t, R), nII(t, R), and f(R') were directly measured and/or carefully estimated by a number of numerical simulations using Eq. (1) (5). We found that the cross-sectional profiles (Gaussian or flat shape) and the spatial width of n_I(t, R) and n_II(t, R) have almost no influence on skewing of the images. The shape and the volume of the laser beam had no effect on the images, while the position of the laser beam relative to the scattering center was the most crucial quantity for an accurate analysis. Therefore, the variation of images for different laser beam positions were experimentally examined. With the obtained experimental parameters, the v_LAB-dependent detection efficiency (apparatus function) was computed.

The signal intensity of the observed images is concentrated on an annulus of the Newton circle because of the narrow velocity spread of the molecular beams. The DCSs were obtained from this annulus as follows. We extracted one-dimensional angular distribution of the signal intensity at a radius r with a width r, R(₂D; r, r), from both the observed image and the corresponding apparatus function (6). Here, ₂D denotes the angle at the circumference measured with respect to the collision axis in the image. The DCS can be extracted by the following equation,

(2)

d / d (_cm) = R(_2D; r, )_observed / R(_2D; r, )_app.funct.

which is based on two approximations; (i) ₂D = _cm and (ii) a zero velocity spread of the molecular beams.

With the angular resolution of our apparatus, the approximation of _2D = _cm did not significantly affect the quality of the observed images, except for that an angular resolution diminishes near the poles (_cm = 0° and 180°). The experimental uncertainty in the extracted DCSs arises from the statistics (signal to noise ratio) of R( _2D; r, r )_observed and an error in estimation of R( _2D; r, r )_{app. funct.}. We have performed a number of simulations in which the parameters for the molecular beams [n_I(t, R) and n_II(t, R)] and the laser beam [f(R')] were systematically varied (5), and we confirmed that the uncertainty of our apparatus function is mostly ±10%. This means that the uncertainty in each DCS was mostly ±10% at each _cm for strong inelastic transitions. On the other hand, it is likely that the DCSs obtained for weak transitions contain slightly larger uncertainty, although it is difficult to quantify its error. The angular resolution was estimated as 8° except for _cm ~ 0° or 180°, which is limited predominantly by blurring of the Newton diagram due to the velocity spread of 30 m/s (FWHM) of the incident molecular beams. Because we performed no angular averaging on R( _2D;
r, r )_observed, any structures in the DCSs finer than 8° in Figs. 2 and 3 are regarded as noises. The DCSs obtained independently from the left and right sides of each image should be identical because of the cylindrical symmetry of the collision process. In practice, however, the DCS obtained from the left and right halves showed some discrepancies. Therefore, we have taken the right side of each image for determining the final DCS presented in Figs. 2 and 3, whenever the signal to noise level was sufficiently high, because this part was less sensitive to the error of an apparatus function. For 6 final (j', ') states, the signal to noise level in the right sides was not sufficient, so the DCSs were determined from the left sides of these images.

References and Notes

1. In the present collision geometry, sideway scattering (cm = 90°) of NO into some (j', ') states provides |vcm| ˜|u| (430 m/s) or |vLAB| ˜0. The asymmetry was most significant in these image.
2. G. C. McBane, in Imaging in Chemical Dynamics, A. G. Suits, R. E. Continetti, Eds. (American Chemical Society, Wahington, DC, 2000), pp.215-229.
3. K. T. Lorenz, M. S. Westley, D. W. Chandler, Phys. Chem. Chem. Phys. 2, 481 (2000).
4. In order to avoid any complication due to molecular alignment effect, we used the [1+1] resonantly enhanced multiphoton ionization (REMPI) via A²⁺ with linearly polarized light. It is well known that the A-X transition in this one-color REMPI scheme is easily saturated and practically insensitive to the molecular alignment at the laser power of several tens mJ/cm² used in this work. [D. C. Jacobs, R. J. Madix, R. N. Zare, J. Chem. Phys. 85, 5469 (1986)]. For some high-j' and low j'-states, we have confirmed that parallel and perpendicular laser polarization with respect to the collision plane provided the same image (or DCS) within our experimental error.
5. The time profile in n_I(t, R) and n_II(t, R) was measured by detecting NO(j = 0.5, 1/2) in the incident beams at various laser fire timings changed by 20 s steps. For the Ar beam, 5% NO was added as a tracer. Other input parameters considered in the simulation (along with the range of values) were; the spatial width of n_I(t, R) and n_II (t, R) (0.8 - 3.2 mm), the length of f(R') along the laser propagation (6 - 18 mm), the length across the laser propagation (0.3 - 1.2 mm), and the relative position of f(R') to the scattering center (-1 mm - +1 mm). It is noted that the shift of the relative position of f(R') by 1 mm was found to change locally the DCS around _cm = 90° by the factor of 3 by the simulation.
6. Depending on the (j', ') state, = 8 - 12 (pixels) were used for r = 72 - 153 (pixels), where one pixel-size on the image corresponded to 3 m/s in the velocity. This range of r did not affect significantly the angular resolution of the DCSs.