Rotational Rainbow Research Articles

Calculations for the scattering of N 2 at a Ag surface

Motivated by experiments of Sitz, Kummel and Zare in 1987–1989 a theoretical study for the scattering of N 2 at a Ag surface is presented. It is examined how the experimentally observed two peaks in the rotational state distribution can be accounted for; a potential is suggested for which only the high energy peak is to be interpreted as a rotational rainbow. The excitation of phonons and its anticorrelation with the excitation of rotation is discussed. The angular distribution of scattered particles is described assuming normal energy scaling and parallel momentum conservation; with that the dependence of the detected rotational state distribution on the detection angle can be explained. The alignment data are a measure of the surface corrugation.

Rotational rainbows in the inelastic scattering of N2 from Au(111)

Resonantly enhanced multi-photon ionization (REMPI) laser spectroscopic and molecular beam-surface scattering techniques are coupled to study the rotationally inelastic scattering of N2 from Au(111). The scattered N2 rotational distributions are markedly non-Boltzmann and exhibit pronounced maxima at large angular momentum transfers ( Delta J=20 to 30). These maxima are indicative of direct rotational rainbow scattering and strongly depend upon the translational energy of the incident N2 molecular beam. The translational energy dependence of the rainbow rotational energy yeilds a value of 4.1 kcal mol-1 for the depth of the laterally averaged N2-Au(111) physisorption potential.

Velocity selective rotational rainbows for normal incidence/normal detection gas–surface scattering

Velocity selected rotational state distributions of N2 molecules scattered from Ag(111) have been measured. These measurements have been made at normal incidence and normal detection for incident energies of 0.25 and 0.75 eV. This new technique allows us to rapidly study the cross correlation between the exit rotational state and its velocity distribution. A pronounced difference between the rotational distributions of fast and slow scattered molecules is observed, and this difference is much more pronounced for high incident kinetic energy.

Collisions of NO(X 2Π) with a Ag(111) surface: New quantum scattering studies based on a semiempirical potential energy surface

We report the results of fully quantum close-coupled studies of collisions of NO(X 2Π) with a Ag(111) surface. The recent corrected effective medium potential energy surfaces (PES) of DePristo and Alexander [J. Chem. Phys. 94, 8454 (1991)] were used. The final state rotational distributions show evidence of at least four rotational rainbows, corresponding to scattering on (and interference between) the two PES which arise when the degeneracy of the NO molecule is lifted upon approach to the surface. A strong tendency is seen to populate the lower spin–orbit manifold at low to moderate final J, which disappears as J rises beyond 30.5 and the final states are better described in Hund’s case (b). Simultaneously, there exists a propensity to populate those Λ-doublet levels in which the electronic–rotational wave function is antisymmetric (ΠA″) with respect to reflection of the electronic coordinates in the plane of rotation of the scattered NO molecule. This feature is opposite to what has been seen experimentally. An approximate averaging over the lateral position of the NO molecule above the surface showed that although the rainbow oscillations are strongly sensitive to surface corrugation, the fine-structure propensities are not. This suggests that these latter are reflective of some fundamental characteristic of the NO–Ag interaction which is independent of the position of the NO molecule above the Ag(111) unit cell.

Rotational rainbow structures in collisions of electrons with highly symmetric polyatomic molecules

Following the studies of diatomic and highly symmetric planar molecules in collision-induced rotational excitation (discussed in a spectator model) an extension to highly symmetric spatial targets is developed. For the calculated cases all molecular atoms are treated as equivalent (with the liberty of a different atom at the centre of mass). A compact closed-form expression for the state-to-state differential cross sections is derived. Selection rules and rotational rainbow structures are discussed.

Differential cross sections for rotationally state-resolved inelastic scattering of HF by argon

We present differential cross section (DCS) measurements for scattering of HF by Ar. These crossed-beam experiments employ rotational state sensitivity, allowing determination of the DCS as a function of the scattered HF rotational state. The initial HF rotational distribution is generated by nozzle expansion, without further state selection. Its composition is mostly J=0 and J=1, with small admixtures for J>1. The DCS for each final state J′ is measured using a stabilized cw HF chemical laser, in conjunction with a rotatable liquid He-cooled bolometer. Measurable signals are obtained for scattering into 0≤J′≤5, where J′=6 is the thermodynamic limit for our collision energy of 120 meV. The measured DCS’s show a strong forward peak, largely from elastic scattering. In addition, the DCS’s evolve from a broad shoulder in the θ≊25°–40° region for J′=0—through a flattening of the wide-angle scattering for J′=2 and J′=3—to an increase in the scattering beyond ∼40° for J′=4. The DCS for scattering into J′=5 also shows increased intensity at wide scattering angles, but its onset is delayed until ∼70°. These features are shown to be independent of the laboratory → center-of-mass kinematic transformation. The wide-angle scattering into J′=4 and J′=5 corresponds to transferring up to 40% and 60%, respectively, of the available kinetic energy into HF rotation. Since the center-of-mass scattering angles are up to ∼110°, we interpret the observed features for J′=4–5 in terms of rotational rainbow scattering from the hard core of the HF+Ar potential energy surface. The origin of the shoulder for J′=0 scattering is less clear, but it may arise from the strongly anisotropic nature of the HF+Ar van der Waals attraction.

Angularly resolved rotational energy transfer in highly vibrationally excited states: Na2(v=31)–Ne

The scattering of high vibrationally excited sodium molecules Na2(v=31) with Ne atoms at 180 meV collision energy is investigated in a crossed molecular beam arrangement using laser optical methods. Angularly resolved rotationally inelastic and vibrationally elastic cross sections ji→jf are measured for ji =5, 7, and 9≤jf ≤25. Pronounced rotational rainbow maxima are observed, the angular position of which indicates a significantly larger anisotropy of the interaction potential compared to that of Na2(v=0)–Ne. Except for the increase of the anisotropy the vibrational excitation has little effect on the dynamics of rotational energy transfer. Good agreement with a new semiempirical Na2–Ne potential surface V(r,R,γ) is found.

Rainbow scattering from solid surfaces

Rainbow scattering from solid surfaces is discussed for beams of particles with energies from thermal to more than 1 MeV. The phenomenon of rainbow scattering is introduced. It is demonstrated to be a singularity in the classical scattering cross section. Both in two- and more-dimensional scattering problems rainbows are observed. In the latter case casastrophe theory can be used as a means to classify the rainbows. At thermal energies the rainbows are observed in scattering of notably He and Ne from corrugated surfaces. At hyperthermal energies rainbows are observed in both atom and ion scattering. At energies above 100 eV penetration of the solid by the beam particles occurs and rainbow scattering is not as pronounced and also less studied. In the case of scattering of molecules also rotational rainbows are observed. In this case the rainbow refers to a singularity in the classical excitation function for rotational excitation.

Inelastic scattering of NO from Ag(111): Internal state, angle, and velocity resolved measurements

We have determined the velocity distributions of individual quantum states of NO scattering from Ag(111) at specific scattering angles θf using molecular beam techniques to control the incidence energy Ei and angle θi. We find that the mean energies of scattered species Ef depend weakly on θf at low collision energies, but become increasingly independent of this parameter as Ei approaches 1.0 eV. This is true for all final rotation states J. The previously reported insensitivity of the final kinetic energy to J is found to apply at all scattering angles, so that Ef vs θf curves for high J fall only slightly below those for low J. This system is highly translationally inelastic at high incidence energies, with up to 55% of Ei being lost to phonons at Ei=1.0 eV. Angular distributions are relatively insensitive to J at low Ei, but for high Ei the peak flux is found to shift away from the surface normal as Ei increases. The effect of the surface temperature only becomes apparent at low incidence energies. A search for supernumerary rotational rainbows reveals no discernible oscillations even for the lowest surface temperatures. We believe that these supernumerary oscillations may be damped by ‘‘surface corrugation’’ effects for this system. Discussion focuses on the observed anticorrelation between kinetic energy transfer to phonons and to rotation, the extent to which parallel momentum is conserved in this system, and energy-angle scaling laws for energy transfer. In this latter case we show that energy transfer in this system scales approximately with the quantity Ei cos θi, over the full range of conditions covered.

Rotational excitation in electron-molecule scattering at intermediate collision energy: A two-centre scattering model

A simple two-centre scattering model is discussed, which leads to compact closed form differential cross sections for rotationally inelastic scattering from diatomic molecules. The model elucidates in a simple way the rotational rainbow structure of the cross sections for both initially rotating and nonrotating molecules. Surprisingly good agreement with extensive computations and experimental measurements fore-Na2 scattering at 150–300 eV is observed.

Collisional induced rotational excitation ofN-atomic molecules

A simple model for collisional rotational excitation of diatomic molecules presented previously (in excellent agreement with experiments and computations for electron-Na2 collisions at about 300 eV) is extended toN-atomic molecules. Special attention is given to highly symmetric cases, where all molecular atoms are equivalent (with the possible exception of a different atom in the centre). Selection rules, propensity rules and rotational rainbow structures in differential state-to-state cross sections are discussed.

Rotational excitation by electron impact: a state selective study using laser-induced fluorescence

Angularly resolved pure rotational excitation of sodium dimers by intermediate energy electrons (150 to 300 eV) is studied using laser state selection technique, under nonresonant collision conditions. The experimental procedure, which is new to electron scattering, is described in detail. The experimental data show large rotational transitions and rotational rainbow structures. The relative cross sections agree well with the close coupling calculations and also with the spectator model predictions.

A crossed-beam study of the state-resolved dynamics of CH(X 2Π)+D2. I. The inelastic scattering channel

The state-to-state integral cross sections for the inelastic scattering of CH(X 2Π) and D2 to produce rotationally excited CH(X 2Π) product have been measured in a crossed-beam apparatus by the laser-induced fluorescence method. Two types of measurements were performed: (1) the translational energy dependence of an individual quantum state of the product and (2) the state distribution of the products at fixed and well-defined translational energy. For the inelastic scattering channel, the cross sections gradually increased from a dynamical threshold to a broad maximum and then slowly decreased as the translational energy increases. Evidence for multiple-impact rotational rainbows was found and a possible frequency-locking phenomenon between the two receding rotors resulted. Moderate orbital alignment was observed except for the highest rotational levels of the product. By comparing and contrasting the kinematically similar system CH(X 2Π)+He, the influence of a strongly attractive potential energy surface on the inelastic scattering of CH+D2 was inferred. Combining the results of the inelastic scattering and the isotopic exchange channels (the following paper) provide an unprecedented look into the dynamics of collisions between CH(X 2Π) and D2.

Surface scattering of NO from graphite: A statistical description of energy distributions

In the present theoretical study, inelastic scattering of NO from graphite surfaces is analyzed with a statistical model. The results are in good agreement with previous classical trajectory calculations by Pettersson et al. (1988). Angular distributions and the ‘‘rotational cooling’’ effect found in experiments published by Frenckel et al. (1982), Segner et al. (1983), and Häger and Walther (1984) are successfully reproduced. The model describes a small part of the graphite surface together with a scattering diatom as a collision complex, which decomposes in a unimolecular fashion. The surface is assumed to be flat, whereby the diatom angular momentum component along the surface normal and the linear momentum parallel to the surface are conserved. Otherwise the diatom translation and rotation are allowed to exchange energy with the surface, which is characterized by a set of harmonic oscillators. The experimentally observed ‘‘rotational cooling’’ effect is clearly demonstrated to be due to the conservation of the normal component of the angular momentum. The surface oscillator mass and the number of surface oscillators are treated as parameters. The results indicate that on the average one to three surface atoms are directly involved in each molecule-surface collision. ‘‘Rotational rainbow’’-like distributions are observed at high total energies, even though the simulations are purely statistical with no dynamic effect included.

Orientation dependence of rotational excitation in no scattering from Ag(111)

Scattering experiments with oriented NO beams on Ag(111) have been performed. The scattered molecules are detected by a resonantly enhanced multiphoton ionization. A dependence of the rotational rainbow upon the scattering angle has been observed. The first results of the steric effect measured for single rotational ( J) levels of scattered molecules are presented. The J=18.5 state is produced preferentially when the O-end collides first with the surface. For the J=8.5 state we find that the N-down geometry is preferable. These results are in qualitative agreement with theory.

Photofragmentation of the Ne⋅⋅⋅ICl complex: A three-dimensional quantum mechanical study

Converged three-dimensional quantum mechanical calculations for photofragmentation of the Ne⋅⋅ICl van der Waals molecule in the energy region of the electronically excited B(3∏0+) state of ICl are presented and compared with experiments. Lifetimes and final state distributions for the ICl fragments were determined for vibrational predissociation from the lowest van der Waals level in the B(v′=2) channel. Good agreement between theory and experiment was achieved using a sum of atom–atom pairwise potentials. This potential energy surface predicts the equilibrium geometry of the complex to be bent at 140° with the Ne atom towards the Cl end of ICl. The diabatic vibrational golden rule (DVGR) approximation, as well as the rotational infinite order sudden approximation (RIOSA), have been tested again the full 3D calculations. Analysis of the quasibound wave function reveals that the highly inverted rotational distribution of the ICl fragments observed in the experiment, is not due to zero-point bending motion. It is more likely to be due to a rotational rainbow effect enhanced by the favorable initial geometry of the complex. The effect of the excitation of the bending van der Waals mode in the complex has also been studied. As compared with the lowest level, a longer lifetime and a different rotational distribution of the fragments is predicted. The results presented in this work not only elucidates many dynamical aspects of vibrational predissociation for the Ne⋅⋅⋅ICl complex, but also provide benchmark data for the study of other theoretical methods and approximations.

Rotationally inelastic gas–surface scattering: HCl from Au(111)

A quantum-resolved molecular beam–surface scattering study of HCl scattered from Au(111) is described. The HCl is detected in a quantum-resolved manner via (2+1) resonant enhanced multiphoton ionization (REMPI). Greater than 85% of the incident HCl molecules are in a single-quantum state (v=0, J=0) with a narrow velocity distribution (Δυ/υ<0.10). The scattered HCl is strongly peaked about the specular angle, and both its final velocity and rotational distributions are indicative of direct inelastic scattering. The scattered rotational distributions exhibit features characteristic of rotational rainbows and have a mean rotational energy that displays a bilinear dependence upon the incident normal kinetic energy and surface temperature. The final velocity distributions are largely insensitive to the rotational level and indicate that the energy loss to phonons is small (<20%). Analysis of the scattered data indicates an orientation-averaged attractive well depth of ∼5 kcal/mol for the HCl–Au(111) interaction.

Vibrational and rotational excitation in collisions of diatomic molecules with rigid surfaces

The disappearance of the rotational rainbow in the scattering of unexcited diatomic molecules from a smooth rigid surface due to initial vibrational and rotational excitation and surface corrugation has been studied by classical trajectory calculations. The ratio of vibrational to rotational excitation in initially unexcited molecules strongly increases with increasing impact energy and increasing surface hardness.

State-to-state integral cross sections for the inelastic scattering of CH(X 2Π)+He: Rotational rainbow and orbital alignment

The state-to-state integral cross sections for the inelastic scattering of CH(X 2Π) with He were measured in a newly constructed crossed molecular beam machine. Use of laser-induced fluorescence in an unconventional flux mode of detection provided single fine-structure state specific detection of the products. Two types of measurements were performed to further our understanding of the collision dynamics of open shell systems: (1) the product state distribution at a fixed and well-defined collision energy and (2) the dependence on collision energy of product state-resolved cross sections. A qualitative understanding of the collision dynamics can be obtained by properly factoring out features dependent on the fine-structure states, i.e., effects involving individual Λ-doublet states and features dependent on the rotational level alone, i.e., effects remaining after summing over all four fine-structure states associated with a given rotational quantum number. As for the fine-structure effects, a preferential population of product Λ-doublet states with reflection symmetry Π(A″) was observed. The physical origin of this observed electronic orbital alignment can be attributed to a quantum interference phenomenon, as detailed in the accompanying paper. At the rotational level, the dominance of rotational rainbow scattering is unambiguously identified from both the existence of dynamical thresholds and a strong correlation between rotational level distributions at fixed translational energy and level specific excitation functions. These effects combined with other experimental observations lead us to visualize the CH+He scattering dynamics in a novel fashion. The collision can be regarded as a series of approximately independent sequential events each mediated by different regions of the interaction potential during the course of the whole encounter.

Anisotropy inversion model for atom-diatom rotational rainbows at large scattering angles

Two new anisotropy inversion procedures are outlined and compared with inversions using the well known hard-shell (HS) model. The new inversion procedures are based on explicit formulae for positions of primary rotational rainbows, using the more realistic deformed inverse-power potential (DIPP) model in a semiclassical infinite-order sudden theoretical framework. In a numerical study, the authors use the quantal infinite-order sudden theory to compute the rotational rainbow input data from various deformed potential surfaces of known anisotropies, each modelling the He-Na2 system. The authors consider centre-of-mass energies ranging from 0.1 eV up to 0.5 eV and rotational rainbows located at scattering angles theta >70'. Positions of three large-angle rotational rainbow maxima are used at each scattering energy. The inversion procedures then provide the anisotropy for the corresponding equipotential surface of the atom-molecule interaction.