Polarization-dependent Differential Cross Sections Research Articles

Cross-sections of relativistic quantum-mechanical versus those of classical magnetic resonant scattering

Context. Radiative transfer calculations in strong (few ×1012 G) magnetic fields, which are observed in X-ray pulsars, require accurate differential cross-sections of resonant scattering. While such cross-sections exist, their application is cumbersome. Aims. Here, we compare the classical (non-relativistic) with the quantum-mechanical (relativistic) resonant differential scattering cross-sections and offer a prescription for the use of the much simpler classical expressions with impressively accurate results. Methods. We expanded the quantum-mechanical differential cross-sections and kept the terms up to the first order in ϵ ≡ E/mec2 and B ≡ ℬ/ℬcr, where E is the photon energy and ℬcr is the critical magnetic field. We recovered the classical differential cross-sections along with the terms that are due to spin flip, which is a pure quantum-mechanical phenomenon. Results. When adding the spin-flip terms to the polarization-dependent classical differential cross-sections by hand, we find that they are in excellent agreement with the quantum mechanical ones for all energies near resonance and all angles. We plotted both of them and the agreement is impressive. Conclusions. We give a prescription for the use of the classical differential cross-sections for radiative transfer calculations that guarantees accurate results.

Theoretical rate constants and stereodynamics for the O + N2 collision

Quasiclassical trajectories have been integrated to determine the reaction rate constants for O + N2 collision. This reaction is endothermic and the channel only opens up above 10 000 K. When fitting those rate constants to the Arrhenius functions, we obtain the rate expression as: k(T) = 1.95 × 10−11exp(−37372/T) cm3 s−1 molecule−1. The distributions Pθr,ϕr for the title reaction at collision energies 3.6–10 eV are calculated. The results show that the angular momentum j of the product is not only perpendicular to the direction of the incident velocity vector, but also oriented along the negative direction of the Y-axis. Also, four polarization-dependent differential cross-sections (PDDCSs) are provided. PDDCS00 shows that the NO products change from forward scattering to backward scattering with increasing of the collision energy. Reaction cross sections and probabilities have been discussed and, the variation of nuclear distances indicate that the title reaction undergoes both the direct and indirect reaction mechanisms.

Differential cross sections and collision-induced rotational alignment in inelastic scattering of NO(X) by Xe

Fully Λ-doublet resolved differential cross sections and collision-induced rotational alignment moments have been measured for the NO(X)–Xe collision system at a collision energy of 519 cm−1. The experiments combine initial quantum state selection, employing a hexapole inhomogeneous electric field, with quantum state resolved detection, using (1+1′) resonantly enhanced multiphoton ionization and velocity map ion imaging. The differential cross sections and polarization dependent differential cross sections are shown to agree well with quantum mechanical scattering calculations performed on ab initio potential energy surfaces [J. Kłos et al. J. Chem. Phys. 137, 014312 (2012)]. By comparison with quasi-classical trajectory calculations, quantum mechanical scattering calculations on a hard-shell potential, and kinematic apse model calculations, the effects of the attractive part of the potential on the measured differential cross sections and collision-induced rotational alignment moments are assessed.

Active stereo-control of the Cl + CH4(ν3 = 1) reaction: a three-dimensional perspective.

The transition state in Cl + CH4 is of Cl-H-C collinear geometry. As the reactant CH4 is vibrationally excited by a linearly polarized infrared (IR) light to the antisymmetric-stretching state of ν3 = 1, all four C-H bonds are collectively excited and any one of the H-atoms can be reactive. Yet, a strong alignment of the excited CH4(ν3 = 1), as evidenced from the striking stereo-specificity in the Cl + CH4 reaction, was clearly revealed in a previous, exploratory study. Reported here is the full account of that investigation at two collisional energies of Ec = 4.8 and 2.7 kcal mol-1, using a crossed molecular-beam, product-imaging approach. By active control of the polarization direction of an IR laser under judiciously chosen beam-geometries, a complete set of polarization-dependent differential cross sections is disentangled from the CH3(00) product images. To our surprise, the quantitative results appear nearly identical to those obtained for the isotope-substituted reaction of Cl + CHD3(ν1 = 1) → HCl(ν) + CD3(00). A detailed discussion is presented to elucidate the underlying physics for such an intriguing similarity in stereo-reactivity between a spherical-top and a symmetric-top reactant.

Effect of isotope on state-to-state dynamics for reactive collision reactions and in ground state 12A″ and first excited 12A′ potential energy surfaces**Project supported by the National Natural Science Foundation of China (Grant No. 11504206) and the Shandong Jiaotong University PhD Research Start-up Fund, China.

We carry out quantum scattering dynamics and quasi-classical trajectory (QCT) calculations for the reactive collision in the ground (12A″) and first excited (12A′) potential energy surface. We calculate the reaction probabilities of and reaction for total angular momentum J = 0. The results calculated by QCT are consistent with those from quantum mechanical wave packet. Using the QCT method, we generate in the center-of-mass frame the product state-resolved integral cross-sections (ICSs); two commonly used generalized polarization-dependent differential cross-sections (PDDCSs), (2π/σ)(dσ00/dωt), (2π/σ)(dσ20/dωt); and three angular distributions of the product rotational vectors, P(θr), P(ϕr), and P(θr, ϕr). We discuss the influence on the scalar and vector properties of the potential energy surface, the collision energy, and the isotope mass. Since there are deep potential wells in these two potential energy surfaces, their kinetic characteristics are similar to each other and the isotopic effect is not obvious. However, the well depths and configurations of the two potential energy surfaces are different, so the effects of isotopic substitution on the integral cross-section and the rotational polarization of product are different.

Quasi-classical trajectory study of H+LiH (v = 0, 1, 2, j = 0) Li+H2 reaction on a new global potential energy surface

Quasi-classical trajectory (QCT) calculations are reported for the H+LiH (v = 0–2, j = 0) Li+H2 reaction on a new ground electronic state global potential energy surface (PES) of the LiH2 system. Reaction probability and integral cross sections (ICSs) are calculated for collision energies in the range of 0 eV–0.5 eV. Reasonable agreement is found in the comparison between present results and previous available theoretical results. We carried out statistical analyses with all the trajectories and found two main distinct reaction mechanisms in the collision process, in which the stripping mechanism (i.e., without roaming process) is dominated over the collision energy range. The polarization dependent differential cross sections (PDDCSs) indicate that forward scattering dominates the reaction due to the dominated mechanism. Furthermore, the reactant vibration leads to a reduction of the reactivity because of the barrierless and attractive features of PES and mass combination of the system.

Imaging inelastic scattering of CO with argon: polarization dependent differential cross sections.

Rotationally inelastic scattering of carbon monoxide (CO) with Argon at a collision energy of 700 cm-1 has been investigated by measuring polarization dependent differential scattering cross sections (PDDCSs) for rotationally excited CO molecules using a crossed molecular beam apparatus coupled with velocity-map ion imaging. A simple and robust (1 + 1' + 1'') VUV (Vacuum Ultra-Violet) REMPI (Resonance Enhanced Multi Photon Ionization) scheme is used and images are obtained by setting the VUV light polarization direction parallel or perpendicular to the scattering plane. Clear differences between the images for the two polarizations are observed, indicating strong collision induced alignment of the rotational angular momentum of scattered CO. A direct image analysis procedure as described in our previously published paper (A. G. Suits et al., J. Phys. Chem. A., 2015, 119, 5925), is employed to extract the fully quantum state resolved alignment-free differential cross sections (DCSs) and the state-to-state angle-dependent alignment moments for each final rotational state. The experimental results are compared with advanced theory, in particular with the predictions of CC QM (Close-Coupling Quantum Mechanical) and QCT (Quasi-Classical Trajectory) calculations. The agreement between experiment and theory is generally found to be quite good throughout the entire scattering angle range for all the final states probed, showing the reliability of the experiment and use of the direct extraction method, as well as the accuracy of the potential surface over the studied collision energy range. A classical kinematic apse (hard shell) model was found to be useful in interpreting the measured collision induced alignment moments.

The dynamics of the Br + HgBr (v = 0, j = 0) → Br2 + Hg reaction based on quasi-classical trajectory calculations

To investigate the dynamics mechanism of the Br + HgBr → Br2 + Hg reaction, the quasi-classical trajectory calculations are performed on Balabanov’s potential energy surface (PES) of ground electronic state. Both the scalar and vector properties are investigated to recognize the dynamics of the title reaction. Reaction probability for the total angular momentum quantum number J = 0 is determined at the collision energies (denoted as Ec) in a range of 1–25 kcal/mol, and the product vibrational distributions are given and compared between Ec = 20 and 40 kcal/mol. Other calculation values characterizing product polarizations including polarization-dependent differential cross sections (PDDCSs), distributions of P(θr), P([Formula: see text]), and P(θr, [Formula: see text]), are all discussed and compared between the two different collision energies in detail to analyze the alignment and orientation characteristics. It is revealed that the products prefer forward scattering and the PDDCSs are anisotropic in the whole range of the scattering angle. The product rotational angular momentum j′ shows a tendency to align perpendicular to the reagent relative velocity k. In fact, the product polarization of the title reaction is weak at both collision energies. In terms of horizontal comparison, the alignment is slightly stronger but the orientation is even less remarkable at higher collision energy.

Low-energy stereodynamics in the ion–molecule reactions D+ + D2/HD and H+ + H2/HD: reagent vibrational excitation effect and mass factor effect

The low-energy chemical stereodynamics of these ion–molecule reactions D+ + D2 → D2 + D+, D+ + HD → DH + D+/D2 + H+, H+ + HD → H2 + D+/HD + H+ and H+ + H2 → H2 + H+ has been investigated by performing the quasi-classical trajectory calculations on the ground electronic state potential energy surface (11A′ PES). The polarization-dependent differential cross sections and the probability distributions of P(θ r ) and P(ϕ r ) have been calculated, compared and discussed within the center-of-mass frame and at the low collision energy of 100 meV. The calculation results indicate that the product scattering and the polarization of the product rotational angular momentum are sensitive to the reactant vibrational quantum number. Such sensitivity increases with the increasing value of mass factor. Moreover, the products from the reaction H+ + HD → HD + H+ show quite different scattering and polarization behaviors with reagent vibration excitation.

Quasi-classical trajectory insight into stereodynamics for reaction $$\hbox {He}+\hbox {D}_{2}^{+}\rightarrow \hbox {HeD}^{+}+\hbox {D}$$ He + D 2 + → HeD + + D with the improved potential energy surface

In this work, a theoretical study on the detailed vector correlation for the reaction \(\hbox {He}+\hbox {D}_{2}^{+}\rightarrow \hbox {HeD}^{+}+\hbox {D}~(\hbox {v}, \hbox {j})\) has been carried out at the collision energy of 23.06 kcal/mol with different rotational states of j \(=\) 0–5 and vibrational states v \(=\) 1–5 by use of the quasi-classical trajectory calculation on an improved potential energy surface. The features of \({{\varvec{P}}}\mathbf{(}{\varvec{\theta }}_{{\varvec{r}}}{} \mathbf{)}\) distributions describing k and \(\mathbf{j}^\prime \) correlations were discussed. In addition, in order to get full knowledge of stereodynamics of the system, the distributions of dihedral angle \({{\varvec{P}}}\mathbf{(}{\varvec{\varphi }}_{{\varvec{r}}}{} \mathbf{)}\) and the polarization-dependent differential cross-sections (PDDCSs) were also reported. It has been demonstrated that both product alignment and polarization are sensitive to the reagent vibrational and rotational number. Furthermore, the dynamics behavior of the reaction is independently changed with respect to j under a certain v except for the product alignment effect \(({\varvec{P}}({\varvec{\theta }}_{{\varvec{r}}}))\), while it exhibits a generally regular trend concerning v when j is invariable.

Influence of collision energy on the stereodynamics of the H+CH+→C++H2 reaction

The reactive cross section and stereodynamics at selected collision energies for the H(2S)+CH+(X1Σ+)→C+(2P)+H2(X1Σg+) reaction on a globally smooth ab initio potential surface of the 2A' state are calculated in detail by the quasi-classical trajectory(QCT) method. The calculated cross section decreases with the increase of the collision energy, which is found to be in overall good agreement with the previous time-dependent quantum results in the high collision energy regime (Ec>20 meV). The discrepancy between the QCT and previous quantum cross section below 20 meV can be attributed to the limitations of the classical trajectory method, because the QCT method cannot handle the effect of zero point energy. In general, QCT results show qualitative agreement with the quantum results, which confirmsthe validity of the QCT method. The research shows that the product rotational angular momentum vector is aligned and oriented. The alignment of the product rotational angular momentum vector j' depends very sensitively on the collision energy. With the increase of the collision energy, the alignment effect recedesin the low collision energy region (1500 meV), while it is enhanced in the high collision energy region (500-1000 meV). Moreover, the k-k'-j' distributions tend to be asymmetric with respect to the k-k' scattering plane (or about φr=180°), with two peaks appearing at φr=90° and φr=270°, respectively. This indicates that the product rotational angular momentum is not only in the Y-axis direction but also along the positive Y-axis direction. The peak intensity decreases with the collision energy increasing from 1 meV to 100 meV, while it increases with collision energy increasing from 100 meV to 1000 meV. Therefore the Y-axis orientation effect turns weak with the enhancement of the collision energy in the low energy region, while it becomes strong in the high energy region. In addition, the polarization dependent differential cross sections (PDDCSs) (2π/σ)(dσ00/dωt) and (2π/σ)(dσ20/dωt) are calculated. PDDCS (2π/σ)(dσ00/dωt) results indicate that the products have almost symmetrically scattered forward and backward, and the intensity of the scattering increases with the increase of the collision energy. The PDDCS (2π/σ)(dσ20/dωt) shows that the alignment effect of the rotational angular momentum of the products is stronger at the terminal of the scattering angle than at the other directions.

The complex-forming dynamics of Ne + NeH(D)+ (v = 0, 1; j = 0) reactions

The quasi-classical trajectory method has been employed to investigate the dynamics of the title reactions based on an accurate potential energy surface. The agreement between the QCT and quantum wave packet results is good for the integral cross sections and the reaction probabilities. The influences of the collision energy on the average lifetime of the complex-forming are also predicted. The polarization dependent differential cross sections of the title reactions are also calculated to uncover the reaction mechanism. The calculated results indicated that the collision energies play an important role in the complex-forming dynamics of the title reactions.

Isotope Effect of the Stereodynamics for the Reactions N(<sup>4</sup>S)+HD→NH+D and N(<sup>4</sup>S)+HD→ND+H

Quasi-classical trajectory calculations have been employed to investigate the influence of isotope effect on the stereodynamics of the title reactions N(4S)+HD→NH+D and N(4S)+HD→ND+H on the 4A" double many-body expansion (DMBE) potential energy surface (PES) newly constructed by L. A. Poveda et al. [Phys. Chem. Chem. Phys. 7 (2005) 2867]. The generalized polarization-dependent differential cross sections (PDDCSs) and the three angular distributions of P(θr), P(φr) and P(θr,φr) are presented and discussed. It is revealed isotope effect exert a substantial influence on the product polarizations.

Differential steric effects in Cl reactions with aligned CHD3(v1 = 1) by the R(0) and Q(1) transitions. II. Abstracting the unexcited D-atoms.

A complete set of four polarization-dependent differential cross sections in the reactions of Cl + aligned-CHD3(v1=1,jK)→DCl(v=0)+CHD2(v1=1) is reported here for two different, rotationally polarized states with j = 1: specifically the jK=10 state prepared via the R(0) excitation and the 1±1 state via Q(1). In stark contrast to the complicated situation of the HCl(v) + CD3(v = 0) channel reported in Paper-I, the stereo-requirement of this isotopic channel for both polarized reactants appears quite straightforward and consistent with a direct rebound mechanism. The extent of steric effects is moderate and relatively smaller than the alternative H-atom abstraction channel. All major findings reported here can qualitatively be understood by first noting that the present reaction invokes abstracting a D-atom, which is the spectator in the IR-excitation process. Next, it is recognized that the directional properties of two polarized states of CHD3(v1=1, jK) should manifest primarily in the IR-excited C-H bond, leaving secondary imprints in the unexcited CD3-moiety. The stereo-specificity of the DCl + CHD2 product channel is further reduced by the fact that the abstraction can occur with any one of the three spatially distinct D-atoms.

Differential steric effects in Cl reactions with aligned CHD3(v1 = 1) by the R(0) and Q(1) transitions. I. Attacking the excited C-H bond.

When a CHD3 molecule is pumped to the C-H stretching-excited state by absorbing a linearly polarized infrared (IR) photon via the R(0) branch of the v1 = 1←0 transition, the rotational angular momentum j of the prepared state jK=10 preferentially lies in a plane perpendicular to the IR polarization axis εIR. By way of contrast, when the Q(1) branch is used, the state of jK=1±1 is prepared with j aligned along the direction of εIR. Reported here is a detailed study of the title reaction by actively controlling the collision geometries under these two IR-excitation schemes at collision energy Ec = 8.6 kcal mol-1, using a crossed molecular beam, product imaging approach. We found that under the R(0) excitation, the polarization-dependent differential cross sections for the HCl(v = 0) + CD3(00) channel can largely be understood by invoking dual reaction mechanisms. The forward-scattered products are most likely mediated by a time-delayed resonance mechanism-as the formation of the HCl(v = 1) + CD3(00) channel, whereas the backward/sideways scattered products are governed by a direct abstraction mechanism. Compared to the previous results at lower Ec of 3.8 kcal mol-1, the sighting of opening-up the attack angle at the transition state of the direct pathway is proposed. Results under the Q(1) excitation are, however, perplexing and bear no obvious correlation to the corresponding ones for the R(0) excitation, defying simple intuitive interpretation. Possible reasons are put forward, which call for theoretical investigations for deeper insights. The results on the alternative isotope channel, DCl + CHD2, will be reported in the following paper.

Intrinsic product polarization and branch ratio in the S(1D, 3P)+HD reaction on three electronic states

The intrinsic product polarization and intramolecular isotope effect of the S(1D, 3P)+HD reaction have been investigated on both the lowest singlet state (1A′) and the triplet state (3A′ and 3A″) potential energy surfaces by using quasi-classical trajectory and quantum mechanical methods. The calculations indicate that intramolecular isotope effects are different on the three electronic states. The stereodynamics study shows that the P(θr) distributions, P(ϕr) distributions, and polarization-dependent differential cross sections (PDDCSs) (00) are sensitive to mass factor and the product angular momentum vectors are not only aligned but also oriented.

Influence of Reactant Vibrational Excitation on Stereodynamics of C + NO → CN + O Reaction

Stereodynamics of the title reaction at different reagent vibrational states (v=0-2) are obtained using the quasiclassical trajectory method at collision energy of 0.06 eV on an accurate 4A” potential energy surface. The vector properties including angular momentum P(θr) and P(φr) distributions as well as polarization-dependent differential cross sections (PDDCS) of the product CN are presented. Furthermore, the influence of vibrational excitation on the product vector properties has also been studied in present work. PACS: 34.35.+a, 34.50.-s

Product rotational angular momentum polarization of H+FCl (v=0-5; j=0, 3, 6, 9) → HF+Cl and HCl+F at Erel=0.5-20 kcal mol(-1).

The rotational angular momentum polarizations of product molecules of the title reactions on the ground potential energy surface 1 (2)A' of DHTSN [Deskevic et al. J Chem Phys 2006, 124, 224303] have been studied using the quasi-classical trajectory method. Reaction dynamic results of the HF product channel comparing with another channel of HCl with 100,000 trajectories can be accurately resolved. We show the value of the polar p(ϑr) in the range of 0° ≤ ϑr ≤ 180(°), azimuthal p(φr) in the range of 0° ≤ φr ≤ 360(°), and dihedral p(ϑr, φr) in the ranges of 0(°) ≤ ϑr ≤ 180(°) and 0(°) ≤ φr ≤ 360(°); the angular distributions of the product molecules HF and HCl at relative Erel = 0.5, 1, 2, 5, 10, 15, and 20 kcal mol(-1); and four polarization-dependent differential cross sections (PDDCSs) of HF and HCl at Erel = 0.5, 1, 2, 5, 10, and 15 kcal mol(-1). p(φr) distributions at v = 0-5, and j = 0, 3, 6, 9 at every Erel are plotted cylindrically together. The stereo dynamic transformation reaction dependent upon the rovibrational states of the reactant molecule FCl and its relative translational energies around 0.5-5 kcal mol(-1) can be significantly differentiated. Translational and rovibrational enhancements of the title reactions on both early barrier potential energy surfaces have been shown in great detail and clarified. Reaction mechanisms of forward and backward scattering of the product molecules HF and HCl, respectively, have been obtained. Graphical Abstract H + FCl → either HF + Cl (left) or HCl + F (right) is moving along a trajectory on the respective PES.

Inelastic Scattering of CO with He: Polarization Dependent Differential State-to-State Cross Sections.

A joint theoretical and experimental study of state-to-state rotationally inelastic polarization dependent differential cross sections (PDDCSs) for CO (v = 0, j = 0, 1, 2) molecules colliding with helium is reported for collision energies of 513 and 840 cm(-1). In a crossed molecular beam experiment, velocity map imaging (VMI) with state-selective detection by (2 + 1) and (1 + 1') resonance enhanced multiphoton ionization (REMPI) is used to probe rotational excitation of CO due to scattering. By taking account of the known fractions of the j = 0, 1, and 2 states of CO in the rotationally cold molecular beam (Trot ≈ 3 K), close-coupling theory based on high-quality ab initio potential energy surfaces for the CO-He interaction is used to simulate the differential cross sections for the mixed initial states. With polarization-sensitive 1 + 1' REMPI detection and a direct analysis procedure described by Suits et al. ( J. Phys, Chem. A 2015 , 119 , 5925 ), alignment moments are extracted from the images and the latter are compared with images simulated by theory using the calculated DCS and alignment moments. In general, good agreement of theory with the experimental results is found, indicating the reliability of the experiment in reproducing state-to-state differential and polarization-dependent differential cross sections.

Theoretical study of product polarization of O(1D) + HCl(v = 0; j = 0) → ClO + H and its isotope exchange reaction

O(1D) + HCl(v = 0; j = 0) → ClO + H and its isotope exchange reaction O(1D) + DCl(v = 0; j = 0) → ClO + D are studied in the collision energy range 14.0–20.0 kcal/mol based on the potential energy surface 1[Formula: see text] state. Reaction probabilities, integral cross sections, the two angular distribution functions (concerning the initial/final velocity vector, and the product rotational momentum vector), and the product rotational alignment parameters are calculated as a function of the collision energy for the two reactions. The four generalized polarization dependent differential cross sections are presented to manifest the polarization characters. Also, the effect of the collision energy and the kinetic isotope effect are studied.