Statistical theory of angular momentum polarization in chemical reactions

A general statistical treatment applicable to any vector property of reactive scattering is derived from angular correlation theory. This pertains to the usual experimental situation in which two or three vector directions are observed but numerous other vectors are random or unobserved, particularly various angular momentum vectors. The dependence of the cross section on the angles relating the observed vectors is expanded as a Legendre polynomial series, with coefficients which represent averages of angular momentum functions over the unobserved vectors. An algorithm for calculating these angular correlation coefficients is provided by the statistical theory. All non-vanishing terms involve only even-order Legendre polynomials. In many experiments, one or two terms are predominant. Classical and quantal versions give the same algorithm in the correspondence principle limit, which often holds for chemical reactions. The angular correlations involving the initial and final relative velocity vector directions [kcirc] and [kcirc]′ and the product rotational angular momentum j′ are treated in detail, including both pairwise and triple correlations. Explicit formulae are given for three choices of the quantization axis : along [kcirc], along [kcirc]′, and along [kcirc] × [kcirc]′. Coefficients for the ([kcirc], [kcirc]′, j′) correlations are tabulated for seven reactions as examples and comparison made with recent experimental measurements of the spatial orientation or polarization of j′ in reactions of alkali atoms with hydrogen halides and with methyl iodide.     

反应N(2D)+H2→NH+H和N(2D)+D2→ND+D的立体动力学理论研究.

基于Ho等人的精确势能面(J. Chem. Phys. 119, 3063(2003))研究,运用准经典轨线方法计算了21.3 kJ/mol碰撞能下反应N(²D)+H₂→NH+H和N(²D)+D₂→ND+D的产物与反应物之间的矢量相关．发现两个反应的产物角分布都是前向和后向呈现峰值分布,产物的转动角动量矢量j′不仅是取向的,而且是在y轴负方向上定向的．两个反应显示出的同位素效应主要归因于同位素质量的差别．     

Quasi-classical trajectory approach to the O（1D）＋HBr ）OH＋Br reaction stereo-dynamics on X1A＇ potential energy surface

The vector properties of reaction O(¹D)+HBr→ OH+Br on the potential energy surface (PES) of X¹A′ ground singlet state are studied by using the quasi-classical trajectory (QCT) theory. The polarization-dependent differential cross sections (PDDCSs), the average rotational alignment factor 2(j′· k)>, as well as the distributions reflecting vector correlations are also computed. The analysis of the results shows that the alignment and the orientation distribution of the rotation angular momentum vector of product molecule OH is influenced by both the effect of heavy-light-heavy (HLH) type mass combination and the deep well of PES.     

Isotope effect on the stereodynamics for the collision reaction H＋LiF（v = 0, j = 0） → HF＋Li

Stereodynamics for the reaction H+LiF(v=0, j=0) → HF+Li and its isotopic variants on the ground-state (1 2 A′) potential energy surface (PES) are studied by employing the quasi-classical trajectory (QCT) method. At a collision energy of 1.0 eV, product rotational angular momentum distributions P (θr), P (φr), and P (θr ,φr), are calculated in the center-of-mass (CM) frame. The results demonstrate that the product rotational angular momentum j′ is not only aligned along the direction perpendicular to the reagent relative velocity vector k, but also oriented along the negative y axis. The four generalized polarization-dependent differential cross sections (PDDCSs) are also computed. The PDDCS 00 distribution shows a preferential forward scattering for the product angular distribution in each of the three isotopic reactions, which indicates that the title collision reaction is a direct reaction mechanism. The isotope effect on the stereodynamics is revealed and discussed in detail.     

Theoretical study of the stereodynamics for the reaction F+HBr

The vector correlation between products and reagents for exothermic reaction F + HBr → HF + Br has been studied using a quasi-classical trajectory (QCT) method on the latest extended Lond–Eyring–Polanyi–Sato (LEPS) potential energy surface at three collision energies of 0.1 eV, 0.2 eV and 0.3 eV. Four polarization- dependent generalized differential cross-sections (2π/σ)(dσ₀₀/dω _t ), (2π/σ)(dσ₂₀/dω _t ), (2π/σ)(dσ₂₂₊/dω _t ), (2π/σ)(dσ_21?/dω _t ) have been presented in the centre of mass frame, respectively. The distribution of dihedral angle P(φ _r ), the distribution of angle between k and j ′ , P(θ _r ), are calculated. Both the influence of the collision energy and the influence of the reagent rotation on the product polarization have been studied in the present work, and the results indicate that the product rotational angular momentum j ′ is not only aligned, but also oriented along the direction perpendicular to the scattering plane. The orientation of the HF product rotational angular momentum vector j ′ depends very sensitively on the reagent rotation and also effected by the collision energy.     

Directional distribution of high-energy gamma rays from hot 125, 127Cs and 171Lu compound nuclei

Gamma-gamma directional correlations have been measured following ¹²C + ^113,115In and ¹⁵⁹Tb reactions at 100 MeV. Directional distribution coefficients have been determined for the gamma-ray spectrum between 2 and 16 MeV. For energies above 10 MeV the results for both reactions are in disagreement with interpretations in terms of the decay of the giant dipole resonance statistically excited either in a permanently deformed prolate compound nucleus undergoing collective rotation or an oblate nucleus with the angular momentum along the axis of symmetry. For the reaction on Tb indications of a quadrupole component at energies around 10 MeV are presented.     

Simple numerical method of computing the probabilities of angular momentum Jν occurring among possible vectors J Resulting from n angular momentum jμ summation (μ=1?n)

Arelatively simple numerical method of summing angular momentum vectors with maintaining space quantization rules of each summed angular momentum has been presented. The method enables the calculation of the values of probability (p j _μ) of finding a definite angular momentum J _μ among all vectors J being the results of quantum summation of n angular momentum vectors j _μ(μ=1-n. It may be used, e.g., in the calculations of angular momentum of many-particle states. The significance of the paper is connected with the possibility of taking into account, in a simple way, the angular momentum conservation principle for a system which consists of an arbitrary number of excitons. From Yadernaya Fizika, Vol. 67, No. 11, 2004, pp. 2123–2128. Original English Text Copyright ? 2004 by Kaczmarczyk. This article was submitted by the author in English.     

Product polarization distribution: Stereodynamics of the reaction of atom H and radical NH

The product angular momentum polarization of the reaction of H+NH is calculated via the quasiclassical trajectory method (QCT) based on the extended London-Eyring-Polanyi-Sato (LEPS) potential energy surface (PES) at a collision energy of 5.1 kcal/mol. The calculated results of the vector correlations are denoted by using the angular distribution functions. The polarization-dependent differential cross sections (PDDCSs) demonstrate that the rotational angular momentum of the product H2 is aligned and oriented along the direction perpendicular to the scattering plane. Vector correlation shows that the angular momentum of the product H₂ is aligned in the plane perpendicular to the velocity vector. It suggests that the reaction proceeds preferentially when the reactant velocity vector lies in a plane containing all three atoms. The orientation and alignment of the product angular momentum affects the scattering direction of the product molecules. The polarization-dependent differential cross sections (PDDCSs) reveal that scattering is predominantly in the backward hemisphere.      

Vector correlations of the reaction O(1D) + H2 on the DK potential energy surface

This paper reports on the angular momentum polarization of the products of the reaction O(¹D₂) + H₂ via the quasiclassical trajectory (QCT) calculation on the DK (Dobbyn and Knowles) potential energy surface (PES). The four polarization-dependent differential cross-sections (PDDCS) (0, 0), (2, 0), (2, 2), (2, ?1) were calculated at different collision energies. The vector correlation between reagent velocity and product angular moment, the vector correlation between reagent, product velocity and product angular moment were studied. From the calculations, it can be obtained that the OH products are produced mainly in the plane of H–O–H plane. The changes of OH products angular momentum j ′ direction along with the increasing collision energies were ascribed to the existence of a new reaction mechanism.     

Q- and Z-dependence of angular momentum transfer in deeply inelastic collisions of 86Kr with 209Bi

The dependence of the in-plane and out-of-plane angular correlations of fragments from fissioning heavy products on the kinetic energy and Z of the light reaction partner have been measured. From the dependence of the angular correlations on Q-value and hence energy loss, together with existing data from which the total angle-integrated cross section as a function of energy loss can be extracted, we have determined the dependence of the angular momentum transferred to the heavy product on the initial orbital angular momentum or impact parameter. The resulting dependence is qualitatively consistent with the sticking limit for a reaction intermediate of touching deformed fragments. More specific nuclear models generally underestimate the angular momentum transfer, although the one-body proximity-friction model accounts for the major fraction of the angular momentum transfer. A recent model incorporating both one-body proximity friction and collective excitations accounts quite well for the observed angular momentum transfer. The Z-dependendence of the anisotropy shows the importance of angular momentum fractionation for the less probable events, where the Z of the fissioning system is appreciably less than that of the target. The transferred angular momentum is shown to be fairly strongly aligned along the perpendicular to the reaction plane, with alignment values of 0.6 to 0.8. The component of angular momentum not along the perpendicular to the reaction plane is found to be primarily oriented perpendicular rather than parallel to the recoil direction. The absolute fission probabilities are found to be qualitatively consistent with J-dependent calculations using the J-values deduced from the angular correlations.     

Dynamics of reaction of O with H2 and its isotopic variants in different rotational excited states

Scalar properties and vector correlations of the reactions O+H₂ →OH+H, O+HD→OH+D, O+DH→OD+H, and O+D₂ →OD+D at collision energies of 25 and 34.6 kcal/mole have been studied via quasi-classical-trajectory (QCT) method based on a BMS1 potential energy surface (PES). Generalized polarization-dependent differential cross section and the distributions of the dihedral angle at the collision energy of 34.6 kacl/mole are presented. The calculated results indicate that both reagent rotational angular momentum and the mass factor have a significant influence on the scalar properties and vector correlations of the title reactions.     

On Calculations of Legendre Functions and Associated Legendre Functions of the First Kind of Complex Degree

Formulas for calculating Legendre functions and associated Legendre functions of the first kind of complex degree using an Ermakov—Lewis invariant are presented. These formulas are straight-forward to implement numerically and are motivated by the lack of computational routines in standard university tools like those of MatLab and Maple. Angular waves propagating in opposite directions are also obtained. The results are particularly useful in complex angular momentum theories and nearside/farside analysis of spin-dependent angular scattering from central potentials.     

Definitions and units in mechanics

With displacement, time, and force as basic undefined physical quantities, other physical quantities are defined as combinations of two vector quantities and one scalar quantity. Combinations include multiplication and division of vectors by vectors, scalars by vectors, and scalars by scalars. Defined quantities are vectors, scalars or quaternions, depending on directions of vectors in the definitions. Division of a vector by a vector is equivalent to multiplication of vectors divided by a scalar. The unit of a vector (or scalar) is itself a vector (or scalar) quantity. Thesquare meter (a vector) differs from meter ² (a scalar), and the cubic meter (a scalar) is different frommeter ³ . The characteristics of displacement, time, and force are considered known from experience.     

Two-Body T-Matrices Without Angular-Momentum Decomposition: Energy and Momentum Dependences

The two-body T-matrix is calculated directly as function of two vector momenta for different Malfliet-Tjon-type potentials. At a few hundred MeV projectile energy the total amplitude is quite a smooth function showing only a strong peak in forward direction. In contrast, the corresponding partial-wave contributions, whose number increases with increasing energy, become more and more oscillatory with increasing energy. The angular and momentum dependence of the full amplitude is studied and displayed on as well as off the energy shell as function of positive and negative energies. The behaviour of the T-matrix in the vicinity of bound-state poles and resonance poles in the second energy sheet is studied. It is found that the angular dependence of T exhibits very characteristic properties in the vicinity of those poles, which are given by the Legendre function corresponding to the quantum number either of the bound state or the resonance (or virtual) state. This behaviour is illustrated along numerical examples. Received May 29, 1997; revised October 17, 1997; accepted for publication December 28, 1997     

Line frequency expressions for triply degenerate fundamentals of spherical top molecules appropriate for large angular momentum

For states of high angular momentum we obtain expressions for the transition frequencies in the triply degenerate fundamental of a “heavy” spherical top, that are accurate to spectroscopic resolution, without tedious calculation of octahedral (or tetrahedral) vector coupling coefficients or diagonalization of matrices. Our calculations are based upon the physical assumption that as the molecule rotates with larger angular momentum it behaves like a symmetric top with the coupled pure rotational angular momentum quantized along either the four- or threefold body-fixed axes. We find that the approximate calculation which involves only 3-J symbols quite accurately reproduces the result of a complete diagonalization of the Hamiltonian, so that for all but the most accurate saturated absorption spectra theoretical analysis may be carried out without reference to the octahedral (or tetrahedral) subgroup of the rotation group.     

Exclusive (e,e′pp) knockout reactions

Exclusive cross sections of the ¹⁶O(e,e′pp) ¹⁴C knockout reaction are calculated for transitions to the low-lying discrete final states of the residual nucleus. Short-range correlations and two-body currents, due to the excitation of a Δ resonance in the intermediate state, are included in the calculations. Final-state interactions are taken into account by means of phenomenological spin-dependent optical potentials. Recoil-momentum distributions for transitions to states with different angular momentum exhibit different shapes, which are basically determined by the cm orbital angular momentum of the initial proton pair.     

The ideal relativistic spinning gas: Polarization and spectra

We study the physics of the ideal relativistic rotating gas at thermodynamical equilibrium and provide analytical expressions of the momentum spectra and polarization vector for the case of massive particles with spin 1/2 and 1. We show that the finite angular momentum J entails an anisotropy in momentum spectra, with particles emitted orthogonally to J having, on average, a larger momentum than along its direction. Unlike in the non-relativistic case, the proper polarization vector turns out not to be aligned with the total angular momentum with a non-trivial momentum dependence.     

Light-particle emission in the reaction 32S + 27A1 AT 135 and 190 MeV

The properties oflight particles emitted by the ³²S + ²⁷Al reaction at 135and 190 MeV bombarding energies were studied by means of a coincidence spectrometer. The spectrometer consisted of two large-area ionization chambers which measured the energy, momentum, mass and nuclear charge of the heavy reaction products. By requiring the conservation of those quantities the energy, momentum, mass and charge deficits were determined which are representative of the unobserved light particles. The analysis of the momentum deficit in the event plane did not yield an indication for a fast, direct process of light-particle emission. The alternative analysis in the fragments rest system confirmed the statistical nature of the emission process. The out-of-plane angular correlations were used to determine the spins of the particle-emitting fragments.     

Product polarization distribution: Stereodynamics of the reaction of atom H and radical NH

The product angular momentum polarization of the reaction of H+NH is calculated via the quasiclassical trajectory method (QCT) based on the extended London-Eyring-Polanyi-Sato (LEPS) potential energy surface (PES) at a collision energy of 5.1 kcal/mol. The calculated results of the vector correlations are denoted by using the angular distribution functions. The polarization-dependent differential cross sections (PDDCSs) demonstrate that the rotational angular momentum of the product H₂ is aligned and oriented along the direction perpendicular to the scattering plane. Vector correlation shows that the angular momentum of the product H₂ is aligned in the plane perpendicular to the velocity vector. It suggests that the reaction proceeds preferentially when the reactant velocity vector lies in a plane containing all three atoms. The orientation and alignment of the product angular momentum affects the scattering direction of the product molecules. The polarization-dependent differential cross sections (PDD-CSs) reveal that scattering is predominantly in the backward hemisphere.     

Monazite (monoclinic LaPO4) slip systems at room temperature

Polycrystalline monazite (monoclinic LaPO₄) was deformed by spherical indentation at room temperature. Slip systems were identified using TEM of thin sections prepared parallel and close to the indented surface. Dislocation Burgers vectors (b) were identified by Burgers circuit closure in high resolution TEM images, supplemented by diffraction contrast where possible. A total of 441 b determinations were made in 97 grains. The most common slip systems were [001]/(010), [100]/(010) and [010]/(100). Slip on (001) was less common. Many other less common slip systems and Burgers vectors were also identified, including b = [101], [101], [011], [110] and [111]. b = [101] dislocations dissociate into ½[101] partials, and b = [101] dislocations are inferred to dissociate to ½[101] partials, with a low energy stacking fault of ～30 mJ/m². b = [100] dislocations may dissociate into ¼[210] + ¼[210] partials. b = [010] may sometimes dissociate to ½[010] + ½[010] partials. Other types of partial dislocations were also observed and discussed. All partial dislocations were climb dissociated. The line energies of monazite dislocations and their partials were calculated, and stacking fault structures for partial dislocations are analyzed. Satisfaction of the Von Mises criterion for full ductility most likely involves [101]/(111) and ?011?/{011} or {111} slip, but other combinations that require both b = [101] and ?011? or ?110? are possible. If deformation twinning is active, slip systems with b = ?011? or ?110? may not be necessary for full ductility.