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.     

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.     

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．     

He+H+2及其同位素取代反应的立体动力学研究

运用准经典轨线方法,基于AQUILANTI势能面, 在碰撞能为145 kJ/mol时,对He+H⁺₂及其同位素取代反应的立体动力学性质进行了理论研究.对k-j′两矢量相关和k-k′-j′三矢量相关的分布函数,以及产物转动取向参数进行了详细的讨论.结果表明,He+HH⁺→HeH⁺+H,He+HD⁺→HeH⁺+ 关键词： 矢量相关 立体动力学 准经典轨线方法     

Theoretical study of stereodynamics for the O（3P） ＋ H2 → OH ＋ H reaction

This paper employs the quasi-classical trajectory calculations to study the influence of collision energy on the title reaction on the potential energy surface of the ground ³A' triplet state developed by Rogers et al. (J. Phys. Chem. A 2000 104 2308). It calculates the product angular distribution of P(θ_r), P(φ_r) and P(θ_r, φ_r) which reflects vector correlation. The distribution P(θ_r) shows that product rotational angular momentum vectors j' of the products are strongly aligned along the relative velocity direction k. The distribution of P(φ_r) implies a preference for left-handed product rotation in planes parallel to the scattering plane. Four different polarisation-dependent cross-sections are also presented in the centre-of-mass frame. Results indicate that OH is sensitively affected by collision energies of H₂.     

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.     

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.      

Structure of dislocations and stacking faults in the complex intermetallic ξ′-(Al–Pd–Mn) phase

We present the results of an electron microscopy study of defects in plastically deformed single crystals of the intermetallic ξ^′-(Al–Pd–Mn) phase. Pure edge dislocations with two different Burgers vector directions and four different Burgers vector magnitudes were found. All Burgers vector magnitudes observed can be described in terms of irrational fractions of the unit-cell parameters, and we have observed Burgers vector directions that can be indexed using irrational indices. The stacking faults observed have displacement vectors whose magnitudes and directions are incompatible with the unit cell of the ξ^′ phase. A comparison of the Burgers vectors observed in this study with those reported for the corresponding icosahedral quasicrystal shows that they are equivalent with respect to their directions and lengths. This leads to the conclusion that local order rather than long-range periodic (or quasiperiodic) order governs the structure of defects in these materials.     

An orthonormalization method: Theory and application to the j-j coupling of fermions in a single j-shell

An orthonormalization method suitable to deal with an overcomplete and non-orthogonal set of vectors is derived. It is then used to obtain the orthonormal set of states connected with the angular momentum coupling of N fermions in a single j-shell.     

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.     

The doubling of the anomalous magnetic moment of electron in a very strong constant homogeneous electric field

Calculated by the author previously [8], the anomalous magnetic moment (AMM) of the electron in an intense constant electric field changes nonmonotonically as the field increases, passing through a minimum and tending to the doubled Schwinger value for very strong fields. In the present paper, it is supposed that the AMM is related by the Lande factor to the angular momentum of a virtual electron accompanied by a virtual photon. This factor changes its effective value because of the influence of the external field on the motion of the virtual electron and its self-action. With increase of the electric field strength, the virtual electron can successively occupy the excited states l = 1, j = 1/2 and l = 1, j = 3/2 in addition to the original state with the orbital angular momentum l = 0 and the total angular momentum j = 1/2. The first of these excited states decreases the AMM and the second increases and doubles it if only this state is occupied for a very strong field. The latter condition is equivalent to the alignment of the spin and the orbital angular momentum of the electron along the field, while the total angular momentum of the entire system of the virtual electron and the virtual photon remains equal to 1/2.     

Amorphous rare earth alloys: Random-anisotropy ferromagnetism in the pair approximation

A self-consistent two-spin cluster approximation is used to calculate magnetization curves and correlation functions for a ferromagnetic amorphous alloy within the random-anisotropy model for the case of unit total angular momentum (j = 1) and correlations between the anisotropy axes.     

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.     

On the “vector coupling” of unitary spins

The problem of the vector coupling of unitary spins, i.e., the decomposition of the direct product of the irreducible representations D^[o] × D^[w] of the group U(3) into irreducible components D^[1] is considered. Using the projection operators for the group U(3), we have devised a procedure which allows one to construct systematically a complete set of vectors belonging to the irreducible representation D^[j], including those cases where this direct product D^[5] × D^[w] is not simply reducible.     

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.     

A FURTHER EXPLORATION FOR MECHANISM OF ANGULAR MOMENTUM TRANSFER IN HEAVY ION REACTION

The difficulty of the classical dynamics in studing the mechanism of angular momentum transfer is discussed. According to the nuclear rotational concept formed in the study of the nuclear structure for a long time, we point out that it is convenient to divide the total angular momentum I transfered to the target and the projectile into two components-the nuclear collective angular momentum I_R and the rotational alignment angular momentum I_j.     

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.