J. A. R. Coope

Irreducible Cartesian Tensors

This paper considers certain simple and practically useful properties of Cartesian tensors in three‐dimensional space which are irreducible under the three‐dimensional rotation group. Ordinary tensor algebra is emphasized throughout and particular use is made of natural tensors having the least rank consistent with belonging to a particular irreducible representation of the rotation group. An arbitrary tensor of rank n may be reduced by first deriving from the tensor all its linearly independent tensors in natural form, and then by embedding these lower‐rank tensors in the tensor space of rank n. An explicit reduction of third‐rank tensors is given as well as a convenient specification of fourth‐ and fifth‐rank isotropic tensors. A particular classification of the natural tensors is through a Cartesian parentage scheme, which is developed. Some applications of isotropic tensors are given.

Zero‐Field Splittings and Spin Densities in Ground‐State Triplet Aromatic Nitrenes

This paper outlines a simple semiempirical theory of the zero‐field splitting parameters of aromatic nitrenes. The theory is based on a Huckel LCAO description of the aromatic system, and takes advantage of the vanishing of odd orbital mutual atom polarizabilities in odd alternant molecules. The observed parameter for the parent imine radical is used as an empirical one‐center integral. Calculated parameters agree rather well with electron spin resonance results. The relation between the zero‐field parameters and general π‐electron spin densities is also outlined.

Determination of signs of hyperfine coupling constants for an S=½ system through E.P.R., ENDOR and double ENDOR

A systematic method of obtaining relative signs of hyperfine coupling constants is described. It applies to systems consisting of (a) a set of one or more nuclei coupled fairly strongly to the electron spin, and possessing a two-fold (or higher) axis of symmetry, together with (b) a set of weakly coupled nuclei defining superhyperfine transitions. ENDOR measurements for several E.P.R. hyperfine transitions, with the field oriented along the symmetry axis, give relative signs of hyperfine components for this direction. Signs for the other directions can then be obtained through ENDOR measurements on a single hyperfine transition at various field orientations. Additional double ENDOR measurements may be necessary for very weakly coupled nuclei. This method can complement double ENDOR studies in favourable cases. It is illustrated by the determination of signs of coupling constants of protons and of 75As in the AsO4 4- radical in KH2AsO4.

Collisionally Uncoupled Model for the Contribution of Orientational Polarizations to Transport Properties of Dilute Gases. II. The Shear Viscosity

The collisionally uncoupled model formulated in Paper I is applied to the calculation of the shear viscosity. The expressions obtained for the internal state contribution to the viscosity coefficients involve a summation, or integration, over the internal state frequency spectrum for each energy shell. The theory is first developed in a general way, and then a number of particular cases are considered. The expressions simplify in the commonly studied case of low magnetic fields, that is in the linear Zeemann region (and, equally, in the case of a linear Stark effect), because the frequency spectrum for each multiplet is then merely the Larmor frequency and its harmonics. As an example of this case, it is shown that the model gives a satisfactory one‐parameter description of the transverse viscosity coefficients of oxygen gas in the low pressure limit. The multiplet dependence of orientational relaxation times appears to be an important factor. The systematic choice of symmetry adapted phenomenological coe...

Electron spin resonance study of ClO2 and Cl2 - adsorbed on zeolites

An electron spin resonance study of ClO2 and Cl2 - adsorbed on faujasite and mordenite has shown that the surface environment can cause significant changes in the spin hamiltonian parameters describing these systems. In the case of faujasite, two distinct adsorption sites are observed. It is shown that the experimental hyperfine shifts can be explained by the presence of strong electric fields, presumably due to the cations in the zeolite structure.

Collisionally Uncoupled Model for the Contribution of Orientational Polarizations to Transport Properties of Dilute Gases. I

A simple collisionally uncoupled model of the contribution of orientational polarizations to the transport properties of dilute gases is proposed. This takes detailed account of the molecular free motions between collisions, and so encompasses a variety of phenomena associated with the decoupling of angular momenta by relatively strong external fields.

On the contribution of angular momentum polarization to gas phase thermal conductivity

The qualitative theory of the Senftleben effects [Coope and Snider, J. Chem. Phys. 57, 4277)], as applied here to the thermal conductivity, is used to stress the generality with which the contribution of angular momentum polarization to the thermal conductivity can be determined from the field dependence of the thermal conductivity.

Low‐Pressure Senftleben Effects for Molecular Oxygen

The effect of a magnetic field on the thermal conductivity and viscosity coefficients of paramagnetic Hunds case (b) multiplet Σ molecules, and in particular 3Σ oxygen, is investigated theoretically for the low‐pressure region, where the effects are functions of H / p. In this region the fields involved are correspondingly low, so that the total angular momentum J is well defined, and the distribution‐function density matrix may be assumed to commute with J2. This fact is exploited in the solution of the kinetic equation. The present treatment makes only very weak assumptions about the nature of the collisions, basically only that the collision (super)operator R is spin independent, which makes it possible to choose a basis in which the R matrix is diagonally dominant. Through detailed analysis of the dependence of all quantities involved on the mean rotational quantum number, N = 〈[N(N + 1)]1/2〉, information about the transport coefficients is obtained in a manner independent of the precise structure of...

Formulation of Resonance Conditions

A resonance condition is a condition on the variables of a system (external field, orientation, etc.) such that two energy levels differ by a fixed energy δ. This paper discusses the formulation of exact theoretical resonance conditions by methods which do not require knowledge of the energy levels themselves. An exact resonance condition for an n‐level system can be written as an algebraic equation of degree ½n(n−1) in δ2. The determination of this equation from the characteristic equation for the energies is an old algebraic problem discussed by Lagrange and by Cayley, the formulation of the equation of differences. Relevant properties are reviewed, and polynomial and determinantal formulas are given. Application is made to a particle of spin S≦52, with a second‐order tensor zero‐field term, in a uniform external magnetic field.A second approach is to construct an operator F={(K1×12−11×K2)2}, in a ½n(n−1)‐dimensional space of antisymmetric two‐particle states, whose eigenvalues are the squares of the di...

Theory of the Senftleben-Beenakker effects for NH3 and ND3: Collision integrals

The distorted wave Born approximation is used as a basis for understanding the cross sections which govern the production and relaxation of internal state polarizations, in gaseous ammonia, in the presence of shear flow or thermal gradients. For viscous flow, it is shown that the dominant dipole-dipole interaction leads to the production of [W](2)J polarization, but not to any appreciable [J](2) polarization. The essential absence of the usual [J](2) polarization, as produced by the anisotropic dispersion interaction (P2-potential), is consistent with the smallness of the anisotropy of the dipole polarizability in ammonia and with the suppression of the effect of such shorter ranged interactions (e.g. the P2 potential) in the presence of a dominant longer ranged interaction (there the dipole-dipole potential). For thermal conduction, it is shown that the dominant dipole-dipole interaction apparently produces predominantly a WJ(W2−52) polarization. The shorter ranged dipole-quadrupole interaction as well as higher multipole potentials produce a W[J](2) polarization, while a cross term between the dipole-dipole and quadrupole-dipole potentials can produce a velocity-independent dipolar polarization, denoted here as Jh1.