R. F. Snider

Quantum‐Mechanical Modified Boltzmann Equation for Degenerate Internal States

A modified quantum‐mechanical Boltzmann equation has been derived for the general case in which the molecules have degenerate internal states. This is an equation of the Boltzmann type for a quantity which is simultaneously a Wigner distribution function in molecular phase space, and a density matrix in internal state space. In particular, the nondiagonal terms of this density matrix have been included in the formalism, resulting in the collision term being modified from the usual Boltzmann expression. Thus the collisions are described in terms of combinations of the Lippmann‐Schwinger scattering matrix rather than the collision cross section. For nondegenerate states the usual collision term is obtained again.

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.

On the Evaluation of Kinetic Theory Collision Integrals: Diamagnetic Diatomic Molecules

A partial evaluation of the collision integrals for the Waldmann‐Snider collision superoperator is accomplished. This involves an exact integration over the center of mass coordinates and a distorted wave Born approximation treating the nonspherical part of the intermolecular potential as small. Certain exact relations between collision integrals are obtained after center of mass integration and further approximate relations are obtained within the DWB approximation. The general method of organizing the collision integrals has been considered according to their coupling (k ≠ 0) of angular momentum and velocity directions and uncoupling (k=0). In particular, for the latter, approximate formulas are obtained for the j dependence of the relaxation cross sections. For all this work, the collision cross sections are expressed in terms of averages of the nonspherical potential over the collision dynamics due to the spherical potential.

Generalized Boltzmann Equation for Molecules with Internal States

The general form of the Boltzmann collision operator is discussed. It is argued that this should be the natural transition (super‐) operator for a binary collision that is defined analogous to the ordinary transition operator in collision theory. Properties of the collision term are discussed, in particular pointing out its general nonhermiticity and nondefiniteness as well as its behavior under parity, rotations, and time reversal. In general, there is no H theorem.

Definitions and properties of generalized collision cross sections

A unified classification of binary collision cross sections is presented. This is accomplished with the aid of a collision superoperator, in terms of which both differential and kinetic theory cross sections are defined. Connections and comparisons are then made with standard expressions for chemical reaction cross sections and with those ’’spectral’’ cross sections describing collisional widths, shifts and overlapping of spectral lines. A detailed treatment of rotational invariance is included. Finally, it is shown how this general framework encompasses the more complicated cross sections that arise in the study of the field dependence of gas transport coefficients.

Senftleben—Beenakker Effect for the Viscosity of a Dilute Gas of Diamagnetic Diatomic Molecules

The effect of an external magnetic field on the viscosity of a dilute gas of rotating diamagnetic diatomic molecules is presented. A complete tensorial development in irreducible Cartesian tensors is given, the rotational states being treated quantum mechanically and the effect of inelastic collisions included. The magnetic‐field dependence is given in terms of reduced square‐bracket integrals and the variable H/p, where H is the magnetic‐field strength and p, the equilibrium pressure of the gas. In this way it has been possible to include the cross effects existing between shear and bulk viscosities and which arise only in the presence of a magnetic field.

Irreversible Thermodynamics of a Fluid System with Spin

The linear irreversible processes occurring in a fluid system whose molecules possess internal angular momentum are completely classified by means of irreducible Cartesian tensors in terms of the symmetry properties of the local equilibrium state of the fluid. It is assumed that this local thermodynamic state is characterized by the specific internal angular momentum s as well as by the mass density ρ and the specific entropy S. Even with the resulting tensorial phenomenological coefficients, it is found, for example, that there can be no contribution to the heat flux from the velocity gradient.

Thermal Conductivity of a Gas of Rotating Diamagnetic Molecules in an Applied Magnetic Field

The effect of an applied magnetic field on the thermal conductivity of a gas of rotating diamagnetic molecules is determined with the rotational states treated quantum mechanically. The anisotropic part of the thermal conductivity tensor λ is given in terms of square‐bracket integrals and the ratio of the magnetic field H to the equilibrium pressure p of the gas. The square‐bracket integrals to be determined are expressed in terms of reduced relative and center‐of‐mass velocities. The ratio of the saturation values of the anisotropic thermal conductivity when the applied field is parallel to and perpendicular to the temperature gradient is found to be ⅔. Also, a transverse heat flux or Righi—Leduc effect for gases is discussed briefly, and shown to vanish at infinite and zero fields, attaining a maximum somewhere between.

Kinetic theory collision integrals for diatomic molecules

Abstract The derivation of a number of exact and approximate relations between collision integrals for homonuclear and “nearly homonuclear” diatomic molecules is indicated. The validity of the approximations is discussed and tables of certain collision cross sections commonly occurring in the Senftleben-Beenakker effects are given.

Thermal Conductivity of a Gas with Rotational States

When degenerate internal states are present, the most convenient description of the average one‐particle state is by means of a Wigner distribution function‐density matrix which is a Wigner distribution function for translational motion and a density matrix in internal states. This matrix is assumed to be diagonal in energy but may in general be nondiagonal in degenerate states. The Boltzmann equation appropriate for this quantity is solved by the Chapman—Enskog method to obtain an expression for the thermal conductivity in terms of generalized quantum‐mechanical cross sections.