William M. MacDonald

The contribution to the Fokker-Planck equation for the distribution function for gases, due to particle-particle interactions in which the fundamental two-body force obeys an inverse square law, is investigated. The coefficients in the equation, $〈\ensuremath{\Delta}\mathrm{v}〉$ (the average change in velocity in a short time) and $〈\ensuremath{\Delta}\mathrm{v}\ensuremath{\Delta}\mathrm{v}〉$, are obtained in terms of two fundamental integrals which are dependent on the distribution function itself. The transformation of the equation to polar coordinates in a case of axial symmetry is carried out. By expanding the distribution function in Legendre functions of the angle, the equation is cast into the form of an infinite set of one-dimensional coupled nonlinear integro-differential equations. If the distribution function is approximated by a finite series, the resultant Fokker-Planck equations may be treated numerically using a computing machine. Keeping only one or two terms in the series corresponds to the approximations of Chandrasekhar, and Cohen, Spitzer and McRoutly, respectively.

The relaxation to a Maxwellian distribution of a system of particles interacting through inverse-square-law forces is investigated in the approximation of two-particle interactions resulting in small-angle deflections of particle trajectories. The time required for the relaxation of the distribution in the neighborhood of the average energy is found to agree with the self-collision time defined by Spitzer. The time required for the distribution to become Maxwellian throughout the range from zero energy to several times the average energy is found to be nearly ten times the self-collision time. Filling of the high-energy portion of the Maxwell distribution is also discussed.

Once introduced into captive orbits, protons and electrons should be strictly trapped in the earth's dipole magnetic field. However, various mechanisms exist which limit their lifetimes, such as collisions with atoms and ions in the earth's outer atmosphere, charge exchange, and scattering by hydromagnetic waves. This paper considers only the effect of the scattering of these particles by the ionized hydrogen and electron components of the outer atmosphere. However, the effect of scattering from neutral atoms can be qualitatively taken into account by using the radius of the atom in place of the Debye shielding length in the scattering formulas. The Fokker-Planck equation has been used to derive an expression for the change in the distribution function due to small-angle, single-particle Coulomb collisions. Upper lifetime limits, as determined by this mechanism, of both protons and electrons are derived as functions of their initial energies.

A $K$-matrix formulation is given for the shell-model approach to a unified reaction theory. This $K$ matrix differs from the $R$ matrix of Wigner and Eisenbud in using distorted waves of a diffuse potential. The penetrabilities and channel phase shifts are therefore those for a diffuse potential rather than those for a square well with hard-sphere boundary conditions at some radius. This $K$ matrix is used to discuss the fine structure produced by a doorway-hallway system, and a result obtained earlier by Ferrell and MacDonald is rederived. A resonance expansion is then found which explicitly exhibits the widths and resonance energies in a multilevel formula. The distribution of the widths is found, a sum rule is derived, and the average cross section is derived. A general resonance formula is then derived for the case of any number of hallway and doorway states. This is used to generalize the previously obtained doorway-hallway results to the case of hallways coupled to the continuum. In certain cases, a characteristic asymmetry is shown to result.