Magnetic Field Reversal by Relativistic Electrons Which Slow Down While Circulating in a Uniform Impressed Magnetic Field

Abstract

The prior restriction to conservative motion of the electrons has been removed, but scattering has still been neglected. The rate of energy loss has been used to derive the stationary distribution over a range of energy (or momentum) and canonical angular momentum for an arbitrary magnetic field distribution. A necessary ingredient is the retardation of the electrons; the radiative component of this has been estimated, but not included; the dynamical friction component, arising from motion through a uniform, fully ionized plasma, has been used as the sole contributor to energy loss. This friction also affects the canonical angular momentum and the latter's change in one cycle relative to the energy change has been formulated. Maxwell's equation connecting magnetic field with current density, applied to the stationary distribution, has permitted the formulation of the relation between field and the other variables. This completes the number of relations required to determine the magnetic field (eigenfunction) arising from a given rate of electron injection (eigenvalue): The trajectory equations have been derived. It has been necessary to picture physically the complete evolution of a trajectory from injection to stopping, both to understand what the mathematics says and to devise an adequate machine program. When field reversal occurs, then just before the electrons die, the electron swarm divides into two counter-rotating eddies. Two iterative solution techniques have been used to calculate the field distributions for a range of initial energies, field strengths, and injection rates. A striking result is that once reversal of field occurs by increasing the injection rate, it is generally true that the reversed field varies little, but the layer becomes progressively thinner as the injection rate increases further. At some rate the layer thickness passes through zero and thereafter the injection radius lies not at the apocenter of the initial trajectory but at the pericenter, the layer protruding from its prior position. For full-energy radii of gyration which are less than about one-third the injection radius, anomaolus solutions appear which are unstable. A number of aspects of the problem of incorporating an E layer into a thermonuclear device are briefly discussed, namely, the effect of scattering, the possibility of two-beam instability, the injection of the electrons, effects of end reflection in a finite-length layer, electrodynamic effects, and plasma diamagnetism.