Radiative Equilibrium of a Free-Electron Gas in a Uniform Magnetic Field

Abstract

In this paper the subject of radiative equilibrium of a free-electron gas in a uniform magnetic field is studied on the basis of Landau's quantized particle motion of the electrons and the "Golden Rule" of time-dependent perturbation theory. The analysis presented here is somewhat analogous to the quantum theory of blackbody radiation. It is shown that the radiative equilibrium between electrons and their radiation field arises as a consequence of a balance between the two competing processes: photon emission (spontaneous plus induced or stimulated emission) and photon absorption. A coupled set of equations, one describing the time evolution of the photon distribution function and the other describing the time evolution of the electron distribution function, is thus derived. A simple closed-form solution of the equation for the time rate of change of the photon distribution function is presented for situations where absorption exceeeds the stimulated emission. This solution yields equations for the steady-state photon number density and the radiative relaxation time. It is shown that the steady-state photon number density is the familiar distribution function corresponding to Bose-Einstein statistics for "complete thermodynamic equilibrium." The conditions for the existence of overstabilities near the cyclotron frequency and its harmonics are also discussed. The equation for the time rate of change of the electron distribution function reduces, in the classical limit, to a Fokker-Planck equation in which there appear the usual "diffusion" and "dynamical friction" terms. In the classical limit, it is shown that this coupled set of equations (one describing the time evolution of the electromagnetic energy density and the other describing the time evolution of the electron distribution function) is self-consistent, by proving that the average rate of loss of $z$ momentum of the electrons as predicted by one equation is equal to the average rate of gain of $z$ momentum of their radiation field as predicted by the other equation. The $z$ axis is chosen to be along the uniform magnetic field.