Boltzmann Equation for an Electron Guide Field Accelerator. I. Quasi-Stationary Solutions for an Electron Beam

A simplified model for dePackh's version of an electron guide field accelerator is set up by substituting for the actual external quadrupole or solenoidal focusing magnetic field an azimuthally symmetric focusing field, and by replacing the actual toroidal geometry by a cylindrical geometry with periodic boundary conditions. The Boltzmann equation for an electron beam in this system is studied, and a set of solutions is obtained which contain just enough parameters to represent the quasi‐stationary behavior of the beam realistically. The values of these parameters are related to the initial conditions of the beam by the adiabatic invariance of linear charge density and of the radial and azimuthal action integrals in the absence of collisions and radiation. Thus, the quasi‐stationary development of the beam in time is determined without an explicit time dependence in the Boltzmann equation. While the electron energy is being increased by a betatron field, the beam passes from a condition in which its electrons are in almost neutral equilibrium with respect to displacement from the axis (low temperature or `` $\frac{3}{2}$ '' regime) to a condition in which each electron is hardly affected by the other electrons (high temperature or betatron or ``½'' regime), as predicted by dePackh. If the beam is initially isothermal, the temperature becomes a monotonic decreasing function of distance from the axis in the course of electron energy increase.