Electron Orbits

Abstract

We now consider the unperturbed motions of individual electrons in magnetic fields such as the simple magnetic-mirror configuration discussed in Chapter 2. In most situations where electrons are confined for long times by a static magnetic field there are three well-separated time scales that can provide a convenient albeit approximate description of these motions; namely, gyration about magnetic lines of force, bounce along field lines and between magnetic mirrors, and drift across lines of force. We begin with a nonrelativistic description which will be generalized later to include relativistic effects. Electron GyromotionThe most rapid of the three motions is the (right-handed) electron gyration about magnetic lines of force. This gyration is caused by the Lorentz forceð3:1ÞIn general, since the magnetic intensity varies along the trajectory of the electron, this equation of motion, Eq. (3.1), is intrinsically nonlinear. Rigorous treatments of this problem using expansions in the small parameter (m/e) have been published by several authors [1]. To obtain a useful approximate integration of this equation of motion we simply assume that within a region large enough to contain the electron for many periods of its gyromotion we may treat the local magnetic intensity, B, as spatially uniform and in the z-direction to write this equation asHere u z is a unit vector in the z-direction. This approach relies for its validity on the typical magnetic confinement situation in which the magnetic intensity varies negligibly over the extent of an electrons orbit, as discussed, for example, by Alfv en and F€ althammar as well as many others [1]. The two components of the equation of