On ion—electron beams emitted by a plane

Abstract

The non-linear one-dimensional steady-state equations which govern the flow of an ion—electron beam emitted from a plane are solved in the phase plane, and it is shown that a perfectly neutralized beam follows for a large range of injection velocities of the electrons. When the velocity of the ions is less than the electron sound speed the transition region for the neutralization has a length of the order of a Debye length λD = (kT)½ (4πNe2)–½, which is a typical plasma sheath. The maximum velocity of injection of the electrons for which neutralization is predicted is, in this case, the sound speed of the electrons. If the electrons are injected with a supersonic speed, they cannot be decelerated continuously to the subsonic speed corresponding to the velocity of the ions. No bound is set on the electron injection velocity from below. When the velocity of the ion beam is greater than the electron sound speed, oscillations with an amplitude which depends on the velocity of injection of the electrons, and a wavelength which depends on the ratio of the ion velocity to the electron speed of sound, are found. In this case the injection speed of the electrons needed to obtain the steady-state oscillatory solution is bounded both from above and from below. Subsonic electrons cannot be accelerated continuously to the supersonic velocity required to match the velocity of the ions, and within the supersonic range there is shown to be a limit (depending on the ratio of the ion velocity to the electron speed of sound, so that faster electrons cannot be decelerated continuously to match the (supersonic) ion velocity.