Acceleration of electrons in plasmas

Abstract

When an applied electric field greatly exceeds the value Ec = e/D2 approximately equal to 10-8n/TV cm-1 (where D is the Debye length) the rate of momentum transfer in electron-ion encounters is negligible. One would therefore suppose that the electrons and ions of a plasma should be freely accelerated. The experimental results from plasma betatrons have shown, however, that with electric fields of E0 >> Ec the accelerated electron currents are still extremely small. Evidently the theory of runaway electrons is valid for E0 << Ec, but is inadequate when E0Ec, and a more sophisticated theory is required which takes into account the momentum transfer processes in collective interactions. In a finite plasma there are no exponentially growing longitudinal plasma oscillations when Dk' > 0.755, where k' depends upon the boundary conditions. In a cylindrical plasma of radius a, k' is approximately equal to a-1. In such a plasma E0 >> Ec is a sufficient condition for the free acceleration of the electrons and ions. However, the current that is achieved when the electrons approach the velocity of light is small, and is typically less than 0.1 amp. If it is possible to accommodate a very wide energy spread in a circular machine it is shown that the electrons can then be accelerated free from longitudinal instabilities when E0 << Ec. But in machines such as plasma betatrons only a relatively narrow energy spread can be tolerated, and the question is then how large must the applied field be to accelerate the electrons in spite of the electrostatic instability? Approaching the problem from a phenomenological point of view, one finds that the sufficient condition is that E0 > Ec, where Ec = 4 nek'-1 approximately equal to 10-6nk' V cm-1 is now the maximum possible fluctuating electric field. The value of Ec is many orders of magnitude larger than Ec. Thus, in order to force the electrons out of their thermal state in a machine designed to yield several kilo-amperes of relativistic electrons requires accelerating electric fields not of a few volts per cm but of the order of 106 V cm-1.