On the High-Frequency Oscillations of the Electronic Plasma
For the purpose of this note an electronic plasma is defined as a gas of classical, non-relativistic electrons immersed in a constant charge-neutralizing background. The plasma is assumed to be spatially limitless and free of externally applied fields. An exact analysis of the general oscillatory behavior of the plasma is forbiddingly difficult because it requires a detailed knowledge of the collision mechanism and ultimately leads to an intractable integro-differential equation. However, there are two extreme cases that are simple enough to be handled mathematically. One of these limiting cases occurs when the collisions are so frequent that the electronic distribution is Maxwellian in every volume element and local equilibrium is established. Then the behavior of the plasma is determined by macroscopic hydrodynamical equations which lead to the dispersion relation ω^2 = ω_(p)^(2) + (5/3)((ℋT)/m) k^2 where ω_p is the plasma frequency given by ω_(p)^(2) = (ne^2)/(mε_o) with T denoting the equilibrium temperature, ℋ Boltzmann's constant, k the wave number, m and e the electronic mass and charge respectively, and ε_o the dielectric constant of free space (M.K.S. system). This dispersion relation does not agree with the dispersion relation derived by the Thomsons, by Bailey, by Borgnis, and others. The reasons for this discrepancy have been reported by Van Kampen. In the other limiting case the collisions of the electrons with the ions and with each other are negligible and the collision term of Boltzmann's equation can be set equal to zero. This state is physically approximated when the frequency of oscillation is sufficiently high. Under special circumstances the dispersion relation in this case is approximately given by ω^2 = ω_(p)^(2) + 3((ℋT)/m) k^2. In this lecture we shall critically examine the theory of the high-frequency case, placing in evidence the tacit assumptions and hypotheses upon which the theory is based.
Additional InformationResearch supported by the U. S. Air Force Office of Scientific Research.
Submitted - TR000395.pdf