Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior

TLDR

The plasma frequency ωP, electron temperature T, wavelength λ, and oscillation frequency ω are defined. The study develops a theory to determine the frequencies of organized steady‑state oscillations in an unbounded plasma with uniform ion density, accounting for thermal motion but neglecting collisions. The authors derive a linear dispersion relation and extend the theory to describe how oscillations are initiated and how the system evolves toward organized behavior. The analysis yields a dispersion relation for long wavelengths, shows that sub‑Debye oscillations are forbidden, reveals that arbitrary frequencies exist but only the dispersion‑relation frequencies sustain macroscopic oscillations, and finds nonlinear trapped‑particle and traveling‑pulse solutions.

Abstract

A theory of electron oscillations of an unbounded plasma of uniform ion density is developed, taking into account the effects of random thermal motions, but neglecting collisions.The first problem considered is that of finding the frequencies at which a plasma can undergo organized steady-state oscillations of small enough amplitude so that a linear approximation applies. It is found that long wave-length oscillations of plasmas with a Maxwell distribution of electron velocities are characterized by the steady-state dispersion relation ${\ensuremath{\omega}}^{2}=\ensuremath{\omega}_{P}^{}{}_{}{}^{2}+(/frac{3\ensuremath{\kappa}T}{m}){(/frac{2\ensuremath{\pi}}{\ensuremath{\lambda}})}^{2}$. Here ${\ensuremath{\omega}}_{P}$ is the plasma frequency, $T$ the absolute temperature of the electron gas, $\ensuremath{\lambda}$ the wave-length, and $\ensuremath{\omega}$ the angular frequency of oscillation. It is also shown that organized oscillations of wave-lengths smaller than the Debye length for the electron gas are not possible.The theory is then extended to describe the processes by which oscillations are set up. It is found that, for a given wave-length, a plasma can oscillate with arbitrary frequency, but that those frequencies not given by the steady-state dispersion relation describe motions in which, after some time, there is no contribution to macroscopic averages. These additional frequencies lead asymptotically only to microscopic fluctuations of the charge density about the organized oscillation of the plasma. In this way, one can describe the manner in which the system develops organized behavior.The treatment is then applied to large steady-state oscillations for which the equations are non-linear. One obtains solutions in which particles close to the wave velocity are trapped in the trough of the potential, oscillating back and forth about a mean velocity equal to that of the wave. One can also obtain non-linear traveling pulse solutions in which a group of particles, moving as a pulse, creates a reaction on the surrounding charge, which traps the particles and holds them together.