Collective Oscillations in a Cold Plasma

The collective motions of a fully ionized cold plasma in a uniform external magnetic field are treated by standard small amplitude theory. Finite temperature and collision effects are neglected. Specializing the analysis to a neutral plasma of uniform unperturbed density containing electrons and ions of one species, one obtains a dielectric tensor and dispersion relation which is a special example of results previously given by Aström. A detailed discussion of the exact dispersion relation is given for the entire frequency spectrum, and completeness theorems are presented with the aid of scalar potentials representing the electromagnetic field quantities. It is found that when the Alfvén dielectric constant α = 4πn(m + M)c2/B02 becomes comparable in magnitude to the ion-to-electron mass ratio, the plasma space charge may play an important role in determining the nature of collective oscillations. In particular, if the axial wavelength of the perturbation is sufficiently large, the singularities of the effective dielectric constant become displaced from the neighborhood of the particle cyclotron frequencies to hybrid frequencies, which, in the limit of high plasma density, become equal to the geometric mean of the cyclotron frequencies and the plasma frequency, respectively. The last two sections discuss particle orbits in idealized oscillatory modes and simplified boundary value problems associated with plasma resonance.