Instabilities in a Plasma-Beam System Immersed in a Magnetic Field

Abstract

The interaction of an electron or an ion beam moving with velocity $\ensuremath{\beta}c$ through a stationary plasma in the presence of a static magnetic field is investigated theoretically under the assumption that the plasma is cold, the intensity of the beam is very small, and that the beam, as well as the waves resulting from the interaction, are aligned along the direction of the magnetic field. It is well known that an electron beam moving with velocity exceeding the phase velocity of electromagnetic waves in a stationary plasma is capable of exciting such waves and these waves (circularly polarized) rotate in the same direction and at the same angular frequency as the gyroelectrons of the beam. It is shown in this investigation that there is an apparent reversal in the direction of rotation of these gyroelectrons as seen by the stationary observer. Consequently, the excited wave has a circular motion in the same direction as perturbed stationary ions. The frequency $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}$ of the excited wave satisfies the inequality $|\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}|l{\ensuremath{\Omega}}_{i}$, where ${\ensuremath{\Omega}}_{i}$ is the gyrofrequency of perturbed stationary ions. Similarly, a wave excited by an ion beam has a circular motion in the same direction as perturbed stationary electrons. The frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{i}$ of the latter wave satisfies the inequality $|{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}^{i}|l{\ensuremath{\Omega}}_{e}$, where ${\ensuremath{\Omega}}_{e}$ is the gyrofrequency of perturbed stationary electrons. It is shown that an electron (ion) beam moving in the direction of the magnetic field can excite only a wave having negative (positive) helicity. The reverse situation occurs for a beam moving against the magnetic field. General relationships are formulated and illustrated graphically for determining the frequency and growth rate of waves which can be excited by an electron or an ion beam in any magnetized cold plasma. A particular case is illustrated in which an incident ion beam interacts with a relative dense plasma such as occurring in thermonuclear discharges, ionosphere, and interstellar space. It is shown that in such cases there is an excitation of hydromagnetic waves for a wide range of velocities of the ion beam.