Interaction of a Plasma with a "Helical" Electron Beam

The behavior of a plasma-beam system immersed in a magnetic field ${\mathbf{B}}_0$ is investigated theoretically under the assumption that the electrons comprised in the beam have helical trajectories coaxial with ${\mathbf{B}}_0$. The beam is designated as "helical." The plasma-beam system is examined for instabilities which result in growing transverse waves aligned in a direction parallel to that of ${\mathbf{B}}_0$. Two classifications of instabilities are introduced. In accordance with one of these, the instabilities can be "superluminous," "subluminous," or "counterstreaming," depending on direction and velocity of the excited wave. The other classification differentiates between excited "$P$" and "$B$" waves and is based on the comparison of some of the waves excited by the beam and the corresponding waves which can be radiated by a single particle in the unperturbed plasma. If the beam is linear (not helical), i.e., if the electron trajectories in the beam are parallel to ${\mathbf{B}}_0$, there is only one type of instability ("$P$" instability) in which the frequencies of the waves excited by the beam differ very little from the frequency of the corresponding waves which can be radiated by a single particle in the undisturbed plasma. On the other hand, if the beam is helical there is an additional instability ("$B$" instability). The latter instability occupies a relatively wide frequency range in the neighborhood of the corresponding "$P$" instability and represents a continuation of the "$P$" instability along the frequency axis. When the intensity of the helical beam is sufficiently small, the "$P$" instability is "strong" and the "$B$" instability is "weak," i.e., the rate of growth of the "$B$" waves is relatively small. Physical conditions under which different types of instability may occur are discussed and the above types of plasma-beam instabilities are investigated with particular reference to the frequency ranges and rates of growth of the excited waves. It is shown that a "$P$" instability is always multiple, i.e., there are at least two excited waves having different frequencies which are aligned in the direction of the magnetic field. (A multiple "$P$" instability does not occur if the beam is linear.) The character of the multiple "$P$" instability is analyzed for various transverse and longitudal velocity components of the electrons in the beam. Using a graphical analysis of the dispersion equation for the plasma-beam system in the $\omega -k$ plane (where $\omega$ is the frequency and $k$ is the wave number), it is found that superluminous and subluminous instabilities are convective, whereas counterstreaming instabilities are nonconvective.