Electromagnetic Interaction of a Beam of Charged Particles with Plasma

The plasma-beam instability has been studied by Akhiezer and Fainberg under the assumption that $\theta =0\mathrm{}$, where $\theta$ is the angle formed by the direction of the beam and the direction of the growing wave resulting from the instability. Under these conditions the interaction is electrostatic, i.e., the wave is longitudinal. In this investigation the above assumption is generalized so as to include the case of $\theta \ne 0$ and the effect of electromagnetic interaction. For $\omega \sim {\omega }_1$, where ${\omega }_1$ is the Langmuir frequency of the plasma, the interaction is electrostatic for all values of $\theta$ and the resulting instability which produces a longitudinal wave increases exponentially in accordance with the term $\exp {\left(\frac{3\sqrt{3}{{\omega }_0}^{2}k{v}_{0}cos\theta }{8}\right)}^{\frac{1}{3}}$ (where ${\omega }_0$ is the Langmuir frequency of the beam). For a frequency range below ${\omega }_1$ the instability is less pronounced. However, this instability is significant, since the interaction is electromagnetic and the "growing wave" resulting from this interaction is characterized by an electric vector having both transverse and longitudinal components. In investigating the above instabilities, an assumption was made that the density of the incident beam is small and the results cover all values of $\theta$ except those in the immediate neighborhood of $\frac{\pi }{2}$. For $\theta$ in the neighborhood of $\frac{\pi }{2}$ the assumption is more general and the results apply to any density of the beam.