Application of the Vlasov Equation to the High-Frequency Stability of Finite Relativistic Beam-Plasma Systems

The response of a finite relativistic beam‐plasma system to small disturbances is analysed using the coupled Vlasov‐Maxwell equations. It is shown that when ${\mathrm{|\omega -kV}}_0{|}^2{\mathrm{\gg \omega }}_{\beta }^2$ , where ω_β is the betatron frequency and V₀ the drift velocity of the beam, straight‐line orbits are sufficiently accurate to describe the beam‐particle trajectories. Dispersion relations are derived for a finite beam and for both infinite and finite plasma configurations. The effect of a finite but large beam radius is shown to increase the growth of the electrostatic instability.