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 |ω−kV0|2≫ωβ2, where ωβ 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.