Particle simulations of the low-α Pierce diode

The evolution of small initial perturbations of the uniform equilibrium of the ‘‘classical’’ Pierce diode [J. Pierce, J. Appl. Phys. 15, 721 (1944)] is studied using particle simulations. These simulations have been performed with the new bounded-plasma code PDW1 [Wm. S. Lawson (private communication)] and cover the parameter range 0<α<3π, where α=ωpL/v0. In the linear regime, three stages (initial transit, adjustment, and dominant eigenmode) are distinguished; oscillation frequencies, growth/damping rates, and potential profiles of the dominant eigenmode as well as oscillation frequencies of the next-to-dominant eigenmode are recovered and shown to agree quantitatively with recent analytical results. In the linearly unstable cases, the system evolves nonlinearly to a final state which may be either a new, nonuniform dc equilibrium, or a state of large-amplitude oscillations. In particular, for α=1.5π the character of the final state is found to depend on the details of the initial conditions.