An integral formulation is presented taking into account ion dynamics in the Pierce diode problem. Earlier restrictions on the ion drift velocity are lifted and generalized to an arbitrary ion injection velocity. An analysis of equilibrium solutions reveals a new quality of the diode as the number of nonuniform solutions is finite and reduces pairwise when the ion beam control parameter μ^≡${\mathit{m}}_{\mathit{e}}$${\mathit{v}}_{0\mathit{e}}^2$/${\mathit{m}}_{\mathit{i}}$${\mathit{v}}_{0\mathit{i}}^2$ is increased. Beyond μ^=0.2172 no solution exists. The consequences on the ballistic ion beam transport in the ion-beam neutralization problem are discussed.

• Received 4 December 1992

DOI:https://doi.org/10.1103/PhysRevLett.70.1105