Counterstreaming electrons and ions in Pierce-like diodes.

The dynamics of Pierce-like diodes is investigated for ions moving with arbitrary velocities opposite to as well as in the direction of electron propagation. By application of an integral formalism that is able to account for mobile streaming ions, equilibrium solutions for the diode in an external circuit are derived. The stability of the uniform equilibrium for the short-circuited diode and for counterstreaming species is investigated. In this case, new oscillatory unstable branches appear that destabilize the diode for all values of \ensuremath{\alpha}, where \ensuremath{\alpha} is the Pierce parameter. This contrasts with the diode with co-moving ions being stable for sufficiently small values of \ensuremath{\alpha}. These new branches coincide with the Pierce-Buneman modes for initially resting ions. If \ensuremath{\Vert}${\mathit{v}}_{\mathit{i}0}$\ensuremath{\Vert}/${\mathit{v}}_{\mathit{e}0}$ is increased, where ${\mathit{v}}_{\mathit{s}0}$ are the injection velocities (s=e,i), a stronger destabilization is observed for counterstreaming in comparison to co-streaming with maximum growth rate in the range of ${\mathit{v}}_{\mathit{i}0}$\ensuremath{\approxeq}-2${\mathit{v}}_{\mathit{e}0}$ for a hydrogen plasma. It is furthermore shown that the familiar picture of waves (Fourier modes) must be substantially modified to meet the physics of bounded plasmas.