Magnetically insulated electron flow with ions with application to the rod-pinch diode

A one-dimensional, steady-state, relativistic electron-flow model is developed that describes magnetically insulated electron flow in the presence of ions produced by space-charge-limited emission from the anode. The model is applied to the rod-pinch diode which is a cylindrical pinched-beam diode consisting of a small radius anode rod extending through the hole of an annular cathode. The diode is designed to run at critical current so that electrons emitted from the cathode are magnetically insulated and flow axially along the anode rod until they pinch radially onto the rod tip. Ions are emitted along the length of rod and flow radially outward. Without these ions, magnetically insulated electron flow cannot be established and electrons cannot propagate to the rod tip. Both fluid and Vlasov treatments of the electrons are considered. An analytic expression for the critical current is derived and is compared with the critical current determined from experimental data and particle-in-cell simulations. Reasonable agreement is obtained when the electron flow is nonlaminar, suggesting that a significant radial electron pressure is required to correctly describe the flow.