Poisson's equation, a third-order differential equation, is formulated for the steady-state flow of electron current in a parallel-plane diode with an arbitrary distributed source of positive ions throughout the interelectrode region. Solutions are obtained for one particular class of ion distribution, and there is no upper limit to the electron current density J that can flow between the electrodes, provided there is sufficient ion production. The solution within this class that is most efficient in neutralization requires a total ion current density J/sub +/ that is related to J/sub -/ and the charge masses through J/sub +/J/sub -/ = 1.07 (m/sub -//m/sub +/)/sup 1/2/. For electron currents a few multiples of the Child-Langmuir limit, this ion source distribution has a single peak in the central region of the interelectrode space. As J/sub -/ is further increased, this peak moves progressively towards the cathode, increasing in height and decreasing in width. Methods of establishing the optimum ion distribution by ionization of a gas filling are considered. Dependent on the diode anode voltage, this ionization can be achieved either by collisions with the cathode electrons or by interaction with a transversely injected electron or laser beam. >