Space Charge in Ionization Detectors

The effect on the operation of a vacuum diode of a spatially varying distribution of free electrons, commonly called space charge, was first considered by Child [1] and Langmuir [2]. In that example electrons are released from a hot cathode at low velocity, and then drift freely under the influence of the local electric field as modified by the drifting electrons. In this problem, deduce the steady-state electric field distribution E = E(x) x̂ in an ionization detector (whose nominal field, E0 x̂, is uniform over the length D of the detector) as affected by the positive-ion space-charge density ρ(x). Suppose that ionization is created uniformly throughout the volume of the detector (by cosmic rays or by secondary particles from a particle accelerator) at a volume rate of K ion pairs/volume/sec. The ionization electrons are quickly collected at the anode and the associated electron space-charge density can be ignored. That is, you may assume the positive-ion space-charge density is not so great as to prevent the ionization electrons from reaching the anode. The positive ions drift slowly to the cathode with a velocity given by