Fit Functions for Relativistic, Single-Species and Bipolar, and Space-Charge-Limited Current Densities

This technical note reviews solutions for planar, relativistic space-charge-limited (SCL) flow both for the single species case with electrons and for bipolar flow with electrons and protons. In most cases, numerical integration or evaluation of special functions is needed to obtain values for these solutions. Here, fit functions for these solutions are derived so that they can be used to calculate 1-D SCL electron and proton current densities and the bipolar enhancement factor for any voltage, including the range transitioning from nonrelativistic to relativistic behavior (approximately 0.1-2 MV, γ = 1.2-5 for electrons and 102-105 MV, γ = 1.1-100 for protons). These fit functions reproduce the exact values to within a few percent, or with an error correction, to within a fraction of 1%. These functions should be useful for evaluating SCL current densities for a range of problems where relativistic effects are important and the 1-D approximation is applicable.