The Knee of the Transition from TL to FSCL Operation Space Charge Effects on Thermionic Emission

Thermionic emission cathodes are widely used in microwave tubes. In the temperature limited (TL) region, the Richardson-Dushman equation, corrected for the Schottky effect applies. In the fully space charge limited (FSCL) region, the Child-Langmuir law applies. Longo found that the following simple relation described the emission from an impregnated cathode over a wide range of voltages and temperatures: 1/J_OP = 1/J_FSCL + 1/J _TL (1) However, the knee of the transition curve from TL to FSCL operation described by Longo's equation is much broader than that found for most cathodes. Vaughan proposed the relation 1/J_OPⁿ = 1/J_FSCLⁿ + 1/J_TLⁿ (2) where n can be adjusted to provide the best fit to data. A value of 6 to 10 was given as being appropriate for good electron guns. Fortgang employed a simple model based on the cathode having a work function distribution function to fit the experimental data measured from TL to FSCL regime. He found that a single work function distribution could fit the I-V curve over a range of cathode temperatures very well. In this work, we study the space charge effects on thermionic emission via a self-consistent approach. The thermionic emission process is described by the Richardson-Dushman equation, corrected for the Schottky effect. Space charge effects are described by Poisson's equation including relativistic effects. The approach enables us to make detailed calculations over a wide range of voltages and temperatures. The knee of the transition curve from TL to FSCL operation is obtained self-consistently.