Theoretical assessment of the transition between electron emission mechanisms in a collisional crossed-field gap

Understanding the transition to the limiting current under vacuum and collisional conditions for thermal-field emission in a crossed-field gap is critical for characterizing the operating regime for numerous applications, such as high-power microwaves and Hall thrusters. By deriving the exact solution for the current density as a function of cathode temperature (and, by extension, initial velocity), cathode electric field, applied voltage, collision frequency, and magnetic field, we determine the transition from the canonical current source (thermal-field emission) to the vacuum and collisional limiting currents under different operating conditions. We further show that the exact solution recovers the asymptotic solutions for the canonical and limiting currents for vacuum and collisional crossed-field gaps under appropriate limits. The exact solution demonstrates that the critical current decreases with both increasing magnetic field strength and increasing collisionality. Nexus phase space plots constructed by matching the individual asymptotic solutions provide the conditions for the transitions between mechanisms, elucidating when a single mechanistic equation would suffice and when theories combining multiple mechanisms are necessary.