Analytical model for estimating plasma parameters in a planar diode

We report the analytical model for estimating the electron temperature and number density in the plasma of a planar diode by considering the energy and particle balance equations. In a planar diode, two groups of electrons are present: an electron beam emitted from the cathode and cold electrons trapped in the plasma bulk. An electron beam emitted from a cathode excites plasma waves which can heat cold electrons in the bulk via particle-wave interactions. The resulting equation for the cold electron temperature contains six terms, including the energy exchange in the elastic, and inelastic (excitation and ionization) collisions between the cold electrons and neutrals, the energy exchange in the Coulomb collisions between cold and beam electrons, wave heating, and the wall losses of energetic electrons to the anode. We compare the results of the analytical theory with that of the particle-in-cell simulations using EDIPIC code and find a good agreement. The analytical model also shows that the heating term due to the Coulomb collisions with beam electrons and the energy loss to the anode are dominant, and they mostly balance each other, whereas the energy exchange due to the collisions with neutrals and wave heating are small and can be omitted. By coupling the energy balance equation with the particle balance equation, we can solve for both the number density and electron temperature as functions of the current density, electrode distance, pressure, and applied voltage. This model has been validated against past experiments and a good agreement was found.