Analytical model for estimating the equilibrium plasma quantities in an electron beam-plasma system

We develop an analytical model for estimating the equilibrium quantities, such as electron temperature and number density, in an electron beam-plasma interaction system. This model provides a convenient way to calculate the effective electron temperature and density by considering the energy balance of the bulk cold electrons. Six energy sources/losses terms relevant to the cold electrons are accounted for, where quasi-linear theory is applied for estimating wave heating at equilibrium. We compare this calculation with the particle-in-cell (PIC) simulation results and find good agreement. Based on these results, we then consider two situations where we can simplify our model. The first is dominated by the balance between electron-electron Coulomb collisions and loss to the anode, which is mostly relevant to the conduction phase of plasma switches. The second is dominated by wave heating balanced by the anode loss, relevant to the electron beam-plasma discharge systems. We then couple our simplified energy balance model with the ion diffusion model and solve both the number density and the electron temperature as functions of the current density, electrode distance, pressure, and applied voltage, where a nice agreement is also obtained when comparing to PIC simulations.