Adiabatic thermal Child-Langmuir flow

Abstract

A simulation model is presented for the verification of the recently developed steady-state one-dimensional adiabatic thermal Child-Langmuir flow theory. In this theory, a self-consistent Poisson equation is developed through the use of the fluid-Maxwell equations and an adiabatic equation of state. The adiabatic equation of state is also a statement of normalized rms thermal emittance conservation. Solving the self-consistent Poisson equation with the appropriate boundary conditions yields the current density, electrostatic potential, fluid velocity, equilibrium density, temperature, and pressure profiles at a given cathode temperature. A one-dimensional simulation model has been developed. It consists of the initial loading, the charged-sheet model algorithm, and the post-processing of the results. Great care has been taken in the initial loading of the beam, with the beam loaded as close to the equilibrium values as possible. Because there is no known solution for the interface problem between the quantum mechanical flow of electrons inside the solid material and the classical flow of electrons in the cathode vacuum, a reinjection scheme is proposed in which the initial phase space near the cathode be maintained throughout the simulation. Three one-dimensional beams are simulated at dimensionless cathode temperatures of 0.1, 0.01, and 0.001. Great success is achieved at validating the theory at the dimensionless cathode temperature of 0.1. The simulation results for the dimensionless cathode temperature of 0.01 and 0.001 cases are consistent with the theoretical prediction. Because of the good agreement between the simulation and theory, the use of the adiabatic equation of state is justified. A strategy to extend the one-dimensional adiabatic thermal Child-Langmuir theory into two-dimensions is presented. Because two-dimensional adiabatic equation(s) of state are currently unknown, a two-dimensional simulation is used to both investigate and help formulate the adiabatic equation(s) of state. A two-dimensional simulation model has been developed to simulate flows in a Pierce gun slab geometry. The two-dimensional simulation consists of the meshing of the domain, the initial loading, the particle-in-cell algorithm, and the post-processing of the results. An estimate on the two-dimensional form of the equilibrium density is used to initially load the beam. Like the one-dimensional case, the proposed particle boundary condition for the two-dimensional simulation is that the initial phase space near the cathode be preserved throughout the simulation. Preliminary results from the two-dimensional simulation model are presented.