The XMGUN Particle Path FEM Code

This work reports some initial results of a 2-D electron gun design code (XMGUN) based on the finite-element method (FEM). Using the Galerkin weak formulation, the nodal analysis, and the first-order elements, the Poisson equation was solved for the electron gun electrostatic potential. The nonrelativistic particle paths were numerically calculated by a fourth-order Runge-Kutta method. An iterative scheme was repeated until the electron paths convergence was achieved under full space-charge limited condition. In order to validate the algorithm, the focusing properties of a 2-D Pierce electron gun with planar symmetry were studied. The quality of the beam was evaluated based on the particle´s final position, the transit time, and the particle energy evaluations. Using these three parameters, a good agreement was found between the theoretical and calculated results. Absolute current density errors of less than 1% were found, even with a coarse discretization of the domain. The XMGUN tool was used to design a 30-kV, 7.1-A, and 1.37-μPerv axis-symmetric high-power electron gun for use in vacuum microwave devices. The figure of merit used as reference to measure the quality of the electron beam was the normalized transverse velocity.