An analysis of the basic space-charge-limited emission algorithm in a finite-element electrostatic gun code

Beam currents computed with a general-purpose gun code can exhibit puzzling behavior as the mesh is refined. To understand such behavior, we analyze the convergence, with respect to element size, of the basic space-charge-limited (SCL) emission algorithm in a one-dimensional (1-D) finite-element electrostatic gun model. With the current density fixed at the Child´s law value, we find that the relative error of the potential at the first vertex adjacent to the cathode does not converge to zero, but rather increases as the mesh is refined. Convergence of the basic SCL emission algorithm, which depends on said error, is due instead to the increasing sensitivity of the potential to the current density. The current density converges slowly from above to the Child´s law value, with a maximum error of 2.7% and ultimately with a sublinear convergence rate of 2/3. Tests on a three-dimensional parallel plate geometry with unstructured meshes of tetrahedral elements demonstrate that insight from the 1-D model applies to a general-purpose code. The behavior is similar to the 1-D model, but with a maximum error of 4.3%. Thus, using an unstructured mesh instead of a smooth structured mesh introduces only a modest additional error to the beam current. Based on the analysis of the 1-D model, we present two scaled SCL emission algorithms. The first exhibits linear convergence from below. The second limits the maximum error to 0.9%. Similar scalings can be employed in general-purpose gun codes to improve the accuracy of the computed beam current.