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Unformatted text preview: IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 32, NO. 3, JUNE 2004 1223 An Analysis of the Basic Space-Charge-Limited Emission Algorithm in a Finite-Element Electrostatic Gun Code Eric M. Nelson , Member, IEEE and John J. Petillo , Member, IEEE Abstract— Beam currents computed with a general-purpose gun code can exhibit puzzling behavior as the mesh is refined. To un- derstand 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 ad- jacent to the cathode does not converge to zero, but rather increases as the mesh is refined. Convergence of the basic SCL emission algo- rithm, which depends on said error, is due instead to the increasing sensitivity of the potential to the current density. The current den- sity converges slowly from above to the Child’s law value, with a maximum error of 2.7% and ultimately with a sublinear conver- gence rate of 2 3. Tests on a three-dimensional parallel plate geometry with un- structured meshes of tetrahedral elements demonstrate that in- sight 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 struc- tured 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. Index Terms— Convergence of numerical methods, electron gun codes, finite-element methods, space-charge-limited (SCL) emis- sion. I. INTRODUCTION A GUN code attempts to self-consistently compute the steady-state emission and transport of charged particles in an electrostatic field and perhaps also a magnetostatic field. The charged particles contribute to the static field, and the static field contributes to the forces acting on the charged particles. Early examples of gun codes are described in . Gun codes are often employed to design and model electron guns. Two-dimensional (2-D) gun codes have been very suc- cessful for the design and modeling of axisymmetric sources. However, recent interest in modeling complicated source ge- ometries, such as gridded, multibeam, and sheet beam guns, has Manuscript received August 29, 2003; revised January 15, 2004. This work was supported by the Office of Naval Research....
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This note was uploaded on 06/08/2011 for the course ELECTRICAL 124 taught by Professor Ghjk during the Spring '11 term at Institute of Technology.
- Spring '11