The advantages of using linear and non‐linear second‐order finite element methods in electron lens design

Abstract

SUMMARYPowerful personal computers (PCs) and advanced software now enable electron optical designers to set up a proposed design, revise it and finally obtain a rapid overview of its electron optical properties. The finite element method (FEM) in general is useful for the computer modelling of electron lenses as the mesh layout is well adapted to the shape of electrodes and polepieces. The second‐order FEM is especially relevant to curved boundaries, for example in spherical‐double and single‐pole lenses. In the case of high‐resolution magnetic objective lenses, where marked saturation of the polepieces is an essential feature of the design, special care is needed to reduce the computational errors. Since the difference in saturation magnetization of soft iron and a typical iron–cobalt is around 10%, this means that for a meaningful comparison between the two materials, the computation errors should not exceed 2%. This level of accuracy can be achieved easily with second‐order, non‐linear computations even when very low mesh density is used. In first‐order, non‐linear computations a very high mesh density is needed, and this requires more care and effort to reach even lower levels of accuracy.