Comment on the validity of first-order paraxial theory for electron and ion sources with needle-type or pointed geometry

An essential feature of first-order paraxial theory is the assumption that if the potential along the axis of an axially symmetric system is known, then the potential (fields) near the axis can be obtained by a power-series expansion about points lying on axis. However, in traditional first-order theory, which commonly assumes systems with cylindrical symmetry, only terms up to second-order in the coordinate transverse to the beam axis are retained. In this letter we argue that a consequence of this restriction is that traditional first-order paraxial theory should not be applicable to electron and ion sources with pointed or needle-type geometries. In order to treat non-parallel trajectories which occur in the pointed geometries present, say, in field emission liquid metal ion sources, a modified paraxial theory is suggested which describes two-dimensional particle dynamics.This work has been supported in part by the Division of Materials Research, National Science Foundation, Grant No. DMR-81008829