On the location of the maximum field enhancement factor in field electron emitters

Apex field enhancement factor is widely used to characterize electrostatic field enhancement in field electron emitters. However, the maximum electrostatic field at the surface, which is expected to largely determine where the highest local electron emission occurs for emitters with a spatially uniform work function, does not necessarily occur at the geometrical apex when electrostatic interaction breaks the local axial symmetry. In this work, a general analytical formulation is developed to determine the normal electrostatic field at the surface of a perfectly conducting sphere from the local harmonic expansion of a prescribed external potential. The formulation is first applied to a single floating conducting sphere, recovering the classical apex field enhancement factor within the floating-sphere-at-emitter-plate-potential (FSEPP) approximation. It is then combined with a self-consistent charge–dipole FSEPP model for two interacting floating conducting spheres, yielding analytical expressions for the local polarizing field, the location of the maximum electrostatic field at the surface, and the corresponding maximum field enhancement factor. Our results show that electrostatic interaction generates a transverse component of the local polarizing field that displaces the maximum field away from the geometrical apex toward the outer side of each emitter. This displacement is a first-order effect in the transverse polarization, whereas the increase in the maximum field is a second-order correction. Comparison with finite-element solutions of Laplace’s equation shows close quantitative agreement for both the apex and maximum field enhancement factors. More importantly, the formulation provides an analytical interpretation of off-apex field maxima previously observed primarily through numerical simulations.