Some comments on models for field enhancement

To estimate the apex field-enhancement factor γ aassociated with a pointed protrusion on a flat planar surface, the simple physical models of a ‘floating sphere at emitter-plane potential’ and a ‘hemisphere on a post’ are often discussed. The corresponding mathematical expressions have the form: γ a= m+ h/ ρ, where ρ is the sphere or hemisphere radius, h is its ‘height above the emitter plane’, and m is a constant variously taken as 0, 2 or 3. Recent numerical simulations for the ‘hemisphere on a post’ model, reported elsewhere by two of us (CJE and GV) and by Kokkaris, Modinos and Xanthakis, have shown that all of these simple formulae significantly overpredict γ a if h/ ρ is large. This article first reexamines the basis of these simple formulae and confirms that they are less secure than is sometimes thought. The formulae reported elsewhere as fits to the numerical results are then quoted and compared with the simple formulae, and with the known exact analytical result for the ‘hemi-ellipsoid on a plane’ model. Discrepancies can be rationalised. Some general conclusions are drawn.