Fowler-Nordheim theory for a spherical emitting surface

In tests on a field emitter whose dimensions and work function were known, Fowler-Nordheim (F-N) theory as usually stated for a planar emitter was found to give poor agreement with observations. The effect of curvature of the emitting surface has been modelled by including (a) non-linear variation of potential with distance from the surface; (b) the consequent changes in the exponent and pre-exponential terms in the F-N expression for current; and (c) the variation of current density over the surface, modelled by an effective solid angle. Application of the resulting expression to the measured data gives estimates for apex radius which agree much more closely with the measured value than the value from planar theory does.