  The field enhancement factor (FEF) is an important quantity in field emission
calculations since the tunneling electron current depends very sensitively on
its magnitude. The exact dependence of FEF on the emitter height $h$, the
radius of curvature at the apex $R_a$, as well as the shape of the emitter base
is still largely unknown. In this work, a universal formula for the field
enhancement factor is derived. It depends on the ratio $h/R_a$ and has the form
$\gamma_a = (2h/R_a)/[\alpha_1 \log(4h/R_a) - \alpha_2 ]$ where $\alpha_1$,
$\alpha_2$ depend on the charge distribution on the emitter. Numerical results
show that a simpler form $\gamma_a = (2h/R_a)/[\log(4h/R_a) - \alpha]$ is
equally valid with $\alpha$ depending on the class of emitter and indicative of
the shielding by the emitter-base. For the hyperboloid, conical and ellipsoid
emitters, the value of $\alpha$ is $0, 0.88$ and $2$ while for the cylindrical
base where shielding is minimum, $\alpha \simeq 2.6$.

Applied Physics;Mesoscale and Nanoscale Physics;Plasma Physics