Shielding effects in random large area field emitters, the field enhancement factor distribution and current calculation

A finite-size uniform random distribution of vertically aligned field emitters on a planar surface is studied under the assumption that the asymptotic field is uniform and parallel to the emitter axis. A formula for field enhancement factor is first derived for a 2-emitter system and this is then generalized for $N$-emitters placed arbitrarily (line, array or random). It is found that geometric effects dominate the shielding of field lines. The distribution of field enhancement factor for a uniform random distribution of emitter locations is found to be closely approximated by an extreme value (Gumbel-minimum) distribution when the mean separation is greater than the emitter height but is better approximated by a Gaussian for mean separations close to the emitter height. It is shown that these distributions can be used to accurately predict the current emitted from a large area field emitter.