The cosine law of field enhancement factor variation: generic emitter shapes

TL;DR

This paper analytically derives the cosine law governing the variation of the field enhancement factor for a broad class of smooth, vertical emitter shapes, confirming its validity near the apex where field emission occurs.

Contribution

It provides a general analytical derivation of the cosine law for smooth vertical emitters, extending previous numerical and specific shape analyses.

Findings

Cosine law holds for general smooth vertical emitters near the apex.

Analytical derivation confirms previous numerical results.

Law applies in the region where field emission predominantly occurs.

Abstract

The cosine law of field enhancement factor variation was recently derived for a hemi-ellipsoidal emitter and numerically established for other smooth emitter shapes (Biswas et al, Ultramicroscopy, 185, 1 (2018)). An analytical derivation is provided here for general smooth vertical emitter shapes aligned in the direction of the asymptotic electrostatic field. The law is found to hold in the neighbourhood of the emitter apex from where field emisson pre-dominantly occurs.