# Variation of field enhancement factor near the emitter tip

## TL;DR

This paper analytically and numerically investigates how the field enhancement factor varies near the tip of axially symmetric emitters, revealing a cosine dependence that influences emitter current calculations.

## Contribution

It introduces a simple cosine-based model for the variation of the field enhancement factor near emitter tips, validated across different shapes.

## Key findings

- Variation follows a cosine dependence in normalized coordinates.
- Analytical model confirmed for hemi-ellipsoidal emitters.
- Numerical validation extends to other shapes with quadratic tips.

## Abstract

The field enhancement factor at the emitter tip and its variation in a close neighbourhood determines the emitter current in a Fowler-Nordheim like formulation. For an axially symmetric emitter with a smooth tip, it is shown that the variation can be accounted by a $\cos{\tilde{\theta}}$ factor in appropriately defined normalized co-ordinates. This is shown analytically for a hemi-ellipsoidal emitter and confirmed numerically for other emitter shapes with locally quadratic tips.