The Schottky Conjecture and beyond

TL;DR

This paper investigates the Schottky Conjecture for compound electrostatic structures, demonstrating its validity under certain conditions, proposing a corrected version for broader cases, and analyzing its implications for field emission.

Contribution

The paper introduces the Corrected Schottky Conjecture (CSC), extending the original conjecture's applicability to more general structures and providing a quantitative approximation method.

Findings

Schottky Conjecture holds when apex radius is much larger than crown height.

The corrected conjecture accounts for averaged field enhancement, improving accuracy.

Error within 1-3% for small crown heights, applicable to 3-primitive structures.

Abstract

The `Schottky Conjecture' deals with the electrostatic field enhancement at the tip of compound structures such as a hemiellipsoid on top of a hemisphere. For such a 2-primitive compound structure, the apex field enhancement factor $\gamma_a^{(C)}$ is conjectured to be multiplicative ($\gamma_a^{(C)} = \gamma_a^{(1)} \gamma_a^{(2)}$) provided the structure at the base (labelled 1, e.g. the hemisphere) is much larger than the structure on top (referred to as crown and labelled 2, e.g. the hemi-ellipsoid). We first demonstrate numerically that for generic smooth structures, the conjecture holds in the limiting sense when the apex radius of curvature of the primitive-base $R_a^{(1)}$, is much larger than the height of the crown $h_2$ (i.e. $h_2/R_a^{(1)} \rightarrow 0$). If the condition is somewhat relaxed, we show that it is the electric field above the primitive-base (i.e. in the absence of the crown), averaged over the height of the crown, that gets magnified instead of the field at the apex of the primitive-base. This observation leads to the Corrected Schottky Conjecture (CSC), which for 2-primitive structures reads as $\gamma_a^{(C)}\simeq \langle \gamma_a^{(1)}\rangle\gamma_a^{(2)}$ where $\langle . \rangle$ denotes the average value over the height of the crown. For small protrusions ($h_2/h_1$ typically less than 0.2), $\langle \gamma_a^{(1)}\rangle$ can be approximately determined using the Line Charge Model so that $\gamma_a^{(C)} \simeq \gamma_a^{(1)}\gamma_a^{(2)} (2R_a^{(1)}/h_2)\ln(1 + h_2/2R_a^{(1)})$. The error is found to be within $1\%$ for $h_2/R_a^{(1)} < 0.05$, increasing to about $3\%$ (or less) for $h_2/R_a^{(1)} = 0.1$ and bounded below $5\%$ for $h_2/R_a^{(1)}$ as large as 0.5. The CSC is also found to give good results for 3-primitive compound structures. The relevance of the Corrected Schottky Conjecture for field emission is discussed.