The Schottky Conjecture and beyond

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.