Analytical Advances about the apex Field Enhancement Factor of a Hemisphere on a Post Model

TL;DR

This paper develops an analytical method to estimate the apex field enhancement factor (FEF) for various axially symmetric conducting emitter shapes, including complex models like hemisphere on a post, providing insights into charge distribution and emitter interactions.

Contribution

It introduces a novel analytical approach to calculate the apex FEF for general axial-symmetric shapes, extending previous results and analyzing emitter interactions from first principles.

Findings

Confirmed the known FEF of 3 for a hemisphere on a flat plate.

Derived the dependence of FEF on aspect ratio for the hemisphere on a post.

Established the power-law decay of FEF change with emitter separation.

Abstract

In this dissertation, we analytically study the apex field enhancement factor (FEF), $\gamma_a$, by constructing a method which consists in minimizing an error function defined as to measure the deviation of the potential at the boundary, yielding approximate axial multipole coefficients of a general axial-symmetric conducting emitter shape, on which the apex FEF can be calculated by summing its respective Legendre series. Such method is analytically applied for a conducting hemisphere on a flat plate, confirming the known result of $\gamma_a = 3$. Also, it is applied on a hemi-ellipsoid on a plate where the values of the apex FEF are compared with the ones extracted from the analytical expression. Then, the method is applied for the hemisphere on a cylindrical post (HCP) model. In this case, to analytically estimate the apex FEF from first principles is a problem of considerable complexity. Despite the slow convergence of the apex FEF, useful analytical conclusions are drawn and explored, such as, it is confirmed that all even multipole contributions of the HCP model are zero, which in turn leads to restrictions on the charge density distribution: it will be shown the surface charge density must be an odd function with respect to height in an equivalent system. Also, expressions found for the apex FEF depend explicitly on the aspect ratio, that is, the ratio of height by base radius. Using the dominant multipole contribution, the dipole, at sufficient large distances, it is shown that, for two interacting emitters, as their separation distance increases, the fractional change in apex FEF, $\delta$, decreases following a power law with exponent $-3$. The result is extended for conducting emitters having an arbitrary axially-symmetric shape, where it is also shown $\delta$ has a pre-factor depending on geometry, confirming the tendency observed in recent analytical and numerical results.