Relativistic Correction of the Field Emission Current in the Fowler-Nordheim Formalism

TL;DR

This paper extends the Fowler-Nordheim model to include relativistic effects, providing analytical formulas for high electric fields and demonstrating that relativistic corrections are generally small but significant in extreme astrophysical environments.

Contribution

It offers a novel analytical generalization of the Fowler-Nordheim equation incorporating relativistic corrections for high electric fields and large interelectrode distances.

Findings

Relativistic correction is usually less than 0.1%.

In pulsar environments, correction can reach about 10%.

Derived analytical expression for field emission current density.

Abstract

As the title implies the article describes the possibility of taking into account the relativistic correction to the field current density of the field emission of electrons from the metal. The article provides the reader with some analytic generalization of the Fowler-Nordheim equation with the relativistic correction. The relativistic correction to the Fowler-Nordheim equation makes it possible to take into account the influence of the relativism on the field emission current. It is especially noted that the consideration of this correction is necessary in the case of sufficiently strong electric fields and relatively large interelectrode distances. It should be stressed that this correction is valid for fixed interelectrode distances that decrease with increasing electric field strength. It means that for the electric field strength of 0.1 to 1 GV/m the interelectrode distance should not exceed values of 1 to 0.1 cm. First in the article it is spoken in detail about finding of the electron wave function. Next the field emission current calculations are given. As a result the transmission coefficient of the potential step from the Klein-Gordon equation within the framework of the Fowler-Nordheim approximation is found. It is shown that in the case of the interelectrode distance less than 1 cm, an analytical expression for the field electron emission current density is obtained. The conclusion that usually relativistic correction does not exceed a tenth of a percent is made. But in the case of the field electron emission from pulsars (where the work function and electric field strength are much higher) the contribution of the relativistic correction about 10 % has been established.