About the screening of the charge of a proton migrating in a metal

TL;DR

This paper calculates the screening of a migrating proton in a metal using various theoretical methods, finding partial screening inconsistent with some previous theories, and clarifies the theoretical basis for electromigration forces.

Contribution

It provides new simplified linear-response expressions for the electromigration driving force, reconciling previous conflicting theories and clarifying the screening effect on protons in metals.

Findings

Screening of a proton in a metal is approximately 15%

New linear-response expression matches previous complex calculations

Bosvieux and Friedel's formula aligns with established linear response theory

Abstract

The amount of screening of a proton in a metal, migrating under the influence of an applied electric field, is calculated using different theoretical formulations. First the lowest order screening expression derived by Sham (1975) is evaluated. In addition 'exact' expressions are evaluated which were derived according to different approaches. For a proton in a metal modeled as a jellium the screening appears to be 15 +/- 10 %, which is neither negligible not reconcilable with the controversial full-screening point of view of Bosvieux and Friedel (1962). In reconsidering the theory of electromigration, a new simplified linear-response expression for the driving force is shown to lead to essentially the same result as found by Sorbello (1985), who has used a rather complicated technique. The expressions allow for a reduction such that only the scattering phase shifts of the migrating impurity are required. Finally it is shown that the starting formula for the driving force of Bosvieux and Friedel leads exactly to the zero-temperature limit of well-established linear response descriptions, by which the sting of the controversy has been removed.