Second-order Born approximation for the scattering phase shifts: Application to the Friedel sum rule

TL;DR

This paper revisits the second-order Born approximation for quantum scattering phase shifts and applies it to derive an improved Friedel sum rule, enhancing the accuracy of screening parameter calculations in degenerate electron gases.

Contribution

It introduces a second-order Born correction to the Friedel sum rule, providing a more accurate analytical expression for the screening parameter in ion-electron interactions.

Findings

B2 screening parameters match exact solutions at high and moderate densities.

B2 approximation outperforms B1 at lower densities.

Developed a Padé approximant that improves perturbative results.

Abstract

Screening effects are important to understand various aspects of ion-solid interactions and, in particular, play a crucial role in the stopping of ions in solids. In this paper the phase shifts and scattering amplitudes for the quantum-mechanical elastic scattering within up to the second-order Born (B2) approximation are revisited for an arbitrary spherically-symmetric electron-ion interaction potential. The B2 phase shifts and scattering amplitudes are then used to derive the Friedel sum rule (FSR) involving the second-order Born corrections. This results in a simple equation for the B2 perturbative screening parameter of an impurity ion immersed in a fully degenerate electron gas which, as expected, turns out to depend on the ion atomic number $Z_{1}$ unlike the first-order Born (B1) screening parameter reported earlier by some authors. Furthermore, our analytical results for the Yukawa, hydrogenic, Hulth\'{e}n, and Mensing potentials are compared, for both positive and negative ions and a wide range of one-electron radii, to the exact screening parameters calculated self-consistently by imposing the FSR requirement. It is shown that the B2 screening parameters agree excellently with the exact values at large and moderate densities of the degenerate electron gas, while at lower densities they progressively deviate from the exact numerical solutions but are nevertheless more accurate than the prediction of the B1 approximation. In addition, a simple Pad\'{e} approximant to the Born series has been developed that improves the performance of the perturbative FSR for any negative ion as well as for $Z_{1}=+1$.