Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

TL;DR

This paper investigates the focusing effect of a homogeneously charged sphere's electrostatic field on high-energy electrons, comparing classical and quantum solutions to validate the effective momentum approximation in electron scattering.

Contribution

It provides a classical calculation of electron focusing in a Coulomb field and compares it with Dirac equation solutions to support the effective momentum approximation.

Findings

Effective momentum approximation is valid at high energies.

Classical focusing results agree with Dirac solutions in the relativistic limit.

Supports use of approximation for Coulomb corrections in scattering experiments.

Abstract

The impact of the strongly attractive electromagnetic field of heavy nuclei on electrons in quasi-elastic (e,e') scattering is often accounted for by the effective momentum approximation. This method is a plane wave Born approximation which takes the twofold effect of the attractive nucleus on initial and final state electrons into account, namely the modification of the electron momentum in the vicinity of the nucleus, and the focusing of electrons towards the nuclear region leading to an enhancement of the corresponding wave function amplitudes. The focusing effect due to the attractive Coulomb field of a homogeneously charged sphere on a classical ensemble of charged particles incident on the field is calculated in the highly relativistic limit and compared to results obtained from exact solutions of the Dirac equation. The result is relevant for the theoretical foundation of the effective momentum approximation and describes the high energy behavior of the amplitude of continuum Dirac waves in the potential of a homogeneously charged sphere. Our findings indicate that the effective momentum approximation is a useful approximation for the calculation of Coulomb corrections in (e,e') scattering off heavy nuclei for sufficiently high electron energies and momentum transfer.