Low-Energy-Electron Scattering Amplitudes. II. Validity of Free-Electron-Gas Exchange Approximations Applied to Electrons Incident on Hg

For the test case of 300-eV electrons incident on Hg the influence of the exchange of the incoming electron with the atomic electrons is studied. Four different approximations for the derivation of an exchange potential based on the free-electron-gas description are discussed. These approximate exchange potentials were employed in calculations of electron scattering factors. The comparison of those results with a rigorous Hartree—Fock (HF) treatment of the total scattering problem ($N+1\mathrm{}$ particles) shows that the exchange given by Kohn and Sham is too large, especially at large atomic distances. By Gombás's introduction of a cutoff at the radius of the Thomas—Fermi—Dirac atom, the long tail of the potential was avoided. While the truncated results of Gombás decrease the deviations significantly, a different treatment of the self-exchange as suggested by Lindgren and Rosen brings the best agreement with the HF calculation. The theoretical results show that precise relative measurements in the valleys of the different cross sections are an extremely sensitive tool for testing scattering theory.