Key Takeaway: The simplified Thomas-Fermi screening function provides a simple analytical formula for the Born scattering amplitude at zero angle, which agrees within 30% with the expression calculated according to Hartree-Fock.

Abstract

Abstract A simple analytical formula χ(x) = Sh2βa/Sh2β(a + x) of the Thomas-Fermi screening function is presented, where β = 0.4/Z1/3, a = 1.8, x = 1.13 Z1/3 r, r - is the distance from atomic nucleus, e = m = phi; = 1, φ(r) = Z χ(x)/r is the atomic potential, Z - is the atomic number (the atomic units are used). The Thomas-Fermi function χτφ (x) is formally valid only for φ ≥ 10, or for r<0.2 Z 1/3, mainly due to neglect of the exchange effects. The proposed function χ(x) is valid for φ ≥ 1, and for r—about two times larger, because the exchange effects were considered. The proposed function χ(x) agrees with the Thomas-Fermi-Dirac function χTFD and Hartree-Fock function within 2% in the region of validity. With the help of this screening function one can easily obtain a simple analytical formula for the Born scattering amplitude at zero angle, which agrees within 30% with the expression calculated according to Hartree-Fock. The mean potential of a substance is expressed by means of this value (in the add...