Approximate Analytical Solutions of the Thomas–Fermi Equation for Positive Ions

Abstract

Approximate analytical solutions of the Thomas–Fermi (TF) differential equation for positive ions are obtained in two different forms. In the Sommerfeld-type approximation the solution is expressed as φS = φ0(1 + ηS), and in the Fermi-type approximation as φF = φ0 + ηF. The function φ0 is an approximate analytical solution of the TF equation for neutral atoms, obtained previously by making use of a variational principle, and the functions ηS and ηF are determined by an iterative solution of second-order ordinary linear differential equations with the boundary and subsidiary (normalization) conditions of the TF theory of positive ions. The approximate solutions φS and φF are tested by calculating the second ionization energies of Ne, Ar, Kr, Xe, and Rn. The calculated values are found in better agreement with the experimental values than those obtained from the exact solution of the TF equation for ions. The reason for this is explained.