A new method for solving the Z>137 problem for energy levels of hydrogen-like atoms

TL;DR

This paper introduces a numerical method that accounts for finite nuclear size to resolve the Z>137 problem in Dirac equation solutions for hydrogen-like atoms, ensuring physically consistent energy levels across all Z.

Contribution

A novel boundary condition at the nucleus boundary is proposed, removing the Z>137 catastrophe and enabling accurate energy level calculations for all nuclei.

Findings

Energy levels match standard solutions for Z ≤ 137.

Energy level functions are monotone and smooth for Z > 105.

Electron 'drop' occurs at Z=178.

Abstract

The "catastrophe" in solving the Dirac equation for an electron in the field of a point electric charge, which emerges for the charge numbers Z > 137, is removed in this work by effective accounting of finite dimensions of nuclei. For this purpose, in numerical solutions of equations for Dirac radial wave functions, we introduce a boundary condition at the nucleus boundary such that the components of the electron current density is zero. As a result, for all nuclei of the periodic table the calculated energy levels practically coincide with the energy levels in standard solutions of the Dirac equation in the external field of the Coulomb potential of a point charge. Further, for Z > 105, the calculated energy level functions E(Z) are monotone and smooth.The lower energy level reaches the energy E=-mc^{2} (the electron "drop" on a nuclei) at Zc = 178. The proposed method of accounting of the finite size of nuclei can be easily used in numerical calculations of energy levels of many-electron atoms.