Universal Approximate Analytical Solution of the Thomas-Fermi Equation for Ions

Abstract

The great utility of the Thomas---Fermi (TF) theory of atoms lies in the fact that it gives a solution of the TF equation that is valid for for all neutral atoms. In contrast to this situation, the TF equation for positive ions has to be solved separately for each degree of ionization of each atom. Using a previously obtained approximate analytical solution of the TF equation for neutral atoms (which is based on a variational principle), and a series-expansion approach (which relates the parameters in the ionic solution to those in the neutral-atom solution), the present paper derives a universal approximate analytical solution of the TF equation for ions. This solution, in addition to being valid for positive ions, is also applicable for negative ions for which the original TF theory does not furnish a solution. Furthermore, the approximate solution obtained here is not associated with finite ionic radii, and gives an exponentially decreasing radial electron density, which is not the case in the original TF theory. For this reason, the electron densities and potentials of ions that result from the present work are expected to be in better agreement with the quantum-mechanical data for these quantities than the electron densities and potentials obtained with the original TF theory. To show that this is so, the accuracy of the universal approximate analytical solution is investigated by calculating the diamagnetic susceptibilities of singly and doubly charged positive and negative ions of noble-gas electron configurations. It is found that, in most cases, calculated and experimental values for ions of the Ar, Kr, and Xe electron configurations agree to within a factor of 2, while for ions of the Ne electron configuration the agreement is somewhat worse. This is about the same accuracy as that found using the Lenz-Jensen approximation to the TF theory which, like the universal approximation, also makes use of variational electron density but, unlike the universal approximation, has to be separately obtained for each particular ion. As a further check on the usefulness of the universal ionic solution, the diamagnetic susceptibilities of the isoelectronic ions ${\mathrm{Ga}}^{3+}$, ${\mathrm{Ge}}^{4+}$, and ${\mathrm{As}}^{5+}$ are also calculated and found to agree only slightly worse than within a factor of 2 with the experimental data. After further comparison with data obtained from more-refined statistical models, the conclusion is that the universal solution for ions may be useful in a variety of problems where quantum-mechanical accuracy may be traded for a simpler approach.