Estimating the D-line Energy Splitting of Alkali Metals Using a Modified Perturbation Method

Abstract

We report here on how a known method from standard perturbation theory for estimating the energy of the D-line splitting in hydrogen can be modified to effectively approximate this quantity for all of the alkali metals. The approach utilizes a Rayleigh- Schrodinger perturbation theory first order correction to the energy. The perturbing Hamiltonian is the standard relativistically corrected spin-orbit Hamiltonian. From this, one derives an energy difference between the doublet lines that is theoretically appropriate for any one electron atom. This energy difference is written in terms of the Bohr energy. The results are good for hydrogen but, as expected, are significantly off from the experimental values for the multi-electron alkali metals. We show here that by replacing the Bohr energy with a first ionization potential, that the theory then gives greatly improved values for the D level splitting energy. However, this approach overestimates the splitting energy for the light alkali metals and underestimates it for the heavy ones. The best result was for Rb where the estimate only varied from the experimentally reported value by 3.2%. The modified theory is shown to yield accurate results for all of the alkali metals when the original Bohr energy is adjusted with an appropriate screening constant. Screening constants generated using the Slater scheme however, which yield accurate estimates for ionization potentials, do not give the correct results for the D line splitting energies. A method is given whereby effective screening constants can be computed. These screening constants are found to be a function of the atomic number with constant coefficients that can be estimated by the ratio of cation to atomic radii. We conclude that the discrepancy in screening constants is due to electron-relaxation, a phenomenon that occurs during ionization.