Theory of Zeeman Effect in the Ground Multiplets of Rare-Earth Atoms

A number of corrections are made to the simple Landé formula for the $g$ values of levels deriving from the ground term of configurations of the type $4f^n$. These include (a) the Schwinger correction, to give an accurate value of the gyromagnetic ratio for the electron spin; (b) a correction to allow for the deviations from perfect $\mathrm{RS}$ coupling; (c) a relativistic correction, which is directly related to the kinetic energy of the electrons; (d) a diamagnetic correction, depending on the electron density of the core. In order to calculate (b), the spin-orbit coupling constants and the Coulomb integrals $F_k$ are estimated either from existing spectroscopic data or from a process of interpolation or extrapolation. An argument is presented for taking ratios of the integrals $F_k$ corresponding to a hydrogenic eigenfunction. The various radial integrals required in the calculation of (c) and (d) are derived from a modified hydrogenic eigenfunction of the form $r^n{e}^{-ar}cosh\left[\kappa \mathrm{}\right(ar-n\left)\right]$. The parameter $\kappa$ in this expression is estimated to be approximately 0.42 over the rare-earth series by comparison with available Hartree self-consistent field eigenfunctions. The second parameter $a$ is chosen to give a fit with the spin-orbit coupling constants. The result of calculating (a), (b), (c), and (d) is to give atomic $g$ values which agree remarkably well with the experimental data. This confirms that the ground configurations of Pri, Ndi, Pmi, Smi, Eui, Dyi, Hoi, Eri, and Tmi are of the type $4f^n$, and that such a configuration is very low-lying in Tbi. Tables of spin-orbit coupling constants and $〈r^{-3\mathrm{}}〉$ for both neutral and triply ionized rare earth atoms are given as well as of other radial integrals.