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

A number of corrections are made to the simple Land\'e 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 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₊ 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₊ 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^ne^-arcosh (ar-n). The parameter 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〉 for both neutral and triply ionized rare earth atoms are given as well as of other radial integrals.