Theoretical Investigation of Some Magnetic and Spectroscopic Properties of Rare-Earth Ions

Investigations of some magnetic and spectroscopic properties of rare-earth ions based on approximate Hartree-Fock calculations are reported. First, a set of conventional, nonrelativistic Hartree-Fock wave functions were obtained for ${\mathrm{Ce}}^{3+}$, ${\mathrm{Pr}}^{3+}$, ${\mathrm{Nd}}^{3+}$, ${\mathrm{Sm}}^{3+}$, ${\mathrm{Eu}}^{2+}$, ${\mathrm{Gd}}^{3+}$, ${\mathrm{Dy}}^{3+}$, ${\mathrm{Er}}^{3+}$, and ${\mathrm{Yb}}^{3+}$; second, calculations for ${\mathrm{Ce}}^{3+}$ were carried out in which spin-orbit coupling was directly included in the conventional Hartree-Fock equations in order to obtain some estimate of wave-function dependence on $J$ and the resulting effects on experimental quantities. These results are then used to discuss spin-orbit splittings, hyperfine interactions, and the determination of nuclear magnetic moments, the Slater $F^k(4f, 4f)$ integrals, and the crystal-field parameters, ${V_n}^m={A_n}^m〈r^n〉$, all of which depend fairly critically on the precise form of the $4f$ wave functions. Comparisons are made with experiment and with the result of previous theoretical investigations which relied on either Hartree or modified hydrogenic wave functions or on semiempirical parametrizations. The usual spin-orbit formula, $〈\frac{r^{-1\mathrm{}}\mathrm{dV}}{\mathrm{dr}}〉$, is found not to give agreement with experiment; the reasons for this are discussed, and some evidence is described which indicates the importance of including spin-orbit exchange terms between the $4f$ electrons and the core. The implications of this result for efforts to relate $〈r^{-3\mathrm{}}〉$ integrals to experimentally observed spin-orbit coupling parameters are discussed, as is the relation (and use) of $〈r^{-3\mathrm{}}〉$ integrals to the determination of nuclear magnetic moments. Our $〈r^{-3\mathrm{}}〉$ values agree very closely (i.e., to within 5%) with Bleaney's parametrized values, and, hence, so do our estimates for the hyperfine interactions. A sampling of estimated rare-earth nuclear magnetic moments, based on the conventional Hartree-Fock $〈r^{-3\mathrm{}}〉$ data, is given; comparison with previous estimates are made; and several causes of the uncertainty in these and all other estimates are discussed. The spectroscopic properties of these ions in a crystalline field are interpreted on the basis of the simple crystal-field theory. The $〈r^n〉$ integrals are found to be in good agreement for $n=2, 4, \mathrm{and} 6$ with the Elliott and Stevens parametrization formula, but the assumption of the constancy with $Z$ of the ${A_n}^m$ is not valid, as is shown by analysis of the available trichloride and ethyl-sulfate data. Systematic discrepancies between experimental and theoretical $F^k(4f, 4f)$ have been found which are similar to but greater than what has been previously observed for smaller ions. Finally, the role of spin polarization and aspherical distortions (of the closed shells and the $4f$ electrons) is indicated, particularly from the "unrestricted" Hartree-Fock point of view, and an estimate of the field due to polarization of the core electrons is given for all the ions. Results for smaller ions and their implications for the interpretation of observed rare-earth magnetic and spectroscopic properties are sketched.