Zero-Field Splitting of S-State Ions. I. Point-Multipole Model

The zero-field splitting terms in the spin Hamiltonian for an $S$-state ion, $D[3{{S}_{z}}^{2}\ensuremath{-}S(S+1)]$ and $E({{S}_{x}}^{2}\ensuremath{-}{{S}_{y}}^{2})$, are computed using a point-multipole model. Various contributions to $D$ and $E$ are considered and quantitative results are given for the most important mechanisms. Specific application is made to ${\mathrm{Mn}}^{2+}$: Zn${\mathrm{F}}_{2}$ and Mn${\mathrm{F}}_{2}$, where accurate values for $D$ and $E$ are known from electron-spin-resonance experiments. The most important contribution for these cases comes from the "Blume-Orbach" mechanism involving the first-order matrix element of the axial and rhombic fields between excited quartet states which have been admixed into one another by the cubic component of the crystalline field. This term contributes results of the correct sign, and of nearly the correct magnitude, to explain the entirety of the axial and rhombic field splitting of ${\mathrm{Mn}}^{2+}$ in these hosts. The next most important contribution arises from the spin-spin interaction (the "Pryce" mechanism) involving again the first-order matrix element of the crystalline field, but this time between an excited configuration and the ground state. Instead of the usual perturbation approach, the Schr\"odinger equation containing the crystal-field potential is integrated numerically. It is found that the term considered by Pryce, the $d\ensuremath{\rightarrow}s$ admixture, is small, and of the opposite sign from the more important contribution of the $d\ensuremath{\rightarrow}d$ admixture. The net contribution from the spin-spin mechanism yields results for $D$ of the wrong sign and of roughly one-third the magnitude of the Blume-Orbach term, and for $E$ a constribution of the correct sign but an order of magnitude smaller than the Blume-Orbach term. The configuration-interaction contribution of Orbach, Das, and Sharma is shown to be next in decreasing order of importance, followed by the contribution originally computed by Watanabe.