Influence of lattice structure on multipole interactions in ${\mathrm{\Gamma }}_3$ non-Kramers doublet systems

We study the multipole interactions between $f^2$ ions with the ${\mathrm{\Gamma }}_3$ non-Kramers doublet ground state under a cubic crystalline electric field. We construct the ${\mathrm{\Gamma }}_3$ doublet state of the electrons with the total angular momentum $j=5/2$. By applying the second-order perturbation theory with respect to the intersite hopping, we derive the multipole interactions. We obtain a quadrupole interaction for a simple cubic lattice, an octupole interaction for a bcc lattice, and both quadrupole and octupole interactions for an fcc lattice. We also discuss general tendencies of the multipole interactions depending on the lattice structure by comparing the results with those for the ${\mathrm{\Gamma }}_8$ quartet systems of $f^1$ ions.

Figure 1

Electron configurations for (a) ${\mathrm{\Gamma }}_7$ state of $f^1$, (b) ${\mathrm{\Gamma }}_3$ state of $f^2$, and (c) ${\mathrm{\Gamma }}_6$ state of $f^3$. The bold lines denote spin singlets composed of the ${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_8$ orbitals.

Figure 2

Schematic figures of the electronic states on the nearest-neighboring sites preferred by the interaction (a) along the $z$ direction (antiferroarrangement of the $O_2^0$ moments) and (b) along [111] direction (antiferroarrangement of the $T_{xyz}$ moments). The gradation of color in (b) indicates the anisotropic distribution of the dipole moment.