Quadrupole Susceptibility of Gd-Based Filled Skutterudite Compounds

It is shown that quadrupole susceptibility can be detected in Gd compounds contrary to our textbook knowledge that Gd$^{3+}$ ion induces pure spin moment due to the Hund's rules in an $LS$ coupling scheme. The ground-state multiplet of Gd$^{3+}$ is always characterized by $J$=7/2, where $J$ denotes total angular momentum, but in a $j$-$j$ coupling scheme, one $f$ electron in $j$=7/2 octet carries quadrupole moment, while other six electrons fully occupy $j$=5/2 sextet, where $j$ denotes one-electron total angular momentum. For realistic values of Coulomb interaction and spin-orbit coupling, the ground-state wavefunction is found to contain significant amount of the $j$-$j$ coupling component. From the evaluation of quadrupole susceptibility in a simple mean-field approximation, we point out a possibility to detect the softening of elastic constant in Gd-based filled skutterudites.