Lanthanide Crystal Field Theory in the J-Mixing Regime: $f^n$ Configurations

The low-energy sector of the lanthanide $f^n$ shell is to first approximation governed by the interplay of three interactions: Coulomb repulsion, spin-same-orbit coupling and the ligand crystal field potential. To rationalize the energy landscape in different chemical environments, it is desirable to extract the phenomenological crystal field parameters from experimental data; however, in practice, high-dimensional fitting procedures complicate their routine determination. Here, we present a computational extraction procedure based on real tensorial sets and the $U(7)$ canonical chain that, unlike existing operator equivalent methods, takes all $J$ multiplets of the lowest-energy $LS$ term into account, ensuring meaningful results even in the presence of considerable $J$-mixing. Two series of lanthanide compounds, [Ln(III)(C$_8$H$_8$)$_2$]$^-$} and doped Ln(III):Y$_2$O$_3$} (Ln $\in$ {Ce, Pr, Nd, Sm, Eu}) serve as examples for high- ($D_{8h}$) and low-symmetry ($C_2$) scenarios. Consequences for the generalisation toward actinides are briefly discussed.