Eigenstate analysis of the crystal electric field at low-symmetry sites: Application for an orthogonal site in the tetragonal crystal Ce2Pd2Pb

We have established an analyzing method to determine eigenstates of $4f$ electrons under the crystal electric field (CEF) at low-symmetry sites. This analysis requires macroscopic physical properties only, namely the saturation moments along several symmetrical directions of the crystal in addition to the magnetic susceptibility and specific heat, instead of the spectrum of the inelastic neutron scattering frequently used for such a complicated circumstance. As a successful case, the eigenstates of a CEF Hamiltonian under an orthogonal point group ${C}_{2v}$ are determined for the $J=\frac{5}{2}$ state of the cerium ion in a tetragonal compound ${\mathrm{Ce}}_{2}{\mathrm{Pd}}_{2}\mathrm{Pb}$. During the analysis, an angular momentum operator along the arbitrary direction is deduced from the space rotation operator. A matrix representation of the projection operator that transforms the basis of wave functions from $|{J}_{z}\ensuremath{\rangle}$ to $|{J}_{x}\ensuremath{\rangle}$ or $|{J}_{y}\ensuremath{\rangle}$ is also shown for $J=\frac{5}{2}$. This analysis reveals the anisotropic moments within highly symmetrical structures, and will contribute to understanding anisotropic field responses in rare-earth compounds.