4f^N-15d energy levels of lanthanides: a quasi-angular-momentum approach and its application to Cs₂NaYF₆:Er³⁺

The spin--orbit interaction of the 5d electron needs to be taken into account to give the proper energy structure for the 4f^N-15d configuration of heavy lanthanide ions occupying a site with ligands forming an octahedron. This paper derives theoretical results for the energy structure by treating the t₂ orbitals as quasi p orbitals and then using angular-momentum coupling techniques. An analytic expression for the electric dipole absorption line strengths between 4f^N multiplets and 4f^N-15d states is given in terms of various angular-momentum quantum numbers and re-coupling coefficients. The result is then applied to interpret the excitation spectrum of Cs₂NaYF₆:Er³⁺. The high-spin and low-spin states of Cs₂NaYF₆:Er³⁺ are discussed in terms of the wavefunctions obtained by using the developed theoretical model.