Rare-earth fourth-order multipole moment in cubic ErCo₂Probed by linear dichroism in core-level photoemission

We developed a method to experimentally quantify the fourth-order multipole moment of the rare-earth 4f orbital. Linear dichroism (LD) in the Er 3d₅=₂ core-level photoemission spectra of cubic ErCo₂ was measured using bulk-sensitive hard X-ray photoemission spectroscopy. Theoretical calculation reproduced the observed LD, and the result showed that the observed result does not contradict the suggested³₈ ground state. Theoretical calculation further showed a linear relationship between the LD size and the size of the fourth-order multipole moment of the Er³⁺ ion, which is proportional to the expectation value hO⁰₄ þ 5O⁴₄i, where O^m_n are the Stevens operators. These analyses indicate that the LD in 3d photoemission spectra can be used to quantify the average fourth-order multipole moment of rare-earth atoms in a cubic crystal electric field.