Visualization of spin-orbit-entangled 4f electrons in crystalline materials.

Lanthanide 4f electrons are strongly influenced by spin-orbit coupling, resulting in well-defined J multiplets, which are further split by the crystalline electric field in condensed matter. While the anisotropy of 4f electrons is closely linked to material properties, direct experimental observation of the 4f electron distribution in real space remains a significant challenge. Here, we present an approach for visualizing the anisotropic distribution of lanthanide 4f electrons in pyrochlore oxides by combining high-photon-energy X-ray diffraction and valence electron density (VED) analysis based on the core differential Fourier synthesis (CDFS) method. The observed VED distributions around the lanthanide site reveal the parameters of the ground-state wavefunction, which roughly agree with point-charge calculations for the trigonal crystal electric field under the LS coupling scheme. This CDFS-based VED observation method not only provides insights into the anisotropic nature of 4f electrons but also opens a pathway for studying the 4f states in a wide range of crystalline materials.