Resonant X-ray spectroscopies on Chromium $3\textit{d}$ orbitals in CrSBr
We investigate the Cr electronic structure and excitations in CrSBr, a layered magnetic semiconductor, using a combination of resonant x-ray spectroscopic techniques. X-ray absorption spectroscopy (XAS) and resonant inelastic x-ray scattering (RIXS) spectra collected at the Cr $L_{2,3}$ edges reveal significant linear dichroism, which arises from the distorted octahedral environment surrounding the Cr$^{3+}$ ions. The origin of the bright excitons observed in this compound is examined through a comparison of the d-d excitations identified in the RIXS spectra, the x-ray excited optical luminescence (XEOL) spectra, and previously reported optical spectroscopic and theoretical studies. To further understand these phenomena, we develop a multiplet model based on a crystal electric field (CEF) approach that accounts for the local environment of Cr ions. This model successfully reproduces several experimental features, while also suggesting strong hybridization effects between Cr $3\textit{d}$ orbitals and ligands that are not fully captured by the present framework. These findings advance our understanding of the electronic structure and excitonic behavior in CrSBr and provide a foundation for future $\textit{in-situ}$ and $\textit{operando}$ studies of CrSBr-based devices for spintronic and optoelectronic applications.