Interpretation of3dN-ion photoemission spectra. II. A general formalism for the core levels

Abstract

The object of this paper is to provide a formalism for the interpretation of the core-level photoemission spectra of $3{d}^{N}$ ions embedded in a cubic environment. The formalism is first developed in the general case of the photoemission of a ${n}^{\ensuremath{'}}{l}^{\ensuremath{'}}$ electron from an inner shell of a $n{l}^{N}$ ion in arbitrary symmetry $G$. This general formalism may be thought of as a coherent combination of the theory of level splitting (for the determination of the position of the core photo-peaks) and of the theory of the photoemission intensity (for the determination of the intensity of the core photopeaks). The interactions taken into consideration here are the Coulomb, the spin-orbit, and the crystal-field interactions both for the photoemission initial Hamiltonian and the photoemission final Hamiltonian. General formulas for the matrix elements of the involved interactions are given for the initial configuration $n{l}^{N}$ and the final configuration ${n}^{\ensuremath{'}}{l}^{\ensuremath{'}4{l}^{\ensuremath{'}}+1}n{l}^{N}$ in a weak-field basis adapted to the group $G$. The determination of the photopeak position follows from the diagonalization of the final Hamiltonian within the entire ${n}^{\ensuremath{'}}{l}^{\ensuremath{'}4{l}^{\ensuremath{'}}+1}n{l}^{N}$ manifold. The determination of the photopeak intensity requires the knowledge of the ground-state eigenvector of the initial Hamiltonian and of (all) the eigenvectors of the final Hamiltonian. General formulas are given for the intensity of the transitions $n{l}^{N}\ensuremath{\rightarrow}{n}^{\ensuremath{'}}{l}^{\ensuremath{'}4{l}^{\ensuremath{'}}+1}n{l}^{N}$ in a weak-field basis adapted to the group $G$. Finally, the general formalism is particularized to the case of iron-group ions ($\mathrm{nl}\ensuremath{\equiv}3d$) in octahedral symmetry ($G\ensuremath{\equiv}O$). Only those particular points which are specific to the case of $3{d}^{N}$ ions in octahedral symmetry are examined. In particular, the ejection process may concern the shells ${n}^{\ensuremath{'}}{l}^{\ensuremath{'}}\ensuremath{\equiv}{n}^{\ensuremath{'}}p$ or ${n}^{\ensuremath{'}}{l}^{\ensuremath{'}}\ensuremath{\equiv}{n}^{\ensuremath{'}}s$.