Electron spectroscopies for Ce compounds in the impurity model

Abstract

We present a method for calculating the core-level x-ray photoemission (XPS), the $3d\ensuremath{\rightarrow}4f$ x-ray absorption (XAS), the valence photoemission, and the bremsstrahlung isochromat spectra in a slightly modified Anderson impurity model of a Ce compound at zero temperature. Both the spin and orbital degeneracies of the $f$ level are included and the Coulomb interaction between the $f$ electrons is taken into account. The spectra are expressed in terms of a resolvent operator. A many-electron basis set is introduced, and the resolvent is obtained from a matrix inversion. The particular form of the Anderson model allows us to find a small but sufficiently complete basis set, if the degeneracy ${N}_{f}$ of the $f$ level is large. In particular, we consider the limit ${N}_{f}\ensuremath{\rightarrow}\ensuremath{\infty}$, and show that the method is exact for the XPS, XAS, and valence photoemission spectra in this limit. It is also demonstrated that for ${N}_{f}\ensuremath{\gtrsim}6$, the method provides accurate spectra. Analytical results are obtained for the valence photoemission spectrum ${\ensuremath{\rho}}_{v}(\ensuremath{\epsilon})$. The spectrum has a sharp rise close to the Fermi energy ${\ensuremath{\epsilon}}_{F}$, which goes over to a "Kondo peak" in the spin-fluctuation limit. An exact relation between ${\ensuremath{\rho}}_{v}({\ensuremath{\epsilon}}_{F})$ and the $f$-level occupancy ${n}_{f}$ is shown to be satisfied to within 10% for ${N}_{f}\ensuremath{\ge}6$. We discuss how core-level XPS spectra can be used to estimate the $f$-level occupancy ${n}_{f}$ and the coupling $\ensuremath{\Delta}$ between the $f$ level and the conduction states. We find that the values of ${n}_{f}$ and $\ensuremath{\Delta}$ obtained from core-level XPS are basically consistent with the other spectroscopies and the static, $T=0$ susceptibility. It is, therefore, possible to describe these experiments in the Anderson model, using essentially the same set of parameters for all the experiments. Typically, we find ${n}_{f}>0.7$ and $\ensuremath{\Delta}\ensuremath{\sim}0.1$ eV.