Electron spectroscopies for Ce compounds in the impurity model

We present a method for calculating the core-level x-ray photoemission (XPS), the $3d\to 4f$ 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 $N_f\to \infty$, and show that the method is exact for the XPS, XAS, and valence photoemission spectra in this limit. It is also demonstrated that for $N_f\gtrsim 6$, the method provides accurate spectra. Analytical results are obtained for the valence photoemission spectrum ${\rho }_v\left(\epsilon \right)$. The spectrum has a sharp rise close to the Fermi energy ${\epsilon }_F$, which goes over to a "Kondo peak" in the spin-fluctuation limit. An exact relation between ${\rho }_v\left({\epsilon }_F\right)$ and the $f$-level occupancy $n_f$ is shown to be satisfied to within 10% for $N_f\ge 6$. We discuss how core-level XPS spectra can be used to estimate the $f$-level occupancy $n_f$ and the coupling $\Delta$ between the $f$ level and the conduction states. We find that the values of $n_f$ and $\Delta$ obtained from core-level XPS are basically consistent with the other spectroscopies and the static, $T=0\mathrm{}$ 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 $n_f>0.7\mathrm{}$ and $\Delta \sim 0.1$ eV.