Ultrashort lifetime expansion for indirect resonant inelastic x-ray scattering

In indirect resonant inelastic x-ray scattering (RIXS) an intermediate state is created with a core-hole that has an ultrashort lifetime. The core-hole potential therefore acts as a femtosecond pulse on the valence electrons. We show that this fact can be exploited to integrate out the intermediate states from the expression for the scattering cross section. By doing so we obtain an effective scattering cross section that only contains the initial and final scattering states. This effective cross section turns out to be a linear combination of the charge response function S(q,ω) and the dynamic longitudinal spin density correlation function, both with a resonant prefactor. This result is asymptotically exact for both strong and weak local core-hole potentials and ultrashort lifetimes. The resonant scattering prefactor is shown to be weakly temperature dependent. We also derive a sum rule for the total scattering intensity and generalize the results to multiband systems. One of the remarkable outcomes is that one can change the relative charge and spin contribution to the inelastic spectral weight by varying the incident photon energy.