  Absorbing a focused, femtosecond X-ray pulse from a Free-Electron Laser (FEL)
can lead to extreme electronic excitations in solids. This excitation drives
changes of the electronic system over the course of the pulse duration and the
overall absorption of the pulse becomes fluence-dependent. Thus,
fluence-dependent non-linear X-ray Absorption Near Edge Spectroscopy (XANES) is
sensitive to the valence excitation dynamics around the Fermi level on the
few-femtosecond timescale. Here we present a simplified rate model based on
well-established physical mechanisms to describe the evolution of the
electronic system. We construct temporal and spatial differentials for the
processes of resonant absorption, stimulated emission, non-resonant absorption,
Auger decay, valence band thermalization and scattering cascades of free
electrons. The phenomenological rate model approach provides a direct
understanding how each physical process contributes to the fluence-dependent
changes observed in XANES measurements. Without accounting for
fluence-dependent changes to the density of states, the model shows good
agreement with experimental results on metallic nickel over more than three
orders of magnitude in fluence, establishing electron redistribution as the
main driver of non-linear absorption changes at high fluences. Although in the
closest vicinity of the resonance, more complex approaches are necessary to
describe remaining discrepancies of the fluence-dependence changes, the
demonstrated capability to describe spectral changes up to extreme fluences
yields fundamental insights into the complex dynamics after intense core
excitation and provides an important tool for the design and evaluation of
future FEL experiments, in particular for the development of non-linear X-ray
spectroscopy.
