  We present a combined theoretical and experimental study of X-ray optical
wave mixing. This class of nonlinear phenomena combines the strengths of
spectroscopic techniques from the optical domain, with the high-resolution
capabilities of X-rays. In particular, the spectroscopic sensitivity of these
phenomena can be exploited to selectively probe valence dynamics. Specifically,
we focus on the effect of X-ray parametric down-conversion. We present a
theoretical description of the process, from which we deduce the observable
nonlinear response of valence charges. Subsequently, we simulate scattering
patterns for realistic conditions and identify characteristic signatures of the
nonlinear conversion. For the observation of this signature, we present a
dedicated experimental setup and results of a detailed investigation. However,
we do not find evidence of the nonlinear effect. This finding stands in strong
contradiction to previous claims of proof-of-principle demonstrations.
Nevertheless, we are optimistic to employ related X-ray optical wave mixing
processes on the basis of the methods presented here for probing valence
dynamics in the future.
