  We show that the nonlinear interactions between x rays and longer wavelengths
in crystals depend strongly on the band structure and related properties.
Consequently, these types of interactions can be used as a powerful probe for
fundamental properties of crystalline bulk materials. In contrast to previous
work that highlighted that these types of nonlinear interactions can provide
microscopic information on the valence electrons at the atomic scale
resolution, we show that these interactions also contain information that is
related to the periodic potential of the crystal. We explain how it is possible
to distinguish between the two contributions. Our work indicates on the
possibility for the development of novel multi-dimensional pump-probe metrology
techniques that will provide spectroscopic information combined with structural
information including ultrafast dynamics at the atomic scale.
