Using the Kohn–Sham density
functional theory, we numerically
study the four-wave mixing response of a carbon atom model system
exposed to a train of femtosecond two color ω–3ω
random phase coherent X-ray pulses near the K-edge. The phase-sensitivity
cancellation of the 5ω anti-Stokes component previously described
in two- and three-level systems in the infrared and optical regions
is extended into the X-ray. Resonances with the absorption lines in
the XANES and EXAFS regions produce 5ω peak intensities that
increase near the phase-sensitivity cancellation frequencies. Based
on this effect, we predict that highly selective intense X-ray 5ω
photon energies can be achieved in real systems. The high localization
of the ω–3ω four-wave mixing nonlinear technique
that we address entails a new valuable tool in X-ray spectroscopies
of chemical species as it can readily be extended to different photon
energies in other atomic absorption edges, with broad applications.