  We report the observation and the theoretical explanation of the parametric
down-conversion nonlinear susceptibility at the $K$-absorption edge of diamond
and at the $L_{23}$-absorption edge of a silicon crystal. Using arguments
similar to those invoked to successfully predict resonant inelastic x-ray
spectra, we derive an expression for the renormalization term of the non-linear
susceptibility at the x-ray edges, which can be evaluated by using
first-principles calculations of the atomic scattering factor $f_1$. Our model
is shown to reproduce the observed enhancement of the parametric
down-conversion at the diamond $K$ and the Si $L_{23}$ edges rather than the
suppression previously claimed.
