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Explaining the x-ray nonlinear susceptibility of diamond and silicon near absorption edges

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.