Analytical And Numerical Approaches For Diffraction Efficiency In Low-curvature Curved Crystals

Crystals with curved diffraction planes (CDPC) are an emerging technology in X-ray optics. CDPC allow manipulating the trajectories of high-energy photons with efficiency near the unity in a broad energy range. An elective application of CDPC is the construction of hard X-ray lenses. Up to now, the impossibility to focalize hard X-rays left the observation of the sky in this energy range to direct-view instruments, featuring low sensitivity and resolution. In fact, only the spectra of few and strongest sources is known above 70 keV. Mosaic crystals have already been implemented for the construction of focusing optics, but they show low reproducibility in the fabrication, and diffraction efficiency is physically limited to 50% at most. The theory of diffraction in curved crystals was developed in the past half century in the frame of the dynamical theory of diffraction, with particular contribution by C. Malgrange If the curvature is quite strong, is possible to find a simple expression to quantitatively determine the fraction of diffracted photons. To date, it exists no analytical theory that quantitatively calculates the diffraction efficiency for crystals of low curvature. Indeed, the applications of CDPC sometimes requires a curvature radius in this range. For this reason, we developed a model, which is able to produce realistic previsions of the diffraction efficiency of a thick crystal in Laue geometry for any curvature radius. The model agrees with the results of the dynamical theory when this latter is applicable. It also leads to the same results of the flat crystal case when the curvature radius is very large, and gives a realistic and quantitative description of diffraction efficiency when these two cases are not applicable.