Modelisation by finite elements of sagittal focusing

Abstract

To lower the anticlastic curvature of a sagittally bent crystal to focus horizontal divergence, we performed numerical simulations by finite elements modelling. The results of simulations on a classical crystal with ribs demonstrated that the radius of the sagittal curvature (Rs) oscillates periodically according to the ribs distribution. These waves prevented the diffraction to propagate on the whole reflective surface of the crystal; to minimize this effect, we studied the influence of the geometrical parameters of the ribs on the oscillations of Rs. The output model of the simulation does not fit machining requirements: the thickness of both the reflective area and the ribs are not large enough to prevent the crystal of breaking during machining or under bending. A new model was then designed with flying buttresses in place of the ribs.The numerical simulations on this new model demonstrated that the radius of the sagittal curvature does not oscillate on the diffracting surface of the crystal. The sagittal focusing is perfectly monotonous. We found an optimum solution with a range of validity extending from 4 to 25 keV for the Si(111) and Si(311) Bragg reflections.