X-ray reflectivity of bent perfect crystals in Bragg and Laue geometry

The X-ray reflectivity of bent perfect crystals is calculated from a model where the crystal is approximated by a stack of perfect-crystal lamellae which have a gradually changing orientation. A computer program is developed for calculation of the reflectivity of the composite crystal from the dynamical theory of diffraction. An approximate solution is also given where an analytical formula for the reflectivity of a non-absorbing lamella is used and the effects of absorption are calculated separately. Typically, in the Bragg case, the reflectivity curve has a steep edge and an exponentially falling slope, while in the Laue case the curve is almost rectangular if the absorption is not too large. The width of the curve is inversely proportional to the bending radius in both cases. Reflectivity curves were measured for the 111 and 400 reflections of Si with Mo Kα₁ radiation. The agreement with analytical and computer calculations is good, particularly at small bending radii where the kinematical limit is approached.