Directional dark-field simulation results for three different relative orientations of two strongly oriented sample layers.

The upper row and left column both show the dark-field signal with respect to the orientation of two sample layers containing 250 cylinders each. The cylinders were randomly distributed over a plane perpendicular to the x-ray beam. We examined different relative orientations between the layers . For each relative orientation the plots show the dark-field signals of the separate layers and both of them together. Clearly visible is the dependence for each single layer and the superposition of both of them. For the single layer results we calculated the corresponding fit curves shown as solid lines. The product of both fit functions is shown as well as a solid line. It perfectly agrees with the simulation results for both layers together, although in the upper left plot the variance of the sum signal with respect to the actual microstate of the cylinder ensemble is quite large. In the lower right plot the first sample layer contained only 125 cylinders to simulate a layer of half the thickness. The shape of the signals does not change. From these results we derived that the superposition signal is simply the product (logarithmic scale) of the two single layers. As a simple consequence for a relative angle of the oscillation almost completely vanishes and the whole sample appears nearly perfectly isotropic.