An algebraic algorithm for generation of three-dimensional grain maps based on diffraction with a wide beam of hard X-rays

A reconstruction method is presented for generation of three-dimensional maps of the grain boundaries within powders or polycrystals. The grains are assumed to have a mosaic spread below 1°. They are mapped by diffraction with a wide beam of hard X-rays, using a setup similar to that of parallel-beam absorption contrast tomography. First the diffraction spots are sorted with respect to grain of origin. Next, for each grain the reconstruction is performed by an algebraic algorithm known as three-dimensional ART. From simulations it is found that reconstructions with a spatial accuracy better than the pixel size of the detector can be obtained from as few as five diffraction spots. The results are superior to three-dimensional reconstructions based on the same data using a variant of the filtered back-projection algorithm. In comparison with layer-by-layer type reconstructions based on the two-dimensional ART algorithm, as introduced by Poulsen & Fu [J. Appl. Cryst. (2003), 36, 1062-1068], the quality of the maps is found to be similar, provided that five to ten spots are available for analysis, while data acquisition with the three-dimensional method is much faster. The three-dimensional ART methodology is validated on experimental data. With state-of-the-art detectors, the spatial accuracy is estimated to be 5 µm.