X-ray diffraction of bent crystals in Bragg geometry. II. Non-ideally imperfect crystals, modelling and results

An earlier model for finite curved perfect crystals is extended to the non-ideally imperfect regime, allowing for mosaic structure from dislocations, vacancies or phase boundaries. Effects of Johann crystal mounting and depth penetration in the Bragg geometry are included. The model estimates diffraction shifts for mosaic crystals with regular local structure. Integrated reflectivities, diffraction widths, shifts and profiles against several parameters demonstrate agreement with the earlier model as an extreme and hence agreement with the literature. The theory is applied to first- and fourth-order spectra in differential quantum electrodynamic (QED) measurements and to pentaerythritol 002 crystals. A study of widths, reflectivities and shifts shows that comparison of profiles from wavelengths differing by large factors can yield the mean mosaic block thickness, angular misorientation half-width, incident polarization and beam divergence and can provide sensitive experimental tests of theory and modelling. Results for ammonium dihydrogen phosphate 101 and silicon 111 crystals agree with experiment for parameters investigated. Qualitative contributions to shifts and other parameters are identified. Results for precision QED measurements of iron and germanium Lyman αand Balmer β radiation are presented. Uncertainties in shifts due to input parameters are provided for each crystal. Crystal thickness can be a major variable in the determination of diffraction shifts, and differences between perfect and mosaic crystals are reduced for curved crystals.