Performance of multi-crystal Bragg X-ray spectrometers under the influence of angular misalignments

X-ray beams have long been analyzed by series of Bragg reflections from crystals arranged in various configurations. The dependence of reflectivity and selected wavelength on the beam's incidence angle makes proper crystal alignment crucial, particularly as the number of crystals increases. A list of possible misalignments and a method for calculating their effects on the selected wavelengths, output beam positions, and throughputs are presented here. The following goes beyond the DuMond diagram to include rolls and rotation axis misalignments. These are described for a single crystal, and a ray-tracing method is derived for multiple crystals assuming that the misalignment angles are small. The angles are classified by their lowest-order term. Reflectivities for each ray are calculated by the equations of plane-wave dynamical diffraction theory. Then, as a practical application, sample calculations are discussed for the (+--+)(+--+) four-crystal monochromator being designed for beamline I20 at the Diamond Light Source.