Simultaneous reflections and the mosaic spread in a crystal plate

The intensity of an X-ray (Bragg) reflection from a mosaic crystal plate under the conditions of multiple diffractions is discussed theoretically. The set of simultaneous differential equations is solved exactly for the two-beam and the three-beam cases. A second order approximation is given for the multiple beam case, and a third order approximation for the triple beam case. The fact that the ratio R of the peak intensity of a single diffracted beam to the intensity of the same beam under conditions of multiple diffraction depends on the mosaic spread η of the crystal provides a method to obtain this magnitude from the experiment. Preliminary measurements were performed on Si and Ge single-crystal plates under conditions of multiple diffraction. From them experimental values of R were obtained for different planes, which then were used to calculate the mosaic spread of the crystals. In this method only relative intensity measurements have to be used and most of the corrections found to be necessary in other procedures used to determine mosaic spread are not needed. The polarization factor for the case of a double reflection preceded by a monochromator is analyzed and the results are given in an Appendix.