Multiplicity distribution of reflections in Laue diffraction

If a crystal is illuminated by a polychromatic beam of X-rays, then many orders of each Bragg reflection may be stimulated simultaneously, and overlap exactly in scattering angle. The overlap of these multiple orders along a ray (a central line in reciprocal space) poses a problem for Laue methods. A theory of the distribution of multiple orders as a function of the relevant experimental parameters is presented, with the following conclusions: (1) If the angular acceptance of the detector is unrestricted, then a remarkably large proportion (72.8%) of all Bragg reflections occur on single rays for the case of an infinite range of incident wavelengths. (2) This proportion increases to greater than 83% when more realistic experimental values of λ_max and λ_min are used. (3) This proportion depends only on the ratio of λ_max to λ_min and not on the space group, unit-cell dimensions, crystal orientation or the limiting resolution of the crystal d*_max (provided d*_max < 2/λ_max). (4) The total number of single rays, like the total number of all stimulated Bragg reflections, is approximately proportional to the wavelength range. (5) The proportion of reflections at a given resolution d* that lie on single or double rays depends markedly on d*, and on the ratio of λ_max to λ_min; it is generally lower at low resolution than at high. (6) Restricted angular acceptance of the detector can reduce significantly both the proportion and the total number of single rays. (7) Agreement between the theoretical distributions and those derived from analysis of X-ray Laue photographs of macromolecular crystals, and from extensive computer simulations, is good. It is evident that, under a wide variety of experimental conditions, the effect of multiple orders is not a serious limitation on the use of the Laue method for structure determination. The analysis presented has some relevance to polychromatic neutron diffraction.