Calculation of the instrumental function in X-ray powder diffraction

Abstract

A new method for calculating the total instrumental function of a conventional Bragg–Brentano diffractometer has been developed. The method is based on an exact analytical solution, derived from diffraction optics, for the contribution of each incident ray to the intensity registered by a detector of limited size. Because an incident ray is determined by two points (one is related to the source of the X-rays and the other to the sample) the effects of the coupling of specific instrumental functions, for example, equatorial and axial divergence instrumental functions, are treated together automatically. The intensity at any arbitrary point of the total instrumental profile is calculated by integrating the intensities over two simple rectangular regions: possible point positions on the source and possible point positions on the sample. The effects of Soller slits, a monochromator and sample absorption can also be taken into account. The main difference between the proposed method and the convolutive approach (in which the line profile is synthesized by convolving the specific instrumental functions) lies in the fact that the former provides an exact solution for the total instrumental function (exact solutions for specific instrumental functions can be obtained as special cases), whereas the latter is based on the approximations for the specific instrumental functions, and their coupling effects after the convolution are unknown. Unlike the ray-tracing method, in the proposed method the diffracted rays contributing to the registered intensity are considered as combined (part of the diffracted cone) and, correspondingly, the contribution to the instrumental line profile is obtained analytically for this part of the diffracted cone and not for a diffracted unit ray as in ray-tracing simulations.