Peak profile function for synchrotron X-ray diffractometry

A formula of the instrumental function for a high-resolution synchrotron X-ray diffractometer, equipped with a flat crystal analyser and a set of Soller slits for limiting the axial divergence of the diffracted beam, has been derived. The formula incorporates the effects of (i) the axial divergence of the diffracted beam limited by the Soller slits, (ii) the Bragg angle of the flat crystal analyser, and (iii) the tilt angle defined as the deviation of the normal direction of the analyser face from the goniometer plane. The model profile function given by the convolution of a Lorentzian function with the instrumental function has been applied to fit the experimental diffraction peak profiles of standard Si powder (NIST SRM640b) measured with a high-resolution synchrotron X-ray diffractometer, MDS, on beamline BL4B2 at the Photon Factory in Tsukuba. The convolution has been calculated by applying an efficient algorithm for numerical integration. The profile function reproduces not only the experimental profiles measured with a well aligned crystal analyser, but also significantly distorted profiles arising from misalignment of the analyser, with R_p values within 1.4%, by varying only the instrumental parameter for the tilt angle. It is suggested that further convolution with a Gaussian distribution is practically not necessary for the model instrumental function to fit the data collected with MDS. More rapid computation can be achieved by applying an analytical formula of the profile function, when the tilt angle of the crystal analyser is within about 0.2°.