A Statistical Method for the Fitting of Double-Crystal X-ray Rocking Curves

A statistical method has been developed for the simulation, optimization and analysis of double-crystal X-ray rocking curves for epitaxial thin films in the Bragg case. This method is based on the approach of Bartels, Hornstra & Lobeek [Acta Cryst. (1986), A42, 539-545], where the diffracting medium is modeled as a set of lamellae on a semi-infinite substrate. The expressions of Bartels et al. have been recast in terms of a Möbius transform of the type G[X] - (aX + b)/(cX + d), where the constants a, b, c and d are equivalent to the complex quantities used by Bartels et al. and X is the amplitude ratio. The use of these parameters leads to a set of constraints on the fitting process, thus permitting the procedure to be automated. Convergence to a solution is determined by examination of a statistical criterion for goodness of fit; factors that contribute to the net error between the calculated and experimental rocking curves have been identified and have been incorporated into the method. The noise associated with an experimental rocking curve has been found to be very important to the fitting process, and a method is described for explicitly including noise in the procedure presented here. Statistically significant fits to a variety of experimental rocking curves (obtained by fully automated fitting with minimal user interventions) have demonstrated good agreement with experimental data.