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

Abstract

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 Co, st. (1986) 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.

Diffractometer Noise Characteristics: Experimental Determination and Application to Statistical Rocking-Curve Analysis

“…The term O'gran in the definition of %~, which we term the 'granularity error' (Staley & Matyi, 1997), represents the error associated with estimating the correspondence between two continuous curves by a set of discrete data points; in the results discussed here, ~'gra, is negligible in all cases except the unweighted fit, for which it significantly reduces the calculated residual.…”

“…Rocking-curve analysis was performed using a custom-designed software package based on the Takagi-Taupin model for dynamical X-ray diffraction; this package allows automatic optimization of the sample structural parameters without user intervention [see Staley & Matyi (1997) for a full description of this procedure]. In order to account for the effects of beam divergence, a Gaussian instrument function with standard deviation of approximately 6.5 arcseconds was convolved with all calculated patterns before comparison to the data; the instrument function width was individually optimized for each pattern.…”

“…A fuller discussion of the weight normalization, with numerical examples, can be found in Staley & Matyi (1997). Using the weighted error as the criterion in a leastsquares optimization will tend to equalize the relative uncertainty at each point, thereby fitting the entire pattern to an optimum degree of precision, rather than a total optimum error.…”

“…An unweighted fit was also obtained to serve as an additional standard for comparison. In the optimization routine [which, again, is described in detail in Staley & Matyi (1997)], the dynamical diffraction equations for a lamellar crystal are decomposed into a set of independently variable functional parameters and a set of initial values for these parameters is chosen [the same set of initial conditions was used for all fits described herein; these were derived from an initial structure factor estimate for the AIGaAs layer from the International Tables for X-ray Crystallography (1965) and a layer thickness and peak positions estimated from an examination of the data]. The criterion Sfit is then minimized by an automatic multiparameter gradient descent method until a best fit is achieved.…”

“…In a previous paper, the authors have described a procedure for the optimization and statistical quality assessment of fits between theoretical and experimental rocking-curve data-sets (Staley & Matyi, 1997). One of the salient features of this method is its use of quantitative estimates of the experimental noise in the data analysis procedure.…”