Numerical Data Analysis

A measured reflection profile or diffraction pattern essentially consists of a set of N data couples of intensities 1; collected at scattering angles 2O. All physical parameters derived from it, like unit cell edges, crystallite size or average microstrain, are based on the adaptation of the data to a physical or a microstructural model. The aspects of structural modeling are shown schematically in Fig. i4.1. Often the experimentalist is interestedin the preparation of thin films with optimized physical properties (lower right box),which are governed by the structural and microstructural properties of the layer-substrate system (upper box). To the same extent the x-ray(x ray protection) scattering pattern (lower left box) is determined by the specimen's structure. In order to elicit certain material properties the structure has to be deduced from the pattern, which can be considered as the inverse problem of thin-film analysis by x-ray scattering.
The model should allow for the formulation(disabled supplies) of a theoretical intensity function y(20, x) depending on a set of P parameters x,. The Cauchy function with parameters x- (10, 2Bo, 2w) or any other of the profile functions of Section 3.1 may serve as examples. It is realized from their analytical expressions that the parameters may be linearly (10) or nonlinearly (280, 2w) related to the model function. A model function for microstructural analysis by whole pattern fitting may comprise the parameter vector x- (a0, pt, D0, Iny, pd).If only one nonlinear parameter occurs in x, nonlinear techniques are required for the solution. The task to be solved is the identification of parameter values x} that optimally describe the set of measured intensities 1;. If the errors of the measurement are equally distributed "optimum" parameters follow from the minimization of the chi-square function.