Determining X-ray Diffraction Data at Varying Depths – a Preliminary Theoretical Approach

A theoretical method for extracting diffractograms from a range of depths will be described. The method is non-destructive and requires only a series of traditional data collections, where the incident angle is varied. The measured spectra are transformed to give calculated X-ray diffraction data arising from different depths. This is in contrast to conventional approaches where either a specific feature is examined, or the inherent depth averaging of grazing angle Xray diffraction is tolerated. The resultant calculated spectra may be analysed by any of the current techniques to provide information about any number of different characteristics. In order to obtain X-ray diffraction patterns from specific depths within samples, a Fredholm integral equation of the first kind is solved using regularisation techniques. Using known solutions of the Fredholm integral equation to create pseudo-experimental data, the method was validated by transforming this data and thus recovering the solution used. INTRODUCTION Many current methods of structural depth profiling are destructive, where a small amount of material is removed from the surface of the sample between each analysis. This is not suitable or desirable in many cases as additional stresses, for example, may be introduced into the sample with each layer of material removed [1]. Alternatively, non-destructive methods may be used which reduce the probe incident angle, thus reducing the depth of analysis. However, this method only gives information about the characteristics of the sample averaged over the penetration depth at each incident angle. The method introduced here aims to produce data from specific depths by transforming experimental data collected over a range of different incident angles. Li et al [2] and Wu et al [3] have previously proposed a similar approach to this problem. However, the diffraction data recovered at varying depths was produced on the scale of absorption depth, whereas the method outlined in this study gives the diffraction data on a directdepth scale. Further, in the work presented by Li et al, a single 2θ value was used for analysis, whereas in the work presented here, we extended this to a range of 2θ. Predecki [4] described a method of transforming profiles of quantities that are functions of the penetration depth into quantity profiles that are functions of the depth below the surface of the sample. Laplace transforms were used to perform this transformation, but a function was also required to be fitted to the observed quantity profile. The method described in our work makes no assumptions regarding the depth distribution of structural features. In this paper, pseudo-experimental data was used to test and refine our methodologies. Copyright ©JCPDS International Centre for Diffraction Data 2004, Advances in X-ray Analysis, Volume 47. 187