Evaluation of stress gradients ??ij(z) from their discrete laplace transforms ??ij(τk) obtained by x-ray diffraction performed in the scattered vector mode

Due to the absorption, X-ray diffraction analysis of residual stress gradients yields only exponentially weighted averages (“Laplace profiles”) of the strains and stresses, εψφ(τ) and σij(τ), with respect to the penetration depth τ of the X-rays. However, because the main interest is focussed on the actual depth profiles of the stresses, σij(z), the τ-profiles have to be inverted into those of the (original) z-space by suitable methods of the inverse Laplace transform. For discrete profiles σij(τk) obtained in the scattering vector mode, where τ is adjusted by rotating the sample around the normal of the reflecting lattice planes, Nh, the calculations may be performed numerically by the methods of orthogonal polynomials. It will be shown on examples of simulated profiles that the use of Jacobi polynomials gives the best results, because they offer the maximum flexibility with regard to the selection of the discrete values τk of the penetration depth.