Evaluation of stress gradients sigma(ij)(z) from their discrete laplace transforms sigma(ij)(tau(k)) 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, epsilon(phi psi)(tau) and sigma(ij)(tau), with respect to the penetration depth tau of the X-rays. However, because the main interest is focussed on the actual depth profiles of the stresses, sigma(ij)(z), the tau-profiles have to be inverted into those of the (original) z-space by suitable methods of the inverse Laplace transform. For discrete profiles sigma(ij)(tau(k)) obtained in the scattering vector mode, where tau is adjusted by rotating the sample around the normal of the reflecting lattice planes, N-h, 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 tau(k) of the penetration depth.