Sin²ψ-based residual stress gradient analysis by energy-dispersive synchrotron diffraction constrained by small gauge volumes. I. Theoretical concept

The influence of the gauge volume size and shape on the analysis of steep near-surface residual stress gradients by means of energy-dispersive synchrotron diffraction is studied theoretically. Cases are considered where the irradiated sample volume is confined by narrow-slit systems, in both the primary and the diffracted beam, to dimensions comparable to the `natural' 1/e information depth τ_1/e of the X-rays. It is shown that the ratio between τ_1/e, defined by the material's absorption, and the immersion depth h^GV of the gauge volume into the sample is the crucial parameter that shapes the d_ψ^hkl or [epsilon]_ψ^hkl versus sin²ψ distributions obtained in the Ψ mode of X-ray stress analysis. Since the actual information depth 〈z〉^GV to which the measured X-ray signal has to be assigned is a superposition of geometrical and exponential weighting functions, ambiguities in the conventional plot of the Laplace stresses versus 〈z〉^GV may occur for measurements performed using narrow-slit configurations. To avoid conflicts in data analysis in these cases, a modified formalism is proposed for the evaluation of the real space residual stress profiles σ_||(z), which is based on a two-dimensional least-squares fit procedure.