X-ray phase-contrast topography to measure the surface stress and bulk strain in a silicon crystal

Abstract

The measurement of the Si lattice parameter by X-ray interferometry assumes the use of strain-free crystals, which might not be true because of intrinsic stresses due to surface relaxation, reconstruction and oxidation. X-ray phase-contrast topography was used to investigate the strain sensitivity to the finishing, annealing and coating of interferometer crystals. The topography capabilities were assessed by measuring the lattice strain due to films of copper deposited on the interferometer mirror crystal. A by-product has been the measurement of the surface stresses after complete relaxation of the coatings.

Crystal bending in triple-Laue X-ray interferometry. Part I. Theory

“…Equation ( 1) is not strictly valid in the presence of anisotropy, unpaired surface stresses and Dirichlet boundary conditions imposed at the crystal base. Our finite element analyses and experimental verifications are given by Massa et al (2023). In particular, we observed that a copper film coated on one of the surfaces bends the crystal in such a way that its opposite, naked, surface lies in the neutral, Àz ¼ z ¼ plane.…”

“…Sections 7 and 8 deal with the wavefields leaving a bent crystal and a triple-Laue interferometer having the splitting or recombining crystal bent. In the conclusion, we outline predictions that have been verified by the phase-contrast topography of a monolithic interferometer having one of its crystals bent by a thin copper film (Massa et al, 2023).…”

“…To realize the kilogram, an essential assumption is that the measured lattice spacing is the bulk value of the unstrained analyser (a blade, typically 1 mm thick). However, surface relaxation, reconstruction and oxidation might cause lattice strains (Melis et al, 2015(Melis et al, , 2016Massa et al, 2020a). Furthermore, analytical and numerical studies of the X-ray propagation in a bent crystal (e.g.…”

Crystal bending in triple-Laue X-ray interferometry. Part II. Phase-contrast topography

“…Specifically, we noted that the interferometer sees the displacement fields of the splitter's and analyser's inner sides. This result and, in turn, the prediction that opposite bending will result in the observation of opposite (compressive or tensile) strains opened the way to an experimental investigation by the phase-contrast imaging of crystal interferometers (Bonse & Hart, 1966;Bonse et al, 1976;Ohler et al, 1999;Bergamin et al, 2000;Fodchuk & Raransky, 2003;Massa et al, 2009Massa et al, , 2020aDrmeyan et al, 2013Drmeyan et al, , 2017, which is the subject of this paper.…”

“…Bent silicon crystals have been extensively studied because they are used as optics for the conditioning of X-ray beams and analysers for X-ray spectroscopy (Nesterets & Wilkins, 2008;Qi et al, 2021;Kaganer et al, 2020;Guigay & Sanchez del Rio, 2022), and to infer the stresses in thin films and devices on substrates (Vaudin et al, 2011). We are motivated by the search for systematic errors in the measurement of the silicon lattice parameter by crystal X-ray interferometry and the realization of the kilogram by counting silicon atoms (Massa et al, 2011(Massa et al, , 2015(Massa et al, , 2020aKessler et al, 2017;Yang et al, 2020). Therefore, our concern is the phase of the diffracted waves.…”