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

Abstract

The measured value of the (220) lattice-plane spacing of silicon 28 using scanning X-ray interferometry is essential to realize the kilogram by counting 28Si atoms. An assumption made is that the measured lattice spacing is the bulk value of an unstrained crystal forming the analyser of the interferometer. However, analytical and numerical studies of the X-ray propagation in bent crystals suggest that the measured lattice spacing might refer to the analyser surface. To confirm the result of these studies and to support experimental investigations of the matter by phase-contrast topography, a comprehensive analytical model is given of the operation of a triple-Laue interferometer having the splitting or recombining crystal bent.

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

“…According to the dynamical theory model of a triple-Laue interferometer (Sasso et al, 2023), the displacement fields u S ðx; yÞ and u A ðx; yÞ refer to the (splitter and analyser) inner sides (see Fig. 1) and u Mi ðx; yÞ are the means of the displacement fields of the two mirror surfaces (see Fig.…”

“…In (1), we neglected minor contributions coming from the phase of the crystals' reflection and transmission coefficients; this phase is sensitive to deviations of the crystal surfaces from being plane and parallel and to the misalignment and spacing of the diffracting planes. These contributions are discussed by Mana & Vittone (1997), Bergamin et al (2000) and Sasso et al (2023) and amount to a few per cent of a period. Also, since we subtracted the phase map of the naked interferometer, the interferometer and phase modulator geometry and intrinsic strains are irrelevant.…”

“…The y = const. sections of (5a) were assumed in solving the Takagi-Taupin equations for the dynamical X-ray propagation (Sasso et al, 2023). The second bending occurs about an axis parallel to ẑ z in the same x ¼ x 0 plane and has curvature z ¼ az 0 .…”

“…Secondly, when one of the interferometer crystals is bent, the rays interfering parallelly leave the source from different points and propagate in different directions (Sasso et al, 2023). In turn, these differences cause optical path differences and raise questions about source coherence.…”

“…Therefore, in a previous paper (Sasso et al, 2023), we studied the operation of a triple-Laue X-ray interferometer having one of its splitting and recombining crystals cylindrically bent. Specifically, we noted that the interferometer sees the displacement fields of the splitter's and analyser's inner sides.…”