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

Abstract

In a previous paper [Sasso et al. (2023). J. Appl. Cryst.56, 707-715], the operation of a triple-Laue X-ray interferometer having the splitting or recombining crystal cylindrically bent was studied. It was predicted that the phase-contrast topography of the interferometer detects the displacement field of the inner crystal surfaces. Therefore, opposite bendings result in the observation of opposite (compressive or tensile) strains. This paper reports on the experimental confirmation of this prediction, where opposite bendings were obtained by copper deposition on one or the other of the crystal sides.

Keywords: bent crystals; crystal X-ray interferometry; crystal strains; moiré images; phase-contrast topography; thin films.

Figure 1

X-ray phase-contrast topography. The X-ray paths are drawn in red (RRT path) and blue (TRR path). The Bragg angle is out of scale. The phase delay of each reflection is given. The X-ray crossings with the mirror are spaced by 4 mm. Adapted from Massa et al. (2020a ▸).

Figure 2

Surface stress modelling by forces per unit length applied orthogonally to edges and lying in the crystal surfaces. Black: forces acting on the coated z = 0 mm surface. Red: forces acting on the crystal rim. The boundary conditions specify a displacement field equal to zero at the bottom, y = 0 mm, surface.

Figure 3

Finite element analysis of a bent crystal: displacement field (top), (middle) and (bottom). The z = 0 mm surface is coated, and the surface stress is 6 N m⁻¹. The colour scale is from −1.4 nm (blue) to 1.4 nm (red). White lines are contours of constant displacement (solid) and the diffracting planes with displacements magnified (dashed). The rectangle indicates the imaged area.

Figure 4

Top: mean displacement field, see (8), predicted when only one side of the mirror is coated. Middle and bottom: observed displacements. The colour scale is from −89 pm (blue) to 89 pm (red). White lines are contours of constant displacement (solid) and the diffracting planes with displacements magnified (dashed). The white pixels indicate outliers excluded from the analysis.

Figure 5

Top: predicted observation of the displacement field when the inner side of the splitter or the analyser is coated. Middle and bottom: observed displacements. The colour scale is from −1 nm (blue) to 1 nm (red). White lines are contours of constant displacement (solid) and the diffracting planes with displacements magnified (dashed). The white pixels indicate outliers excluded from the analysis.

Figure 6

Top: predicted observation of the displacement field when the outer side of the splitter or the analyser is coated. Middle and bottom: observed displacements. The colour scale is from −110 pm (blue) to 110 pm (red). White lines are contours of constant displacement (solid) and the diffracting planes with displacements magnified (dashed). The white pixels indicate outliers excluded from the analysis.

Figure 7

Predicted (orange lines) and observed (black crosses) displacement fields of the splitter and analyser when their outer side is coated. The displacements are averaged over the 10 mm height of the imaged area. Since they are expected to be identical, the observed (splitter and analyser) displacements were averaged. The bars indicate the uncertainties, set equal to two standard deviations. The orange lines make reference to the surface stress applied or not applied to the crystal rim.

Figure 8

Predicted observation of the displacement field when the outer side of the splitter or the analyser is coated and the elastic anisotropy of silicon is neglected. The colour scale is from −83 pm (blue) to 83 pm (red). Comparison with Fig. 6 ▸ shows that, neglecting the anisotropy, we fail to predict correctly the result of the phase-contrast topography.

Figure 9

Geometry of the bent crystal; const. section. The axis x is normal to the diffracting planes, and the axis z is normal to the crystal surfaces. The orange line indicates the Cu coating. The rear surface lies in the neutral plane, .

Figure 10

Example of the measurement sequence. Top: front and rear topographic surveys are carried out to infer the differential displacements. Middle: front and rear surveys are carried out to determine the surface stress induced by the Cu coating. Bottom: front and rear surveys are carried out to observe the displacements yielded by the Cu coating on the external surface of the splitter and analyser. The red lines indicate the coated surfaces. The orange marker indicates the interferometer front and rear orientations.