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Six-dimensional real and reciprocal space small-angle X-ray scattering tomography


When used in combination with raster scanning, small-angle X-ray scattering (SAXS) has proven to be a valuable imaging technique of the nanoscale1, for example of bone, teeth and brain matter2,3,4,5. Although two-dimensional projection imaging has been used to characterize various materials successfully, its three-dimensional extension, SAXS computed tomography, poses substantial challenges, which have yet to be overcome. Previous work6,7,8,9,10,11 using SAXS computed tomography was unable to preserve oriented SAXS signals during reconstruction. Here we present a solution to this problem and obtain a complete SAXS computed tomography, which preserves oriented scattering information. By introducing virtual tomography axes, we take advantage of the two-dimensional SAXS information recorded on an area detector and use it to reconstruct the full three-dimensional scattering distribution in reciprocal space for each voxel of the three-dimensional object in real space. The presented method could be of interest for a combined six-dimensional real and reciprocal space characterization of mesoscopic materials with hierarchically structured features with length scales ranging from a few nanometres to a few millimetres—for example, biomaterials such as bone or teeth, or functional materials such as fuel-cell or battery components.

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Figure 1: Experimental set-up and raster-scanning technique.
Figure 2: Virtual SAXS patterns extracted from the reconstructed 3D volume.
Figure 3: 3D visualization of collagen fibre orientation within the tooth sample.


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The SAXS experiments were performed at the cSAXS beamline of the Swiss Light Source (SLS) at the Paul Scherrer Institut (PSI), Villigen, Switzerland. We are grateful for travel support that was granted by the EU access program CALIPSO. We acknowledge financial support through the DFG Cluster of Excellence Munich-Centre for Advanced Photonics (MAP) and the DFG Gottfried Wilhelm Leibniz program. P.Z. is grateful for funding of the DFG (German Research Foundation) through SPP1420. F.S. and C.J. thank the TUM Graduate School for support of their studies. F.P. acknowledges support through the TUM Institute for Advanced Studies (TUM-IAS). We thank A. Fehringer for developing the GPU projectors used for the reconstruction.

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Authors and Affiliations



F.P., M.B. and F.S. conceived the experiment. M.L. and M.G.-S. developed the sample stage used in the experiment. M.B., C.J., M.L., M.G.-S. and F.S. performed the experiment at cSAXS/PSI. P.Z. prepared the tooth sample. Data analysis was performed by F.S. with input and discussion from M.B., F.P. and P.Z. F.S. wrote the manuscript with contributions from all co-authors.

Corresponding authors

Correspondence to Florian Schaff, Martin Bech or Franz Pfeiffer.

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Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Comparison of measured and reconstructed values for one projection.

Left, reprojected q data for two different orientations (90° and 152°) is compared to the measured data. Right, azimuthal values for both the reconstructed and measured data are given for three select points, indicated by the dashed lines in the left panel. For the chosen |q| range, distinct collagen peaks are reconstructed correctly for points 1, 2 and 3 at around 15°, 0° and 90°, respectively. A good agreement between reconstruction and measurement is seen. Animations of this and further projections showing all q orientations are provided in Supplementary Videos 3, 4, 5.

Extended Data Figure 2 Coordinate system used for the experiment.

The sample orientation is described using three Euler angles θ, φ and ψ. With ψ = 90°, θ represents a tilt of the tomography axis around xlab and φ describes a rotation around this tilted axis. The sample is scanned along xlab and ylab, with a diffraction pattern collected at each point. The detector coordinates are given by dx and dy.

Extended Data Figure 3 Rotational invariance for a standard SAXS computed tomography with a vertical tomography axis.

a, Absorption image acquired from the diode data with two vertical slices marked. b, c, Rotational invariance as defined in the text for both slices. In both cases, rotational invariance is present for all pixels that correspond to scattering orientations parallel to the vertical rotation axis. The collagen fibres in the top part of the sample, shown in b, are mainly vertical. Owing to this symmetry, rotational invariance is also present for pixels not on the vertical axis. Without this symmetry, rotational invariance exists only for the vertical direction, as seen in c. The white bars in the lower half of the images are areas between the individual detector modules.

Extended Data Figure 4 Generalized rotational invariance.

a, b, Rotational invariance shown for different virtual tomography axes: (a) and (b). Radially integrated data are used. Compared to the standard case, shown in Extended Data Fig. 3, a line-wise integration is not possible in the general case. Instead, the standard deviation and mean are calculated from the projection-wise sum of all integrated SAXS patterns. A shift of the azimuthal angle so that the scattering orientation parallel to t is at 0° was applied. As can be seen, rotational invariance is also achieved in the general case for scattering orientations parallel to t.

Supplementary information

SAXS-patterns extracted from the reconstructed data at three different points.

All slices from top to bottom are shown. Animation of Figure 2. (MP4 1436 kb)

Three-dimensional visualization of collagen orientation and scattering strength within the tooth sample.

All slices of the sample from front to back are shown. Animation of Figure 3. (MP4 14801 kb)

A comparison of measured and reconstructed scattering intensities.

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = 0.0° and φ = 346.9°. Animation of Extended Figure 1. (MP4 1016 kb)

A comparison of measured and reconstructed scattering intensities.

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = -52.0° and φ = 216.9°. Animation of Extended Figure 1. (MP4 882 kb)

Comparison of measured and reconstructed scattering intensities.

Azimuthal scattering orientations over 180° are shown for the projection recorded at θ = -68.0° and φ = 120.0°. Animation of Extended Figure 1. (MP4 890 kb)

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Schaff, F., Bech, M., Zaslansky, P. et al. Six-dimensional real and reciprocal space small-angle X-ray scattering tomography. Nature 527, 353–356 (2015).

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