Orientational mapping of minerals in Pierre shale using X-ray diffraction tensor tomography

X-ray diffraction tensor tomography is used to map clay mineral orientation and high-density inclusions in a sample of Pierre shale.

Beam attenuation artefacts were observed in the reconstructed tomograms in both the isotropic XRD-CT reconstruction and in the orientational XRDTT reconstruction. As no attenuation measurements were done during the experiment, we devised a simple attenuation correction based on the reconstructed cross-sections. The attenuation correction was applied for the reconstruction of all sinograms, except where the diffraction patterns had been filtered before integration. An uncorrected reconstructed cross-section of the illite 001 scattering is shown in Figure S5a. From the line plot in Figure S5c, it is seen that for the uncorrected tomogram (having the attenuation coefficient µ = 0), lower intensities occur close to the sample centre, where the beam is most attenuated upon traversing the sample.
The sample attenuation correction of the sample was done by: Performing an initial reconstruction to make a sample mask. (ii) Assigning an effective attenuation coefficient µ, constant for the whole sample volume. (iii) Forward projection of the simulated transmission sinogram. (iv) Correcting the sinogram by the simulated transmission sinogram.
The attenuation correction was applied for both isotropic XRD-CT and for XRDTT. Examples are shown in Figure 4.2. The absorption correction in isotropic XRD-CT was applied separately for each q. An attenuation coefficient of µ = 0.005 vox -1 was found to mitigate the absorption artefacts, cf.  intensity projection (α = 0), using an absorption coefficient of µ = 0.005 vox -1 . The maximum absorption in the projection is estimated to be approximately 20%.

S6. XRDTT reconstruction
XRDTT reconstruction was done by following the approach described by (Liebi et al., 2018) and by using the small-angle scattering tensor tomography (SASTT) package developed by the coherent X-Ray Scattering Group at the Paul Scherrer Institute, Villigen, Switzerland. An additional rotation matrix was used to align the sample z'-axis with the laboratory y-axis, see Fig. 1 and Eqs. 2 and 3.
Parameters used for XRDTT reconstruction of the clinochlore 001 and clinochlore 002 peaks are found in Table S4. The parameter names refer to entries in the SASTT MATLAB package. Supporting information, sup-11

S7. XRDTT of hydroxyapatite crystallites in bone using a single tomography axis
To further strengthen our assumption of using a single tomography axis for tensor tomography to determine the clay mineral orientation in shale, we demonstrate the approach on a sample of bone, measured using two tomography axes. The primary constituent of bone is the mineralized collagen fibre matrix, where the mineralized phase consists of hydroxyapatite (HA) crystallites (Stock, 2015), located within and outside the collagen fibre bundles. The textured wide-angle X-ray scattering from bone originates primarily from the electron density periodicity within the HA crystallites, and the preferred orientation of the crystallites with respect to the trabecula gives information about the collagen fibre orientation, which is closely tied to the mechanical properties of bone (Wenk & Heidelbach, 1999;Stock, 2015).
We have in a previous study used XRDTT to determine the 3D c-axis orientation of bone mineral hydroxyapatite crystallites of the same sample (Mürer et al., 2021). Measurements of the bone and cartilage sample were done by a tensorial tomography measurement scheme utilizing two sample rotation axes (Liebi et al., 2018). α denotes the fast-axis tomography rotation and β the tilt axis. β = 0º corresponds to single-axis tomography, as was used for the shale sample.   (Hughes et al., 1989). The HA002 and HA004 peaks displayed texture, apparent as azimuthal intensity variations in Fig. S7.2e, consistent with a preferred orientation of the HA c-axis (Wenk & Heidelbach, 1999). Texture could not be seen in the other Debye-Scherrer rings, because of the many overlapping Bragg peaks, consistent with previous reports from bone (Meneghini et al., 2003). There was no HA scattering originating from the cartilage, with the notable exception of the mineralized cartilage zone close to the bone-cartilage interface. Fig. S7.2f shows the dominant scattering direction of the HA002 peak obtained for a randomly chosen projection with (α, β) = (60º, 0º), with each pixel (x,y) in Fig. S7.2f corresponding to a single measured diffraction pattern, all revealing HA002 scattering in approximately the same direction perpendicular to the bone-cartilage interface. The irregularities at the bottom and right edge are due to artefacts introduced by cutting the brittle bone with a surgical blade, as confirmed by comparison with the PPC-CT tomograms. Based on the anisotropy of the HA002 Bragg peak we used XRDTT to reconstruct the HA crystallite c-axis orientation in the bone-cartilage sample. The XRDTT reconstructions were based on (i) all 259 measured projections, and (ii) only the 61 projections obtained with β = 0, corresponding to the measurement scheme in conventional attenuation-contrast CT (Kak, Avinash C., Slaney, 1987) and single-axis XRD-CT (Harding et al., 1987;Stock et al., 2008;Kleuker et al., 1998). 2D cross-sections of the reconstructed tomograms are shown in Fig. S7.3a and S.7b, demonstrating a close resemblance between the two reconstructions. A quantitative comparison of the reconstructed preferred orientation reconstructed using all tomography axes and only a single tomography axis was done by calculating the angular difference in orientation γ(r') reconstructed from the two datasets, defined as From the maps of γ(r') and ΔS(r') we observed the difference of the XRDTT tomograms reconstructed from all tomography axes and one tomography axes to be mainly restricted to the sample edges and close to gaps between the trabecula in the sample, cf. Fig. S7.3c and S7.3d. The analysis demonstrates that at least for the present dataset, the local HA c-axis orientation can be reliably reconstructed using a single tomography axis.

S8. XRDTT reconstructions of clinochlore 001
By using the same strategy as for reconstruction of the clinochlore 002/kaolinite 002 Bragg peak as shown in Fig. 4 in the main article, XRDTT maps were generated using the broad clinochlore 001 peak, shown in Fig. S8. Similar orientation features were found as for the clinochlore 002/kaolinite 002 peak, although differing slightly due to Bragg peak overlap.

Figure S8
XRDTT reconstructions of clinochlore 001. Figure S9 shows the scattering resulting scattering intensity distributions reconstructed for two selected voxels of the sample. The scattering intensities are calculated by using Eq. 1.