Texture tomography, a versatile framework to study crystalline texture in 3D

TL;DR

TexTOM addresses the challenge of characterizing local crystallographic texture with high spatial and angular resolution in polycrystalline materials by modeling diffraction data in full reciprocal space with an orientation distribution function (ODF) expressed via hyperspherical harmonics. It combines a detailed forward model that maps voxel ODF coefficients to diffraction patterns and a constrained, gradient-based inversion that enforces physical positivity, yielding 3D, quantitative texture reconstructions without heavy a priori assumptions. The method is validated on simulated data and demonstrated on helicoidal silica-witherite biomorphs, showing accurate mean orientations, observable variance, and rapid reconstructions, with benchmarking under reduced angular sampling. TexTOM promises fast, information-driven texture analysis in 3D, enabling deeper insights into nanostructural organization in natural and technical materials and opening avenues for multi-phase and live, in situ studies.

Abstract

The crystallographic texture is a key organization feature of many technical and biological materials. In these materials, especially hierarchically structured ones, the preferential alignment of the nano constituents is heavily influencing the macroscopic behaviour of the material. In order to study local crystallographic texture with both high spatial and angular resolution, we developed Texture tomography (TexTOM). This approach allows to model the diffraction data of polycrystalline materials by using the full reciprocal space of the ensemble of crystals and describe the texture in each voxel via a orientation distribution function. This means, it provides 3D reconstructions of the local texture by measuring the probabilities of all crystal orientations. The TexTOM approach addresses limitations associated with existing models: It correlates the intensities from several Bragg reflections, thus reduces ambiguities resulting from symmetry. Further, it yields quantitative probability distributions of local real space crystal orientations without further assumptions on the sample structure. Finally, its efficient mathematical formulation enables reconstructions faster than the time-scale of the experiment. In this manuscript, we present the mathematical model, the inversion strategy and its current experimental implementation. We show characterizations of simulated data as well as experimental data obtained from a synthetic, inorganic model sample, the silica-witherite biomorph. In conclusion, Tex-TOM provides a versatile framework to reconstruct 3D quantitative texture information for polycrystalline samples. In this way, it opens the door for unprecedented insights into the nanostructural makeup of natural and technical materials.

Figures (6)

• Figure 1: Schematics of acquiring experimental and simulated diffraction patterns. a) The sample is raster-scanned using a focused X-ray beam in the $y'/z'$ direction for various rotation ($\phi$) and tilt ($\varkappa$) angles. At each point, a full diffraction pattern is collected, parametrized by the momentum transfer $q$ and the azimuthal component $\chi$ of diffraction. b) A simulated diffraction pattern originates from an ODF and a crystal structure. The ODF is parametrized by orientations the three angles ($\omega,\theta,\phi$), which describe axis-angle rotations in the sample coordinate system ($x,y,z$). The shown ODF is color-coded so that brighter colors mean higher probability of the respective orientation. Each crystal orientation yields a different single crystal diffraction pattern and the resulting image is the sum over all of them weighted by the ODF.
• Figure 2: a) A sample with HSH expansion coefficients in each voxel, represented by bar diagrams. These are weighted by the beam intensity for a given configuration and result in a projection of these coefficients. A summation of diffractlets weighted by these coefficients results in a diffraction pattern. b) is a selection of diffractlets of order 4 sHSHs with a BaCO$_3$ structure factor.
• Figure 3: a) TexTOM reconstruction of the simulated sample for testing the reconstruction algorithm. The sticks represent the reconstructed preferred orientation of the crystal c-axis in each voxel, color coded by the angular deviation $dg$ from the simulation. A corner was cut out for showing the interior of the sample. b) histogram of $dg$ for the same sample. The distribution of the standard deviations is shown in c).
• Figure 4: a) SEM image of a helicoidal silica biomorph. b) Arrangement of BaCO$_3$ nanorods (black) in amorphous silica (white) as seen by TEM. For reference, size of the shown region is given by the red rectangle in a). The red circle corresponds to the dimension of the X-ray beam. c) TexTOM reconstructions of a 60µm long piece of a helix. Sticks represent the preferred orientation of the indicated crystal axes. The volume is cut in transverse (c-axis) and longitudinal (a- and b-axes) directions, thus showing the interior of the sample. Images were produced by Paraview ParaView
• Figure 5: Plot of the minimum description length (MDL) for the reconstruction of the helicoidal biomorph in function of the HSH order $n$.