Theoretical Analysis of Topotomography Using Small Intragranular Strain Approximations

TL;DR

Topo-Tomography (TT) is analyzed to determine how intragranular crystal orientation fields can be reconstructed from near-field TT diffraction data. The authors derive a sequence of forward models, from a 3D undeformed grain to 5D normal-variation and 6D elastic-deformation representations, and show that integrating TT spots over base tilt yields projections of a pseudo-distorted grain volume. Fourier analysis reveals TT captures only a subset of the grain's Fourier components, motivating multi-scan strategies such as using opposite scattering vectors and varying detector distances to enhance coverage. Simulations with a cubic grain phantom validate that joint TT acquisitions improve orientation accuracy and provide practical lower bounds for orientation-sampling resolution, guiding experimental design. Overall, the work provides a foundational theoretical framework that informs TT acquisition parameters and emphasizes the interplay between projection geometry, intragranular strain, and reconstructable information.

Abstract

Topo-Tomography (TT) is a synchrotron-based X-ray diffraction imaging technique used to characterize grain shape and crystal orientation in polycrystalline samples. This work aims to provide a decisive and fundamental understanding of 3D grain shape and orientation field reconstruction from TT diffraction data. We derive mathematical expressions for the TT projection geometry, considering grain shape, intragranular lattice rotations, and elastic strains, under the assumption of kinematical diffraction. These expressions are simplified using approximations for small strain variations and grain size. The simplified expressions show that integrated TT projection images correspond to projections of a "pseudo" distorted grain volume. Its Fourier analysis provides insights into the feasibility of orientation field reconstruction from TT scans. We propose methods to expand data coverage, including using opposite scattering vectors and varying detector distance. A lower bound for orientation sampling resolution is derived and validated through simulations.

Figures (11)

• Figure 1: Sketch of Topo-Tomography (TT) setup and associated coordinates.
• Figure 2: Sketches of simplified coordinates used for formulation: (a) reconstruction coordinates used for formulation are equivalent to the rotation of laboratory coordinates based on $\phi$ and $\omega$; (b) reference coordinates used for formulation are equivalent to the rotation of laboratory coordinates around base tilt by -$\theta$.
• Figure 3: Sketches depicting the projections of the voxels in the reference coordinates onto the detector coordinates. The $\hat{\mathbf{y}}_f$-$\hat{\mathbf{z}}_f$ plane is parallel to the plane of the detector.
• Figure 4: Sketches depicting the diffraction condition and the projection position of a voxel which has undergone some lattice rotation.
• Figure 5: The sketch depicting the meaning of the coefficient $C_p$.