Inferring Crystal Strain Distributions from X-ray Diffraction Data: Analytical and Computational Advances

By harnessing the powers of polycrystalline materials an unprecedented, technologically-driven age has taken form
over the last 100 years. The polycrystal is today, arguably, the most central building block for electronics, renewable
energy, transportation and medical equipment industries. This material class is the reason for the emergence of the
powerful microprocessors, semiconductors, memory chips and integrated circuits driving the computing revolution,
which seem to be ever accelerating. To advance our efforts in all of these areas a deep understanding of the mechanics
that govern the polycrystal is crucial. Today, state-of-the-art far-field X-ray microscopy techniques can non-destructively
record diffraction from the individual crystal grains in a polycrystalline aggregate. By analysing the recorded diffraction
patterns, volume averaged orientations and strains of the individual crystal grains can be mapped. In this thesis we
upgrade these microscopy techniques beyond the recording of grain averaged properties to also extract information on
the strain tensor fields as they vary across the crystal domains. While we exclusively focus this thesis around the Three
Dimensional X-ray Diffraction microscope (3DXRD) and its variants, other avenues of application are possible, including
microscopes that use neutron and electron based techniques.

We show that data generated by well established scanning diffraction techniques can be used to reconstruct, in 3D,
intragranular strain tensor maps. The inference of the strain tensor field is made possible by emphasising the
tomographic aspects of the inverse problem. Multiple regression methods are derived, featuring both the popular least
squares maximum likelihood estimator as well as Bayesian inference alternatives. The presented regression methods
are enriched with constraints that can be used to stabilise the strain inversion when data are scarce, noisy and/or non-
uniformly sampled. The developed algorithms are implemented and validated by use of synthetic, simulated, diffraction
data. Additionally, applications to state of the art real world synchrotron diffraction data are presented.
As the number of data grow, the prospect of an increased spatial resolution in strain follows and the computational
aspects of the inverse problem become a pressing concern. With the help of matrix algebra we bring the problem closer
to an algebraic absorption tomography setting. Our result shortens the reconstruction compute time, simplifies the
computer implementation and provides easier access to GPU acceleration.

For the popular case of full-field measurements, when the beam probe is intentionally taken wider than the crystal
diameter, the spatial resolution in strain is lost. For these acquisition geometries we provide alternative methods that
can recover the probability distribution of the intragranular strain tensor fields (strain PDF). That is; the probability of
encountering any one particular strain tensor as a result of randomly (uniformly) traversing a point in the grain. In the
case of Gaussian strain PDFs, analytical results from this research allow for a parametric description of the null-space
of the inversion problem and a closed form solution is derived. In the general case, when the strain PDFs is non-
Gaussian, we provide an iterative finite basis expansion scheme.

Beyond the development of methods for strain inversion we derive diffraction simulation models that pave the way for
the next generation of diffraction strain estimation techniques. These models are infused by analytical solutions to a
time-dependent version of the Laue equations that unlocks exploration of optimal data acquisition strategies.

In summary, this thesis introduces a collection of mathematical advancements that enhance the capabilities of well-
established X-ray diffraction microscopy techniques. By employing the developed algorithms, polycrystalline
deformation mechanisms can be studied simultaneously at the inter- and intragranular levels.