At synchrotron sources, such as the Advanced Photon Source (APS), it is now routinely possible to collect x-ray diffraction data from single crystals that contain thousands, and even tens of thousands, of Brillouin Zones, in well under half an hour. This means that detailed parametric studies, e.g., as a function of temperature or magnetic field, can be completed in well under a day. These capabilities have arisen from the coupling of high x-ray brightness with new generations of photon-counting area detectors that combine fast frame rates with high dynamic range and low backgrounds.

Each rotation contains both Bragg peaks, which can be used to determine the average crystal structure, and finely-meshed diffuse scattering, which results from local disorder,  measured in many thousands of Brillouin zones (Fig. 1). The data therefore contain thermodynamic averages of all the structural correlations, both long- and short-range (~2 to 200Å), that characterize the material and generate its physical properties. The technique is therefore extremely powerful, with the potential to reveal subtle structural features that are hidden from more limited measurements, but the sheer size of the data sets and the speed of their measurements make conventional approaches to data analysis impractical. 

AXMAS is a multi-disciplinary project to use advanced computational tools to address the challenges and realize the opportunities provided by this x-ray data. The project, entitled “Structural Signatures of Hidden Order in Spin-Orbit Coupled Systems” was funded by the Department of Energy, Office of Basic Energy Sciences, to investigate materials with strong spin-orbit coupling that are predicted to display novel kinds of multipolar/nematic order that have not always been detected in structural studies in the past. To accomplish this, we have developed tools to interrogate large data volumes from the APS using advanced computational methods, which we are making available to the materials science community through open-source libraries. These are being incorporated into the data analysis workflow, NXRefine, that has been developed for experiments at the APS and CHESS. 

Machine Learning

The experimental methods are ideal for investigating the temperature dependence of structural correlations, whether short- or long-range. For example, if there is a structural phase transition, below which new superlattice peaks emerge owing to a reduction in symmetry, these experiments will contain information on the temperature dependence both of the order parameter, i.e., the superlattice peak intensities, and the critical fluctuations above the transition. However, ensuring that all components of the order parameter, including secondary order parameters, or all relevant fluctuations have been identified is not generally possible by manual inspection alone.  

As part of AXMAS, a group based at Cornell University have developed a machine learning approach to address this issue. Most analyses of complex experimental data that use machine learning have tended to emphasize supervised learning using hypothesis-driven synthetic data. 

However, this requires extensive training sets that will bias the results and make them uninterpretable. Our approach is to exploit the fact that our experimental data is collected at many temperatures. We can then use unsupervised machine learning to cluster the data into qualitatively distinct temperature “trajectories”. By using unsupervised learning, we are able to Identify systematic trends and correlations in thousand of Brillouin Zones without selection bias. 

The application, XRD Temperature Clustering (X-TEC ), uses the Gaussian Mixture Model to extract from the billions of recorded pixels a reduced set of temperature trajectories that correspond to distinct physical processes in the material.  The trajectories are rescaled so that we can compare trajectories at different intensities scales, focusing on their temperature dependence rather than the absolute scale. A technique known as label smoothing averages cluster assignments of neighboring pixels to enforce local correlations. As shown in Fig. 2, X-TEC is able to extract both the temperature dependence and the Q-dependence of emergent order parameters, such as charge-density-wave modulations, without any prior input. It has also been used to separate superlattice peaks from the critical fluctuations that surround them. 

Spectral Analysis

The measurement of large contiguous volumes of reciprocal space enables new modes of analysis based on transformations of the data into real-space. Pair-distribution-functions (PDF) are widely utilized in the analysis of powder diffraction data, but Thomas Weber and colleagues at ETH Zürich have shown that it is possible to extend the PDF technique to three dimensions, e.g., see T. Weber and A. Simonov, Zeit Krist 227, 238 (2012). The result is a Patterson function, containing the summed probabilities of interatomic vectors in the crystal with contributions from both the average structure and local disorder. By excluding Bragg scattering from the real-space transform in a process known as “punch-and-fill”, the resulting PDF function only includes those interatomic vectors whose probabilities differ from the average (3D-ΔPDF). The Patterson maps can be used to identify atomic distortions and the length scale of their correlations without prior modeling. 

Applied mathematicians at Argonne have been exploring methods of making the  3D-ΔPDF method more efficient and robust. Although one- and two-dimensional spectral analysis is a very mature field, the extension to three-dimensions is not trivial. The choice of appropriate multi-taper functions to minimize leakage artifacts, efficient methods of interpolation using Laplacian and Matern functions when removing the Bragg peaks, and novel Fourier filtering methods are all being explored. For example, Figure 3 shows PDF maps produced by applying tiled Gaussian filters situated on the superlattice peaks identified by machine learning in Fig. 2. This is more effective than the “punch-and-fill” method in removing thermal diffuse scattering, revealing only those fluctuations associated with charge-density-wave ordering. These methods are made available as modules written in Julia, available on Github, which can then be incorporated into Python packages. 

As these methods are developed, they are being implemented in the Python-based data reduction workflow, NXRefine, which is used to analyze the data in real-time during experiments. With the efficiency improvements we have achieved already, we are able to transform the data into reciprocal space coordinates automatically after each scan is completed and convert the data into 3D-ΔPDF maps in parallel with the data collection, ensuring that scientists can make informed decisions during the experiment.