🔬 materials science

Connecting Diffuse Scattering to Atomic-Site-Resolved Occupancy and Displacement Fields through Fourier Filtering

The paper introduces MOSAIC, a computational framework that utilizes Fourier filtering on scattering amplitudes to map specific diffuse scattering features in reciprocal space directly to atomic-site-resolved occupancy and displacement fields, thereby enabling the identification of local structural correlations in large-scale models derived from simulations or experimental data.

Original authors: Maksim Eremenko, Victor Krayzman, Matthew G. Tucker, Igor Levin

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Maksim Eremenko, Victor Krayzman, Matthew G. Tucker, Igor Levin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a giant, messy jigsaw puzzle of a crystal. From far away, the picture looks like a neat, repeating pattern. But if you zoom in, you see that the pieces are slightly jiggled, swapped, or pushed out of place. These tiny, local messes create a faint, fuzzy glow in the background of the X-ray or neutron images scientists take. This glow is called diffuse scattering.

For a long time, trying to figure out exactly which jumbled pieces caused that specific fuzzy glow was like trying to guess which specific person in a crowded stadium made a single, tiny cheer, just by listening to the roar of the whole crowd. It's incredibly hard to tell who did what.

Enter MOSAIC, a new computer tool created by Maksim Eremenko and his team. Think of MOSAIC as a magical pair of "noise-canceling headphones" for crystal images.

The Magic Filter

Here is how it works, using a simple analogy:

Imagine the crystal's fuzzy glow is a complex song made of many different instruments playing at once. Some instruments represent atoms swapping places (chemical ordering), while others represent atoms wiggling out of position (displacements). Usually, these sounds are all mixed together, making it impossible to hear the individual instruments.

MOSAIC takes a known map of the crystal (a "configuration" built from experiments or computer simulations) and calculates the song it should make. Then, it uses a Fourier filter—which is like a super-precise radio tuner—to isolate just one specific frequency of the song.

Once the tool isolates that specific frequency, it turns the signal back into a picture. Suddenly, the fuzzy glow isn't just a blur anymore; it transforms into a clear map showing exactly which atoms are wiggling or which ones have swapped seats.

What MOSAIC Can (and Can't) Do

The paper is very clear about what this tool is and isn't.

It is NOT a magic wand that builds a crystal from scratch.
The authors explicitly state that MOSAIC does not take a blurry image and magically reconstruct a unique atomic structure out of thin air. You must already have a "structural model" (a guess or a simulation of how the atoms are arranged) to start with. MOSAIC is a detective that checks your model: "If your model is right, does it explain this specific part of the fuzzy glow?" It connects the dots between a known arrangement and the specific scattering features it creates.

It IS a precise decoder for specific patterns.
The tool successfully separates two things that usually get mixed up:

Chemical Ordering: When different types of atoms (like Lithium and Iron) decide to sit next to each other in a specific pattern.
Displacements: When atoms physically push or pull away from their perfect spots.

In a test, the team simulated a crystal with two overlapping problems: some atoms were rotating like gears, while others were breathing (expanding and contracting). Without MOSAIC, these two movements looked like one big mess. But by using different "masks" (like tuning into different radio stations), MOSAIC successfully separated the "rotating" signal from the "breathing" signal, showing exactly which atoms were doing what.

Real-World Tests

The team didn't just stop at simulations. They tested MOSAIC on real, complex materials called relaxor ferroelectrics (specifically PMN and PMN-PT). These materials are famous for being messy, with atoms that don't like to sit still.

The Chemical Test: They looked at how Magnesium and Niobium atoms arrange themselves. In the raw data, it looked like a chaotic mess. But when MOSAIC filtered the signal, it revealed "chessboard-like" patches of order hidden inside the chaos, showing exactly where the atoms were lining up and where they were breaking the rules.
The Displacement Test: They looked at how Lead atoms wiggle. The raw data was so noisy it looked like static on an old TV. After MOSAIC filtered out the noise, clear "swirling" patterns of polarization appeared, showing that the atoms were actually moving in coordinated, vortex-like groups.

The Scale of the Operation

This isn't a tool for small, simple puzzles. The paper highlights that MOSAIC is built to handle millions of atoms (specifically mentioning configurations with $10^6$ atoms) and massive amounts of data. It uses a clever computing trick called "Map–Reduce" and a special math engine (NUFFT) to crunch these numbers without running out of memory, which would happen if you tried to use standard methods.

The Bottom Line

The paper suggests that MOSAIC is a powerful new way to "interrogate" large structural models. It doesn't claim to have solved the mystery of every crystal in the universe, but it provides a robust, quantitative way to ask: "If I look at this specific part of the fuzzy glow, what specific atomic movements or swaps are causing it?"

By turning vague, fuzzy signals into sharp, site-specific maps, MOSAIC helps scientists finally see the hidden stories of disorder that drive the special properties of advanced materials. It turns the "static" of the atomic world into a clear, readable story.

Technical Summary: Connecting Diffuse Scattering to Atomic-Site-Resolved Occupancy and Displacement Fields through Fourier Filtering

Problem Statement
Local structural correlations in crystalline materials, such as chemical ordering and atomic displacements, are encoded in diffuse scattering patterns observed in X-ray, neutron, or electron diffraction. While traditional crystallographic analysis using Bragg diffraction captures the average structure, it largely fails to access the local deviations responsible for functional properties. A significant challenge exists in interpreting diffuse scattering: even when an atomistic structural model (derived from Reverse Monte Carlo refinements, molecular dynamics, or simulations) successfully reproduces the experimental diffuse intensity, it remains difficult to identify which specific correlated atomic motifs within that model generate particular diffuse features. Furthermore, distinguishing between overlapping contributions from chemical ordering and static displacements, or mapping these features to specific atomic sites in large-scale models, is non-trivial. Existing methods, such as 3D- $\Delta$ PDF or qualitative Fourier filtering of microscopy images, often lack a quantitative means to determine the underlying chemical or displacement correlation fields with site resolution.

Methodology: The MOSAIC Framework
The authors introduce MOSAIC (Mapping of Occupancy and Site-resolved Atomic Interactions via Correlation), a computational framework designed to link specific diffuse-scattering features to site-resolved occupancy and displacement fields in a known atomic configuration. The method does not reconstruct a structure from intensity data alone; rather, it interrogates a supplied structural model to compute the scattering amplitude and isolate specific features.

The workflow proceeds as follows:

Scattering Amplitude Calculation: Starting from an atomic configuration, the total complex scattering amplitude $A_{tot}(\mathbf{Q})$ is calculated. The diffuse scattering amplitude, $\Delta A(\mathbf{Q})$ , is obtained by subtracting the contribution of the average crystallographic structure.
Fourier Filtering: Specific regions of reciprocal space (e.g., diffuse peaks, rods, or lobes) are isolated using phase-preserving masks ( $W_F(\mathbf{Q})$ ). The masked amplitude is then inverse Fourier transformed to generate real-space fields.
Mode Separation: The framework employs two distinct modes to disentangle chemical and displacive contributions:
- Chemical Mode: A reference configuration is constructed where all atoms are placed at their average crystallographic positions, but the chemical species distribution is retained. The diffuse amplitude calculated from this "Chemical Mode" configuration isolates scattering arising solely from occupancy variations (chemical ordering), suppressing displacement effects.
- Displacement Mode: This mode targets the displacement-sensitive part of the diffuse amplitude. Because the inverse-transformed field is a convolution of discrete displacements with the mask's point-spread function, the authors employ a linear "M-decoder." This data-driven linear operator maps local patches of the filtered real-space field to site-resolved displacement vectors. The decoder is trained on paired data (filtered field vs. known true displacement) for the specific configuration and then applied uniformly.
Scalability: To handle large atomic ensembles (millions of atoms) and extensive reciprocal-space regions, the implementation utilizes a Map–Reduce architecture and Non-Uniform Fast Fourier Transforms (NUFFTs). This avoids the memory bottlenecks of global grid-based FFTs and allows for efficient processing of arbitrary Q-point meshes.

Key Results and Validation
The paper validates the MOSAIC framework through simulations and applications to experimental models:

Chemical Ordering Separation: In a simulated $\gamma$ -LiFeO $_2$ system containing coexisting short-range order (SRO) and nanoscale L1 $_1$ ordering, MOSAIC successfully separated the two effects. By applying masks to specific reciprocal-space regions (the L-point reflections vs. the continuous diffuse surface), the method reconstructed distinct real-space maps: one highlighting the L1 $_1$ nanoclusters and the other revealing the matrix SRO, despite both contributing to the same overall scattering pattern.
Disentangling Distortions: In a simulated ReO $_3$ -type perovskite with coexisting octahedral rotations and breathing distortions, the framework separated the overlapping diffuse features. Cylindrical masks isolated the rotational disorder (diffuse rods), while spherical masks isolated the breathing mode (superlattice peaks). The linear additivity of the method was confirmed, as the sum of the separated displacement fields reconstructed the original total distortion field with negligible residual error.
Experimental Application: The framework was applied to RMC-refined models of relaxor ferroelectrics PbMg $_{1/3}$ $_{1/3}$ Nb $_{2/3}$ $_{2/3}$ O $_3$ $_{3}$ (PMN) and PMN–PbTiO $_3$ $_{3}$ (PMN-PT).
- Chemical: The Chemical Mode revealed continuous variations in the degree of rocksalt-type ordering and identified anti-phase boundaries, providing a site-resolved view of chemical environments that was obscured in the raw models.
- Displacement: The Displacement Mode filtered out uncorrelated noise from Pb displacement fields, revealing nanoscale polar textures and swirling polarization patterns consistent with previous observations.

Significance and Claims
The authors claim that MOSAIC provides a robust, quantitative method for attributing specific diffuse-scattering features to underlying atomic-site-resolved occupancy and displacement fields. Key contributions include:

Quantitative Decoupling: The ability to separate chemical and displacive correlations that produce overlapping scattering features, a task difficult with traditional analysis.
Linearity and Additivity: The framework ensures strict linearity, meaning that contributions from disjoint reciprocal-space regions add exactly in real space. This allows for the rigorous decomposition of complex scattering patterns into constituent structural motifs.
Scalability: The use of NUFFTs and Map–Reduce processing enables the analysis of large-scale structural models (e.g., from RMC or MD) that were previously computationally prohibitive for such detailed feature filtering.
Versatility: The method is applicable to both 3D atomic configurations derived from scattering data or simulations and 2D structural projections from atomic-resolution electron microscopy.

The paper emphasizes that MOSAIC is a tool for analyzing known configurations to understand the origins of specific scattering features, rather than a direct inversion method for solving structures from intensity-only data. It offers a pathway to quantitatively link local structural motifs to macroscopic functional responses in complex materials.

The Magic Filter

What MOSAIC Can (and Can't) Do

Real-World Tests

The Scale of the Operation

The Bottom Line

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