STEM Pair Distribution Function Mapping

Electron pair distribution function analysis (ePDF) in transmission electron microscopy, particularly when combined with 4D-STEM acquisition, provides a powerful framework for probing local atomic structure at the nanometer scale. Unlike X-ray or neutron PDF techniques that average over bulk volumes, STEM-ePDF exploits the strong interaction between electrons and matter to extract pair-correlation information from extremely small sampling regions. This enables spatially resolved mapping of structural heterogeneity in amorphous and nanostructured materials with nanometer resolution.

The ePDF workflow begins with electron diffraction patterns that are detector-corrected and azimuthally integrated to produce radial intensity profiles I(k), where k is the electron scattering vector. After normalization by total elastic scattering factors, the structure function S(k) is obtained. Its oscillatory component is Fourier sine transformed to yield the real-space PDF G(r), which describes deviations of local atomic density as a function of interatomic distance. A central challenge in ePDF is strong multiple elastic scattering, which remains significant even in thin amorphous foils. These effects introduce thickness-dependent diffuse backgrounds, artificial low-r intensity, systematic shifts in nearest-neighbor peak positions, and attenuation of medium-range order oscillations.

Figure 1. 4D-STEM PDF mapping of nanoglass multilayers. (a) Schematic of 4D-STEM PDF acquisition: a nanobeam is scanned across the sample, a diffraction pattern recorded at each probe position, and the resulting stack is converted to a three dimensional PDF cube with two real-space coordinates (x, y) and an interatomic distance axis r. (b) Example PDFs from a ZrFe-based nanoglass multilayer, showing pair correlations (FeZr & ZrO2 layers and Fe3O4 interface) that distinguish the local atomic packing in different amorphous environments. (f) PDF based phase mapping of a ZrFe/Fe3O4/ZrO2 multilayer: HADDF image (left) and coefficient maps for the different amorphous phases (center) with an RGB composite (right) that visualizes their nanoscale spatial distribution. (d–f) Principal component analysis (PCA) of the PDF cube: scree plot of eigenvalues (d), first three principal component PDFs (e), and their spatial maps (f), which already reflect the presence of three structurally distinct regions. (g) Independent component PDFs obtained by applying independent component analysis (ICA) to the PCA results, separating FeZr, ZrO₂, and an additional interfacial phase. (h) Comparison of the third independent component with a simulated PDF and partial PDFs of Fe₃O₄, identifying Fe–O and Fe–Fe correlations characteristic of an iron oxide interlayer. (i) Spatial maps of the ICA coefficients, revealing the distribution of FeZr, ZrO₂, and Fe₃O₄-like interfacial regions and their correspondence with the virtual ADF image. Figures (a–d) adapted from Mu et al. 1.  Figures adapted from Mu et al. 6

To correct for these artifacts, analytical deconvolution is combined with low-order polynomial baseline subtraction. The deconvolution framework models the measured diffraction signal as a Poisson-weighted convolution series and recovers a distribution dominated by single scattering. Subsequent polynomial background correction removes smooth residual curvature without distorting genuine structural oscillations. When properly applied, this combined treatment yields thickness-independent peak positions and substantially stabilizes peak height and width, enabling quantitative interpretation of interatomic distances. The fidelity of ePDF further depends on electron-optical conditions: moderate convergence angles preserve angular resolution, while adequate detector sampling and an optimized k-range prevent truncation and aliasing artifacts.

Figure 2. STEM-PDF and ICA mapping of local atomic structure around a shear band of a Vitreloy 105 metallic glass. (a) STEM-HAADF image of a deformed section, indicating the region used for 4D-STEM PDF acquisition and the position of the shear band. (b) Independent-component PDFs IC1 and IC2 obtained by ICA of the STEM-PDF cube, corresponding to Zr-rich (IC1) and Zr-poor (IC2, Cu/Ni/Ti/Al-rich) local configurations. (c, d) Spatial weight maps of IC1 and IC2, revealing a sharp structural contrast across the shear band (yellow dashed line). (e) Line profile of the IC weight fractions and fitting error along the direction marked in (a), showing antisymmetric Zr-rich / Zr-poor SBAZs on either side of the shear band and a narrow region of enhanced misfit at the shear band core. (f) PDF line scan perpendicular to the shear band extracted from the STEM-PDF cube. (g) Magnified view of the line scan highlighting changes in the Zr–Zr nearest-neighbor distance and the second-shell feature associated with face-sharing tetrahedra; right-hand traces show intensity profiles in the Zr-depleted SBAZ, shear band and Zr-enriched SBAZ. (h) PDFs averaged over these three regions, illustrating the progressive change in nearest-neighbor correlations and the abrupt reduction of the face-sharing contribution in the shear band. (i) Map of the population of face-sharing tetrahedral motifs derived from the PDF amplitude at √8/3 RNN, showing depletion of geometrically favored motifs along the shear band. Panels adapted from Mu et al.Integration with 4D-STEM extends ePDF from point analysis to spatially resolved structural mapping. A nanometer-sized probe is rastered across the specimen while a diffraction pattern is recorded at each position, generating a four-dimensional dataset. Each pattern is converted into a local PDF, and stacking these results produces a three-dimensional PDF cube (x, y, r). This approach enables direct visualization of nanoscale phase separation, interfacial structure, and local packing variations that are inaccessible to conventional imaging. Multivariate statistical tools such as principal component analysis (PCA) and independent component analysis (ICA) can separate overlapping amorphous environments, revealing hidden nanophases and their bonding geometry even when projected in the same imaging volume.

Machine-learning approaches further enhance interpretation. Nonnegative matrix factorization (NMF) decomposes the PDF cube into physically meaningful structural bases that correspond to distinct local atomic configurations. In metallic glasses, these bases can be interpreted as “liquid-like” and “solid-like” motifs, allowing quantitative tracking of structural population changes during thermal processing. Such analysis demonstrates that annealing redistributes existing configurations rather than creating new motifs, providing direct nanoscale insight into glass relaxation mechanisms.

Figure 3. Large-angle Lorentz 4D-STEM mapping of magnetic field, strain, density and local structure around shear bands in a metallic glass. (a) ADF-STEM image of a lamella showing two SBs; the green frame marks the LA-Ltz-4D-STEM scan area. (b) Map of the in-plane magnetic field Bobtained from the shift of the first amorphous diffraction ring; the color encodes the field direction and brightness its magnitude. (c) Map of the compressive principal strain εcom; the color code indicates the strain direction. (d) Relative density map Δρ derived from the integrated intensity of the first diffraction ring, highlighting local dilatation and compaction around the shear bands. (e) Combined view of magnetic and strain fields: white arrows show Band colored sticks εcom, visualizing strong magnetoelastic coupling in the highly strained regions. (f, g) Representative 2D histogram of the tensile strain field and corresponding binarized map, illustrating two preferred strain orientations set by the shear-band geometry. (h, i) Analogous 2D histogram and binarized map for the magnetic field, revealing domains that follow the strain-induced anisotropy and domains dominated by magnetostatic interactions. (j) PDF line scan perpendicular to one shear band, marking the left and right SBAZs and the band core. (k) PDFs averaged over the left and right SBAZs, showing subtle but systematic differences in nearest-neighbor distances and peak amplitudes that reflect asymmetric atomic packing across the shear band. Panels adapted from Kang et al9.

Advanced STEM-PDF applications include the study of shear bands in metallic glasses, where spatially resolved PDFs reveal asymmetric structural changes, compositional segregation, and the breakdown of geometrically favored packing motifs. Extensions of large-angle Lorentz 4D-STEM further enable simultaneous mapping of strain, density, and magnetic fields from the same dataset. By correlating functional fields with local pair correlations, this multimodal framework links atomic-scale packing to magnetoelastic behavior and deformation physics.

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More details on this work have been published at:

  1. X. Mu, A. Mazilkin, C. Sprau, A. Colsmann and C. Kübel: Mapping structure and morphology of amorphous organic thin films by 4D-STEM pair distribution function analysis. Microscopy 68, 301 (2019).
  2. X. Mu, S. Neelamraju, W. Sigle, C.T. Koch, N. Toto, J.C. Schön, A. Bach, D. Fischer, M. Jansen and P.A. van Aken: Evolution of order in amorphous-to-crystalline phase transformation of MgF2. Applied Crystallography 46, 1105 (2013).
  3. G. Anstis, Z. Liu and M. Lake: Investigation of amorphous materials by electron diffraction—The effects of multiple scattering. Ultramicroscopy 26, 65 (1988).
  4. J. Ankele, J. Mayer, P. Lamparter and S. Steeb: Quantitative electron diffraction data of amorphous materials. Zeitschrift für Naturforschung A 60, 459 (2005).
  5. S. Kang, H. Cho, M. Töllner, V. Wollersen, D. Wang, H. Baik, M.J. Perera, O. Adjaoud, K. Albe and C. Kübel: Validating Electron Pair Distribution Function Analysis: The Role of Multiple Scattering, Beam, Measurement, and Processing Parameters. Ultramicroscopy, 114295 (2025).
  6. X. Mu, L. Chen, R. Mikut, H. Hahn and C. Kübel: Unveiling local atomic bonding and packing of amorphous nanophases via independent component analysis facilitated pair distribution function. Acta Materialia 212, 116932 (2021).
  7. X. Mu, D. Wang, T. Feng and C. Kübel: Radial distribution function imaging by STEM diffraction: Phase mapping and analysis of heterogeneous nanostructured glasses. Ultramicroscopy 168, 1 (2016).
  8. S. Kang, V. Wollersen, C. Minnert, K. Durst, H.-S. Kim, C. Kübel and X. Mu: Mapping local atomic structure of metallic glasses using machine learning aided 4D-STEM. Acta Materialia 263, 119495 (2024).
  9. S. Kang, M. Töllner, D. Wang, C. Minnert, K. Durst, A. Caron, R.E. Dunin-Borkowski, J. McCord, C. Kübel and X. Mu: Large-angle Lorentz Four-dimensional scanning transmission electron microscopy for simultaneous local magnetization, strain and structure mapping. Nature communications 16, 1305 (2025).