4D Electrocatalytic Atomic-Resolution Tomography

• The paper introduces a framework that repeatedly reconstructs the same electrocatalyst in 3D across electrochemical cycles, achieving R-factors between 5–9%.
• It integrates atomic electron tomography, 4D-STEM, and multislice inversion to precisely track atomic migration, chemical identity, and structural evolution.
• This technique yields quantitative insights into nanoscale phenomena such as surface reconstruction, atom leaching, and strain mapping in Pd–Pt nanoparticles.

Searching arXiv for the named topic and key supporting methods. Four-dimensional electrocatalytic atomic-resolution electron tomography denotes a framework in which the full three-dimensional atomic structure and chemical identities of an individual electrocatalyst are reconstructed repeatedly through an electrocatalytic life cycle, so that structure is resolved in three spatial dimensions and followed along a fourth dimension given by time or electrochemical state (Xie et al., 22 Sep 2025). In the current literature, the term is most explicitly instantiated by tracking identical Pd–Pt nanoparticles through electrocatalytic cycles, while its broader methodological basis draws on atomic electron tomography, 4D-STEM, multislice inversion, open environmental TEM, and dose-efficient multimodal imaging (Xie et al., 22 Sep 2025).

1. Definition and conceptual scope

In this context, “four-dimensional” refers to three-dimensional space plus a temporal or state variable. One formulation used for atomic electron tomography is a sequence of maps

$\rho(\mathbf{r}, t_k),\quad \mathbf{r} = (x,y,z),$

where each $t_k$ denotes a specified stage such as 0, 100, 500, 1500, or 2500 electrocatalytic cycles, and each map is converted into atomic coordinates and chemical identities for direct comparison across time (Xie et al., 22 Sep 2025). A closely related generalization, proposed from the perspective of multiple-section local-orbital tomography, is to represent the object as $x(\mathbf{r},t)$ or $x(\mathbf{r},\lambda)$, where the fourth dimension is time $t$ or electrochemical state $\lambda$, and to reconstruct coupled 3D states with temporal or state regularization (Mao et al., 2024).

The phrase also sits at the intersection of several distinct uses of “4D” in electron microscopy. In 4D-STEM nanobeam diffraction, the four dimensions are two scan coordinates and two reciprocal-space coordinates, yielding datasets of the form $I(x,y,k_x,k_y)$ (Mukherjee et al., 2020). In 4D-STEM focal-series reconstruction, the experimental input can be written as $I(\mathbf{g},\mathbf{R},\Delta f)$, with reciprocal-space coordinates $\mathbf{g}$, scan coordinates $\mathbf{R}$, and defocus $t_k$0, and can be used for single-projection three-dimensional reconstruction (Brown et al., 2020). These usages are methodologically distinct from 3D-plus-time electrocatalytic tomography, but they provide technical routes toward it.

A persistent misconception is that any four-dimensional electron dataset is automatically four-dimensional tomography. The literature distinguishes between 4D-STEM characterization, 3D atomic electron tomography, and true 3D-plus-time tracking of the same catalyst. The strain-mapping work on Rh@Pt nanocubes, for example, explicitly does not perform tomography in the strict sense, even though it is a genuine four-dimensional electron characterization method (Mukherjee et al., 2020). By contrast, the Pd–Pt electrocatalytic study reconstructs the same nanoparticle repeatedly and is explicitly presented as four-dimensional electrocatalytic atomic-resolution electron tomography (Xie et al., 22 Sep 2025).

2. Historical and methodological foundations

Atomic-resolution electron tomography emerged from STEM-based reconstruction strategies that overcame the projection-alignment barrier at atomic scale. A key early milestone was the demonstration of a general electron tomography method that achieved $t_k$1 resolution on a $t_k$2 gold nanoparticle by combining annular dark-field STEM, center-of-mass alignment, and equally sloped tomography (Scott et al., 2011). That work established that 3D lattice structure, grain morphology, and twin boundaries could be recovered without assuming a rigid lattice, but it also exposed the practical limits imposed by missing wedge, thickness, non-linear scattering, and dose (Scott et al., 2011).

Subsequent work addressed the under-determined nature of atomic reconstruction from sparse data. “Atomic Super-Resolution Tomography” formulated a grid-free discrete tomography problem in which atoms occupy continuous positions rather than fixed lattice sites, and augmented the data term with an atomic interaction potential such as a Lennard–Jones prior (Ganguly et al., 2020). This was directly relevant to defect-rich catalyst particles because it allowed continuous deviations from ideal lattice positions, vacancies, interstitials, and dislocations, rather than forcing atoms onto a coarse grid (Ganguly et al., 2020).

Another major line of development addressed missing-wedge artifacts and dose limits with learning-based reconstruction. Deep-learning-aided information recovery was reported to improve tomographic resolution to $t_k$3 by recovering Fourier components up to the $t_k$4 Bragg spots in a gold crystal, and to make reconstruction possible from $t_k$5 to $t_k$6 data with a stated 44% reduction of dosage compared to a $t_k$7 to $t_k$8 full tilt series (Wang et al., 2020). This suggests a route to temporally repeated, lower-dose acquisitions, although the same work also makes clear that reconstruction quality depends strongly on the similarity between training data and the target structures (Wang et al., 2020).

A separate foundation concerns the depth-of-field problem in aberration-corrected STEM. Multiple-section local-orbital tomography, or nLOT, combines electron tomography with depth sectioning by acquiring multiple defocus images at each tilt angle and modeling section-specific, depth-dependent probes (Mao et al., 2024). In the reported implementation, single-section tomography used a tilt range of $t_k$9 to $x(\mathbf{r},t)$0 with 2° steps, whereas nLOT used the same tilt range with a coarsened 6° step and three defocus images at each tilt, $x(\mathbf{r},t)$1, while matching the total number of images and dose budget (Mao et al., 2024). For a 1,000,000-atom model, nLOT achieved RMSD $x(\mathbf{r},t)$2, with an RMSD slope of $x(\mathbf{r},t)$3, compared with $x(\mathbf{r},t)$4 for adaptive DDI and $x(\mathbf{r},t)$5 for direct projection (Mao et al., 2024). This establishes a scalable 3D basis for four-dimensional extensions.

3. Core reconstruction frameworks

Current four-dimensional electrocatalytic atomic-resolution tomography is built on iterative inverse problems that jointly model imaging physics and atomic structure. In repeated atomic electron tomography of electrocatalysts, a tilt series of ADF-STEM projections is reconstructed into a 3D volume with the Real Space Iterative Reconstruction algorithm, RESIRE, by minimizing projection mismatch, often written conceptually as

$x(\mathbf{r},t)$6

where $x(\mathbf{r},t)$7 is the forward projection operator and $x(\mathbf{r},t)$8 the measured image at tilt $x(\mathbf{r},t)$9 (Xie et al., 22 Sep 2025). In the Pd–Pt study, the final R-factors ranged from about 5–9%, indicating good agreement between reconstructed and measured projections (Xie et al., 22 Sep 2025).

The atomic models are then refined by locating local maxima in the 3D volume using polynomial fitting within a $x(\mathbf{r},\lambda)$0 voxel box, rejecting peaks closer than $x(\mathbf{r},\lambda)$1, and assigning Pd- or Pt-like identities from integrated intensities combined with K-means clustering and local re-classification within a sphere of radius $x(\mathbf{r},\lambda)$2 (Xie et al., 22 Sep 2025). This produces a list of atomic coordinates and species $x(\mathbf{r},\lambda)$3, with $x(\mathbf{r},\lambda)$4, that can be matched across time points by pairing atoms whose positions differ by less than half the first valley of the pair distribution function (Xie et al., 22 Sep 2025).

From the nLOT perspective, the same problem can be written more generally as a nonlinear least-squares optimization over atomic coordinates, intensities, widths, and section-specific probes: $x(\mathbf{r},\lambda)$5 and its 4D generalization can be written

$x(\mathbf{r},\lambda)$6

where $x(\mathbf{r},\lambda)$7 encodes temporal or state regularization (Mao et al., 2024). This suggests a mathematically natural extension from static 3D reconstruction to coupled 3D reconstructions across electrocatalytic states.

A complementary route uses 4D-STEM rather than scalar ADF projections. MultiSlice Electron Tomography, MSET, formulates 3D reconstruction from 4D-STEM tilt series as a multislice inverse problem based on the paraxial Schrödinger equation

$x(\mathbf{r},\lambda)$8

with a 3D potential $x(\mathbf{r},\lambda)$9 recovered by minimizing

$t$0

Because the full diffraction pattern is retained at each scan position and tilt, MSET directly models multiple scattering and avoids the linear projection approximation that limits conventional ADF-STEM tomography in thick or strongly scattering materials (Lee et al., 2022).

Single-projection 3D reconstruction from a focal series of 4D-STEM data takes yet another form. There the raw data are written as $t$1, and a scattering matrix $t$2 is recovered by iterative amplitude flow from intensities

$t$3

followed by optical sectioning of $t$4 to recover depth-dependent structure from a single plan-view orientation (Brown et al., 2020). This is not electrocatalytic tomography in itself, but it offers an alternative strategy where tilting is impractical.

4. Electrocatalytic implementation in Pd–Pt nanoparticles

The most explicit realization of four-dimensional electrocatalytic atomic-resolution electron tomography to date uses Pd–Pt bimetallic nanoparticles as a model system for ethanol oxidation in alkaline medium (Xie et al., 22 Sep 2025). The same gold TEM grids serve as both TEM supports and working electrodes in a three-electrode electrochemical setup, enabling interleaved electrochemical cycling and tomography on the same particles (Xie et al., 22 Sep 2025). Electrochemical durability testing was performed by cyclic voltammetry from 0.05 to 1.1 V vs RHE at $t$5 in Ar-saturated 1 M ethanol / 1 M KOH, and different particles were examined after 0, 100, 200, 500, 1500, or 2500 cycles (Xie et al., 22 Sep 2025).

All four-dimensional electrocatalytic atomic electron tomography datasets were acquired on an aberration-corrected FEI Themis Z in ADF-STEM mode at 300 kV, with a convergence semi-angle of 25 mrad, HAADF inner and outer detector angles of 40.6 mrad and 200 mrad, a pixel size of $t$6, and three frames per tilt angle with $t$7 dwell time (Xie et al., 22 Sep 2025). Each tilt series used a dose rate of $t$8–$t$9, reported as less than 40% of the dose used in earlier 4D-AET heating experiments (Xie et al., 22 Sep 2025). Tilt ranges were approximately $\lambda$0 to $\lambda$1, $\lambda$2, or $\lambda$3 depending on particle and stage, with oversampling ratio 4 and 300 RESIRE iterations (Xie et al., 22 Sep 2025).

The central experimental advance is relocation and repeated reconstruction of identical nanoparticles after electrochemical cycling. This is achieved by recording low-magnification ADF-STEM maps of the local CNT network and using the morphology and orientation of surrounding CNTs as a fingerprint for relocation after each electrochemical run (Xie et al., 22 Sep 2025). This allows atom-by-atom comparison of common positions, disappeared positions, and new positions across states, and separates atoms that remain chemically consistent from atoms that change identity at common positions (Xie et al., 22 Sep 2025).

Several quantitative findings define the method’s operating regime. For particle 1 from 0 to 0.5K cycles, 96.4% of atoms were paired as common positions, and 89.4% of those common positions had consistent species (Xie et al., 22 Sep 2025). For particle 1 from 0.5K to 1.5K cycles, only 49.2% of atoms in Pd@Pt1_0.5K were paired as common positions, and only 36.4% remained consistent in species (Xie et al., 22 Sep 2025). For particle 2, 92% of atoms were consistent between 0 and 0.1K cycles, close to the intrinsic consistency limit, indicating minimal structural change at 100 cycles (Xie et al., 22 Sep 2025).

The study also used separate in situ electrochemical liquid-cell TEM to visualize dynamic behaviors under potential control. Under chronoamperometry at $\lambda$4, transient “liquid-like” clusters appeared near particles, adhered, diffused along particle corners, and disappeared into the electrolyte, whereas no such changes were observed at $\lambda$5 or at zero bias (Xie et al., 22 Sep 2025). This provides an in situ 2D correlate for the atom-leaching stage identified ex situ by 4D tomography.

5. Quantitative observables: migration, order, strain, and chemistry

The four-dimensional framework is valuable because it yields atomically defined observables rather than only projected morphology. One primary observable is atom migration relative to the surface. Distances from atoms to the surface are computed from an alpha-shape surface mesh, and atoms are binned into shells from 0–5 Å up to 20 Å (Xie et al., 22 Sep 2025). In the surface reconstruction stage for particle 1 from 0 to 0.5K cycles, unpaired atoms were predominantly Pd, located within about 2 Å of the surface, and the coordination number distribution shifted from about 6 to about 8, indicating motion from edges and corners toward flatter $\lambda$6-like sites (Xie et al., 22 Sep 2025). In the atom-leaching stage from 0.5K to 1.5K cycles, disappeared atoms extended up to 20 Å from the surface, and more than 4,000 Pd atoms with CN = 12 disappeared, indicating deep leaching into the bulk (Xie et al., 22 Sep 2025).

Another key observable is local chemical environment and short-range order. The local Pd concentration around heterogeneous Pt centers is defined as

$\lambda$7

where the coordination shell extends to the fourth neighbor shell, corresponding to about two atomic layers (Xie et al., 22 Sep 2025). The pairwise multicomponent short-range-order parameter is written as

$\lambda$8

with $\lambda$9 in the electrocatalytic Pd–Pt analysis, where $I(x,y,k_x,k_y)$0 is the probability of finding a $I(x,y,k_x,k_y)$1-type atom around an $I(x,y,k_x,k_y)$2-type atom in shell $I(x,y,k_x,k_y)$3, $I(x,y,k_x,k_y)$4 is the global concentration of $I(x,y,k_x,k_y)$5, and $I(x,y,k_x,k_y)$6 the Kronecker delta (Xie et al., 22 Sep 2025). Positive $I(x,y,k_x,k_y)$7 or positive $I(x,y,k_x,k_y)$8 indicate Pd-abundant local environments (Xie et al., 22 Sep 2025). Orientation-resolved CSROP analysis showed that $I(x,y,k_x,k_y)$9 regions in particle 1 evolved much more strongly than $I(\mathbf{g},\mathbf{R},\Delta f)$0 regions between 0.5K and 1.5K cycles, whereas particle 2 showed negligible CSROP change up to 100 cycles (Xie et al., 22 Sep 2025).

A related but distinct four-dimensional characterization is 4D-STEM nanobeam diffraction for strain metrology in Rh@Pt core@shell nanocubes. There the dataset is $I(\mathbf{g},\mathbf{R},\Delta f)$1, acquired with a reduced convergence semi-angle of about 5 mrad so that diffraction disks remain separated (Mukherjee et al., 2020). By logarithmic transformation, Sobel filtering, cross-correlation with a disk-edge template, and 2D Gaussian fitting of the correlation peaks, local Bragg disk positions are measured with sub-pixel precision, enabling determination of in-plane strain components $I(\mathbf{g},\mathbf{R},\Delta f)$2 (Mukherjee et al., 2020). Cross-validation on preconditioned 4D datasets yielded strain accuracy of about 0.07%, corresponding to sub-picometer precision in lattice parameter for typical fcc metals (Mukherjee et al., 2020). This is not tomography in the strict sense, but it demonstrates that the “fourth dimension” in catalyst electron microscopy can also encode local reciprocal-space information, and it provides a path toward 3D strain mapping when combined with tilt-series approaches.

Chemical information can also be fused across modalities under dose constraints. Fused multi-modal electron microscopy jointly uses HAADF-STEM and EELS or EDX by solving

$I(\mathbf{g},\mathbf{R},\Delta f)$3

where $I(\mathbf{g},\mathbf{R},\Delta f)$4 is the HAADF image, $I(\mathbf{g},\mathbf{R},\Delta f)$5 the spectroscopic map for element $I(\mathbf{g},\mathbf{R},\Delta f)$6, $I(\mathbf{g},\mathbf{R},\Delta f)$7 the unknown chemical distribution, and $I(\mathbf{g},\mathbf{R},\Delta f)$8 typically about 1.7 (Schwartz et al., 2022). The reported dose reduction can exceed one order of magnitude, and quantitative recovery was shown down to $I(\mathbf{g},\mathbf{R},\Delta f)$9 in CoS nanoparticles (Schwartz et al., 2022). A plausible implication is that future electrocatalytic 4D tomography will rely increasingly on multimodal inversion when full spectroscopic tilt series are dose-limited.

6. Reactive environments, scalability, and future directions

Four-dimensional electrocatalytic atomic-resolution tomography depends not only on reconstruction algorithms but also on environmental electron-optical stability. Open gas-cell TEM at 300 keV has been reported to maintain 50 pm resolution at pressures up to 1 mbar in $\mathbf{g}$0, using a four-stage differential pumping system, a 5th order aberration corrector, a monochromatized beam with $\mathbf{g}$1, Nelsonian low electron dose-rate illumination, and direct electron detection (Biran et al., 6 Aug 2025). In reciprocal space, Young fringe experiments extended to about $\mathbf{g}$2 at both 0 mbar and 1 mbar $\mathbf{g}$3, and exit-wave reconstructions used 47 images at 0 mbar and 63 images at 1 mbar from a 100-image focal series with 1 nm defocus step and $\mathbf{g}$4 pixels (Biran et al., 6 Aug 2025). This does not yet constitute 3D-plus-time electrocatalytic tomography in liquid, but it establishes that object-limited, single-atom-sensitive imaging and long optical lifetimes can coexist in a reactive environment (Biran et al., 6 Aug 2025).

Scalability in 3D has improved dramatically. nLOT extends atomic electron tomography to a million atoms with high positional accuracy by combining tomography and depth sectioning, and it reports no theoretical upper bound on object size, only practical computing-resource limits (Mao et al., 2024). MSET demonstrates atomic-resolution 3D reconstruction from 4D-STEM tilt series with improved sensitivity for low-$\mathbf{g}$5 elements and considerably lower electron dose than ADF-STEM-based tomography (Lee et al., 2022). A plausible implication is that four-dimensional electrocatalytic tomography will develop by combining nLOT-like multi-section acquisition, MSET-like multislice inversion, and strong temporal or state regularization.

The field has already moved beyond catalytic cycling in Pd–Pt and thermal evolution in Pd–Ir. Tracking atomic-scale interdiffusion in immiscible PdIr nanoparticles by ex situ four-dimensional atomic-resolution electron tomography combined with in situ STEM revealed surface reconstruction atom hopping at 200°C, surface flattening at 300°C, a critical transition at 400°C with interfacial diffusion coefficient

$\mathbf{g}$6

and collective inward Ir diffusion near a nanoscale melting point of about 900°C (Xie et al., 10 Jun 2026). Although thermal and electrocatalytic trajectories are not identical, this work shows that repeated atomic reconstruction of the same nanoparticle can resolve discrete intermediates and non-bulk phase behavior in four dimensions (Xie et al., 10 Jun 2026).

Important limitations remain. Temporal resolution is still discrete rather than continuous, because each full tilt series is time-consuming and dose-intensive (Xie et al., 22 Sep 2025). Missing wedge and anisotropic resolution remain intrinsic to limited-angle tomography (Scott et al., 2011). In liquid or electrochemical cells, beam-induced radiolysis, drift, and window scattering complicate both data acquisition and forward modeling (Biran et al., 6 Aug 2025). Deep-learning recovery can improve resolution and dose efficiency, but it depends strongly on training-set representativeness and can mis-handle structures outside the training distribution (Wang et al., 2020). For these reasons, present four-dimensional electrocatalytic tomography is best understood as a rapidly advancing synthesis of atomic electron tomography, 4D-STEM, environmental microscopy, and dose-efficient multimodal inversion, rather than as a single settled technique (Mao et al., 2024).

A broader misconception is that the central challenge is merely achieving higher nominal spatial resolution. The literature suggests instead that the defining difficulty is simultaneous control of four constraints: atomic precision in 3D, chemically meaningful species assignment, sufficient dose efficiency for repeated acquisition, and enough environmental realism to preserve electrocatalytic relevance (Xie et al., 22 Sep 2025). In that sense, four-dimensional electrocatalytic atomic-resolution electron tomography is not simply an extension of static atomic tomography by adding time; it is a coordinated inversion problem in which imaging physics, catalyst dynamics, and electrochemical state must all be modeled together.