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Oxygen-diffusion-driven oxidation behavior and tracking areas visualized by X-ray spectro-ptychography with unsupervised learning

Abstract

Oxygen storage and release with oxygen diffusion in the bulk of the cerium–zirconium solid solution oxide Ce2Zr2Ox (x = 7–8), which possesses an atomically ordered arrangement of cerium and zirconium atoms, is the key to three-way exhaust catalysis. Oxygen storage proceeds via heterogeneous oxygen diffusion into the vacant sites of Ce2Zr2O7 particles, but the heterogeneous oxygen diffusion track is erased after oxygen storage in the Ce2Zr2Ox bulk. Here we show three-dimensional hard X-ray spectro-ptychography to clearly visualize the three-dimensional cerium valence map in Ce2Zr2Ox particles, and unsupervised learning reveals the concealed oxygen-diffusion-driven three-dimensional nanoscale cerium oxidation behavior and tracking areas inside individual mixed-oxide particles during the oxygen storage process. The described approach may permit the nanoscale chemical imaging of reaction tracking areas in solid materials.

Introduction

Three-way exhaust catalysis is a key reaction in automobile systems1,2, and cerium-containing mixed oxides have been widely used as support for exhaust catalysts. The oxygen storage and release capacity (OSC) of cerium-containing mixed oxides associated with the reversible oxidation and reduction of Ce3+ and Ce4+ ions enables widening of the operation window of three-way catalysts3,4,5. In particular, Ce2Zr2Ox (denoted CZ-x, where x = 7–8) solid solutions with an ordered arrangement of Ce and Zr atoms exhibit remarkable OSCs6,7. The dynamic structural changes of CZ-x compounds with oxygen diffusion in the bulk during the redox reaction have been investigated using X-ray diffraction8,9, neutron diffraction10, time-resolved X-ray absorption fine structure (XAFS)11, and theoretical calculations12,13. However, the reversible oxygen storage and release processes erase the oxygen diffusion track in the bulk of the CZ-x particles, and consequently the details of the oxygen storage pathways in the CZ-x particles remain unclear.

Visualization of the heterogeneity of solid catalyst particles is a state-of-the-art technology made possible by the rapid progress of X-ray imaging techniques. In particular, the combination of lens-based X-ray microscopy and XAFS spectroscopy to provide chemical state information, which makes it possible to resolve both the chemical distribution and particle structure of thick specimens with less radiation damage than that resulting from transmission electron microscopy with electron energy loss spectroscopy, is breaking new ground in materials science14,15,16. For example, scanning fluorescence/transmission X-ray microscopy has permitted two-dimensional (2D) imaging of oxygen diffusion in individual catalyst particles during oxygen storage and release17 and captured the genesis of an active Fischer–Tropsch synthesis catalyst18. Full-field transmission X-ray microscopy has also revealed the three-dimensional (3D) structure and elemental distribution of an entire individual fluid catalytic cracking particle19. Lens-based X-ray microscopy, however, suffers from limited spatial resolution owing to the fabrication accuracy of the X-ray lens20.

X-ray spectro-ptychography (XSP)21, a combination of XAFS spectroscopy and X-ray ptychography22, is a promising tool for visualizing both the structures and chemical states of bulk materials at the nanoscale beyond the limitation of the X-ray lens. In XSP, coherent X-ray diffraction patterns are collected around an absorption edge, then both the structural images and XAFS are reconstructed from the diffraction patterns via phase retrieval calculations. In this approach, the spatial resolution is determined by the maximum angular extent of the diffracted X-ray photons. 2D XSP was first demonstrated using soft X-rays21. The nanoscale chemical imaging of thin materials has typically been conducted in the soft X-ray realm, where the incident X-rays are mostly attenuated23,24,25. Recently, 2D XSP in the hard X-ray region has been reported, which provided 2D maps of the Ce density and valence state, suggesting the existence of several 2D domains with different oxygen storage behavior in Pt-supported CZ-x (Pt/CZ-x) catalyst particles26. However, significant ambiguities, especially regarding the 3D oxygen-diffusion-driven Ce oxidation tracking areas in individual CZ-x particles during the heterogeneous oxygen storage process, remain in the 2D images derived from projection of the structure and valence state along the optical axis. Recently, 3D XSP combined with computed tomography (CT) has been demonstrated in the soft X-ray region27,28,29. 3D XSP in the hard X-ray region is a promising method for visualizing the chemical states inside bulk materials such as CZ-x particles, although the huge amount of structural and chemical data in real 3D space visualized by 3D XSP necessitate the efficient extraction of key functional parameters to utilize the obtained data.

This increased volume of structural data, together with advances in machine learning and data mining techniques, provides new opportunities for developing techniques to help researchers accelerate the discovery and comprehension of new materials and phenomena30,31. Supervised learning is the major machine learning method for automatically extracting the properties of materials, including their patterns and physical and chemical principles, from a database of reported information (supervisor), and this method has been utilized to construct models and previously unknown materials with optimized physical properties30,31. There have been numerous reports describing the application of structural imaging based on 3D tomography and supervised learning in the field of medical sciences.

The achievement of visualizing not only structures and but also chemical states of materials by spectro-imaging techniques is opening the way to extracting patterns of material properties and functions via the unsupervised learning of big imaging data32. Nd cluster structures in NdFeB magnets were investigated using 2D transmission X-ray microscopy coupled with XAFS and unsupervised learning32, and different modes of crystal growths in cuprous oxide were observed upon visualization of the inhomogeneous internal electronic structures of Cu2O particles by X-ray free-electron laser imaging with manifold learning analysis33. Furthermore, 3D X-ray coherent imaging of the morphology of Ti alloys was applied to predict fatigue crack growth34. The combination of 3D spectro-imaging techniques such as XSP, which afford not only morphological but also spatial information regarding chemical states, and unsupervised learning is a promising strategy for elucidating physical and chemical properties, which play a critical role in materials science.

Herein, we propose the approach of 3D hard X-ray spectro-ptychography (HXSP) imaging coupled with unsupervised learning to achieve the 3D nanoscale chemical imaging of heterogeneous reaction events in bulk solid materials. The 3D HXSP method allowed realization of 3D nanoscale imaging of the structure and valence state inside individual Pt/CZ-x solid solution particles during the oxygen storage process. Unsupervised data mining of the visualized 3D nanoscale chemical maps then successfully revealed the concealed heterogeneous oxygen-diffusion-driven 3D nanoscale Ce oxidation tracking areas inside the individual mixed-oxide particles during the oxygen storage process.

Results

3D HXSP nanoscale imaging of Ce valence state

Pt/CZ-x particles prepared using our previously reported method17 were dispersed on a 200-nm-thick Si3N4 membrane chip. A sample reduced using H2 at 773 K (Pt/CZ-7) was re-oxidized with O2 at 423 K for 1 h. At this temperature, two different types of oxygen storage behavior can be observed via the metastable phase of Ce2Zr2O7.5 (CZ-7.5)17, and 3D HXSP imaging is regarded to reflect the 3D oxidation reaction tracking of the oxidation (oxygen storage) process.

3D HXSP (5.708–5.770 keV, which includes the Ce LIII-edge) was conducted at the SPring-8 BL29XUL beamline, and a schematic representation of the 3D HXSP measurement is presented in Fig. 1. For the CT measurement, the sample was rotated from −75 to 75° via 61 evenly spaced angles at each X-ray energy. The projected amplitude and phase images at each angle were reconstructed using the ePIE algorithm35 with the Kramers–Kronig relation (KKR) constraint36. Then, 3D image reconstruction was accomplished using the filtered back projection (FBP) algorithm at each X-ray energy. The voxel size of the reconstructed 3D HXSP images in real space was as low as 14 nm, in contrast to the size of single CZ-x particles of ~500 nm to 1 μm, enabling the determination of the 3D oxidation reaction tracking areas in individual catalyst particles during the oxygen storage process.

Fig. 1
figure1

Schematic representation of 3D HXSP. Synchrotron X-rays are monochromatized using a Si(111) double-crystal monochromator. The monochromatic X-rays are two-dimensionally focused using a pair of KB mirrors. A sample placed at the focal plane is laterally scanned across the illumination field. Coherent X-ray diffraction patterns are collected as a function of both the incident X-ray energy and angle. The projected amplitude and phase images at each angle and each energy are reconstructed, followed by 3D image reconstruction

Figure 2a shows the isosurface rendering of the reconstructed 3D HXSP phase images of six CZ-x particles in the field of view. The facet structures on the surfaces of the CZ-x particles can be clearly visualized in the 3D HXSP image. The Ce LIII-edge ptychographic-XAFS spectra (Fig. 2b) allowed discrimination of Ce3+ and Ce4+, whose linear combination fitting at each voxel afforded a 3D map of the Ce oxidation state in (x,y,z) real space (Supplementary Movie 1). The resulting 3D cross-sectional images of the Ce valence are presented in Fig. 2c, showing the heterogeneous variation of the Ce oxidation state inside individual CZ-x particles where oxygen storage partially proceeded via the oxidation of CZ-7 with O2 at 423 K for 1 h. The surfaces of the solid particles appeared blue (Ce4+) and wide color variation inside the solid particles was observed in a heterogeneous manner.

Fig. 2
figure2

3D mapping and XAFS analysis. a Isosurface rendering of the reconstructed 3D phase map of partially oxidized Pt/CZ particles. The scale bar is 700 nm. b Three-dimensionally resolved XAFS spectra and their fitted spectra. The green spectra are the results of the linear combination of the XAFS spectra of Pt/CZ-7 (red) and Pt/CZ-8 (blue) normalized at the isosbestic point of 5.7697 keV. The black dots (i), (ii), and (iii) are the XAFS signals extracted from the 56 × 56 × 56 nm3 volumes indicated at (i), (ii), and (iii) in c. c Series of slices of the 3D Ce valence image along the z direction. The black square represents 700 × 700 nm2. d Ce valence distributions for the number of voxels of the particles labeled in a

Histograms of the Ce valence for each of the six CZ-x particles are presented in Fig. 2d. Particles 5 and 6 exhibited a histogram peak around the Ce valence state of +3.5. Particles 1, 2, and 4 displayed similar histograms with a peak around +3.6 in addition to a peak around +4.0. Particle 3 exhibited a continuous broad histogram with a vague peak around +3.6. Most of the particles shown in Fig. 2d afforded histograms with a plateau around +3.85. There were no voxel domains of Ce valence states lower than ~+3.5 for any of the CZ-x particles 1–6. The results for the 3D mapping analysis of the 14 nm voxel domains in Fig. 2c and d reveal the 3D tracking areas of the metastable phase of CZ-7.5 (Ce2Zr2O7.5) with the unique Ce3.5+ mixed valence8,9 and demonstrate the kinetically faster oxidation of the CZ-7 phase (Ce3+) with oxygen vacancies to the CZ-7.5 phase (Ce3.5+), compared with the subsequent oxidation of the CZ-7.5 phase, during the partial oxygen storage process at 423 K for 1 h. This first report of 3D nanoscale HXSP imaging provides a deeper understanding than the previous micrometer-scale (optical depth) averaged analysis of the Ce density–valence relationship by 2D HXSP26.

Unsupervised learning of the 3D nanoscale valence map

The 3D HXSP imaging results provided a 3D nanoscale Ce valence map of the solid particles with 452 × 450 × 136 voxels. We considered each voxel as a data instance, giving a dataset of 27,662,400 Ce valence data instances for the six CZ-x particles for data mining of the patterns of Ce valence in the particles during the oxygen storage process. The process used to find correlations between parameters in the visualized 3D maps was unsupervised learning.

To characterize each voxel, we considered the surrounding binning of 3 × 3 × 3 voxels (42 × 42 × 42 nm3) in the 3D map and used the local mean (m(x,y,z)) and local standard deviation (sd(x,y,z)) of the Ce valence state in each binning domain as descriptors. Note that m(x,y,z) corresponds to the degree of oxygen storage (Ce3+ → Ce4+) and sd(x,y,z) corresponds to the variation of oxygen storage in the local domains. Using these descriptors, we observed a volcano-type correlation between m(x,y,z) and sd(x,y,z), as shown in Fig. 3a. The distribution in the 2D m(x,y,z)‒sd(x,y,z) plot was approximated by a mixture model of K Gaussian distributions, and an expectation–maximization algorithm37 was adopted for parameter estimation. By minimizing the Bayesian information criterion38, the number of components in the 2D m(x,y,z)‒sd(x,y,z) plot was determined.

Fig. 3
figure3

Unsupervised learning analysis of the 3D Ce valence map. a Scatter plot of mean Ce valence (m) and its standard deviation (sd) for 42 × 42 × 42 nm3 (3 × 3 × 3 voxels) domains of partially oxidized Pt/CZ-x particles, and classification of correlation trends using a Gaussian mixture model. Red: G1; orange: G2; green: G3; blue: G4. The white crosses indicate the Gaussian centers for G1G4; μk (k = 1–4): Gaussian center; Σk: covariance matrix. b 3D rendering of the distribution of the four correlation groups between m and sd for 42 × 42 × 42 nm3 (3 × 3 × 3 voxels) domains of partially oxidized Pt/CZ-x particles, and c series of slices showing the 3D distributions of the four correlation groups along the z direction. The scale bar for b and c is 700 nm. d Plot of the proportion of each correlation group in c with respect to distance from the particle surface

Note that the Gaussian mixture model approximating the 2D plot of (m, sd) revealed that the actual distribution of the entire dataset Ɗ was a mixture of four Gaussian components denoted G1, G2, G3, and G4 with respect to the set of the following centers μ = {(3.43, 0.143), (3.61, 0.265), (3.85, 0.224), and (3.97, 0.063)} and covariance matrices  = {[(0.02766, −0.00078), (−0.00078, 0.00221)], [(−0.02076, −0.00351), (−0.00351, 0.00837)], [(0.00478, −0.00441), (−0.00441, 0.00729)], and [(0.00057, −0.00114), (−0.00114, 0.00252)]}, respectively (Fig. 3a). The main significance of the unsupervised data mining procedure was to determine the most likely number of reaction stages during the oxygen storage process from the visualized 3D map. Taking into account that the horizontal axis m(x,y,z) denotes the degree of oxygen storage, the red component G1 corresponds to the domains where oxygen storage did not proceed for the most part but brought about a distribution of Ce valence states with a maximum probability around the CZ-7.5 phase during the oxygen storage process at 423 K for 1 h. The orange component G2 corresponds to the domains where oxygen storage proceeded beyond the CZ-7.5 phase but displayed a large data variation. The green component G3 corresponds to the domains where oxygen storage converged to the final state of CZ-8 containing Ce4+. The blue component G4 corresponds to the domains where oxygen storage was almost complete and displayed the smallest standard deviation. Thus, the 2D scattering plot of the mean Ce valence (m) and its standard deviation (sd) in the 42 × 42 × 42 nm3 (3 × 3 × 3 voxels) domains of partially oxidized Pt/CZ-x particles exhibited a volcano-type pattern as shown in Fig. 3a, which was related to the course of the oxygen storage process during Ce oxidation from Ce3+ to Ce4+ via the G1, G2, and G3 domains in the CZ-x particles concealed in the bulk. The Ce oxidation in the G1 domains proceeds in the CZ-7 phase with a pyrochlore structure, whereas the Ce oxidation in the G2 domains around +3.61 is regarded to occur in the disordered (mixed) phases accompanied by the transformation of the pyrochlore phase to the CZ-8 κ-phase with a fluorite structure and showing a larger sd. Further oxygen storage proceeds in the Ce oxidation states above +3.7 and forms the G3 domains with a maximum Ce valence population around +3.85. The G3 domains, considered nearly a fluorite phase, readily converge to the Ce4+ valence state in the G4 domains and the final CZ-8 phase.

We classified the locations of the four groups (G1, G2, G3, and G4) in (x,y,z) real space and present a 3D map and cross-sectional images in Fig. 3b, c, respectively. These figures clearly demonstrate that the locations of the four groups were dependent on the morphological characteristics of the particles: the domains belonging to G4 (blue) were observed at the outermost surfaces of the particles, whereas those belonging to G1 (red) were predominantly located in the cores of the particles. Similar differences in the 3D Ce valence images of the particles were observed in Fig. 2c. Recently, we reported five different types of correlations between the Ce density and Ce valence (positive, negative, quasi-constant to Ce density, quasi-constant to Ce valence, and no correlation) in 2D HXSP images26; the G1 group can be related to the positive correlation between Ce density and Ce valence that was observed for Ce valences lower than +3.5 and near the centers of the particles, whereas the G3 and G4 groups can be related to the negative correlation that was observed for Ce valences of +3.5 to +4.0 and around the surfaces of the particles in the 2D images. However, the five different types of oxidation behavior in the local domains of the CZ-x particles described in the previous report26 were determined from the correlations in 2D images averaged over the entire depth direction of the local domains (along the optical axis), where the 2D data for the local domains may be merged with and obfuscated by the data at minor heterogeneous sites of the particles, such as boundaries, defects, and interfaces, although the 2D HXSP image analysis successfully revealed the 2D distribution of Ce oxidation states inside the catalyst particles26. The current 3D HXSP image rendering is the first report of the 3D visualization of the nanoscale oxidation tracking areas in CZ-x catalyst particles during the oxygen storage process.

The geometric distance between each domain and the outermost surface of a particle was calculated from the 3D HXSP image and the fractions for the four groups were plotted against the distance from the surface (Fig. 3d). The G1 domains with limited oxidation did not appear within 20 nm of the outer surfaces of the particles and their fraction greatly increased toward the bulk of the particles. The G2 domains exhibited a maximum proportion at 20 ± 17 nm, which gradually decreased over 200 nm toward the bulk. Considering the stable surfaces of the fluorite/pyrochlore structures (e.g., (111) or (110))39,40, the maximum proportion (20 nm) of G2 corresponded to a depth of ~40–50 oxygen vacancy sites from the surface of the particles. The G3 domains with considerable oxidation were located at the surface regions of the particles and their fraction showed a rapid exponential decrease up to 25 nm followed by a gentle decrease over 100 nm toward the core. The G4 domains with a valence of ~+4.0 were located at the surface layer (<20 nm), although it should be noted that these were a minor component and the surface region was composed of greater fractions of the G3 and G2 domains. These results demonstrate that achieving the complete oxidation of Ce3+ to Ce4+ throughout the entire surface region is difficult during the oxygen storage process at 423 K for 1 h.

The parameter sd(x,y,z) corresponds to the spatial gradient of the Ce valence distribution, which is related to oxygen diffusion in the CZ-x particles. As the oxygen is supplied from the surfaces of the particles, the four groups plotted in Fig. 3a reveal the nature of the oxygen diffusion in the CZ-x particles from the surface to the bulk during the oxygen storage process. During the first stage (G1), there are many vacant sites in the CZ-7 particles with the pyrochlore structure and oxygen can diffuse smoothly through the vacant oxygen sites. Following the formation of a heterogeneous distribution of vacant oxygen sites in the intermediate phase of CZ-7.5 (the second stage, G2), further oxygen storage occurs and is completed through the third stage (G3) to the final stage (G4). Although macroscopic analysis of CZ particles during oxygen storage via temperature-programmed oxidation and thermogravimetry previously suggested a two-step oxidation mechanism for the oxygen storage process via CZ-7.58,9, it is noteworthy that the present static 3D imaging and unsupervised data mining have opened up the possibility of visualizing the nanoscale tracking areas of the reaction course with local heterogeneity in the reactivity inside solid catalyst particles.

Discussion

3D XSP in the hard X-ray region permitted successful visualization of the 3D distribution of the Ce oxidation states in Pt/CZ-x exhaust catalyst particles with a 3D sampling pitch of 14 nm. Data mining of the 3D HXSP images of the Ce valence revealed four groups related to the morphological characteristics and local reactivity for oxygen storage, elucidating the oxidation pathways occurring in the solid catalyst. 3D HXSP is expected to be an indispensable tool for determining reaction tracking areas and the relationships between the structure and function of heterogeneous functional materials. In particular, in next-generation synchrotron facilities where fluxes with much higher coherence will be achieved, the present approach will be applied to in situ 3D measurements, which can be expected to significantly accelerate progress in chemistry and materials science.

Methods

Sample preparation

The sample of Pt/CZ-x particles was prepared by the reported method6,17,41,42. Ce2Zr2O8 particles were obtained by a method reported by Toyota Central R&D Labs. Inc., from aqueous solutions of Ce(NO3)3·6H2O and ZrO(NO3)3·2H2O. 1% of Pt (Pt(NH3)4(NO3)2) was impregnated on Ce2Zr2O8 and calcined at 773 K for 1 h to obtain Pt/CZ-8.

About 3 mg of the prepared Pt/CZ-8 was suspended in 10 mL of absolute ethanol by ultrasonic treatment, then 1 μL of the suspension was dispersed on a 200-nm-thick Si3N4 membrane chip (NORCADA Inc.). The Si3N4 membrane with the Pt/CZ-x particles was treated in a H2 flow (99.99999%, 1 bar, 60 mL min−1) at 773 K for 2 h and Pt/CZ-7/Si3N4 was obtained. Then, the obtained Pt/CZ-7/Si3N4 was treated in an O2 flow (99.99999%, 1 bar, 60 mL min−1) at 423 K for 1 h to obtain the partially oxidized Pt/CZ-x sample17. The dispersion of the particles on the membrane was evaluated by SEM (HITACHI, S-4800) (Supplementary Fig. 1a).

3D HXSP measurements

3D HXSP measurements were performed at the BL29XUL undulator beamline at SPring-8, Japan. Twenty-eight X-ray energies between 5.708 and 5.770 keV were used at the Ce-LIII edge with a Si(111) double-crystal monochromator. The finest energy gap was 1 eV (between 5.727 and 5.744 keV). The monochromatized X-rays were two-dimensionally focused to a spot size of 500 nm full-width at half-maximum using a pair of Kirkpatrick–Baez (KB) mirrors. Owing to the achromatic property of the KB mirrors, the focusing position was invariant with respect to the X-ray energy. The sample was mounted on the piezoelectric stages in a vacuum chamber and scanned in an 8 × 8 grid with a 400 nm step size. The diffraction patterns were measured with exposure times ranging from 0.5 to 4.0 s using a pixel array detector (EIGER 1M, Dectris) with a pixel size of 75 μm positioned 2.219 m downstream of the sample. To increase the effective dynamic range of the diffraction intensity, an 800 × 800 × 88 μm3 Si attenuator was placed in front of the central region of the detector. For the CT measurements, the sample was rotated from −75 to 75° via 61 evenly spaced angles for each X-ray energy. The total data acquisition required 5 days, including the energy scan, sample rotation, and experimental interruptions.

3D image reconstruction

The 2D image reconstructions were accomplished using the ePIE algorithm35 with the KKR constraint36 at each orientation (Supplementary Fig. 1b–e). The full-period spatial resolution of the projected images was estimated to be superior to 30 nm at the longest exposure and 50 nm at the shortest exposure on the basis of phase retrieval transfer function (PRTF) analysis (Supplementary Fig. 2a, b). Next, 3D image reconstruction was performed using the FBP algorithm at each X-ray energy. The 3D spatial resolution was better than 60 nm, which was ascertained on the basis of the edge structures (Supplementary Fig. 3). The 3D Ce valence maps of the particles were then estimated by analyzing the HXSP images. The XAFS spectra of Pt/CZ-x (μt)CZ-x can be approximately expressed by a linear combination of the normalized XAFS spectra of Pt/CZ-7 (μt)CZ-7 and Pt/CZ-8 (μt)CZ-8 as follows:

$$\left( {{{\mu t}}} \right)_{{\mathrm{CZ}} - {{x}}} \, = \, {{a}}\left( {{{\mu t}}} \right)_{{\mathrm{CZ}} - 7} \, + \, {{b}}\left( {{{\mu t}}} \right)_{{\mathrm{CZ}} - 8},$$
(1)

where a and b are coefficients. The parameters a and b at each voxel were determined using a least-squares fit in the energy range between 5.718 and 5.749 keV. (3a+4b)/(a+b) corresponds to the Ce valence. The average R factor over all of the voxels in particle 5 was calculated to be 0.0823.

Unsupervised learning

We considered the surrounding binning of 3 × 3 × 3 voxels (42 × 42 × 42 nm3) in the 3D Ce valence map and used the local mean (m(x,y,z)) and local standard deviation (sd(x,y,z)) of the Ce valence state in each binning domain as descriptors. We then obtain a dataset Ɗ that contains the information of all the observed voxels. A voxel with index i in Ɗ is described by the 2D vector \({\mathbf{x}}_i = \left( {{\it{m}}_{\it{i}}{\it{,sd}}_{\it{i}}} \right)\); therefore, the dataset Ɗ is represented using an (n × 2) matrix. The distribution was approximated by a mixture models of K Gaussian distributions37 (Supplementary Fig. 4) as follows:

$${\boldsymbol{p}}\left( {{\mathbf{x}}_{\boldsymbol{i}}|{\boldsymbol{\theta }}} \right) = \mathop {\sum}\limits_{{{k}} = {{1}}}^{{K}} {{\boldsymbol{\pi }}_{\boldsymbol{k}}{\cal{N}}\left( {{\boldsymbol{\mu }}_{\boldsymbol{k}},{\bf{\Sigma }}_{\boldsymbol{k}}} \right)}$$
(2)

where π, μ, and are the weights, centers, and coefficient matrices for the 2D Gaussians, respectively. For a given number of mixture component K, the estimation of the parameter is conducted through an expectation-maximization algorithm37. To determine the number of mixture components, a minimizing Bayesian information criterion38 process is utilized that involves applying several different trials to randomize the initial states. We have tried to approximate the distribution of dataset Ɗ by mixture models of K (= 1, 2, 3, 4, 5, 6) Gaussian distributions. For K (= 1, 2, 3, 4), we can obtain significant improvement in the Bayesian information criterion of the approximation. However, for K (= 4, 5, 6), the Bayesian information criterion of the approximation does not improve significantly. We therefore determine the number of mixture components at four components. As a result, the whole dataset Ɗ on the representation space (m, sd) is divided into four Gaussian components G1, G2, G3, G4 with respect to the set of the following parameters: π = {0.329, 0.319, 0.262, 0.091}, μ = {(3.43, 0.143), (3.61, 0.265), (3.85, 0.224), (3.97, 0.063)},  = {[(0.02766, −0.00078), (−0.00078, 0.00221)], [(0.02076, −0.00351), (−0.00351, 0.00837)], [(0.00478, −0.00441), (−0.00441, 0.00729)], [(0.00057, −0.00114), (−0.00114, 0.00252)]}.

Data availability

The authors declare that all data supporting the findings of this study are available within the paper or are available from the authors upon request.

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Acknowledgements

This work was supported by KAKENHI (Grant Nos. 16K13725, 16K17863, 16H00889, and Kiban B 26288005 and 18H01940), JSPS Fellowships (Grant Nos. 16J00329 and 17J01673), and the SENTAN Project and PRESTO of JST. We thank Prof. Dr. K. Yamauchi (Osaka University), Prof. Dr. T. Ishikawa (RIKEN), and Prof. Dr. M. Takata (Tohoku University/RIKEN) for many stimulating discussions, as well as Dr. Y. Kohmura (RIKEN) for assistance with the experimental setup and Dr. Y. Nagai (Toyota Central R&D Labs. Inc.) for kind discussion regarding the CZ-x samples.

Author information

M.H., K.S., and Y.T. performed the synchrotron experiments. M.H. performed the image reconstructions. M.H. and H.M. prepared the samples. M.H. performed the SEM observations of the samples. D.N.N. and H.C.D. performed the data mining analysis. M.T. and N.I. interpreted the results. M.H., N.I., H.C.D., H.M., M.T., and Y.T. wrote the manuscript. All authors read and approved the manuscript.

Correspondence to Hieu Chi Dam or Mizuki Tada or Yukio Takahashi.

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