Figure 1. Experimental XRD pattern of LaFe₁/₃Co₁/₃Mn₁/₃O₃ synthesized by solid-state reaction at 1250˚C. All reflections are indexed to the orthorhombic Pnma space group (a = 5.509638 Å, b = 7.809736 Å, c = 5.527591 Å). Rietveld refinement confirms single-phase orthorhombic symmetry with wRp = 0.0483, Rp = 0.0384, and χ² = 3.232. The black crosses, red line, green vertical lines, and blue solid line represent the raw data, the model, Bragg peak positions, and difference plot, respectively.

LaFe_1/3Co_1/3Mn_1/3O₃ was synthesized via conventional solid-state reaction. Stoichiometric quantities of La₂O₃, Fe₂O₃, Co₃O₄, and Mn₂O₃ were thoroughly mixed, pelletized, and calcined at 1250˚C in air. Phase identification and structural characterization were performed by powder X-ray diffraction (XRD) using Cu Kα radiation. Rietveld refinement confirmed a single-phase orthorhombic perovskite structure with space group Pnma and lattice parameters a = 5.509638 Å, b = 7.809736 Å, c = 5.527591 Å. Rietveld refinement yielded wRp = 0.0483, Rp = 0.0384, and χ² = 3.232, with a calculated unit cell formula weight of 973.743 g/mol and density of 6.798 g/cm³. The refinement profile is shown in Figure 1 and Pnma structure is shown in Figure 2. The refined fractional coordinates and site multiplicities are listed in Table 1. This experimentally determined orthorhombic structure served as the primary validation benchmark for all ML predictions.

Figure 2. crystallographic picture ofPnma space group. In layered orthorhombic Pnma structures, the symmetry elements dictate that tetrahedral layers (or sheets) stack by pointing in alternating, opposite directions. View of the structure through a) a-axis b) b-axis and c) c-axis.

Table 1.Rietveld-refined atomic positions for LaFe₁/₃Co₁/₃Mn₁/₃O₃ (Pnma, Z = 4).

Note: Lattice: a = 5.509638(51) Å, b = 7.809736(65) Å, c = 5.527591(58) Å. Refinement: wRp = 0.0483, Rp = 0.0384, χ² = 3.232. Formula weight = 973.743 g/mol; density = 6.798 g/cm³. Fe, Co, and Mn co-occupy the 4b site with equal occupancy 1/3 each. B-O bond lengths: Fe/Co/Mn-O1 = 1.967(4) Å; Fe/Co/Mn-O2 = 1.993(24) and 1.994 (25) Å. Tilt angles: (Fe/Co/Mn)-O1-(Fe/Co/Mn) = 166.2 (20)˚; (Fe/Co/Mn)-O2-(Fe/Co/Mn) = 156.3 (12)˚, confirming GdFeO₃-type cooperative octahedral tilting.

A dataset of 27 La-based perovskite oxide entries was compiled from published crystallographic literature, covering the three symmetry classes observed in this compositional family: cubic (Pm-3 m, 11 entries), orthorhombic (Pnma, 9 entries), and rhombohedral (R-3c, 7 entries). Compositions span single B-site end members (LaFeO₃, LaCoO₃, LaMnO₃, LaNiO₃, LaCrO₃, LaAlO₃, LaGaO₃), binary B-site solid solutions across Fe, Co, Mn, and Ni combinations, and ternary B-site compositions. All structural data were taken from peer-reviewed diffraction studies; no computationally generated entries are included in this dataset.

Table 2.Physicochemical descriptors used as ML input features.

While the dataset is intentionally limited to verified experimental values, this approach prioritizes fidelity over scale. We note that expansion using DFT-optimized structures from databases such as the Materials Project [16] or AFLOW [17] would increase training set size and improve boundary resolution between symmetry classes—a direction identified as future work in Section 6.

Seven physicochemical descriptors were computed for each composition (Table 2). Ionic radii were taken from Shannon (1976) [15] for coordination number 6, high-spin states where applicable. The Goldschmidt tolerance factor was calculated as t = (rA + rO)/[√2(rB + rO)], using rA = 1.360 Å (La³⁺, 12-coordinate), rO = 1.400 Å (O²⁻, 6-coordinate), and rB as the composition-weighted average B-site radius. Jahn-Teller activity was treated as a binary descriptor (1 if the composition contains Mn³⁺, Cu²⁺, or other known Jahn-Teller-active ions above a 10% B-site fraction, 0 otherwise).

Three supervised classification algorithms were implemented using scikit-learn [18]: (1) Random Forest (RF, 200 estimators, balanced class weights, Gini impurity criterion); (2) Gradient Boosting (GB, 200 estimators, learning rate 0.05, max depth 3); and (3) Support Vector Machine (SVM, RBF kernel, C = 10, γ = scale, probability calibration enabled). The three-class target variable encodes crystal symmetry as Cubic, Orthorhombic, or Rhombohedral. Feature data were standardized to zero mean and unit variance prior to SVM training; tree-based models were trained on unscaled features.

Model performance was assessed by stratified 5-fold cross-validation (CV) and leave-one-out cross-validation (LOO-CV), the latter being particularly appropriate for small datasets as it maximizes training data at each iteration. Feature importance was quantified by two independent methods: (1) the RF Gini impurity criterion, reported in Table 3; and (2) permutation importance (30 repeats), shown in Figure 3. Both methods produced identical descriptor rankings, providing cross-method validation of the descriptor hierarchy. The target descriptor vector for LaFe_1/3Co_1/3Mn_1/3O₃ was computed solely from Shannon ionic radii.

Random Forest feature importance analysis reveals that crystal symmetry in La-based Fe-Co-Mn perovskites is governed primarily by structural rather than purely compositional descriptors (Figure 3; Table 3). The Goldschmidt tolerance factor ranked first with a Gini importance of 0.42, consistent with its role as the primary predictor of octahedral tilting instability in perovskites. The B-site ionic radius variance σ² ranked second (importance = 0.31), confirming that local lattice strain from B-site ionic mismatch is the second most powerful determinant of symmetry—a factor often overlooked in single-descriptor phase diagrams. The octahedral factor μ ranked third (0.14). Electronegativity variance and Jahn-Teller activity were comparatively less influential (<0.05 each) for this compositional set.

Figure 3. Random Forest permutation feature importance (30 repeats) for crystal symmetry classification in La-based perovskites. The Goldschmidt tolerance factor (t = 0.42) and B-site ionic radius variance (σ² = 0.31) dominate over the octahedral factor, electronegativity variance, and Jahn-Teller activity. Error bars represent ±1 standard deviation. Gini impurity-based importance (Table 3) yield the same ranking.

For LaFe_1/3Co_1/3Mn_1/3O₃ specifically, the three dominant descriptors take the following values: t = 0.970, σ² = 0.00222 Ų (high), and μ = 0.437. The tolerance factor of 0.970 places LaFe_1/3Co_1/3Mn_1/3O₃ within the range where orthorhombic or rhombohedral distortions are commonly observed in La-based perovskites, particularly when Jahn-Teller-active cations are present. The relatively high σ² = 0.00222 Ų reflects the large size contrast between low-spin Co³⁺ (0.545 Å) and Fe³⁺/Mn³⁺ (both 0.645 Å), yielding ⟨rB⟩ = 0.612 Å. This substantial B-site variance imposes unequal strain fields on the surrounding oxygen octahedra, promoting cooperative tilting. Concurrently, the Jahn-Teller activity of high-spin Mn³⁺ (one-third of B-site occupancy) is not fully quenched by configurational averaging and propagates into long-range GdFeO₃-type octahedral tilting—the hallmark of orthorhombic Pnma symmetry.

Table 3. Feature importance ranking from Random Forest analysis and computed descriptor values for the target composition.

Note: Importance values in this table are from the Random Forest Gini impurity criterion (normalized to sum to 1.00); permutation importance (30 repeats) is shown separately in Figure 3. Both methods yield the same ranking. Target values computed from Shannon ionic radii (CN = 6; Fe³⁺ and Mn³⁺ high-spin, Co³⁺ low-spin) and Rietveld-refined lattice parameters a = 5.509638 Å, b = 7.809736 Å, c = 5.527591 Å. ⟨rB⟩ = 0.612 Å; σ² = 0.00222 Ų; t = 0.970; μ = 0.437.

To contextualize the ternary prediction, ML was also applied to the three-constituent end-member perovskites. LaFeO₃ (t = 0.918) and LaMnO₃ (t = 0.924) were correctly predicted as orthorhombic, consistent with their Pbnm ground-state structures arising from cooperative GdFeO₃-type tilting. LaCoO₃ (t = 0.937) was predicted as rhombohedral, consistent with its R-3c room-temperature structure. Upon equimolar mixing, the average t rises to 0.970 and σ² increases to 0.00222 Ų, because low-spin Co³⁺ (0.545 Å) offsets the larger Fe³⁺ and Mn³⁺ (both 0.645 Å), but the residual ionic mismatch between Co³⁺ (0.545 Å) and Fe³⁺ /Mn³⁺ still generates sufficient local strain to favor orthorhombic distortion. The Jahn-Teller activity of Mn³⁺ is not fully quenched by configurational averaging, and cooperative Mn³⁺ distortions persist in the long-range structure. This partial-averaging effect—whereby multi-component B-site mixing moderates but does not eliminate octahedral tilting—is the physical basis for the orthorhombic Pnma structure of LaFe_1/3Co_1/3Mn_1/3O₃.

All three classifiers demonstrated strong performance (Table 4). The Random Forest achieved perfect 5-fold CV and LOO-CV accuracy (1.000). Gradient Boosting attained LOO-CV accuracy of 0.970 and SVM achieved 0.963, indicating robust generalization across all three symmetry classes despite the small training set. The perfect LOO-CV scores for RF and GB likely reflect the strong descriptor-based separation among symmetry classes in this dataset rather than overfitting; larger datasets will be needed to confirm generalizability. Most critically, all three models unanimously predicted orthorhombic symmetry for LaFe_1/3Co_1/3Mn_1/3O₃ with high probabilities (RF: 0.850; GB: 1.000; SVM: 0.824), made without any prior knowledge of the experimental outcome (Figure 4).

Table 4.ML model performance and predictions for LaFe_1/3Co_1/3Mn_1/3O₃.

The unanimous agreement across three independent algorithms, each using a different learning strategy, substantially strengthens confidence in the orthorhombic prediction beyond what any single model could provide. Experimental Rietveld refinement confirms that the orthorhombic Pnma phase (a = 5.510 Å, b = 7.810 Å, c = 5.528 Å) is indeed the room-temperature structure of the solid-state-synthesized material, providing direct and quantitative confirmation of the ML framework.

The ML descriptor framework provides three categories of actionable guidance for ongoing perovskite synthesis at UTTC:

High-throughput pre-screening. Before committing to synthesis, t and σ² can be computed in seconds for any proposed B-site mixture using only Shannon ionic radii. Compositions satisfying 0.950 ≤ t ≤ 0.980 and σ² > 0.002 Ųshould be prioritized as orthorhombic candidates. Compositions with t > 0.990 and σ² < 0.001 Ų are likely cubic and may be deprioritized for applications requiring distorted B-site environments.

Synthesis condition selection. The confirmed orthorhombic structure of LaFe_1/3Co_1/3Mn_1/3O₃ at 1250˚C provides a synthesis reference point. For related compositions expected to adopt orthorhombic distortion, the same solid-state route and calcination temperature can be applied with high confidence, eliminating the need for additional atmosphere or temperature screening.

Diffraction interpretation and property targeting. A Pnma prediction narrows the expected XRD pattern to an orthorhombic cell with characteristic peak splitting (e.g., (200)/(020) doublets) and additional superlattice reflections absent in cubic Pm-3m, simplifying Rietveld refinement against the Pnma model. The orthorhombic distortion arising from cooperative GdFeO₃-type octahedral tilting is consistent with the B-site ionic mismatch between Co³⁺ (0.545 Å) and Fe³⁺ /Mn³⁺ (0.645 Å), confirming that the Jahn-Teller activity of Mn³⁺ is not fully suppressed by configurational averaging in this composition. Characterization efforts can therefore target properties associated with orthorhombic symmetry, including anisotropic electronic conductivity, enhanced catalytic activity at distorted B-site environments, and OER/ORR performance.

Table 5.ML-guided experimental decision framework for perovskite synthesis at UTTC.

Figure 4. Symmetry prediction probabilities for LaFe_1/3Co_1/3Mn_1/3O₃ from all three ML classifiers (Random Forest, Gradient Boosting, SVM). All models assign the highest probability to the orthorhombic (Pnma) class, with probabilities of 0.850, 1.000, and 0.824 for RF, GB, and SVM, respectively. The dashed line at p = 0.50 marks the decision threshold. Experimental Rietveld refinement confirms the orthorhombic Pnma structure, validating all three predictions.

The dominance of the tolerance factor in the feature importance ranking (0.42) confirms that Goldschmidt’s structural rules retain their predictive validity even for high-entropy B-site compositions—a non-trivial result, since the Goldschmidt factor was originally derived for single-cation perovskites. The strong secondary importance of σ² (0.31) is a distinctive ML finding: conventional phase diagrams for perovskites typically plot only t or ⟨rB⟩, yet the ML model identifies radius variance as nearly as determinative as the tolerance factor itself (Table 5). This is physically interpretable: a high σ² implies that different B-site cations impose substantially different strain fields on the surrounding oxygen octahedra, destabilizing the long-range periodicity required for cubic symmetry even when the average t would suggest otherwise (Figure 5).

Figure 5. Phase stability map for La-based perovskites: Goldschmidt tolerance factor (t) versus B-site ionic radius variance (σ²). Training data points (n = 27) are colored and shaped by experimentally confirmed symmetry class: orthorhombic Pnma(orange squares, n = 9), rhombohedral R-3c (blue triangles, n = 7), and cubic Pm-3m (green circles, n = 11). Dashed convex hulls outline each symmetry class domain. The horizontal dotted line marks the σ²= 0.002 Ų distortion threshold. The target composition LaFe_1/3Co_1/3Mn_1/3O₃ (t = 0.970, σ²= 2.22×10⁻² Ų) is marked as a red star within the orthorhombic domain, consistent with its experimentally confirmed Pnma structure.

For LaFe_1/3Co_1/3Mn_1/3O_3,the high σ² = 0.00222 Ų reflects the large size contrast between low-spin Co³⁺ (0.545 Å) and Fe³⁺/Mn³⁺ (both 0.645 Å): although Fe and Mn share identical radii and Co constitutes only one-third of the B-site, the 0.100 Å radius difference between Co³⁺ (LS) and Fe³⁺/Mn³⁺ is sufficient to generate substantial local lattice strain. This B-site mismatch, compounded by the Jahn-Teller activity of Mn³⁺ at one-third B-site occupancy, drives cooperative GdFeO₃-type octahedral tilting that propagates into the long-range orthorhombic Pnma structure. This is the key ML finding: σ² = 0.00222 Ų exceeds the ∼0.002 Ų threshold commonly associated with distorted phases, and the ML model correctly identifies this as determinative of orthorhombic symmetry, as confirmed by Rietveld refinement (wRp = 0.0483, Rp = 0.0384, χ²= 3.232).

Several limitations should be acknowledged. First, the training dataset of 27 entries, while comprising verified experimental values, is small relative to the compositional space available. Boundary resolution between the orthorhombic and rhombohedral classes—which are structurally similar and can be difficult to distinguish by laboratory XRD alone—would improve substantially with larger training sets. Future work should incorporate DFT-optimized structures from the Materials Project and AFLOW databases, which collectively contain hundreds of La-based perovskite entries.

Second, the present model does not capture temperature-dependent phase transitions. LaCoO₃, for example, undergoes a rhombohedral-to-cubic transition above ~500˚C, and similar behavior may occur in related compositions. Extending the descriptor set to include synthesis temperature or incorporating finite-temperature DFT data would address this limitation.

Third, the Shannon ionic radius model assumes average site occupancy (Vegard’s law) and does not account for local B-site ordering, nanoscale domain formation, or short-range correlations that may be present but undetectable by conventional powder XRD. Pair distribution function (PDF) analysis or transmission electron microscopy would be needed to investigate such local structural effects.