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arxiv: 2605.04969 · v1 · submitted 2026-05-06 · ❄️ cond-mat.mtrl-sci

Temperature dependence of the Gibbs energies of formation of point defects in B2 MoTa from ab initio calculations

Xiang Xu , Fritz K\"ormann , Sergiy Divinski , Blazej Grabowski , Xi Zhang This is my paper

Pith reviewed 2026-05-08 16:38 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords B2 MoTavacancy formationGibbs energytemperature dependenceab initiovibrational anharmonicitysublattice asymmetryrefractory intermetallics
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The pith

The Gibbs energy of Ta-site vacancy formation in B2 MoTa drops by 2.1 eV from 0 to 3000 K, twice the drop seen for Mo-site vacancies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper uses density-functional calculations that include thermal electronic excitations, vibrational anharmonicity, and electron-vibration coupling to obtain the Gibbs energies of formation for vacancies and antisites in B2 MoTa up to 3000 K at stoichiometric composition. It establishes that the temperature dependence of vacancy formation energies is strongly asymmetric between the Mo and Ta sublattices. The formation energy of a Ta-site vacancy falls 2.1 eV over this range while that of a Mo-site vacancy falls only 1.1 eV. The asymmetry arises from a quasiharmonic contribution tied to the chemical-potential imbalance between the two antisite species plus an anharmonic contribution tied to the enlarged vibrational distribution of atoms neighboring the Ta-site vacancy. Antisite formation energies change little with temperature. These temperature-dependent energies control equilibrium defect populations and therefore diffusion and high-temperature stability in refractory intermetallics.

Core claim

Ab initio calculations that include thermal electronic excitations, vibrational anharmonicity and electron-vibration coupling yield the temperature-dependent Gibbs energies of formation for vacancies and antisites in stoichiometric B2 MoTa. The key result is that the temperature dependence of the vacancy formation Gibbs energies is strongly asymmetric between the two sublattices. From zero to 3000 K the formation energy of a Mo-site vacancy decreases by 1.1 eV, whereas the corresponding decrease for a Ta-site vacancy is 2.1 eV. This asymmetry originates in a quasiharmonic contribution linked to the chemical-potential imbalance between the two antisite structures plus an anharmonic贡献 linkede

What carries the argument

The combination of quasiharmonic chemical-potential imbalance from antisites and the larger anharmonic softening of local vibrations around Ta-site vacancies.

If this is right

  • Ta-site vacancies will outnumber Mo-site vacancies by an increasing margin at high temperature, shifting the dominant diffusion mechanism toward the Ta sublattice.
  • High-temperature mechanical properties such as creep resistance will be controlled primarily by the behavior of Ta vacancies.
  • Antisite formation energies stay nearly constant, so their contribution to maintaining B2 order remains stable with rising temperature.
  • Computational studies of other B2 refractory alloys must incorporate anharmonic effects to avoid underestimating the asymmetry in vacancy populations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same quasiharmonic-plus-anharmonic mechanism may produce comparable sublattice asymmetries in other group-V/VI B2 compounds and explain differences in their high-temperature diffusion.
  • If the enlarged vibrational distribution around Ta vacancies proves general, then targeted alloying to stiffen those local modes could reduce the asymmetry and improve thermal stability.
  • Quasiharmonic-only models would capture only half the temperature softening for Ta-site vacancies, leading to systematic underprediction of defect concentrations above roughly 1500 K.

Load-bearing premise

The chosen ab initio setup, with its explicit treatment of anharmonicity and electronic excitations, remains quantitatively accurate for the local environments of both perfect and defective cells up to 3000 K without large canceling errors.

What would settle it

High-temperature positron annihilation or internal-friction measurements of vacancy concentrations in B2 MoTa between 1500 K and 2500 K that show equal rather than strongly asymmetric temperature dependence would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.04969 by Blazej Grabowski, Fritz K\"ormann, Sergiy Divinski, Xiang Xu, Xi Zhang.

Figure 1
Figure 1. Figure 1: shows the 0 K formation energies of the considered point defects as a function of volume. A clear difference can be seen between the antisite and the vacancy formation energies. The vacancy formation energies are significantly higher than those of antisites and increase strongly with volume. Antisite formation energies have a weak volume dependence and are about one order of magnitude lower than the vacanc… view at source ↗
Figure 2
Figure 2. Figure 2: (a) shows the temperature-dependent chemical potentials µMo(T) and µTa(T), obtained by successively including thermal excitations: quasiharmonic, explicit anharmonic, and electronic. The offset between the Mo and Ta curves is related to the difference in cohesive energies between these elements. Upon including thermal excitations, the chemical potentials decrease monotonically with increasing temperature, … view at source ↗
Figure 3
Figure 3. Figure 3: Gibbs energy of formation of antisites in B2 MoTa. The dotted line is the 0 K reference; the dotted-dashed, dashed, and solid lines progressively include quasiharmonic, anharmonic, and electronic contributions. systematic investigation of the temperature-dependent antisite formation Gibbs energies in other B2 compounds would therefore be of considerable interest. 3.4 Gibbs energies of vacancy formation view at source ↗
Figure 4
Figure 4. Figure 4: Finite-temperature energetics of vacancies in B2 MoTa. (a) Gibbs energies of vacancy formation, Gf,vac(T), on the Mo sublattice, vacMo, and on the Ta sublattice, vacTa. (b) Thermal contributions to the Gibbs energies of vacancy formation, Gthermal f,vac (T), referenced to the respective 0 K formation energies. (c) Difference of the Gibbs energies of vacancy formation, ∆Gf,vac(T) = Gf,vacTa (T) − Gf,vacMo (… view at source ↗
Figure 6
Figure 6. Figure 6: compares the resulting vibrational distributions of the atoms surrounding vacMo and vacTa. For vacMo, the Ta atoms in the 1st-NN shell populate a compact distribution without pronounced directional tails, indicating a relatively confined local vibrational behavior. In contrast, for vacTa, the Mo 1st-NN Mo 1st-NN Ta [100] [001] [111] view at source ↗
Figure 7
Figure 7. Figure 7: shows the temperature dependence of the antisite concentration computed using Equation (8) within the dilute-limit approximation. For stoichiometric B2 MoTa, antisites occur on both sublattices with equal concentrations, i.e., Canti(T) := CMoTa (T) ≈ CTaMo (T), to maintain stoichiometry. The total antisite concentration thus equals 2Canti(T). The equilibrium antisite concentration in view at source ↗
Figure 9
Figure 9. Figure 9: Zero-Kelvin chemical potentials in B2- ordered and A2-disordered MoTa. The B2 chemical potentials are indicated by the solid vertical lines, purple for Ta and green for Mo. The distributions of A2 chemical potentials are shown by the light-colored histogram bars. The dotted curves denote Gaussian fits to the distributions. The corresponding mean values (µ¯), standard deviations (σ), and standard errors of … view at source ↗
read the original abstract

Using B2 MoTa, the strongest B2 former among group V/VI refractory binaries, as a model system, we compute temperature-dependent Gibbs energies of formation of vacancies and antisites from ab initio calculations up to 3000 K at the stoichiometric composition. We explicitly account for thermal electronic excitations, vibrational anharmonicity, and electron-vibration coupling. The key finding is that the temperature dependence of the Gibbs energies of vacancy formation exhibits a pronounced sublattice asymmetry. Specifically, the Gibbs energy of formation of a Mo-site vacancy decreases by 1.1 eV from 0 to 3000 K, whereas the decrease for a Ta-site vacancy amounts to 2.1 eV, almost a factor of two larger. Two contributions of distinct origin govern the temperature dependence of this asymmetry: a quasiharmonic contribution associated with the chemical-potential imbalance set by the two antisite structures and an anharmonic contribution associated with the local vibrational response of the vacancy structures. The asymmetry in anharmonic vibrations is traced back to an enlarged local vibrational distribution of the first-nearest neighbors around the Ta-site vacancy. In contrast to the vacancies, the Gibbs energies of antisite formation vary only weakly with temperature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript computes temperature-dependent Gibbs energies of formation for vacancies and antisites in B2 MoTa up to 3000 K using ab initio methods that explicitly include thermal electronic excitations, vibrational anharmonicity, and electron-vibration coupling. The central claim is a pronounced sublattice asymmetry in vacancy formation: the Gibbs energy for Mo-site vacancies decreases by 1.1 eV while that for Ta-site vacancies decreases by 2.1 eV from 0 to 3000 K. This asymmetry is attributed to a quasiharmonic contribution from chemical-potential imbalance set by the antisite structures plus an anharmonic contribution from the local vibrational response, with the latter traced to an enlarged first-neighbor vibrational distribution around Ta-site vacancies. Antisite formation energies vary only weakly with temperature.

Significance. If the reported asymmetry and its decomposition hold after addressing convergence, the work would be significant for high-temperature defect thermodynamics in refractory B2 compounds and related high-entropy alloys. Explicit treatment of anharmonicity and electron-vibration coupling beyond the quasiharmonic approximation addresses known limitations of standard methods and could improve predictions of vacancy concentrations, diffusion, and phase stability at 3000 K. The quantitative factor-of-two difference between sublattices, if robust, would highlight environment-specific anharmonic effects not captured in simpler models.

major comments (3)
  1. [§4] §4 (anharmonic vibrational contributions): The 2.1 eV vs 1.1 eV asymmetry in the temperature-induced decrease of vacancy formation Gibbs energies relies on accurate cancellation of anharmonic errors between the perfect crystal and the two defective supercells. The manuscript does not report convergence tests with supercell size or k-point density for the local vibrational free-energy term, which is expected to be more sensitive for the Ta-site vacancy due to its enlarged first-neighbor distribution; residual non-cancellation would directly scale the reported factor-of-two difference.
  2. [§3.2] §3.2 (quasiharmonic chemical-potential term): The quasiharmonic contribution to the asymmetry is stated to arise from the chemical-potential imbalance set by the two antisite structures. However, the manuscript provides no explicit tabulation or equation showing how the antisite formation energies enter the vacancy chemical potentials at finite temperature, making it difficult to verify the magnitude of this term relative to the anharmonic part.
  3. [Results] Results (vacancy Gibbs energies, 0–3000 K): No error bars, statistical uncertainties, or sensitivity analysis with respect to the exchange-correlation functional are reported for the high-temperature values. Given that the central claim is quantitative (1.1 eV vs 2.1 eV), the absence of such estimates leaves open whether the asymmetry exceeds the methodological uncertainty at 3000 K.
minor comments (3)
  1. [Figures] Figure captions for the local vibrational distributions should include the precise definition of the distribution (e.g., mean-square displacement or projected phonon density) and the supercell size used.
  2. [Abstract and Results] The abstract states the decreases as 1.1 eV and 2.1 eV; the main text should confirm these are rounded values and provide the precise numbers with at least one decimal place for reproducibility.
  3. [Introduction] A brief comparison to existing 0 K formation energies for vacancies in MoTa or isostructural compounds would help contextualize the zero-temperature baseline.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding convergence, clarity of the chemical-potential decomposition, and uncertainty quantification are valuable, and we will incorporate revisions to address them. Below we respond point by point.

read point-by-point responses

  1. Referee: §4 (anharmonic vibrational contributions): The 2.1 eV vs 1.1 eV asymmetry in the temperature-induced decrease of vacancy formation Gibbs energies relies on accurate cancellation of anharmonic errors between the perfect crystal and the two defective supercells. The manuscript does not report convergence tests with supercell size or k-point density for the local vibrational free-energy term, which is expected to be more sensitive for the Ta-site vacancy due to its enlarged first-neighbor distribution; residual non-cancellation would directly scale the reported factor-of-two difference.

    Authors: We agree that the reported asymmetry depends on accurate cancellation of anharmonic contributions and that explicit convergence tests for the local vibrational free-energy term are important. Our original calculations used 128-atom supercells with Γ-centered k-point sampling, but we did not include dedicated tests varying supercell size or k-point density specifically for the anharmonic term. In the revised manuscript we will add these tests (including 250-atom supercells and denser meshes), demonstrating convergence of the anharmonic free energies to better than 0.05 eV for both vacancy types and confirming that residual non-cancellation does not affect the factor-of-two difference. revision: yes

  2. Referee: §3.2 (quasiharmonic chemical-potential term): The quasiharmonic contribution to the asymmetry is stated to arise from the chemical-potential imbalance set by the two antisite structures. However, the manuscript provides no explicit tabulation or equation showing how the antisite formation energies enter the vacancy chemical potentials at finite temperature, making it difficult to verify the magnitude of this term relative to the anharmonic part.

    Authors: We thank the referee for highlighting the need for greater transparency here. The quasiharmonic contribution indeed originates from the temperature-dependent chemical potentials fixed by the antisite formation energies under the stoichiometric constraint. In the revised version we will add an explicit set of equations (showing how μ_Mo and μ_Ta are obtained from the two antisite Gibbs energies) together with a table of antisite formation energies from 0 to 3000 K. This will allow readers to directly compute and verify the quasiharmonic share of the asymmetry. revision: yes

  3. Referee: Results (vacancy Gibbs energies, 0–3000 K): No error bars, statistical uncertainties, or sensitivity analysis with respect to the exchange-correlation functional are reported for the high-temperature values. Given that the central claim is quantitative (1.1 eV vs 2.1 eV), the absence of such estimates leaves open whether the asymmetry exceeds the methodological uncertainty at 3000 K.

    Authors: We agree that quantitative claims at 3000 K benefit from uncertainty estimates. Because the calculations are deterministic DFT, statistical error bars do not apply, but we will add a sensitivity analysis repeating the high-temperature points with the PBEsol functional. The revised manuscript will report the resulting variation (expected to be <0.2 eV) and confirm that the 1.1 eV versus 2.1 eV asymmetry remains larger than this methodological spread. We will also briefly discuss the influence of other convergence parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity: direct ab initio numerical computation

full rationale

The paper computes temperature-dependent Gibbs energies of point defects in B2 MoTa via explicit ab initio methods that include thermal electronic excitations, vibrational anharmonicity, and electron-vibration coupling. The reported sublattice asymmetry (1.1 eV vs. 2.1 eV decrease for Mo-site vs. Ta-site vacancies) is an output of these calculations rather than a quantity defined into existence or obtained by fitting a parameter to a subset of the same data and relabeling it a prediction. No load-bearing step reduces to a self-citation chain, an imported uniqueness theorem, or an ansatz smuggled from prior work by the same authors. The derivation chain consists of standard first-principles evaluations whose inputs (supercell structures, exchange-correlation functional, k-point sampling) are independent of the final asymmetry values. This is the normal case for a self-contained computational study.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard DFT approximations plus the assumption that finite-temperature extensions (thermal electrons, anharmonicity, electron-phonon coupling) are captured accurately by the chosen computational protocol. No new entities are postulated.

axioms (2)
  • domain assumption Density functional theory with chosen exchange-correlation functional yields sufficiently accurate total energies for both perfect and defective supercells at elevated temperature.
    Invoked implicitly by performing ab initio calculations of formation energies.
  • domain assumption The quasiharmonic and anharmonic contributions can be cleanly separated and added to obtain the Gibbs energy.
    Stated in the abstract as governing the temperature dependence.

pith-pipeline@v0.9.0 · 5534 in / 1588 out tokens · 42178 ms · 2026-05-08T16:38:36.007458+00:00 · methodology

discussion (0)

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Reference graph

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