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Accurate Prediction of the $\alpha \to \beta$ Phase Transformation Temperature in Tin via Full Anharmonic Treatment

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Anharmonicity, not the functional, sets tin's α→β transition near 288 K.

desk verdict Solid paper showing beta-Sn is strongly anharmonic, but the headline number mixes classical anharmonicity with quantum effects—qualitatively right, quantitatively overstated. read the letter →

arxiv 2607.25978 v1 pith:7SJ64WJV submitted 2026-07-28 cond-mat.mtrl-sci

Petr Šesták , Matous Mrovec , Martin Friák This is my paper · ORCID
classification cond-mat.mtrl-sci PACS 64.70.K63.20.Ry65.40.-b71.15.Mb
keywords tinα-βphasetransitionanharmonicityquasi-harmonicapproximationthermodynamicintegrationmachine-learnedinteratomicpotentialatomicclusterexpansionfreeenergy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the long-standing overestimate of tin's grey-to-white (α→β) transition temperature in first-principles calculations comes mainly from neglected lattice anharmonicity in metallic β-Sn, not from errors in the electronic-structure functional. Using a machine-learned Atomic Cluster Expansion (ACE) potential trained on PBE density-functional data, the authors evaluate free energies of both phases two ways on the same potential energy surface: the quasi-harmonic approximation (QHA) and full thermodynamic integration (TI). QHA predicts a transition at 377 K; TI predicts 288 K, within 2 K of the experimental 286 K. The difference directly quantifies an explicit anharmonic free energy of about −11 meV/atom for β-Sn at room temperature, while α-Sn remains nearly harmonic. If correct, the result shows that capturing explicit phonon–phonon interactions is essential for predicting phase boundaries in soft, polymorphic metals, and it shifts the focus from exchange-correlation functionals to vibrational treatment.

What carries the argument

The central object is a machine-learned Atomic Cluster Expansion (ACE) interatomic potential fitted to PBE density-functional theory data, which reproduces the potential energy surface of both α- and β-Sn and enables large-scale molecular dynamics. Its role is to make thermodynamic integration feasible while keeping the electronic reference fixed. The decisive comparison is between two free energies computed on this same potential: the quasi-harmonic free energy, built from volume-dependent phonon frequencies via Eq. (1), and the thermodynamic-integration free energy, which couples the ACE potential to an Einstein crystal and integrates over the coupling parameter λ (Eqs. 2–3). Their differe

What would settle it

Compute the same QHA-vs-TI comparison using a potential trained on a functional with a smaller static energy difference, such as PBE+U with U≈1–1.5 eV; if the TI transition temperature moves away from 286 K by more than about 10 K, the agreement with experiment partly reflects a cancellation between the static-energy error and the anharmonic correction. More directly, a quantitative calorimetric measurement of β-Sn's heat capacity above 200 K, compared with the TI prediction including the electronic and dilation terms, would falsify the magnitude of the anharmonic free energy if the mismatch e

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Extended reading notes

Core claim

The central discovery is that β-Sn is strongly anharmonic in a temperature-dependent way that stabilizes it relative to α-Sn. The explicit anharmonic free energy, defined as the difference between the thermodynamic-integration and quasi-harmonic free energies (ΔF_anh = F_TI − F_QHA), reaches roughly −11 meV/atom at 300 K for β-Sn and is negligible for α-Sn. This anharmonic stabilization lowers the predicted α→β transformation temperature from 377 K (QHA) to 288 K (TI), almost exactly the experimental 286 K. The claim is corroborated by four independent observations: excess heat capacity above the Dulong–Petit limit, a temperature-driven collapse of a 3–3.5 THz feature in the vibrational dens

Load-bearing premise

The interpretation of ΔF_anh = F_TI − F_QHA as purely 'explicit anharmonic free energy' assumes the two free energies are directly comparable, yet TI is run with classical nuclei while QHA includes zero-point quantum motion, so nuclear-quantum effects are not separately accounted for and may contribute to the 85 K shift.

Editorial extensions

If this is right

  • Predictions of phase boundaries in soft polymorphic metals should include explicit anharmonicity; quasi-harmonic treatments can be off by tens to hundreds of kelvin.
  • The observed stability of white tin at ambient conditions is partly a temperature-driven dynamical effect, not just a static-energy preference.
  • With anharmonicity accounted for, the residual uncertainty in the transition temperature is dominated by the 0 K static energy difference, so improving the exchange-correlation description (e.g., beyond PBE) becomes the next bottleneck.
  • The combination of machine-learned potentials and thermodynamic integration offers a practical route to fully anharmonic free energies at near-ab initio accuracy for materials with shallow potential energy landscapes.
  • The TI–QHA difference provides a quantitative decomposition of heat capacity and free energy into harmonic, dilation, and explicit phonon–phonon contributions, enabling direct comparison with calorimetric data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The near-exact agreement with 286 K may partly result from a cancellation between PBE's overestimated static energy difference and the anharmonic correction, as the paper itself concedes; a test with a functional giving a smaller static ΔE (e.g., PBE+U) would reveal whether the anharmonic mechanism is robust or the match is fortuitous.
  • The finding suggests that other soft metallic phases with shallow potential wells may harbor large hidden anharmonic contributions, implying that quasi-harmonic screening of phase diagrams could be systematically biased for such systems.
  • One could decompose the explicit anharmonic free energy by phonon mode (e.g., via temperature-dependent phonon self-energies) to identify which vibrations drive the stabilization of β-Sn, offering a microscopic design lever for alloying or control of tin pest.
  • The classical-nuclei nature of the thermodynamic integration leaves a small unquantified nuclear-quantum correction; quantifying it (e.g., via path-integral methods) would sharpen the attribution of the 85 K shift purely to anharmonicity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper constructs an Atomic Cluster Expansion (ACE) machine-learned interatomic potential for tin trained on PBE-DFT data and uses it to compute the free energies of α-Sn and β-Sn by two routes: the quasi-harmonic approximation (QHA, Eq. 1) and classical thermodynamic integration (TI, Eqs. 2–3). On the same ACE potential, the QHA transformation temperature is 377 K, whereas TI gives 288 K, in close agreement with the experimental 286 K. The authors attribute the 85 K shift to explicit phonon–phonon anharmonicity, reporting ΔF_anh ≈ −11 meV/atom for β-Sn at 300 K and a much smaller effect for α-Sn. The anharmonicity of β-Sn is further supported by heat capacity, velocity-autocorrelation spectra, and force-deviation metrics. The central claim is that explicit anharmonicity is essential for the phase stability of tin, while the absolute transition temperature remains limited by the underlying 0 K PBE energetics.

Significance. If the central attribution is quantitatively reliable, the work is significant: it demonstrates a transferable MLIP-based strategy for isolating explicit anharmonic contributions to a phase boundary and resolves a long-standing discrepancy for tin without fitting to the experimental transition temperature. The design is sound in that both free-energy routes use the same potential, so the comparison isolates methodological differences rather than potential errors. The microscopic corroboration from heat capacity, VACF spectra, and force residuals is a real strength and makes the qualitative conclusion—that β-Sn is strongly anharmonic—credible. However, the quantitative claim that the −11 meV/atom term is purely anharmonic requires an additional correction or justification because the two free energies being subtracted are not computed at the same statistical-mechanical level.

major comments (3)
  1. [Thermodynamic integration, Eqs. (2)–(3) versus QHA Eq. (1)] The subtraction ΔF_anh = F_TI − F_QHA does not isolate explicit anharmonicity alone. F_QHA (Eq. 1) is a quantum harmonic free energy with zero-point motion and Bose–Einstein occupation, while F_TI from classical MD is a classical free energy. Thus ΔF_anh = (F_TI_classical − F_classical_harmonic) − (F_QHA_quantum − F_classical_harmonic). The second bracket is a nuclear-quantum correction. For Sn's 3–5 THz modes at 286 K, the high-T per-mode correction is roughly (ħω)^2/(24 k_B T) ≈ 0.3–0.7 meV, and the summed phase-dependent correction could be ~1 meV/atom—exactly the quantity entering the claimed 85 K shift. Please quantify this correction (e.g., by adding the harmonic quantum correction to the TI result or by path-integral MD) or explicitly report ΔF_anh as the TI−QHA difference with the nuclear-quantum contribution separated. As written, the title and conclusions state that −11 meV/ato
  2. [Methods, Thermodynamic integration; Fig. 4] It is not specified whether the TI free energies are Helmholtz free energies at fixed 0 K volumes or Gibbs free energies minimized over volume. The QHA curves are explicitly min_V F(V,T) (Methods, QHA), while the TI description lists only temperatures and λ values. If TI was performed at a single volume per phase, the TI−QHA difference includes the thermal-expansion contribution to the free energy in addition to explicit anharmonicity. Please specify the volume protocol (e.g., NPT simulations, or multiple NVT volumes followed by minimization) and report the volumes used at each temperature. Without this, the comparison of the two free-energy curves is not well-defined.
  3. [Results and Fig. 4] No statistical uncertainties are reported for F_TI, ΔF_anh, or the resulting T_TI. The TI result is a numerical quadrature over λ combined with finite MD sampling, and the transition temperature is a crossing of two free-energy curves. The statement that TI gives 288 K, 'within 2 K of experiment,' has no error bar. Please report standard errors (e.g., from block averaging, multiple independent TI runs, or bootstrap over the integration) and propagate them to the transition temperature. This is necessary to assess whether the agreement with experiment is meaningful or fortuitous.
minor comments (6)
  1. [Abstract] The phrase 'existing theoretical approaches over- or underestimating' is grammatically awkward; suggest 'overestimating or underestimating' or a similar rewrite.
  2. [Fig. 6] The VACF spectra are shown at 10 K and 300 K. Since the transition is at 286 K, adding a spectrum just above Tαβ would more directly illustrate the claimed collapse of the 3–3.5 THz feature at the transition.
  3. [Methods, Eqs. (9)–(10)] The harmonic potential energy E_harm = (1/2)u^T Φ u uses displacements u from MD snapshots. Please clarify whether global translations/rotations are projected out before evaluating the harmonic energy, as these can artificially inflate the harmonic term.
  4. [Table 3] The 'Force cosine' column is not defined in the main text or caption. Since it is used as a metric for anharmonicity, a definition (e.g., average cosine of the angle between F_MD and F_harm) should be provided.
  5. [Title and Conclusions] The phrase 'Full Anharmonic Treatment' may be misleading because the TI/MD simulations treat nuclei classically. Consider adding the qualifier 'classical' or noting in the Conclusions that nuclear quantum effects are not included in the TI result.
  6. [Discussion] The comparison with Legrain and Manzhos (Ref. 6) is useful, but the quoted 1 meV/atom estimate is from a different electronic-structure and fitting context. A sentence noting the methodological differences beyond run length (e.g., reference potential and fixed-volume mapping) would improve the comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central result is a model-internal QHA-vs-TI prediction on a DFT-trained potential, not a fitted or self-citational reduction.

full rationale

The derivation chain is: (1) DFT-PBE training data are used to fit an ACE potential; (2) the same ACE potential is used for both quasi-harmonic phonon free energies (Eq. 1) and thermodynamic-integration free energies (Eqs. 2-3); (3) the transformation temperature is identified as the crossing of the two free-energy curves. No experimental transition temperature enters as a fitting target; the paper explicitly contrasts its approach with the Ravelo-Baskes MEAM, where 'the experimental value was used as an explicit fitting target.' The QHA and TI free energies are evaluated on the same potential energy surface by design, so their difference is a well-defined model-internal quantity; calling this difference the 'explicit anharmonic free energy' is an operational definition and a physical interpretation, not a circular derivation. The main substantive concern—that F_TI is computed classically while F_QHA is a quantum phonon free energy including zero-point and Bose-Einstein occupations—is a statistical-mechanics/accuracy limitation in the decomposition, not a circularity: no equation reduces to its own input, and no parameter is fitted to the resulting 288 K prediction. The self-citations to ACE/Pacemaker implementation references are methodological, not load-bearing uniqueness claims, and the paper's own caveat that 'we cannot exclude that the quantitative agreement benefits in part from a cancellation between an overestimated static energy and the anharmonic contribution' is a limitation on the absolute temperature, not evidence that the prediction is equivalent to the inputs used to construct the model. The agreement with the experimental 286 K is therefore a genuine prediction rather than a fitted or definitionally forced outcome.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central calculation rests on a fitted machine-learning potential and on treating classical TI free energies as comparable to quantum QHA free energies. No new physical entities are introduced, and no experimental transition temperature is used as a fitting target.

free parameters (1)
  • ACE potential parameters = Fitted to 4,427 DFT configurations; RMSE 5.7 meV/atom and 33.0 meV/Å on validation set
    All QHA and TI results are evaluated on this fitted potential energy surface. The anharmonic free energy and the 288 K transition temperature depend on how accurately the ACE fit represents PBE-DFT, especially for β-Sn at finite temperature.
assumptions (4)
  • domain assumption PBE-DFT with PAW pseudopotentials gives a sufficiently accurate 0 K energy difference and potential energy surface for tin
    The ACE potential is trained on PBE data, and the absolute transition temperature is directly controlled by the static ΔEαβ ≈ +42 meV/atom (ACE: +38 meV/atom). The authors acknowledge this limits the absolute prediction.
  • domain assumption Classical thermodynamic-integration free energies are directly comparable to quantum quasi-harmonic free energies at 286 K
    Eq. (1) uses quantum phonon occupations and zero-point energy, while Eqs. (2)–(3) use classical MD ensemble averages. The paper treats F_TI − F_QHA as purely anharmonic without quantifying nuclear-quantum corrections.
  • domain assumption The ACE potential transfers accurately to finite-temperature anharmonic configurations of β-Sn, not just training configurations
    Validation uses lattice parameters, elastic constants, and phonon dispersions; the finite-temperature anharmonicity is characterized by ACE-MD metrics, not by direct comparison to DFT-MD forces for β-Sn at 300 K.
  • domain assumption Metastable α-Sn can be thermodynamically integrated above 286 K without transforming to β-Sn during the λ path
    At T > 286 K, α-Sn is metastable. The Methods section describes TI without discussing constraints or barriers that prevent premature transformation during the coupling-parameter integration.

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Cite this review

Pith. "Pith review of Accurate Prediction of the $\alpha \to \beta$ Phase Transformation Temperature in Tin via Full Anharmonic Treatment." pith.science (2026). https://pith.science/paper/7SJ64WJV

@misc{pith2026260725978,
  author       = {Pith},
  title        = {Pith review of: Accurate Prediction of the $\alpha \to \beta$ Phase Transformation Temperature in Tin via Full Anharmonic Treatment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7SJ64WJV}},
  note         = {Machine review of arXiv:2607.25978}
}
abstract

Predicting the $\alpha \to \beta$ (grey-to-white) transition temperature in tin presents a longstanding challenge for atomistic simulations, with existing theoretical approaches over- or underestimating the experimental boundary (286 K) by up to several hundred Kelvin. In this work, we construct an Atomic Cluster Expansion (ACE) potential trained on density functional theory data to evaluate the finite-temperature free energies of both phases. Evaluated on the same potential energy surface, the quasi-harmonic approximation predicts a transformation temperature of 377 K, whereas full thermodynamic integration, which accounts for explicit vibrational anharmonicity, yields 288 K. This shift directly quantifies the explicit anharmonic free energy, which is substantial for metallic $\beta$-Sn but negligible for semiconducting $\alpha$-Sn. The anisotropic anharmonicity in $\beta$-Sn is corroborated by its excess heat capacity, temperature-driven renormalization of its vibrational spectrum, and deviations of its atomic forces and displacements from the harmonic reference. Our results demonstrate that capturing full lattice anharmonicity is essential for predicting the phase stability of tin, while the absolute transition temperature remains limited by the accuracy of the underlying 0 K energetics.

Figures

Figures reproduced from arXiv: 2607.25978 by the authors.

Figure 1
Figure 1. The basic scheme of the workflow used to predict the transformation temperature between α-Sn and β-Sn. Results Potential validation To validate the accuracy of the developed ACE potential, we first compare the optimized lattice parameters and the energy difference ∆Eαβ = Eβ − Eα with selected theoretical and experimental results in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Comparison of phonon band structures (top row) and phonon DOS (bottom row) for α-Sn (left column) and β-Sn (right column) from ACE calculations with available experimental data. Thermodynamic integration The accurate determination of vibrational entropy requires full consideration of explicit anharmonic effects. The ACE free￾energy curves obtained from TI calculations are plotted as solid lines in [PITH_FULL_IMAGE:… view at source ↗
Figure 3
Figure 3. Variations of the phonon densities of states (DOS) for α-Sn (left) and β-Sn (right) as functions of unit cell volumes predicted by ACE for QHA estimations of free energies. The color gradient indicates the volume per cell, with blue representing compressed and red representing expanded volumes [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Temperature dependence of Helmholtz free energies F(T) w.r.t. the energy of α-Sn at 0 K obtained via QHA and TI approaches (left panel) and the difference between TI and QHA contributions (right panel). The vertical dashed and full lines mark the α → β transition tempe…
Figure 5
Figure 5. Figure 5: Calculated QHA constant-pressure heat capacity Cp as a function of temperature for α-Sn (blue line) and β-Sn (red line). The dashed horizontal line represents the classical Dulong-Petit limit of 3R ≈ 24.94J/mol·K. The experimental data are compiled from a detailed comp…
Figure 6
Figure 6. Figure 6: Comparison of VACF and QHA phonon densities of states for α-Sn (left) and β-Sn (right) for selected temperatures. Anharmonicity analysis To further confirm and quantify the anharmonic effects at finite temperatures, we evaluated the vibrational DOS from the velocity au…
Figure 7
Figure 7. Figure 7: Comparison between harmonic forces predicted from the second-order force constants and the instantaneous ACE MD forces. Each point corresponds to one Cartesian component of one atom in one sampled MD configuration. The dashed diagonal indicates perfect agreement betwee…

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Works this paper leans on

33 extracted references · 3 canonical work pages

  1. [1]

    Farrow, R.et al.The growth of metastable, heteroepitaxial films of α-sn by metal beam epitaxy.Journal of Crystal Growth 54, 507–518 (1981)

    1981

  2. [2]

    & Smith, R

    Vnuk, F., De Monte, A. & Smith, R. The effect of pressure on the semiconductor-to-metal transition temperature in tin and in dilute sn-ge alloys.Journal of applied physics55, 4171–4176 (1984). 12/14

    1984

  3. [3]

    & Cohen, M

    Ihm, J. & Cohen, M. L. Equilibrium properties and the phase transition of grey and white tin.Phys. Rev. B23, 1576–1579 (1981). URLhttps://link.aps.org/doi/10.1103/PhysRevB.23.1576

    work page doi:10.1103/physrevb.23.1576 1981

  4. [4]

    & de Gironcoli, S

    Pavone, P., Baroni, S. & de Gironcoli, S. α↔β phase transition in tin: A theoretical study based on density-functional perturbation theory.Phys. Rev. B57, 10421–10423 (1998). URL https://link.aps.org/doi/10.1103/ PhysRevB.57.10421

    1998

  5. [5]

    & Park, C.-H

    Na, S.-H. & Park, C.-H. First-principles study of structural phase transition of sn.Journal of the Korean Physical Society 56, 494–497 (2010)

    2010

  6. [6]

    & Manzhos, S

    Legrain, F. & Manzhos, S. Understanding the difference in cohesive energies between alpha and beta tin in dft calculations. AIP Advances6(2016)

    2016

  7. [7]

    Friák, M.et al.Multi-methodological study of temperature trends in mössbauer effect in sn.Computational Materials Science215, 111780 (2022)

    2022

  8. [8]

    Physical Review Materials7, 053603 (2023)

    Chen, T.et al.Modeling the high-pressure solid and liquid phases of tin from deep potentials with ab initio accuracy. Physical Review Materials7, 053603 (2023). 9.Nitol, M. S.et al.Hybrid interatomic potential for Sn.Physical Review Materials7, 043601 (2023)

    2023

Show all 33 references
  1. [10]

    J.et al.Tin-pest problem as a test of density functionals using high-throughput calculations.Physical Review Materials5, 083608 (2021)

    Mehl, M. J.et al.Tin-pest problem as a test of density functionals using high-throughput calculations.Physical Review Materials5, 083608 (2021)

    2021

  2. [11]

    Houben, K.et al.In situ study of the α-sn to β-sn phase transition in low-dimensional systems: Phonon behavior and thermodynamic properties.Phys. Rev. B100, 075408 (2019). URL https://link.aps.org/doi/10.1103/ PhysRevB.100.075408

    2019

  3. [12]

    P., Burke, K

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple.Physical Review Letters77, 3865–3868 (1996)

    1996

  4. [13]

    & Lee, M

    Ko, W.-S., Kim, D.-H., Kwon, Y .-J. & Lee, M. Atomistic simulations of pure tin based on a new modified embedded-atom method interatomic potential.Metals8, 900 (2018)

    2018

  5. [14]

    & Pasquarello, A

    Haxhijaj, A., Riemelmoser, S. & Pasquarello, A. Lattice dynamics and structural phase stability of group-iv elemental solids with the r2SCAN functional.Phys. Rev. B113, 104105 (2026). URL https://link.aps.org/doi/10. 1103/znkf-3g37

    2026

  6. [15]

    Khvan, A.et al.Thermodynamic properties of tin: Part I experimental investigation, ab-initio modelling of α-, β-phase and a thermodynamic description for pure metal in solid and liquid state from 0 K.Calphad65, 50–72 (2019)

    2019

  7. [16]

    & Järvinen, M

    Merisalo, M. & Järvinen, M. Anharmonicity of lattice vibrations in white tin by x-ray measurement of almost-forbidden reflections.Philosophical Magazine B37, 233–240 (1978)

    1978

  8. [17]

    & Baskes, M

    Ravelo, R. & Baskes, M. Equilibrium and thermodynamic properties of grey, white, and liquid tin.Physical Review Letters 79, 2482–2485 (1997)

    1997

  9. [18]

    R.et al.Structural and dynamic properties of liquid tin from a new modified embedded-atom method force field

    Vella, J. R.et al.Structural and dynamic properties of liquid tin from a new modified embedded-atom method force field. Physical Review B95, 064202 (2017)

    2017

  10. [19]

    A., Baskes, M

    Etesami, S. A., Baskes, M. I., Laradji, M. & Asadi, E. Thermodynamics of solid sn and pbsn liquid mixtures using molecular dynamics simulations.Acta Materialia161, 320–330 (2018)

    2018

  11. [20]

    Chesser, I.et al.The structure and migration of twin boundaries in tetragonal β-Sn: An application of machine learning based interatomic potentials.Acta Materialia299, 121390 (2025)

    2025

  12. [21]

    Atomic cluster expansion for accurate and transferable interatomic potentials.Physical Review B99, 014104 (2019)

    Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials.Physical Review B99, 014104 (2019)

    2019

  13. [22]

    H., Forslund, A., Srinivasan, P

    Jung, J. H., Forslund, A., Srinivasan, P. & Grabowski, B. Dynamically stabilized phases with full ab initio accuracy: Thermodynamics of Ti, Zr, Hf with a focus on the hcp-bcc transition.Physical Review B108(2023)

    2023

  14. [23]

    L., Rowe, J

    Price, D. L., Rowe, J. M. & Nicklow, R. M. Lattice dynamics of grey tin and indium antimonide.Phys. Rev. B3, 1268–1279 (1971). URLhttps://link.aps.org/doi/10.1103/PhysRevB.3.1268

    1971 doi

  15. [24]

    M., Brockhouse, B

    Rowe, J. M., Brockhouse, B. N. & Svensson, E. C. Lattice dynamics of white tin.Phys. Rev. Lett.14, 554–556 (1965). URLhttps://link.aps.org/doi/10.1103/PhysRevLett.14.554. 25.Zdetsis, A. D. On the lattice dynamics of grey tin.Journal of Physics and Chemistry of Solids38, 1113–1...

    1965 doi

  16. [26]

    Rayne, J. A. & Chandrasekhar, B. S. Elastic constants of β-tin from 4.2 K to 300 K.Physical Review122, 1962–1962 (1961)

    1962

  17. [27]

    & Hafner, J

    Kresse, G. & Hafner, J. Ab initiomolecular dynamics for open-shell transition metals.Physical Review B48, 13115–13118 (1993)

    1993

  18. [28]

    & Furthmüller, J

    Kresse, G. & Furthmüller, J. Efficient iterative schemes forab initiototal-energy calculations using a plane-wave basis set. Physical Review B54, 11169–11186 (1996)

    1996

  19. [29]

    & Joubert, D

    Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method.Physical Review B59, 1758–1775 (1999)

    1999

  20. [30]

    Lysogorskiy, Y .et al.Performant implementation of the atomic cluster expansion (PACE) and application to copper and silicon.npj Computational Materials7(2021)

    2021

  21. [31]

    & Drautz, R

    Lysogorskiy, Y ., Bochkarev, A., Mrovec, M. & Drautz, R. Active learning strategies for atomic cluster expansion models. Physical Review Materials7, 043801 (2023)

    2023

  22. [32]

    P.et al.LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales.Computer Physics Communications271, 108171 (2022)

    Thompson, A. P.et al.LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales.Computer Physics Communications271, 108171 (2022)

    2022

  23. [33]

    R.et al.Kokkos 3: Programming model extensions for the exascale era.IEEE Transactions on Parallel and Distributed Systems33, 805–817 (2022)

    Trott, C. R.et al.Kokkos 3: Programming model extensions for the exascale era.IEEE Transactions on Parallel and Distributed Systems33, 805–817 (2022)

    2022

  24. [34]

    & Tanaka, I

    Togo, A., Chaput, L., Tadano, T. & Tanaka, I. Implementation strategies in phonopy and phono3py.J. Phys. Condens. Matter35, 353001 (2023). 35.Togo, A. First-principles phonon calculations with phonopy and phono3py.J. Phys. Soc. Jpn.92, 012001 (2023)

    2023

  25. [36]

    Materials and technologies for sustainable development

    Menon, S., Lysogorskiy, Y ., Rogal, J. & Drautz, R. Automated free-energy calculation from atomistic simulations.Phys. Rev. Mater .5, 103801 (2021). URLhttps://link.aps.org/doi/10.1103/PhysRevMaterials.5.103801. Acknowledgements MF acknowledges financial support from the Czech...

    2021 doi

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