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arxiv: 2604.10561 · v2 · pith:ZEL7HMGOnew · submitted 2026-04-12 · ❄️ cond-mat.stat-mech · cond-mat.mtrl-sci· physics.chem-ph

Location of the liquid-vapor critical point in aluminum

Xuyang Long , Kai Luo This is my paper

Pith reviewed 2026-05-10 15:48 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.mtrl-sciphysics.chem-ph
keywords aluminumliquid-vapor critical pointdeep potential molecular dynamicsspinodal analysisphase coexistenceDFTcritical phenomena
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The pith

The liquid-vapor critical point of aluminum sits at 6531-6576 K, 0.637 g/cm³ and 1.6 kbar.

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

The paper sets out to fix the long-disputed location of the point where liquid aluminum and its vapor become indistinguishable. Earlier estimates of the critical temperature differed by nearly 4000 K. The authors train a deep neural-network potential on accurate electronic-structure calculations, benchmark several density functionals against measured liquid densities, and then apply two independent simulation routes: analysis of the spinodal in the equation of state and large-scale coexistence runs that identify phases with a Gaussian mixture model. The two routes converge on the same narrow interval, reducing the temperature uncertainty to roughly 50 K.

Core claim

Using spinodal analysis of the equation of state and direct coexistence simulations with Gaussian mixture phase identification, the authors determine the liquid-vapor critical point of aluminum to be at 6531-6576 K, 0.637 g/cm³, and 1.6 kbar after selecting the PBEsol exchange-correlation functional for its agreement with experimental liquid densities.

What carries the argument

Deep potential molecular dynamics trained on PBEsol DFT data, combined with spinodal analysis of the equation of state and direct coexistence simulations identified by Gaussian mixture models.

If this is right

  • The reported values can be used directly in models of laser ablation, shock compression, and planetary interiors involving aluminum.
  • The same combination of deep-potential training, spinodal analysis, and Gaussian-mixture coexistence runs supplies a route for locating critical points in other metals.
  • Temperature uncertainty of only ~50 K replaces the previous spread of thousands of kelvin.
  • The framework is stated to be transferable, so the same workflow can be repeated for other metallic elements under extreme conditions.

Where Pith is reading between the lines

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

  • The method could be applied to metals for which no experimental critical-point data exist at all.
  • If the choice of functional remains the dominant uncertainty, systematic tests of additional functionals on the same deep-potential workflow would further tighten the result.
  • The narrowed critical parameters could be inserted into existing hydrodynamic codes to check whether predicted ablation depths or shock Hugoniots shift measurably.

Load-bearing premise

The PBEsol exchange-correlation functional, chosen after benchmarking against experimental liquid densities, together with the deep potential trained on its data, correctly describes the equation of state and phase behavior near the critical point.

What would settle it

An experimental measurement of the critical temperature in aluminum that lies outside the interval 6531-6576 K would show the simulated location to be incorrect.

Figures

Figures reproduced from arXiv: 2604.10561 by Kai Luo, Xuyang Long.

Figure 1
Figure 1. Figure 1: FIG. 1: Bulk liquid density of Al at different temperatures from experiments and theoretical calculations at ambient [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Equation of state isotherms for Al using the DP. The standard deviation has been significantly reduced (see [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Liquid-vapor coexistence curve for Al obtained from DP temperature quench molecular dynamics with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The vapor pressure-temperature curve in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Vapor pressure-temperature relation for Al obtained from liquid-vapor coexistence simulations. Early [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

The precise location of the liquid-vapor critical point in aluminum has remained elusive for decades, with reported critical temperatures spanning nearly 4000 K. Here we resolve this long-standing uncertainty by combining deep potential molecular dynamics with large-scale simulations trained on high-fidelity electronic-structure data. We benchmark multiple exchange-correlation functionals against experimental liquid densities and identify PBEsol as providing the most consistent description. Using complementary approaches -- spinodal analysis of the equation of state and direct coexistence simulations with Gaussian mixture phase identification -- we converge on a critical temperature of 6531-6576 $^\circ$K, a critical density of $0.637$ g/cm$^{3}$, and a critical pressure of $1.6$ kbar. The precision of these values, with temperature uncertainties of $\sim$50 K, represents a marked improvement over previous estimates. Our framework establishes a transferable strategy for predicting critical phenomena in metals, with implications for laser ablation, shock compression, and planetary modeling under extreme conditions.

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

2 major / 1 minor

Summary. The manuscript determines the liquid-vapor critical point of aluminum via deep-potential molecular dynamics trained on PBEsol DFT data. Benchmarking several exchange-correlation functionals against experimental liquid densities selects PBEsol; two complementary large-scale methods—spinodal analysis of the equation of state and direct coexistence simulations with Gaussian-mixture phase identification—converge on Tc = 6531–6576 K, ρc = 0.637 g cm⁻³ and Pc = 1.6 kbar, with temperature uncertainties of ~50 K.

Significance. If the underlying functional remains accurate in the low-density regime, the work supplies the tightest computational bounds yet on aluminum’s critical parameters and demonstrates a transferable workflow for metallic critical phenomena. The dual-method convergence and use of machine-learned potentials trained on independent electronic-structure data constitute clear technical strengths.

major comments (2)
  1. [Benchmarking paragraph] Benchmarking paragraph (abstract and methods): PBEsol is selected solely on the basis of agreement with experimental liquid densities near 2.7 g cm⁻³. The reported critical density (0.637 g cm⁻³) lies far below this regime, where metallic cohesion weakens and possible non-metallic character appears; no additional validation (e.g., vapor pressures, cohesive energies at low density, or comparison with higher-level functionals) is provided to confirm that the same functional error cancellation holds near the spinodal.
  2. [Results section] Results section on spinodal and coexistence: Both the equation-of-state spinodal analysis and the direct-coexistence runs employ the identical PBEsol-trained deep potential. Their numerical agreement therefore tests only internal consistency of the potential, not whether the underlying DFT surface correctly locates the binodal or spinodal at low density. An independent check against a second functional or against experimental proxies in the critical region would be required to support the central claim.
minor comments (1)
  1. [Abstract] Notation: the symbol °K is non-standard; thermodynamic temperatures should be reported in K without the degree symbol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the technical strengths, and constructive comments on functional validation. We address each major comment below and have revised the manuscript with added discussion and clarifications.

read point-by-point responses

  1. Referee: Benchmarking paragraph (abstract and methods): PBEsol is selected solely on the basis of agreement with experimental liquid densities near 2.7 g cm⁻³. The reported critical density (0.637 g cm⁻³) lies far below this regime, where metallic cohesion weakens and possible non-metallic character appears; no additional validation (e.g., vapor pressures, cohesive energies at low density, or comparison with higher-level functionals) is provided to confirm that the same functional error cancellation holds near the spinodal.

    Authors: We appreciate the referee highlighting this limitation. Our selection of PBEsol was driven by the availability of experimental benchmarks for the dense liquid; we have now expanded the Methods section with additional literature comparisons of PBEsol performance for aluminum cohesive energies and other properties across densities. We have also added an explicit limitations paragraph in the Discussion acknowledging that error cancellation may differ near the spinodal and that the critical-point values are conditional on the functional choice. Direct new DFT calculations at critical densities were not feasible within the study scope. revision: partial

  2. Referee: Results section on spinodal and coexistence: Both the equation-of-state spinodal analysis and the direct-coexistence runs employ the identical PBEsol-trained deep potential. Their numerical agreement therefore tests only internal consistency of the potential, not whether the underlying DFT surface correctly locates the binodal or spinodal at low density. An independent check against a second functional or against experimental proxies in the critical region would be required to support the central claim.

    Authors: We agree that the two-method convergence primarily confirms internal consistency of the workflow rather than absolute DFT accuracy at low density. In the revised Results section we now explicitly distinguish these aspects and have added cross-references to prior theoretical studies of aluminum using alternative functionals or methods for context. Performing a full independent check would require new DFT data generation and potential retraining, which lies beyond the present computational resources; we have therefore included a statement on possible systematic uncertainties arising from the functional. revision: partial

Circularity Check

0 steps flagged

No significant circularity in simulation-derived critical parameters

full rationale

The derivation proceeds by first benchmarking several DFT functionals against experimental liquid densities at ambient conditions to select PBEsol, training a deep potential on the resulting DFT data, and then executing two independent large-scale simulation protocols (spinodal analysis of the equation of state and direct coexistence with Gaussian-mixture phase labeling) whose outputs are the reported critical temperature, density, and pressure. These critical values are not used in the functional selection, potential training, or any fitting step; they emerge as predictions from the model. No self-definitional equations, fitted inputs relabeled as predictions, or load-bearing self-citations appear in the chain. Internal consistency between the two simulation methods constitutes a cross-check rather than a reduction to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard DFT approximations and machine-learned potentials without introducing new physical entities or free parameters beyond the choice of functional.

axioms (2)
  • domain assumption PBEsol provides the most consistent description of aluminum liquid densities among tested functionals
    Identified via benchmarking against experimental data as stated in the abstract
  • domain assumption Deep potentials trained on high-fidelity electronic-structure data faithfully reproduce the underlying potential energy surface for large-scale simulations near the critical point
    Implicit in the use of the trained model for spinodal and coexistence simulations

pith-pipeline@v0.9.0 · 5474 in / 1320 out tokens · 34396 ms · 2026-05-10T15:48:16.488545+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    We employ the temperature quench molecular dynamics (TQMD) to obtain the liquid-vapor coexistence curve (see Section II D)

    (circle), Likalter [13] (square), Hess[14] (upper triangle), Singh[15](down triangle), Ray[16] (right triangle), Lomonosov[17] (left triangle), Gordeev[19] (diamond), Povarnitsyn [18] (pentagon), and Faussurier[20](cross). We employ the temperature quench molecular dynamics (TQMD) to obtain the liquid-vapor coexistence curve (see Section II D). We choose ...

    work page 2000

  2. [2]

    A. S. V. Ramana, Molecular dynamics simulation of liquid-vapor coexistence curves of metals, Journal of Physics: Con- ference Series377, 012086 (2012)

    work page 2012

  3. [3]

    T. Ma, T. D¨ oppner, R. W. Falcone, L. Fletcher, C. Fortmann, D. O. Gericke, O. L. Landen, H. J. Lee, A. Pak, J. Vor- berger, K. W¨ unsch, and S. H. Glenzer, X-Ray scattering measurements of strong Ion-Ion correlations in shock-compressed aluminum, Phys. Rev. Lett.110, 065001 (2013)

    work page 2013

  4. [4]

    R. W. Lemke, D. H. Dolan, D. G. Dalton, J. L. Brown, K. Tomlinson, G. R. Robertson, M. D. Knud- son, E. Harding, A. E. Mattsson, J. H. Carpenter, R. R. Drake, K. Cochrane, B. E. Blue, A. C. Robinson, 10 and T. R. Mattsson, Probing off-Hugoniot states in Ta, Cu, and Al to 1000 GPa compression with magneti- cally driven liner implosions, Journal of Applied ...

    work page doi:10.1063/1.4939675/14105290/015904 2016

  5. [5]

    V. I. Mazhukin, O. N. Koroleva, M. M. Demin, A. V. Shapranov, and A. A. Aleksashkina, Atomistic simulation of the coexistence of liquid–vapor phase states for gold and determination of critical parameters, Mathematical Models and Computer Simulations14, 819 (2022)

    work page 2022

  6. [6]

    M. K. Lentz, M. P. Desjarlais, and J. P. Townsend, Liquid–vapor coexistence of platinum from ab initio simulations, The Journal of Chemical Physics164, 094119 (2026)

    work page 2026

  7. [7]

    Vidal, T

    F. Vidal, T. W. Johnston, S. Laville, O. Barth´ elemy, M. Chaker, B. Le Drogoff, J. Margot, and M. Sabsabi, Critical-point phase separation in laser ablation of conductors, Phys. Rev. Lett.86, 2573 (2001)

    work page 2001

  8. [8]

    J. P. Colombier, P. Combis, F. Bonneau, R. Le Harzic, and E. Audouard, Hydrodynamic simulations of metal ablation by femtosecond laser irradiation, Phys. Rev. B71, 165406 (2005)

    work page 2005

  9. [9]

    Fishburn, M

    J. Fishburn, M. Withford, D. Coutts, and J. Piper, Study of the fluence dependent interplay between laser induced material removal mechanisms in metals: Vaporization, melt displacement and melt ejection, Applied Surface Science252, 5182 (2006)

    work page 2006

  10. [10]

    M. D. Knudson and M. P. Desjarlais, Equation of state and transport property measurements of warm dense matter (2009)

    work page 2009

  11. [11]

    Terragni and A

    J. Terragni and A. Miotello, Laser ablation of aluminum near the critical regime: A computational gas-dynamical model with temperature-dependent physical parameters, Micromachines12, 10.3390/mi12030300 (2021)

    work page doi:10.3390/mi12030300 2021

  12. [12]

    Morel, A

    V. Morel, A. Bultel, and B. Ch´ eron, The critical temperature of aluminum, International Journal of Thermophysics30, 1853 (2009)

    work page 2009

  13. [13]

    D. A. Young and B. J. Alder, Critical point of metals from the van der Waals model, Phys. Rev. A3, 364 (1971)

    work page 1971

  14. [14]

    A. A. Likalter, Equation of state of metallic fluids near the critical point of phase transition, Phys. Rev. B53, 4386 (1996)

    work page 1996

  15. [15]

    H. Hess, Z. Metallkd.89(1998)

    work page 1998

  16. [16]

    J. K. Singh, J. Adhikari, and S. K. Kwak, Vapor-liquid phase coexistence curves for Morse fluids, Fluid Phase Equilibria 248, 1 (2006)

    work page 2006

  17. [17]

    A. RAY, M. SRIVASTAVA, G. KONDAYYA, and S. MENON, Improved equation of state of metals in the liquid-vapor region, Laser and Particle Beams24, 437 (2006)

    work page 2006

  18. [18]

    Lomonosov, Multi-phase equation of state for aluminum, Laser and Particle Beams25, 567 (2007)

    I. Lomonosov, Multi-phase equation of state for aluminum, Laser and Particle Beams25, 567 (2007)

    work page 2007

  19. [19]

    Povarnitsyn, K

    M. Povarnitsyn, K. Khishchenko, and P. Levashov, Simulation of shock-induced fragmentation and vaporization in metals, International Journal of Impact Engineering35, 1723 (2008)

    work page 2008

  20. [20]

    D. G. Gordeev, L. F. Gudarenko, M. V. Zhernokletov, V. G. Kudel’kin, and M. A. Mochalov, Semi-empirical equation of state of metals. equation of state of aluminum, Combustion, Explosion, and Shock Waves44, 177 (2008)

    work page 2008

  21. [21]

    Faussurier, C

    G. Faussurier, C. Blancard, and P. L. Silvestrelli, Evaluation of aluminum critical point using an ab initio variational approach, Phys. Rev. B79, 134202 (2009)

    work page 2009

  22. [22]

    L. P. Kadanoff, Scaling laws for Ising models nearT c, Physics Physique Fizika2, 263 (1966)

    work page 1966

  23. [23]

    K. G. Wilson, Renormalization group and critical phenomena. i. renormalization group and the kadanoff scaling picture, Phys. Rev. B4, 3174 (1971)

    work page 1971

  24. [24]

    K. G. Wilson, Renormalization group and critical phenomena. ii. phase-space cell analysis of critical behavior, Phys. Rev. B4, 3184 (1971)

    work page 1971

  25. [25]

    K. G. Wilson, The renormalization group: Critical phenomena and the kondo problem, Rev. Mod. Phys.47, 773 (1975)

    work page 1975

  26. [26]

    Behler and M

    J. Behler and M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy surfaces, Phys. Rev. Lett.98, 146401 (2007)

    work page 2007

  27. [27]

    Behler, Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations, Phys

    J. Behler, Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations, Phys. Chem. Chem. Phys.13, 17930 (2011)

    work page 2011

  28. [28]

    Zhang, H

    L. Zhang, H. Wang, R. Car, and W. E, Phase diagram of a Deep Potential water model, Phys. Rev. Lett.126, 236001 (2021)

    work page 2021

  29. [29]

    Luo, Supplementary: Location of the liquid-vapor critical point in aluminum (2026)

    K. Luo, Supplementary: Location of the liquid-vapor critical point in aluminum (2026)

    work page 2026

  30. [30]

    Chen, G.-C

    M. Chen, G.-C. Guo, and L. He, Systematically improvable optimized atomic basis sets for ab initio calculations, Journal of Physics: Condensed Matter22, 445501 (2010)

    work page 2010

  31. [31]

    D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B88, 085117 (2013)

    work page 2013

  32. [32]

    Kleinman and D

    L. Kleinman and D. M. Bylander, Efficacious form for model pseudopotentials, Phys. Rev. Lett.48, 1425 (1982)

    work page 1982

  33. [33]

    Huang, ABACUS Pseudopot-Nao Square (APNS) project, https://kirk0830.github.io/abacus-pseudopot-nao- square/index.html

    Y. Huang, ABACUS Pseudopot-Nao Square (APNS) project, https://kirk0830.github.io/abacus-pseudopot-nao- square/index.html

    work page

  34. [34]

    J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly constrained and appropriately normed semilocal density functional, Physical Review Letters115, 036402 (2015)

    work page 2015

  35. [35]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett.77, 3865 (1996)

    work page 1996

  36. [36]

    Zhang, J

    L. Zhang, J. Han, H. Wang, R. Car, and W. E, Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics, Phys. Rev. Lett.120, 143001 (2018)

    work page 2018

  37. [37]

    Zhang, H

    Y. Zhang, H. Wang, W. Chen, J. Zeng, L. Zhang, H. Wang, and W. E, DP-GEN: A concurrent learning platform for the generation of reliable deep learning based potential energy models, Computer Physics Communications253, 107206 (2020)

    work page 2020

  38. [38]

    A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS - a flexible 11 simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comp. Phys. Comm. 27...

    work page 2022

  39. [39]

    Y. S. Touloukian,Recommended values of the thermophysical properties of eight alloys, major constituents and their oxides, Tech. Rep. (1966)

    work page 1966

  40. [40]

    Gathers, Thermophysical properties of liquid copper and aluminum, International journal of Thermophysics4, 209 (1983)

    G. Gathers, Thermophysical properties of liquid copper and aluminum, International journal of Thermophysics4, 209 (1983)

    work page 1983

  41. [41]

    E. A. Brandes and G. Brook,Smithells metals reference book(Elsevier, 2013)

    work page 2013

  42. [42]

    M. J. Assael, A. E. Kalyva, K. D. Antoniadis, R. Michael Banish, I. Egry, J. Wu, E. Kaschnitz, and W. A. Wakeham, Reference data for the density and viscosity of liquid copper and liquid tin, Journal of Physical and Chemical Reference Data39, 033105 (2010)

    work page 2010

  43. [43]

    Schmitz, B

    J. Schmitz, B. Hallstedt, J. Brillo, I. Egry, and M. Schick, Density and thermal expansion of liquid Al–Si alloys, Journal of Materials Science47, 3706 (2012)

    work page 2012

  44. [44]

    Leitner, T

    M. Leitner, T. Leitner, A. Schmon, K. Aziz, and G. Pottlacher, Thermophysical properties of liquid aluminum, Metallurgical and Materials Transactions A48, 3036 (2017)

    work page 2017

  45. [45]

    J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Physical review letters100, 136406 (2008)

    work page 2008

  46. [46]

    A. D. Becke, Density-functional thermochemistry. iii. the role of exact exchange, The Journal of chemical physics98, 5648 (1993)

    work page 1993

  47. [47]

    C. Lee, W. Yang, and R. G. Parr, Development of the colle-salvetti correlation-energy formula into a functional of the electron density, Physical review B37, 785 (1988)

    work page 1988

  48. [48]

    Grimme, J

    S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, The Journal of Chemical Physics132, 154104 (2010)

    work page 2010

  49. [49]

    Grimme, S

    S. Grimme, S. Ehrlich, and L. Goerigk, Effect of the damping function in dispersion corrected density functional theory, Journal of Computational Chemistry32, 1456 (2011)

    work page 2011

  50. [50]

    L. D. Gelb and E. A. M¨ uller, Location of phase equilibria by temperature-quench molecular dynamics simulations, Fluid Phase Equilibria203, 1 (2002)

    work page 2002

  51. [51]

    J. S. Rowlinson and B. Widom,Molecular theory of capillarity(Courier Corporation, 2013)

    work page 2013

  52. [52]

    Statt, H

    A. Statt, H. Casademunt, C. P. Brangwynne, and A. Z. Panagiotopoulos, Model for disordered proteins with strongly sequence-dependent liquid phase behavior, The Journal of Chemical Physics152, 075101 (2020)

    work page 2020

  53. [53]

    Zinn-Justin, Precise determination of critical exponents and equation of state by field theory methods, Physics Reports 344, 159 (2001)

    J. Zinn-Justin, Precise determination of critical exponents and equation of state by field theory methods, Physics Reports 344, 159 (2001)

    work page 2001

  54. [54]

    Sengers and J

    J. Sengers and J. L. Sengers, Thermodynamic behavior of fluids near the critical point, Annual Review of Physical Chemistry 37, 189 (1986)

    work page 1986

  55. [55]

    A. J. Liu and M. E. Fisher, The three-dimensional Ising model revisited numerically, Physica A: Statistical Mechanics and its Applications156, 35 (1989)

    work page 1989

  56. [56]

    Bhatt, A

    D. Bhatt, A. W. Jasper, N. E. Schultz, J. I. Siepmann, and D. G. Truhlar, Critical properties of aluminum, Journal of the American Chemical Society128, 4224 (2006), https://doi.org/10.1021/ja0577950

    work page doi:10.1021/ja0577950 2006

  57. [57]

    M. P. Desjarlais, Quantum molecular dynamics simulations for generating equation of state data, AIP Conference Pro- ceedings1161, 32 (2009), https://pubs.aip.org/aip/acp/article-pdf/1161/1/32/12145787/32 1 online.pdf

    work page 2009

  58. [58]

    Luo, Zenodo, https://doi.org/10.5281/zenodo.17213314 (2026)

    K. Luo, Zenodo, https://doi.org/10.5281/zenodo.17213314 (2026)

    work page doi:10.5281/zenodo.17213314 2026

  59. [59]

    N. B. Wilding, Critical-point and coexistence-curve properties of the Lennard-Jones fluid: A finite-size scaling study, Phys. Rev. E52, 602 (1995)

    work page 1995

  60. [60]

    Hultgren and A

    R. Hultgren and A. S. for Metals,Selected Values of the Thermodynamic Properties of the Elements(American Society for Metals, 1973)

    work page 1973

  61. [61]

    D. R. Stull, Inorganic compounds, Industrial & Engineering Chemistry39, 540 (1947), https://doi.org/10.1021/ie50448a023

    work page doi:10.1021/ie50448a023 1947