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REVIEW 3 major objections 6 minor 31 references

Thermoelastic equation of state and melting of Mg metal at high pressure and high temperature

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read One fitted parameter, $\delta_T = 1.5(5)$, describes magnesium's volume up to 20 GPa and 1500 K.

desk verdict Useful Mg p-V-T compaction, but the fit quality and pressure-scale circularity need work before δT = 1.5(5) is a number to rely on. read the letter →

arxiv 1909.00645 v1 pith:ECYNVHY2 submitted 2019-09-02 cond-mat.mtrl-sci

Alexandre Courac , Yann Le Godec , Vladimir L. Solozhenko , Nicolas Guignot , Wilson A. Crichton This is my paper · ORCID
classification cond-mat.mtrl-sci PACS 61.05.cp64.30.Jk
keywords magnesiumequationofstatehighpressuretemperatureAnderson-GrüneisenparameterthermalexpansionmeltingcurvesynchrotronX-raydiffraction
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

This paper reports a pressure-volume-temperature equation of state for magnesium metal, measured by synchrotron X-ray diffraction up to 20 GPa and 1500 K. The central claim is that the whole measured p-V-T surface can be described without new compression or thermal-expansion fits: the authors fix the 300-K compression curve and the ambient-pressure thermal expansion from earlier work and add one fitted parameter, the Anderson-Grüneisen coefficient $\delta_T = 1.5(5)$, which controls how pressure suppresses thermal expansion. With that single parameter, calculated isobars reproduce all experimental volumes down to $V/V_0 = 0.75$, a range in which the previously published equation of state for Mg was thermodynamically inconsistent, with crossing isotherms. The same experiments locate melting by X-ray diffraction and by a sharp electrical-resistance drop, confirming earlier melting data and giving a zero-pressure melting slope of $60(5)$ K/GPa. If correct, the result gives experimentalists a simple, internally consistent pressure-temperature standard for high-pressure synthesis and phase-equilibrium work on magnesium.

What carries the argument

The central object is the Anderson-Grüneisen parameter $\delta_T$, the exponent that controls how quickly thermal expansion shrinks as the material is compressed. The paper uses its integrated form, Eq. (2), which combines a 300-K compression curve and an ambient-pressure thermal-expansion polynomial into one closed expression for $V(p,T)$; here the only unknown is $\delta_T$. The fitted value $\delta_T = 1.5(5)$ makes the equation a complete analytical model of the measured p-V-T domain, while the compression and thermal-expansion anchors are taken from prior determinations rather than refit.

What would settle it

Measure Mg unit-cell volumes along a 1200-K isotherm from 5 to 25 GPa and compare each point with Eq. (2) computed at $\delta_T = 1.5(5)$: a systematic residual that grows as $V/V_0$ falls below 0.75, or a best-fit $\delta_T$ that changes between low- and high-pressure windows, would falsify the constant-$\delta_T$ claim.

Watch

Extended reading notes

Core claim

The paper claims that magnesium's volume between 0.1 MPa and 20 GPa and between 300 K and 1500 K is governed by the integrated Anderson-Grüneisen relation with a constant $\delta_T = 1.5(5)$, using $V(p,300)$ fixed to the 300-K Murnaghan fit with $B_0 = 32.5$ GPa and $B_0' = 3.73(2)$, and $V(0,T)$ fixed to the ambient-pressure polynomial with $a = 25(2)\times10^{-6}$ K$^{-1}$ and $b = 9.4(4)\times10^{-9}$ K$^{-2}$. With only that one fitted parameter, the model reproduces all measured volumes down to a relative volume of 0.75, which the authors take as their test of success. The paper also claims that melting points identified from disappearance of the diffraction pattern and from a drop in furnace resistance agree with the established Mg melting curve, and it reports a zero-pressure melting slope $dT/dp = 60(5)$ K/GPa. In the authors' reading, this replaces the earlier analytical equation of state for Mg that produced crossing isotherms at high temperature.

Load-bearing premise

The fit assumes a single Anderson-Grüneisen exponent $\delta_T$ stays constant over the whole 0-20 GPa, 300-1500 K range; if $\delta_T$ drifts with compression, the calculated isobars will be biased at high pressure, and the paper presents no direct test of that constancy.

Editorial extensions

If this is right

  • The resulting equation of state can refine pressure-temperature conditions in high-pressure experiments that contain Mg, acting as an internal p-T standard up to 20 GPa and 1500 K.
  • The thermodynamically consistent form, with no crossing isobars, provides a more reliable input for phase-equilibrium and synthesis calculations involving Mg-bearing systems at high pressure and temperature.
  • The melting observations from both X-ray diffraction and electrical resistance confirm the established Mg melting curve, including the zero-pressure slope of 60(5) K/GPa.
  • The reported decrease of the melting slope with pressure, to about 38(4) K/GPa near 10 GPa, is consistent with a liquid more compressible than the solid.
  • The combined dataset from three synchrotron facilities establishes a benchmark p-V-T dataset for magnesium across nearly the full solid domain.

Reading between the lines

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

  • Inference: because $\delta_T = 1.0$ and $2.0$ also fit the data almost as well, the present dataset constrains $\delta_T$ only loosely; a sharper determination needs compressions below $V/V_0 = 0.75$ or lower-temperature isotherms, not more points in the same p-T box.
  • Inference: the same fixed-anchor-plus-one-parameter scheme may transfer directly to other hexagonal metals whose 300-K compression and ambient-pressure expansion are already well known, provided their $\delta_T$ is also constant over the studied range.
  • Inference: the sharp resistance drop at melting suggests a cheap and general way to detect melting in metal-carbon high-pressure assemblies, particularly when recrystallization makes diffraction-based melting detection ambiguous.
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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 reports new p-V-T (volume-pressure-temperature) measurements on hcp Mg metal up to 20 GPa and 1500 K, obtained by synchrotron X-ray diffraction in Paris-Edinburgh, multianvil, and large-volume presses, plus electrical-resistivity monitoring of melting. The authors describe the data with a thermodynamically consistent Anderson-Grüneisen equation of state in which the 300-K compression curve and ambient-pressure thermal expansion are taken from prior work, leaving a single fitted Anderson-Grüneisen parameter delta_T = 1.5(5). They further report melting observations that agree, within uncertainties, with previously published melting curves, including a zero-pressure slope of 60(5) K/GPa.

Significance. If the central claim holds, the paper provides a simple, single-parameter thermoelastic description of Mg up to compression V/V0 = 0.75 and to melting temperatures, which is useful for high-pressure experiments and for modeling Mg-bearing planetary and synthesis systems. The study also adds new melting data obtained by two independent methods, supporting previous results. The strength of the paper is the breadth of new experimental data and the use of a thermodynamically consistent model with a minimal number of fitted parameters. However, the key quantitative claim is currently supported mainly by a visual comparison of isobars to data, and the pressure-temperature coordinates of some data are assigned using the very same 300-K Mg equation of state that is fixed in the model, so the independence of the fit is not fully established.

major comments (3)
  1. [Section 2, Section 4, Table 1] The pressure and temperature calibration introduces a circularity that is not quantified. Section 2 states that pressure and temperature were estimated using the 300-K equations of state of Mg and MgO, and that 'knowledge of Mg melting curve also allowed to refine the pressure or temperature values in some cases.' Since Eq. (4) fixes the 300-K compression curve to exactly the Stinton et al. EoS (B0 = 32.5, B0' = 3.73), any high-temperature point whose pressure is assigned with this Mg EoS makes the residual between the measured V/V0 and the model a consistency check rather than an independent test. This is particularly evident in Table 1, where the caption states that pressures for the data of ref. 6 were 'reestimated using Mg equation of state 6.' The manuscript does not report how often each pressure standard was the primary gauge, nor how the fitted delta_T changes when the pressure is assigned from MgO, hBN, or Si-based calibrations instead. Without this sensitivity analysis, the central claim that a single delta_T = 1.5(5) describes all data is not fully supported.
  2. [Section 4, Fig. 3a] The quality of the fit is assessed only graphically, by 'color match' between theoretical domains and symbols in Fig. 3a. No residuals, root-mean-square deviations, or per-point uncertainties are reported for the p-V-T data in Tables 1 and 2, and the symbols in Fig. 3a appear without error bars. As a result, the claim of 'good agreement ... to relative volumes of 0.75' is not quantitatively substantiated. The authors should provide a residual analysis (e.g., V/V0_model - V/V0_obs versus pressure or temperature), state the fitted metric (least-squares, weighted?) and the resulting uncertainty on delta_T, and report how that uncertainty propagates from the data scatter. This is load-bearing because the entire contribution rests on the claim that a one-parameter fit describes the data within uncertainty.
  3. [Section 3, Eq. (2), Section 4] The constancy of the Anderson-Grüneisen parameter delta_T over the entire p-T domain is assumed by the integration of Eq. (1) into Eq. (2), and the paper provides no test of this assumption. The statement that delta_T = 1.0 and 2.0 also give 'reasonable agreement' shows only that the fit is insensitive, which is itself a concern because it implies the data have limited power to constrain delta_T. The authors should test the assumption, for example by fitting delta_T separately in different pressure or compression bins, or by plotting the residuals as a function of V/V0 to look for systematic deviations at high compression. If the constancy assumption fails, the reported delta_T would be an average over the p-T path rather than a physical constant, and the extrapolative use of Eq. (2) would be unjustified.
minor comments (6)
  1. [Abstract and Section 4] The abstract reports B0 = 32.5(1) GPa, whereas Section 4 and Table 1 contextual material give B0 = 32.5(2) GPa; please harmonize the values and their uncertainties.
  2. [Abstract and Section 4] The abstract gives a = 25(2)x10-6 K-1 and b = 9.4(4)x10-9 K-2, but Section 4 lists only central values without uncertainties; the uncertainties should be given consistently at first use.
  3. [Section 3, Eq. (3)] In Eq. (3) the polynomial is written as [1+a(T-273)+b(T-273)^2 - a(300-273)+b(300-273)^2]^3; the sign of the constant term appears to be an error (the subtracted term should be a(27)+b(27)^2, not minus a(27) plus b(27)^2). Please check and correct the expression.
  4. [Table 1 and Table 2] The tables list V/V0 values without any reported uncertainty. Since the paper emphasizes agreement with a model to 'high accuracy of relative volume' for the 300-K EoS, giving at least representative uncertainties would help the reader judge the fit.
  5. [Fig. 3 caption] The phrase 'Grey5grid area' should read 'Grey grid area' and the description of symbols would be clearer if listed with the same symbols as the figure.
  6. [References] Reference 24 contains an apparent DOI typo ('1029b01108' should likely be '1029b01108' or another identifier); please verify and correct.

Circularity Check

3 steps flagged · score 5.0 of 10

Pressure coordinates are partly assigned using the same 300-K Mg EoS that anchors Eq. (4), and Table 1 pressures are explicitly reestimated to agree with observed volumes; the fitted δT comparison is a consistency check rather than an independent prediction.

  1. self definitional [Section 2 (Experimental, PE and ESRF) and Section 4 (Eq. 4 anchor)]
    "Pressure and temperature estimations were made using 300-K equations of states of Mg 6 and MgO17 and temperature calibration curve obtained using Si melting point at HPHT. ... Temperatures and pressures were monitored using the 300-K equations of state of MgO 17 and Mg 6 ... The best fit of experimental data has been obtained using the 300-K equation of state of Stinton et al. 6 (coefficients for Eq. 4 are B0 = 32.5(2) GPa, B'0 = 3.73(2), reproducing the data6 up to 25 GPa...)."

    The p,T coordinates used to constrain the model are partly assigned with the same 300-K Mg EoS (ref. 6) that is then fixed as the 300-K anchor in Eq. (4). A data point whose pressure was read from the Mg gauge therefore inherits the model's compression curve; its residual against Eqs. (2)-(4) is not an independent test. The paper does not state how often the Mg EoS was the primary gauge versus MgO/hBN, so the fitted δT = 1.5(5) may be biased toward the calibration input.

  2. self definitional [Table 1 note]
    "The V/V0 values were evaluated from Fig. 4 of ref. 6, pressure at 300 K (value in parences) was reestimated using Mg equation of state 6 and better agree with observed relative volumes of Mg and with the fact that at 1315 K Mg still remain solid."

    This is an explicit adjustment of the pressure coordinate using the same Mg EoS that is the fixed 300-K input to Eq. (4), with the criterion of 'better agree[ment]' with the observed volumes. Those re-estimated points are then included in the p-V-T dataset used to fit δT and to display 'good agreement' with the isobars; their agreement is imposed by the choice of pressure scale, not measured independently.

1 more flagged steps
  1. other [Section 2 (Experimental) and Section 4 (melting slope)]
    "The knowledge of Mg melting curve also allowed to refine the pressure or temperature values in some cases. ... The in situ observations of melting at different synchrotron facilities (Fig. 2) and by resistance measurements (Fig. 1b) are presented at Fig. 3b. Our experimental value of the zero-pressure melting slope is dT/dp = 60(5) K GPa-1, in agreement with previous resistivity measurements, of 60(2) K GPa-1."

    If the p,T conditions of some melting observations were 'refine[d]' using the known Mg melting curve, then those observations are not fully independent confirmations of that same melting curve. The agreement of the reported slope with previous values is partly seeded by the prior curve used during data reduction, although the paper says only 'in some cases' and combines XRD and resistance data.

full rationale

The derivation of Eqs. (2)-(4) is not itself circular: Eq. (2) is the stated integral of Eq. (1) under constant δT, and the 300-K compression and ambient-pressure thermal expansion are taken from prior external measurements (refs. 6, 11). The paper is also transparent that δT is fitted to all p-V-T data, so displaying the fitted isobars against those same data is a quality-of-fit statement, not an out-of-sample prediction. The circularity burden is in the pressure calibration loop. The experimental section states that pressures/temperatures were estimated using the 300-K EoS of Mg (ref. 6) and MgO, while Eq. (4) is fixed to the same Stinton Mg EoS; the Table 1 note goes further and explicitly re-estimates some literature pressures with that same Mg EoS to 'better agree' with observed volumes. Those data are then used to fit δT and to claim agreement to V/V0 = 0.75. Similarly, melting p,T values were 'refine[d]' using the Mg melting curve in some cases before the paper reports confirmation of that same melting curve. These are calibration feedback loops that partially reduce the central agreement to the choice of pressure/temperature scale. The effect is mitigated by cross-calibration with MgO, hBN and thermocouples, and by independent elastic estimates of δT (1.66 vs 1.5 fitted), so the work is not entirely definitional. Score 5 reflects one explicit pressure re-estimation and a melting-curve refinement loop, not a complete collapse of the EoS derivation.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central model imports four parameters (B0, B0', a, b) from prior fits to other data sets and adds one free parameter (delta_T) fitted here. The structural assumptions are that delta_T is constant, that Murnaghan and the thermal polynomial capture the endpoints, and that the pressure standards are accurate. No invented entities.

free parameters (5)
  • delta_T = 1.5(5)
    The only parameter fitted in this work to the combined p-V-T data set (Tables 1 and 2), via the integrated Anderson-Grueneisen equation.
  • B0 = 32.5(1) or 32.5(2) GPa
    300-K bulk modulus used in the Murnaghan equation (Eq. 4); the abstract says fitted to available data, the text says taken from Stinton et al.; value differs between abstract (32.5(1)) and text (32.5(2)).
  • B0' = 3.73(2)
    Pressure derivative of bulk modulus in Eq. (4); cited as Stinton et al. but lower than Stinton's published 4.05(5), so likely a re-fit of ref. 6 data.
  • a (linear thermal expansion coefficient) = 25(2) x 10^-6 K^-1
    Coefficient in Eq. (3), from ambient-pressure thermal expansion data, attributed to prior work (ref. 11).
  • b (quadratic thermal expansion coefficient) = 9.4(4) x 10^-9 K^-2
    Coefficient in Eq. (3), from ambient-pressure thermal expansion data, attributed to prior work (ref. 11).
assumptions (5)
  • domain assumption delta_T is constant over the studied p-T domain (0.1 MPa to 20 GPa, 300-1500 K)
    Required to integrate Eq. (1) into Eq. (2); the paper states this assumption but does not test it.
  • domain assumption Murnaghan EoS (Eq. 4) accurately represents the 300-K compression of Mg up to 20 GPa
    Used as the isothermal anchor for the model; values come from or are fitted to ref. 6 data.
  • domain assumption Thermal expansion at 0.1 MPa follows the polynomial Eq. (3) up to melting
    Used as the isobaric anchor in Eq. (2); polynomial coefficients from ref. 11.
  • domain assumption The 300-K Mg EoS (ref. 6) and MgO EoS (ref. 17) are valid pressure standards under the experimental conditions
    Pressure determination in Section 2 uses these equations; systematic errors in these standards propagate into all p,T values.
  • domain assumption Quasi-hydrostatic conditions are reached when diffraction peak broadening is no longer remarkable, above about 600 K
    Criterion used to select data for the fit; not independently verified.

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Pith. "Pith review of Thermoelastic equation of state and melting of Mg metal at high pressure and high temperature." pith.science (2026). https://pith.science/paper/ECYNVHY2

@misc{pith2026190900645,
  author       = {Pith},
  title        = {Pith review of: Thermoelastic equation of state and melting of Mg metal at high pressure and high temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECYNVHY2}},
  note         = {Machine review of arXiv:1909.00645}
}
read the original abstract

The p-V-T equation of state of magnesium metal has been measured up to 20 GPa and 1500 K using both multianvil and opposite anvil techniques combined with synchrotron X-ray diffraction. To fit the experimental data, the model of Anderson-Gr\"uneisen has been used with fixed parameter {\delta}T. The 300-K bulk modulus of B0 = 32.5(1) GPa and its first pressure derivative, B0' = 3.73(2), have been obtained by fitting available data up to 20 GPa to Murnaghan equation of state. Thermal expansion at ambient pressure has been described using second order polynomial with coefficients a = 25(2)x10-6 K-1 and b = 9.4(4)x10-9 K-2. The parameter describing simultaneous pressure and temperature impact on thermal expansion coefficient (and, therefore, volume) is {\delta}T = 1.5(5). The good agreement between fitted and experimental isobars has been achieved to relative volumes of 0.75. The Mg melting observed by X-ray diffraction and in situ electrical resistivity measurements confirms previous results and additionally confirms the p-T estimations in the vicinity of melting.

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Reviewed August 14, 2026 · model on record in the stance chip above.