Experiments · E111
Yes. It is a mean-field estimate that runs about 1.3 times high; with our baseline fixed too, the factor of two shrinks to 1.04.
In the log: The reference is mean-field, and the published spread is a factor of three
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E108's "0.54 of the published scale on 74 of 75 compositions" was read as a defect in rung 1. Before repairing anything further, what the reference actually is.
LTVC is a mean-field treatment. ltvc_check.py says so in its own docstring - "an AFLOW
cluster expansion under a mean-field treatment" - and mean field overestimates an
order-disorder temperature by a known amount. For B2 ordering on bcc, which is the
nearest-neighbour Ising model on a bipartite lattice with z = 8, Bragg-Williams gives
k_B T_c = 8J against the Monte Carlo 6.354J: mean field runs 1.26 times high from the
approximation alone.
That is not a textbook argument here, because somebody has measured it on this exact alloy. Huhn & Widom (JOM 65, 1772, 2013) computed both from the same Hamiltonian: mean field 1654 K, Monte Carlo 1280 K. The ratio is 0.774, against the 0.79 the lattice model predicts.
And the published first-principles values for MoNbTaW span a factor of 3.3:
| study | method | T_c (K) |
|---|---|---|
| Huhn & Widom 2013 | mean field | 1654 |
| Huhn & Widom 2013 | Monte Carlo, same Hamiltonian | 1280 |
| Kim & Widom, PRM 7, 063803 (2023) | replica-exchange MC | 1110 |
| Liu et al., arXiv 2011.00698 (2020) | replica-exchange MC, EPI | 870 |
| Ruban, GPM + MC (via Körmann) | static lattice | ~750 |
| Körmann & Sluiter, Entropy 18, 403 (2016) | DFT unrelaxed + config. entropy | 717 |
| Widom et al. 2013 | hybrid MC/MD | "600-1200" |
| del Grosso et al. 2012 | empirical | 600-800 |
| Körmann & Sluiter 2016 | DFT relaxed | 517 |
Two rows matter for us. Körmann's 717 K is the calculation whose approximations are ours - a static ideal lattice, no relaxation, configurational entropy only - and their relaxed number is 517 K, so relaxation costs 30 per cent of the ordering temperature. Our expansion is fitted to unrelaxed MACE on a fixed lattice, so it belongs against 717, not against 1110 and not against a mean-field 1654.
Their enthalpies line up with E109 too. Körmann's unrelaxed A2-B2 enthalpy is 42.8 meV/atom; E109 measured MACE at 50 and our expansion at 65 on the same quantity. Their two rows are exactly proportional - 717/517 = 1.387 against 42.8/30.9 = 1.385 - so in their treatment T_ord tracks the enthalpy linearly, which is the mean-field signature.
Consequence: E108's 0.54 is not one number about rung 1. It factorises into roughly 0.77 for the reference being mean-field and roughly 0.70 for everything else, and that 0.70 sits inside a published spread that is itself 3.3 wide. The task "fix rung 1's factor-of-two scale error" was chasing a defect that is at most partly ours.
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