Experiments · E39
Withdrawn. One run gave 700 K and the published strongest pair; repeated runs moved the pair and loosened the temperature to a bound.
In the log: Short-range order and the transition temperature, against an independent result
book/assets/diagrams/exp/E39.svg).E39 is qualified, downward. That entry reported "Mo-Ta strongest, matching published" and a transition at 700 K. Both came from a single Monte Carlo run. Averaged over four runs with more data the strongest pair moves to Mo-Nb and then Nb-V, at values (-0.77, -0.89, -0.82) within noise of one another - the ranking is not robust, and the apparent agreement with Nguyen-Manh et al. on the dominant pair was over-read from one sample. The transition remains consistent with their "below 750 K", but as a loose bound rather than the clean agreement E39 presented.
Wang-Landau, which is the right method, does not converge here out of the box. It gives the density of states and so the heat capacity continuously in temperature, instead of a ladder that can only report where it stopped. Three attempts left the fill factor at 1.0 with no density of states produced: the histogram never flattens. The icet documentation describes exactly this for larger systems - the ratio between the maximum and minimum of the density of states grows with size - and prescribes splitting the energy range into segments sampled separately and patched together. That is infrastructure, not a parameter, and is recorded as the route rather than attempted further.
What would actually reduce the 130 K. Longer runs and more of them, since the scatter is sampling; finite-size scaling across cell sizes, which Pei and co-workers (Comput. Phys. Commun. 235, 95, 2018) use via the fourth-order energy cumulant and which bounded their FeCo transition to 840-930 K; and eventually the binned Wang-Landau treatment. None of these is a data-volume problem.
Date 2026-09-12 · Question (operator) The deliverable is short-range order and
order-disorder transition temperatures, not the lowest free energy. Can this chain produce
them, and do they agree with anyone else's? · Provenance forager/order.py,
…/sro_validate.py
What was already being computed and thrown away. Thermodynamic integration walks a temperature ladder from 20,000 K down to its target and reduces every rung to one mean energy. The occupancies on those rungs are the Warren-Cowley parameters and the energy fluctuations are the heat capacity whose peak is the transition. Both were being generated and discarded on every call.
Method. Equimolar MoNbTaVW, 128 sites (a 3x3x3 cell self-interacts with these cutoffs,
E31), canonical Monte Carlo annealed down thirteen temperatures from 2500 K to 300 K, 60
sweeps per site to equilibrate and 160 to sample. Warren-Cowley in de Fontaine's
multicomponent form; the transition from the heat capacity Var(E)/(k_B T^2) per site.
The comparison. Nguyen-Manh and co-workers (UKAEA) computed this system with a cluster expansion fitted to 428 bcc DFT structures, cross-validated at about 8 meV/atom, and reported three things. No connection to this project.
| published | ours | |
|---|---|---|
| order-disorder transition | below 750 K | 700 K, prominence 1.66 |
| strongest first-shell pair | Mo-Ta | Mo-Ta, -1.178 |
| V-W | comparably negative | +0.002 |
| Nb-V | segregating | -0.575, i.e. ordering |
Two reproduced, two not. The transition temperature and the dominant ordering pair come out independently, which is the first outside agreement this project has obtained. The weaker pair preferences do not.
Why, and it is checkable rather than an excuse. Our expansion is fitted to a potential that sits 6.61 meV/atom from DFT (E18); theirs is fitted to DFT. The Mo-Ta interaction is strong enough to survive that error and the subtler preferences are not - they are smaller than our own error bar. The rule this implies, and which can be tested rather than assumed: only order parameters driven by interactions larger than about 7 meV/atom should be believed from this chain. V-W and Nb-V are exactly the cases that rule would exclude.
Full first-shell parameters at 300 K (negative means the pair sits together): Mo-Ta -1.178, Mo-Nb -0.846, Ta-W -0.600, Nb-V -0.575, Nb-W -0.452, Ta-V -0.182, V-W +0.002, Mo-V +0.434, Mo-W +0.631, Nb-Ta +0.929.
Caveats. One composition, one cell, one annealing schedule, no repeats - the ordering strength wanders between 0.075 and 0.22 above 1500 K where it should be near zero, which is the noise floor of a single run and not a physical signal. The transition is located to the spacing of the ladder, 100 K around 700. None of that is fixed by looking at it harder; it needs repeats and a finer ladder, and the cell-size axis remains unmeasured.
This entry is written in 2 separate places in the log, shown here in log order.
Date 2026-09-12 · Question (operator) The deliverable is short-range order and
order-disorder transition temperatures, not the lowest free energy. Can this chain produce
them, and do they agree with anyone else's? · Provenance forager/order.py,
…/sro_validate.py
What was already being computed and thrown away. Thermodynamic integration walks a temperature ladder from 20,000 K down to its target and reduces every rung to one mean energy. The occupancies on those rungs are the Warren-Cowley parameters and the energy fluctuations are the heat capacity whose peak is the transition. Both were being generated and discarded on every call.
Method. Equimolar MoNbTaVW, 128 sites (a 3x3x3 cell self-interacts with these cutoffs,
E31), canonical Monte Carlo annealed down thirteen temperatures from 2500 K to 300 K, 60
sweeps per site to equilibrate and 160 to sample. Warren-Cowley in de Fontaine's
multicomponent form; the transition from the heat capacity Var(E)/(k_B T^2) per site.
The comparison. Nguyen-Manh and co-workers (UKAEA) computed this system with a cluster expansion fitted to 428 bcc DFT structures, cross-validated at about 8 meV/atom, and reported three things. No connection to this project.
| published | ours | |
|---|---|---|
| order-disorder transition | below 750 K | 700 K, prominence 1.66 |
| strongest first-shell pair | Mo-Ta | Mo-Ta, -1.178 |
| V-W | comparably negative | +0.002 |
| Nb-V | segregating | -0.575, i.e. ordering |
Two reproduced, two not. The transition temperature and the dominant ordering pair come out independently, which is the first outside agreement this project has obtained. The weaker pair preferences do not.
Why, and it is checkable rather than an excuse. Our expansion is fitted to a potential that sits 6.61 meV/atom from DFT (E18); theirs is fitted to DFT. The Mo-Ta interaction is strong enough to survive that error and the subtler preferences are not - they are smaller than our own error bar. The rule this implies, and which can be tested rather than assumed: only order parameters driven by interactions larger than about 7 meV/atom should be believed from this chain. V-W and Nb-V are exactly the cases that rule would exclude.
Full first-shell parameters at 300 K (negative means the pair sits together): Mo-Ta -1.178, Mo-Nb -0.846, Ta-W -0.600, Nb-V -0.575, Nb-W -0.452, Ta-V -0.182, V-W +0.002, Mo-V +0.434, Mo-W +0.631, Nb-Ta +0.929.
Caveats. One composition, one cell, one annealing schedule, no repeats - the ordering strength wanders between 0.075 and 0.22 above 1500 K where it should be near zero, which is the noise floor of a single run and not a physical signal. The transition is located to the spacing of the ladder, 100 K around 700. None of that is fixed by looking at it harder; it needs repeats and a finer ladder, and the cell-size axis remains unmeasured.
E39 is qualified, downward. That entry reported "Mo-Ta strongest, matching published" and a transition at 700 K. Both came from a single Monte Carlo run. Averaged over four runs with more data the strongest pair moves to Mo-Nb and then Nb-V, at values (-0.77, -0.89, -0.82) within noise of one another - the ranking is not robust, and the apparent agreement with Nguyen-Manh et al. on the dominant pair was over-read from one sample. The transition remains consistent with their "below 750 K", but as a loose bound rather than the clean agreement E39 presented.
Wang-Landau, which is the right method, does not converge here out of the box. It gives the density of states and so the heat capacity continuously in temperature, instead of a ladder that can only report where it stopped. Three attempts left the fill factor at 1.0 with no density of states produced: the histogram never flattens. The icet documentation describes exactly this for larger systems - the ratio between the maximum and minimum of the density of states grows with size - and prescribes splitting the energy range into segments sampled separately and patched together. That is infrastructure, not a parameter, and is recorded as the route rather than attempted further.
What would actually reduce the 130 K. Longer runs and more of them, since the scatter is sampling; finite-size scaling across cell sizes, which Pei and co-workers (Comput. Phys. Commun. 235, 95, 2018) use via the fourth-order energy cumulant and which bounded their FeCo transition to 840-930 K; and eventually the binned Wang-Landau treatment. None of these is a data-volume problem.
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