Can one energy model cover nickel as well as the eight refractory metals?

EXPERIMENTS.md · lines 4531–4632

E83 — Nickel added to the system: a nine-element expansion

Nickel in a refractory high-entropy alloy is ordinary practice, and the ladder is the mechanism for handling a cheap rung that cannot be trusted - that is what the rungs above it are for. E81 read the missing hull phases as a reason not to run; they are a rung of work, not a blocker, and this entry does the work.

Measured first, with MACE, so the design choices rest on numbers:

995 meV/atom is twice vanadium's 505, which was the largest in the system. That single number is why nickel is an addition rather than a base: holding it at refractory spacing costs an electron volt, so a nickel-rich bcc solid solution is not a thing this lattice can represent, while nickel at ten or fifteen per cent rides a Vegard-averaged lattice that has moved to meet it.

The shared lattice stays at 3.2935 A, the eight-element mean, rather than being re-averaged to 3.238 with nickel in it. Re-averaging would move every reference, every anchor and every recorded energy in this file for the sake of a minority element. Keeping it costs nickel a large anchor, and large anchors are now measured exactly rather than modelled (E80), so the cost is bookkeeping rather than error.

Predicted, before fitting:

1. CV RMSE rises, to somewhere between 8 and 15 meV/atom, from 5.68 on eight elements. Nickel roughly doubles the energy scale the expansion has to span and adds a chemistry unlike the other eight.
2. All nine corners pin to within the new CV error, since they are training points with targets that are zero by definition.
3. The eight existing elements keep their present pair interactions - the expansion should not need to repay for what it already knew, so predictions on nickel-free compositions stay within about 15 meV/atom of the eight-element fit.

A failure of 3 is the interesting one: it would mean the nine-element fit cannot serve both systems and that nickel work needs its own expansion rather than a widened one.

Withdrawn. Every number in E83 above is an artefact of a reference-convention error of mine, not a result about nickel.

The expansion is fitted against shared-lattice references - each element evaluated at 3.2935 A - which is why data/ce_8element.npz stores Mo at -10.6158 and not -10.8420, and why a pure element's target is exactly zero. build_ce_nickel.py computed equilibrium references instead, each element relaxed to its own lattice constant, by copying the _equilibrium helper out of build_ce.py without checking which of the two numbers that script actually fits against.

The two conventions differ by the elemental anchor, which averages 226 meV/atom over the eight elements. That is the whole of the discrepancy:

And it was internally inconsistent, which is worse than being on the wrong footing. The nine corners were given targets of exactly zero - true only under shared-lattice references - while the alloy rows carried equilibrium-referenced targets. The fit was asked to satisfy two conventions at once and split the difference, which is the entire explanation for a CV RMSE of 28.41 meV/atom against 5.68, for nickel's corner refusing to pin at +69, and for the 209 meV/atom bias on nickel-free compositions.

So all three predictions are unfalsified rather than falsified. Nothing has yet been learned about whether a nine-element expansion can carry nickel. The three numbers reported against them measure my error.

The tell was there and I read past it. The nickel-free training targets came out at mean +97 with a range of -100 to +461, while the survey compositions those same energies are compared against run -300 to +51. Two populations of the same quantity, offset by the size of the correction that distinguishes the conventions. I checked the distributions, saw 38 meV/atom separating nickel-free from nickel-bearing rows, concluded the scales were comparable, and did not think to compare either of them against the training set already on disk - which took one line and settles it.

Outcome. The controls hold exactly, the verdicts survive, and my expectation of how far the temperatures would move was too strong.

The controls are the ones that mattered and they are exact. The driving force and the activation energy reproduced to the digit on every composition, which is what a refit of the expansion should do to two quantities computed entirely in MACE. Had either moved, the conclusion would have been that I did not know which rung used what.