Direct Methods and the Sign Relation

How often the relation holds

baddeleyite, monoclinic zirconia, P2₁/c — 144 triplets from 24 non-zero coefficients, carrying detail to 0.21 Å.

Each marker is a band of triplets of similar strength, and the curve is Cochran’s prediction for this structure — not a fit to the points. The line at 0.5 is what guessing would give.

The markers sit above the curve, and that is the formula being conservative rather than wrong. Cochran’s derivation assumes the atoms are randomly placed; this projection is far more regular than that, which the statistic ⟨|E² − 1|⟩ measures directly — 0.612 here against 0.968 for atoms at random. Symmetry makes the signs agree more often than chance, so the prediction is a floor and not an estimate. The same fact is why the Wilson plot refuses a verdict for several of these structures.

What that says

The relation holds for 101 of 144 triplets here, 70 per cent against the 50 you would get by guessing. Restrict it to the 19 whose predicted probability is above 0.9 and it holds for every one of them. That is the regime a program works in: it does not use the weak relations, it starts from the strong ones.

This does not solve anything. A direct-methods program fixes an origin, gives a few strong reflections symbolic signs, propagates them through relations like these, and ranks the resulting sign sets by a figure of merit — a search whose answer nothing here could check. What this page does instead is measure the relation the search rests on: the signs in the table are computed from the published coordinates, so every prediction can be marked right or wrong, which is the one thing a solver’s output would not allow.

The strongest relations

Strongest first, which is the order a program uses them in: it starts from the relations it can trust and never reaches the weak ones. Showing 20 of 144; the counts above are over all of them.

Why a product of three

A single sign is not a fact about the crystal. This projection has a centre of symmetry at x = 0 and another at x = ½, and either will do as an origin; move it to the second and F(h) is multiplied by (−1)^h, so the sign of every odd reflection reverses — all 12 of them here. The triple product survives it, because (−1)^h(−1)^k(−1)^h+k = +1 whatever h and k are. That is what makes it a structure invariant, and it is the reason the amplitudes can predict it: an amplitude does not move when the origin does, so no function of amplitudes could ever fix a quantity that does.

The reflections

The E values are normalised in two steps, the same two the Wilson plot uses: divide |F|² by Σg² at each reflection’s own resolution, which removes the form-factor decay exactly, then by the mean over the reflections that are present. The second step makes ⟨E²⟩ equal to 1 by construction rather than by fit, and it is doing real work: without it copper comes out at exactly 2, because half its h00 are systematically absent and the survivors carry their share.

What the prediction was built from

The equivalent scatterer count is the number that decides everything: the relation reads 1/√N, so a structure with one heavy atom among light ones behaves like a much smaller one and gives up its signs easily. That is why the heavy-atom structures fell first, and why a page of equal light atoms is where these methods run out.