Direct Methods and the Sign Relation

How often the relation holds

α-quartz, SiO₂, P3₂21 — 36 triplets from 12 non-zero coefficients, carrying detail to 0.35 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.

2 of the 4 bands here sit above the curve and 2 below it. Cochran’s derivation assumes the atoms are randomly placed; this projection is far more regular than that, which the statistic ⟨|E² − 1|⟩ measures directly — 0.798 here against 0.968 for atoms at random. Symmetry makes the signs agree more often than chance, so on a structure like this the formula is conservative — but it is a probability under that random-atom model, not a bound, and an individual band of a few triplets can and does fall below it. The same fact is why the Wilson plot refuses a verdict for several of these structures.

What that says

The relation holds for 19 of 36 triplets here, 53 per cent against the 50 you would get by guessing. None of them reaches a predicted probability above 0.9, which is itself the result — this projection has too many scatterers of too even a weight for any single relation to be trusted on its own.

The strength of the relation depends on how many scatterers the projection has — this page works in projection, so atoms sharing an x merge into one column, and it is those columns that count, not the atoms in the cell — and that is why direct methods are spoken of as a small-molecule technique. For N equal point atoms Cochran’s prefactor reduces to 1/$\sqrt{N}$, so a triplet of a given normalised magnitude is less likely to hold in a big structure than in a small one — not because the physics changes but because each atom contributes a smaller share of every structure factor, and the sum of many random contributions is what the relation has to beat. Doubling the number of scatterers costs a factor $\sqrt{2}$ in every triple product on this page. That is the whole reason the method solved small molecules in the 1960s and left proteins to isomorphous replacement.

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 36; the counts above are over all of them.

One step of symbolic addition

Nothing here is strong enough to start from: no relation reaches a predicted probability of 0.9, and a procedure that begins by trusting its strongest relation has no first move. That is a result about this structure at this resolution rather than a gap — ask for more terms, or try a structure with fewer scatterers in projection.

Ignore the floor and propagate through every relation regardless of its odds and 10 signs can be derived — of which 4 are wrong. That is the bargain the floor refuses.

This is symbolic addition, which Karle and Karle first applied in 1963 and set out in full in 1966, working from the sign relation Sayre published in 1952. It is three rules: choose the sign the origin makes free, give a letter to the strongest reflection you cannot derive, and take the third sign of a relation from the other two. A modern program does all of it with more reflections, in three dimensions, and ranks the sign sets by a figure of merit instead of asking anyone — but the arithmetic is this arithmetic.

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 6 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/$\sqrt{N}$, 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.