xraytools.

Direct Methods and the Sign Relation

The phases are not measured, but they are not free either: the amplitudes constrain them. For three reflections whose indices add up, the product of their signs is +1 more often the stronger they are — and that one relation is what solved small-molecule crystallography. This counts how often it holds on a structure whose answer is already known.

You supply
One of the named structures, and how far the series should run. Everything else — the amplitudes, the true signs, the predicted probability — is computed from the atoms.
Reading it
This counts the relation, it does not solve anything: the signs it checks against come from the published coordinates. A real program fixes an origin, propagates symbols through relations like these and ranks the results — a search whose answer nothing here could check.

Worked examples: caesium chloride — every relation holds · zirconia — the strong ones hold, the weak ones do not · quartz at 12 terms · quartz at 36 terms — more relations, same rule · albite — where no relation is certain

Input

up to h =

How far the series runs. Blank means 24. A triplet needs h + k inside the range, so the number of relations grows roughly as the square of this.

How often the relation holds

0.50.60.70.80.91.0Triple products from 0.00 to 0.00: the relation holds for 2 of 9 triplets, which is 22 per cent. Cochran predicts 50 per cent.Triple products from 0.00 to 0.00: the relation holds for 2 of 9 triplets, which is 22 per cent. Cochran predicts 50 per cent.Triple products from 0.00 to 0.03: the relation holds for 7 of 9 triplets, which is 78 per cent. Cochran predicts 50 per cent.Triple products from 0.00 to 0.03: the relation holds for 7 of 9 triplets, which is 78 per cent. Cochran predicts 50 per cent.Triple products from 0.08 to 0.23: the relation holds for 4 of 9 triplets, which is 44 per cent. Cochran predicts 51 per cent.Triple products from 0.08 to 0.23: the relation holds for 4 of 9 triplets, which is 44 per cent. Cochran predicts 51 per cent.Triple products from 0.28 to 3.54: the relation holds for 7 of 9 triplets, which is 78 per cent. Cochran predicts 64 per cent.Triple products from 0.28 to 3.54: the relation holds for 7 of 9 triplets, which is 78 per cent. Cochran predicts 64 per cent.0123|E(h) E(k) E(h+k)|how often it holds

Point at a marker for that band’s count and what the theory predicted for it. Click it to keep it; click it again, click empty space, or press Escape to let go.

low albite, NaAlSi3O8, C1 — 36 triplets from 12 non-zero coefficients, carrying detail to 0.30 Å. Another 108 candidate triplets were skipped because at least one member is systematically absent, which is not a sign of zero.

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.

This projection is close to what Cochran’s derivation assumes — atoms at random — so the prediction here is an estimate and the measurement should fall on either side of it. The test of that assumption is ⟨|E2 − 1|⟩, which is 0.929 here against the ideal 0.968 for a centrosymmetric structure of randomly placed atoms.

What that says

The relation holds for 20 of 36 triplets here, 56 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.

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

h k h+k |E1E2E3| predicted signs holds?
6 10 16 3.54 75% + + + yes
4 6 10 3.37 74% + + + yes
6 16 22 2.80 70% + + + yes
6 6 12 2.37 67% + + + yes
10 12 22 2.30 67% + + + yes
4 12 16 1.62 62% + + + yes
12 12 24 0.61 55% + + − no
10 10 20 0.41 53% + + + yes
4 10 14 0.28 52% + + − no
4 16 20 0.23 52% + + + yes
4 18 22 0.17 51% + − + no
6 12 18 0.15 51% + + − no
4 4 8 0.15 51% + + − no
10 14 24 0.14 51% + − − yes
4 8 12 0.12 51% + − + no
4 20 24 0.11 51% + + − no
6 18 24 0.10 51% + − − yes
8 16 24 0.08 51% − + − yes
6 14 20 0.03 50% + − + no
6 8 14 0.02 50% + − − yes

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.

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 — though this structure has no odd reflection left to reverse, its odd h00 being systematically absent. 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

h d / Å |F| E sign sign at the other origin
1 7.276 — — absent absent
2 3.638 0.3 0.00 + +
3 2.425 — — absent absent
4 1.819 130.6 1.26 + +
5 1.455 — — absent absent
6 1.213 123.9 1.56 + +
7 1.039 — — absent absent
8 0.909 5.9 0.09 − −
9 0.808 — — absent absent
10 0.728 84.9 1.72 + +
11 0.661 — — absent absent
12 0.606 38.8 0.98 + +
13 0.560 — — absent absent
14 0.520 4.2 0.13 − −
15 0.485 — — absent absent
16 0.455 37.5 1.32 + +
17 0.428 — — absent absent
18 0.404 2.5 0.10 − −
19 0.383 — — absent absent
20 0.364 3.1 0.14 + +
21 0.346 — — absent absent
22 0.331 28.0 1.36 + +
23 0.316 — — absent absent
24 0.303 12.1 0.64 − −

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

Atoms in the cell52
Distinct projected scatterers 52 — atoms sharing an x project onto one another and act as one
Σ2 = Σg2 5,557.6
Σ3 = Σg3 63,379.6
Σ3Σ2−3/2 0.1530 — the whole prediction is tanh of this times the triple product
Equivalent equal scatterers 42.7, since that coefficient is 1/√N for N equal ones
⟨E2⟩ 1.0000 — 1 by construction, which is what the second normalisation step buys
⟨|E2 − 1|⟩ 0.929 against 0.968 for a centrosymmetric structure of randomly placed atoms — which is the assumption behind the curve
Negative coefficients 4 of 12

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.

Try it

caesium chloride — every relation holds · zirconia — the strong ones hold, the weak ones do not · quartz at 12 terms · quartz at 36 terms — more relations, same rule · albite — where no relation is certain

Where this comes from