Direct Methods and the Sign Relation
https://xraytools.com/direct
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.
- Before this
- The relation is between the signs of three reflections whose indices add up, so what a phase is — where a reflection’s wave sits relative to the cell origin — has to be in place first.
- 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 · aragonite — the origin lands on the other centre
Earlier on the path: The Patterson Function Next on the path: Charge Flipping On From intensities to a structure, step 4 of 6
Notation here: E — what each one means here
See also: The Patterson Function · Charge Flipping
What each input changes
- Terms
- How many reflections are available to form triplets. More reflections means more relations, and the weak ones are where the rule stops being reliable.
Teaching with this page
- Objective
- After this page a learner can explain why a sign relation is a property of a triplet rather than of any one reflection.
- Start from
- this worked example
- Ask first
- For three strong reflections the relation s(h) s(k) s(h+k) = +1 holds. Does that fix their three signs?
- Watch for
- “Yes — one relation for each reflection”
- Then
- Charge Flipping
Check yourself: For three strong reflections the relation s(h) s(k) s(h+k) = +1 holds. Does that fix their three signs?
No — it fixes the product, which four sign sets satisfy Yes — one relation for each reflection
The product is a structure invariant and the individual signs are not: changing the origin changes them and leaves the product alone. That is exactly what makes the relation usable and exactly why it hands you no answer by itself. Direct methods fix a few signs by choosing an origin and then propagate the rest through many relations at once.
Input
The amplitudes know more than they look
A diffractometer records intensities; correcting them gives |F|2 and its square root gives |F|, and no step of that carries a phase. The Fourier page shows what the missing half costs. It does not follow that the phases are unknowable. The amplitudes constrain them, and for a small structure the constraint is tight enough to recover them without ever measuring one — which is what direct methods are, and how most small-molecule structures have been solved since the 1960s.
Summing over h alone makes every phase here a sign — every structure offered below has a symmetry operation sending x to −x, which is what makes F(h00) real — so the constraint takes its simplest form: for three reflections whose indices add up, s(h) s(k) s(h+k) = +1, more often the stronger the three are. Choose a structure and this counts how often it really holds, against the probability the theory predicts.
The signs it checks against are computed from the published coordinates, so every prediction can be marked right or wrong. Nothing here is solved — the answer is known throughout, which is the only reason the count means anything.
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 · aragonite — the origin lands on the other centre
Where this comes from
- The squaring method: a new method for phase determination
D. Sayre, Acta Cryst. 1952, 5, 60–65 · doi:10.1107/S0365110X52000137
The sign relation this page is built on, derived from the requirement that squaring the density leaves its peaks where they were. - An application of a new phase determination procedure to the structure of cyclo(hexaglycyl) hemihydrate
J. Karle and I. L. Karle, Acta Cryst. 1963, 16, 969–975 · doi:10.1107/S0365110X63002607
Symbolic addition first applied to a real structure. - The symbolic addition procedure for phase determination for centrosymmetric and non-centrosymmetric crystals
J. Karle and I. L. Karle, Acta Cryst. 1966, 21, 849–859 · doi:10.1107/S0365110X66004079
The procedure set out in full: assign symbols to a few strong signs and propagate them through the triplets. - Relations between the phases of structure factors
W. Cochran, Acta Cryst. 1955, 8, 473–478 · doi:10.1107/S0365110X55001485
The probability that a triplet relation holds, in closed form. The curve drawn above is this paper’s result. - The theory of sign relations between structure factors
W. Cochran and M. M. Woolfson, Acta Cryst. 1955, 8, 1–12 · doi:10.1107/S0365110X55000017
The centrosymmetric case, which is the one this page is in: with every phase a sign, the relation becomes a statement about a product of three of them. - The probability distribution of X-ray intensities
A. J. C. Wilson, Acta Cryst. 1949, 2, 318–321 · doi:10.1107/S0365110X49000813
Where E values come from, and the distribution whose assumption — atoms at random — is the one these projections violate. See the Wilson plot.