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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.

The amplitudes know more than they look

A diffractometer gives |F| and loses the phase, and the Fourier page shows what that 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, 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

Where this comes from