Crystallography & Symmetry
patterns derived from their groups
Concept

Friedels law — where it appears

That the intensities of a reflection and its opposite are equal, which holds exactly with real scattering factors and fails with anomalous ones. Its holding is what makes the phase unreachable and what adds a centre to every diffraction pattern.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Friedel's law, as an equality rather than a resemblance. Each pair of bars is a reflection and its opposite for a structure of three atoms in no particular arrangement. They are the same height, and not approximately: with real scattering factors, negating the indices conjugates the structure factor, and conjugation does not change a modulus. The test behind this figure requires the largest difference over 40 pairs to be below 10⁻⁹ and it comes back exactly zero. This is why a diffraction pattern is centrosymmetric whatever the crystal is, and why the thirty-two classes collapse to eleven before a structure is even proposed.

The law that hides handedness

With real scattering factors, negating the indices conjugates the structure factor and leaves the intensity exactly alone — so every diffraction pattern is centrosymmetric whatever the crystal is. The escape is an imaginary component that the negation does not touch, and it is how the handedness of a molecule is measured.

diffraction · Friedel's law N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does.

Whether there is a centre is a statistic

Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does.

diffraction · Statistics Two candidates, and the sign that chooses. The phase of reflection (2, 3), recovered from three measurements and no model. The circle is every complex number of the measured amplitude; the isomorphous difference fixes the cosine of the angle between the unknown phase and the heavy atom's, leaving the two candidates marked; the anomalous difference fixes the sine, which picks one. The recovered phase agrees with the true one to fifteen decimal places, and the true phase was never used in the calculation.

One experiment gives the cosine, the other gives the sine

Friedel's law holding exactly is what makes the phase unreachable. Its breaking is what hands it back: an isomorphous difference fixes the cosine of the phase and leaves two candidates, and the anomalous difference fixes the sine, which chooses.

diffraction · Friedel's law Two structures on 8 sites with the same vectors. Two arrangements of 4 atoms on a ring of 8 positions. They are not the same arrangement — no rotation of the ring and no reflection carries one onto the other — and every interatomic vector occurs the same number of times in both. The bars below are the shared vector counts, which is the Patterson function of each: the tall one at the origin is the atom count and carries no information, and everything else is what a diffraction experiment measures. Their diffraction patterns are identical in every intensity, so no measurement of intensities, at any resolution, distinguishes them.

Two structures, one Patterson

Eight arrangements of four atoms on a ring of eight sites, and only seven distinct sets of interatomic vectors between them. Two of the arrangements are genuinely different and no measurement of intensities can tell them apart — at any resolution, for ever.

diffraction · Homometry One number, and it is the fraction. Data simulated from crystals that are nought, a quarter, a half, three quarters and wholly inverted, each fitted for the single parameter. The fitted values sit on the diagonal to better than five parts in a hundred, which is what makes the parameter a measurement of composition rather than a test of a hypothesis: a crystal is allowed to be part one hand and part the other, and a value near a half is a real answer about the specimen rather than a failure of the determination.

How much of it is the other hand

A crystal of one enantiomer is a hypothesis, not an observation. What the diffraction actually measures is a fraction — how much of the specimen is the inverted structure — and the useful part of that measurement is the uncertainty on it.

diffraction · Friedel's law Two plane structures with one Patterson. Two arrangements of 4 atoms on a 4 by 4 torus. No translation and no half-turn carries one onto the other, so they are different structures; every interatomic vector occurs the same number of times in both, so no measurement of intensities distinguishes them. The plane case is not the chain case with an extra index: a mirror in the plane sends a vector set to its mirror image rather than to itself, which is the one place the analogy with a cycle breaks.

Two structures on a torus, and one Patterson

Homometry was settled here on a ring of positions, which is a crystal in one dimension. Moving the same exhaustive search to a torus asks whether the coincidence is commoner or rarer when the vectors have a plane to land in — and the honest answer is that dimension is not what decides it.

diffraction · Homometry

Named alongside it

The objects these essays reach for when they reach for this one.

Anomalous scatteringBijvoet differenceStructure factorAbsolute configurationCentrosymmetryChiralityThe Patterson functionPhase problemAutocorrelationCentral limit theoremCentrosymmetricChinese remainder theorem

All concepts