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⟨cos Φ⟩ against κ, 379 triplets

⟨cos Φ⟩ against κ, 379 triplets

⟨cos Φ⟩ against κ, 379 triplets. The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them.

The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them.

6 essays call phase-relations. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

⟨cos Φ⟩ against κ, 379 triplets. The mean cosine of the triplet, binned by the concentration κ = 2|E₁E₂E₃|/√N, for the 379 triplets of a structure of 24 atoms whose reflections all exceed |E| = 1.2. The curve is Cochran's I₁(κ)/I₀(κ), computed from the distribution and not fitted to anything; the points are measured, with the number of triplets in each bin printed above. They agree to 0.1 root-mean-square. The measured points sit slightly above the curve throughout, which is the finite structure showing: Cochran's derivation assumes atoms placed at random and there are only 24 of them.

How it is known

Three phases that do not move when the origin does

A phase is a property of the description, not of the crystal: shift the origin and every one of them changes. A sum of three phases whose indices add to zero does not change, because the shifts cancel. That sum is the smallest thing about a structure that a diffraction experiment could in principle know, and it is not distributed at random.

$the most consistent answer has 100 per cent of the signs. 60 runs of the sign procedure from 60 different random starts, each plotted at its self-consistency — a figure computed without any knowledge of the answer — against the fraction of its signs that are in fact right. Throwing out the 1 run that reached the uniform solution — every sign the same, perfectly consistent and physically a single peak — the highest consistency belongs to a run with 100 per cent of the signs right. The ranking works here, and the reason it works is that the cell is small. Nothing in the plot's horizontal axis knows the answer, which is the only reason a procedure of this kind is a procedure at all.$ How it is known

The formula that has the answer already

The tangent formula rebuilds each phase from all the others, and the true phase set is very nearly a fixed point of it — hand it the answer and it hands the answer back. Start it anywhere else and it does not arrive. Having a fixed point and finding it are different problems, and the second is where the subject spent twenty years.

B = 3.45 against 3.4, K = 0.37 against 0.37. The mean intensity of each resolution shell of a cell of 1194 reflections, divided by Σf² computed from the cell's content alone, and logged. The points fall on a line whose slope gives B = 3.45 against the 3.4 put in, and whose intercept gives a scale of 0.37 against 0.37 — both recovered before a single atom has been placed. The atoms here are independent, so the line is straight at every resolution; the shells below the cutoff are marked in the second colour.

How it is known

The average that knows the atoms and not where they are

Square a structure factor and average it over a shell of reflections at one resolution. The cross terms — every one of which carries a fact about the arrangement — cancel, and what is left is a sum over the *content* of the cell with no position in it anywhere. A scale and a temperature factor come out of that before a single atom has been placed.

A map from amplitudes alone, 0.59 grid steps out. The density after 150 cycles of flipping, with the atoms that produced the data drawn as rings — moved into the origin and the handedness the solution chose, because a phase set does not fix either and comparing without allowing for them measures the arbitrariness of the description. Every peak of the map is an atom and every atom has a peak. Nothing about the arrangement went into the calculation: the input was a list of amplitudes and a random set of phases.

How it is known

The solver that knows no symmetry

Compute a map from amplitudes and random phases, reverse the sign of everything below a small threshold, transform back and keep the phases. Repeat. The structure appears — and so does its space group, which was never supplied.

The sign turns when the cross terms go weak. Every quartet among the strong reflections of a small structure, sorted by the mean of its three cross terms, with the mean cosine of the phase sum in each bin. Where the cross terms are strong the quartet behaves like a triplet and the cosine is near one. Where they are weakest it is -0.698 — negative — and 93 per cent of those quartets have a cosine below nought. The four reflections of the quartet are equally strong in every bin; what changes is three reflections that are not in the sum at all.

How it is known

The relation that can say no

Every phase relation before this one pushes a sum towards zero, so none of them can contradict another — a phase set satisfying all of them badly is still satisfying them in the same direction. A quartet can be estimated at π instead, and it is when its three cross terms are weak, so the information arrives from the reflections nobody would have thought worth measuring.

An equality, with a known factor in it. Sayre's identity says the structure factor is proportional to the convolution of the structure factors with themselves — an equality, not a probability — with a factor that depends on the atoms and the resolution and on nothing else. Computed for a structure of Gaussian atoms, the ratio of the two sides falls with resolution exactly as predicted: the logarithm of the ratio is a straight line in the squared index, its slope is minus half the width in the atomic factor, and the ratio is very nearly real and positive, so the identity relates the phases and not only the amplitudes.

How it is known

The relation that is an equality

Every relation of this kind so far is a probability — right nine times in ten, useless applied once and decisive applied ten thousand times. One is not. For a structure of equal, resolved atoms the squared density has peaks in the same places, so every structure factor is exactly a convolution of all the others, with a factor that depends only on the atom. It is exact, and on its own it is useless.

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