Wilson Plot and E Statistics

No verdict

This structure cannot be judged this way, and the number below is printed only so that you can see it is not to be believed.

Every atom in this structure sits on a special position, so no atom is free in the cell: the symmetry fixes its coordinates outright, or confines them to a line or a plane it chooses. Wilson's derivation puts atoms at random, and a symmetry-placed atom is the opposite of that — it has fewer copies than the group has operations, and their contributions to F move in step instead of independently. The number below is not evidence about a centre of inversion. Rutile is the case worth trying: it is centrosymmetric and the statistic points the other way.

⟨||E|² − 1|⟩ = 0.828, where the space group P4₂/mnm does contain a centre of inversion.

|E|² is an intensity divided by the mean of its own shell, so it says how strong a reflection is for its resolution and the fall-off is already gone. A centre of inversion forces every phase to 0 or 180°, which spreads the intensities wider than random phases do: ⟨||E|² − 1|⟩ is 0.968 with a centre and 0.736 without one. One term of the full treatment is deliberately missing here: reflections lying in a zone that the symmetry maps onto itself have a mean intensity ε times the general one — ε is 2 or 3 in several zones of quartz, for instance — and a complete normalisation divides by ε before forming |E|. It is left out because it changes the verdict only for structures this page already declines to judge, and an uncheckable correction that buys nothing measurable is worse than a stated omission. Expect a small bias, not a different answer.

The Wilson plot

4 shells, from 1.55 Å down to 0.64 Å. Point at one to read it, or click to keep it.

B = 0.04 Å² from the slope, over 4 resolution shells.

Each point is one resolution shell: the mean intensity divided by the mean of Σf₀², which is what the shell would scatter with stationary atoms. The ratio falls as exp(−2B s²), so the logarithm is a straight line and its slope is −2B, where s = sin θ/λ — the same s the structure factor and the powder pattern use, so a B means the same thing on all three pages. The intercept is the other half of the classical plot: on measured data it gives the scale factor that puts arbitrary detector counts onto the absolute scale of electrons, which is what lets a first model be built at all. This page starts from computed structure factors, so at K = 1 there is nothing for it to recover — put a scale factor in and it recovers that instead, which is the same arithmetic a first data set gets. The temperature factor is the only thing in the calculation that makes the line tilt; the scale is the only thing that lifts it. The recovered value carries a small offset of its own, because Wilson's derivation assumes atoms at random and a real structure is not random; what tracks the temperature factor exactly is the change in the slope, so put in two different values and compare.

N(z): the weak reflections

The other classical test, and it looks at the opposite end of the distribution from everything above: N(z) is the fraction of reflections weaker than z times the mean of their own resolution shell. A centre of inversion forces every phase to 0 or 180°, so an intensity is the square of one number and near-zero values are common; without a centre it is the square of a two-dimensional vector, and a near-zero value needs both components small at once. That is why the two ideals are furthest apart at the bottom of the table: at z = 0.1 they are 0.248 and 0.095, a factor of 2.6. The curve is drawn over 98 reflections, each divided by its own shell mean so the temperature factor is already gone.

This curve decides nothing, for the reason the statistic above decides nothing: every atom here sits on a special position, so the reflections are not the sum of random contributions Wilson’s derivation assumes. The numbers in the table are real and the comparison is not.

What was measured

The effective count is (Σf²)² / Σf⁴: it equals the atom count when every atom is the same element and falls below it as one atom starts to dominate, which is what Wilson’s “atoms of comparable scattering power” is asking for. It is that combination rather than any other because it is the one the fourth moment turns on — ⟨E⁴⟩ is 2 − 1/N_eff for a structure with no centre and 3 − 2/N_eff for one with a centre, and those two relations are what fix it.

The ideal values are in the two columns beside them — 31.7 / 4.6 / 0.3 % with a centre and 36.8 / 1.8 / 0.0 % without, for |E| above t = 1, 2 and 3, computed from erfc(t/$\sqrt{2}$) with a centre and exp(−t²) without rather than taken from a table — but a synthetic data set this size holds only a handful of reflections above |E| = 2 and none above 3 for most structures, so they describe the distribution here without deciding anything. The mean deviation above is what the verdict rests on.