xraytools.

Wilson Plot and E Statistics

https://xraytools.com/wilson

Single-crystal data

The evidence systematic absences cannot give. A centre of inversion forces every phase to 0 or 180°, which leaves a mark on the distribution of the intensities rather than on which of them are missing — so it is measurable exactly where the determination runs out. The same shell averages give the Wilson plot, whose slope is the overall temperature factor.

Before this
The evidence here is the distribution of intensities rather than any one of them, so |F|² and what makes it large or small is assumed — see the structure factor.
You supply
One of the named structures. A full set of reflections is computed from its atoms, so nothing has to be measured — and an overall B can be added to see the plot tilt.
Reading it
Wilson's derivation assumes many atoms of comparable scattering power sitting at random. A structure whose atoms are all on special positions has no free coordinates at all, so the statistic means nothing there — and the page refuses a verdict rather than giving the wrong one.

Worked examples: albite, 52 atoms in the cell — centrosymmetric · quartz, no centre of inversion · cristobalite — the same, from different symmetry · albite with an overall B of 3 Å² · rutile — where the test does not apply · berlinite — where the test is confidently wrong · albite scaled by 1000 — the intercept gives it back · zirconia — where N(z) answers and the statistic cannot

Notation here: s · F, |F| · I · E · U, B · K — what each one means here

Terms here: centring · general position · zone

See also: Intensity Corrections · Space Group from Absences · Structure Factor Calculator

What each input changes
Overall B
An overall temperature factor added to the computed data. It tilts the plot: the slope IS this number, which is what makes the plot readable backwards.

Input

Ų

Added to every atom. Leave it blank for stationary atoms — then put a value in and watch the plot tilt, because the temperature factor is the only thing in the calculation that makes it tilt.

An arbitrary factor on every intensity, as a detector puts one there. It lifts the plot without tilting it, and the intercept is what brings it back — which on measured data is the only way to the absolute scale, and what lets a first model be built at all.

Systematic absences see the centring, the glide planes and the screw axes, and they cannot see a centre of inversion: two groups built from the same centring, glides and screws — C2/c and Cc, Pnma and Pn21a — extinguish exactly the same reflections whether or not one of them has that centre. That is why the determination hands back a family of groups, and this statistic is the independent evidence that separates them.

Has it got a centre of inversion?

Systematic absences cannot answer that, which is why the determination hands back a family of space groups rather than one. The intensities can: a centre of inversion forces every phase to 0 or 180°, and that leaves a mark on the distribution of the intensities which no amount of careful indexing would show.

Choose a structure and this computes a full set of reflections from its atoms, fits the Wilson plot to get the temperature factor back out, and measures ⟨||E|2 − 1|⟩.

Some of the named structures can be judged this way and some cannot — the page says which, and why, rather than answering anyway.

Try it

albite, 52 atoms in the cell — centrosymmetric · quartz, no centre of inversion · cristobalite — the same, from different symmetry · albite with an overall B of 3 Å² · rutile — where the test does not apply · berlinite — where the test is confidently wrong · albite scaled by 1000 — the intercept gives it back · zirconia — where N(z) answers and the statistic cannot

Where this comes from