Wilson Plot and E Statistics

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.

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

Input

Structure

Overall B

Å²

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 do so.

Systematic absences see the centring, the glide planes and the screw axes, and they cannot see a centre of inversion: a centrosymmetric group and its non-centrosymmetric subgroups extinguish exactly the same reflections. 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