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 Ų · rutile — where the test does not apply · berlinite — where the test is confidently wrong
Input
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.
No verdict
This structure defeats the test, and it does so in the misleading direction — the statistic below points confidently at the wrong answer.
The atoms in this cell very nearly repeat at ½ c, a translation that is not a lattice vector — so the cell is close to being half (or a third) of its stated size, and one class of reflections is nearly extinguished while the other carries all the intensity. Wilson's derivation assumes the atoms are at random positions, and a structure that almost repeats is the opposite of that. The distribution comes out wider than either ideal, so the statistic below is not merely inconclusive: read naively it points confidently at a centre of inversion. Berlinite is the case here, and berlinite has no centre of inversion. Separating the two classes and normalising them apart is what a real analysis does; this page does not, and says so rather than answering.
⟨||E|2 − 1|⟩ = 1.113, above 0.968 — the value for a centrosymmetric structure, and the widest distribution Wilson’s assumption can produce. The space group P3121 does not 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.
The Wilson plot
B = -0.09 Ų from the slope, over 12 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. The temperature factor is the only thing in the calculation that makes it tilt. 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.
What was measured
| Space group | P3121, order 6 |
|---|---|
| Atoms in the cell | 18 |
| Effective scatterers | 16.62 |
| Scattering from general positions | 39% |
| Unique reflections | 473 to 0.60 Å |
| Resolution shells | 12 |
| ⟨|E|2⟩ | 1.000 |
| |E| > 1 | 28.8% |
| |E| > 2 | 6.3% |
| |E| > 3 | 0.0% |
The effective count is (Σf)² / Σf²: 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.
The tails have ideal values too, 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.
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 Ų · rutile — where the test does not apply · berlinite — where the test is confidently wrong