Crystallography & Symmetry
patterns derived from their groups
Concept

Resolution — where it appears

The smallest spacing an experiment reaches, which is half the wavelength at best and sets both the number of reflections and the sharpness of a map. The count of unknowns against observations depends on it as a cube and on nothing else.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Everything measurable at λ = 1.54 Å, and it is a finite set. A section through the reciprocal lattice of a cubic cell of 10 Å, with the limiting sphere drawn. Bragg's law reaches a reflection only if its spacing is at least half the wavelength, so the measurable reflections are the lattice points inside a sphere of radius 2/λ and the points outside it are not merely unmeasured — no experiment at this wavelength can reach them. In this section 516 points lie inside; in the full sphere there are 9,092.

How many reflections there are

The reciprocal lattice is infinite and a measurement is not. A wavelength cuts a sphere out of it, the number of points inside is the sphere's volume times the cell's, and symmetry then removes all but a fraction — though never quite the fraction a division would give.

diffraction · Resolution Three atoms, four resolutions. A one-dimensional Fourier synthesis of the same three atoms, cut at four different resolutions. Nothing is approximate except the edge: every amplitude and every phase used is exact, and the only information withheld is the reflections outside the sphere. The peaks broaden as the cut-off comes in, and beside every peak sits a negative ripple that the coarsest map cannot distinguish from a real absence of density. Both effects are the transform of the sphere rather than anything about the structure.

As sharp as the sphere is wide

A map made from a truncated sum is not a blurred picture of the structure. It is the structure convolved with the transform of the sphere — so peaks acquire a width proportional to the resolution, and a negative ripple of twenty-two per cent that no improvement in the data ever reduces.

diffraction · Resolution Observations per unknown, against resolution. Unique reflections divided by refinable parameters, for a triclinic cell, no angle a right angle, with three coordinates and six displacement parameters for every atom and one non-hydrogen atom per 18 ų, in a molecular crystal. The scale is logarithmic because the fall is a cube: 16.3 at 0.8 Å and 0.30 at 3 Å. The line at one is where a determination stops being over-determined, and it is crossed at about 2.0 Å.

The unknowns against the observations

A structure determination is a fit of some number of parameters to some number of measurements, and both counts can be worked out before any data exist. The ratio turns out not to depend on how large the crystal's cell is, or on how symmetric it is — only on the resolution, and on that as a cube.

diffraction · Resolution p4m from 3 orbits of waves — detected p4m. A density built as a sum of 3 symmetry-adapted waves of p4m, each of them the average of a plane wave over the group, shaded from light to dark across one cell. The level set of this density — the darkest points of it — was handed to the same detector the pattern figures use, and it reports p4m, which is exactly the group the waves were built from. The waves are invariant by construction, so the density can never have less symmetry than the group; the interesting direction is the other one.

How many waves a group permits

A pattern can be written as a sum of waves instead of as an orbit of a motif, and then the group ties the coefficients together and forbids some outright. Building a density from the permitted ones and handing it back to the detector closes the same loop through a different door — and at low resolution the density has symmetry the crystal has not.

classification · Seventeen 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.

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.

diffraction · Statistics 66 reflections that no rotation reaches. Every reflection inside the limiting sphere of an orthorhombic cell, 7 × 11 × 13 Å at 1.4 ångström, plotted by its distance from the rotation axis against its height along it. The ones marked are those a rotation about that axis can never bring into diffracting position: turning the crystal moves a point on a circle at fixed height, so a point too close to the axis can never acquire the component along the beam that the Ewald condition demands. The blind region is a cusp about the axis, it is 4.3 per cent of the sphere here, and nothing but remounting the crystal removes it.

What one turn of the crystal reaches

Every reflection inside the limiting sphere is measurable by some orientation. A crystal on a spindle has one axis, and a region around it never reaches the Ewald sphere at all — however patiently the crystal is turned.

diffraction · Resolution The average is the site's orbit, with occupancies. A molecule at a site of symmetry mmm keeping a subgroup of order two takes four orientations, and the average over them is the site group's orbit of each of the molecule's atoms, every image at one over the length of its own orbit. Atoms in general positions give eight images at an eighth each, and give the same eight whichever subgroup the molecule keeps. Atoms on a locus the model keeps give a shorter orbit at a higher occupancy, drawn larger and darker, and those are the only atoms that differ between models. The total scattering is the same for every model, so all of them agree exactly at zero scattering angle.

The molecule size that hides a disorder

Two disorder models with the same occupancy leave averaged structures that differ only in a handful of partial atoms. The difference is 20% in structure factors for a ten-atom molecule and 3% for a sixty-atom one — so the data choose between the models for a small molecule and stop choosing for a large one, and seven pairs are identical at any size.

restriction · Local symmetry

Named alongside it

The objects these essays reach for when they reach for this one.

Limiting sphereMeasurementStructure factorLaue classEnumerationMultiplicityReciprocal latticeRefinementSpecial positionAsymmetric unitAverage structureBlind region

All concepts