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 15 essays across 4 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 What refinement takes back from a wrong disorder model. Every directed pair of disorder models sharing an occupancy, 462 in all: data made from one model's averaged structure, the other model refined against them with positions on its own loci, two-component displacement parameters and fixed occupancies. Across is the residual between the two ideal structures; up is the residual after refinement. Points on the dashed line lost nothing to refinement; the median point lost 58 per cent of its residual. Every point on the floor, where the wrong model fits exactly, is one whose loci contain the true model's.

What refinement takes back from a wrong model

Two disorder models sharing an occupancy leave averaged structures typically a sixth apart. Refine the wrong one against the right one's data and it takes back more than half of that — all of it wherever its loci contain the true ones. Yet the data still choose, even for a molecule of sixty-four atoms, because a likelihood summed over every reflection sees what no single reflection can, and the wrong model's partial atoms give it away through the size of their displacement parameters, not their shape.

restriction · Local symmetry Better counting makes an uncorrected smooth error choose the wrong model. Eighteen pairs of disorder models, each decided by the likelihood ratio at the 0.1 per cent level, at sixty-four and a hundred and twenty-eight atoms, as the random error falls from 3% to 1% to 0.3% of the mean amplitude. In each group, the first bar has random error only, the second adds a smooth scale error of 3% across reciprocal space, and the third adds the same error with a degree-two scale surface refined alongside the structure. With random error only, better counting decides more pairs correctly. With the uncorrected smooth error it stops helping and the number decided for the wrong model grows, to three at a hundred and twenty-eight atoms. With the surface refined, the result is the random-only one in every group.

Better counting, and the wrong model

A smooth error across reciprocal space, as large as the random one, hardly changes which disorder model the data choose: it enters both models' residuals and cancels from their difference. What it changes is where better data lead. With random error alone, counting harder decides more pairs correctly. With the smooth error left in, counting harder stops helping and starts deciding pairs for the model that did not make the data — at the 0.1% level, and more of them the better the counting.

restriction · Local symmetry How many locked phases an experiment of a given resolution can see. For three convex repulsions, the number of steps of the staircase wider than a given share of the chemical-potential axis, as that share is made smaller by factors of ten. A resolution of one part in a million shows 199 steps for the short exponential, 829 for the long one and 41 for the inverse sixth power. Every count is exactly the number of fractions with denominator up to some bound, because every step with one denominator has one width. The short exponential and the inverse power cross: the power law shows fewer steps at every practical resolution and more at one part in a hundred thousand million.

An experiment sees a Farey sequence

A devil's staircase has a step at every fraction, and no experiment sees them all. Which ones it sees is not a matter of luck: every step with the same denominator has the same width, so a finite resolution shows every fraction up to one denominator and nothing beyond it — a Farey sequence, whose length is a sum of Euler's totient. How that denominator grows with the resolution is the one thing the interaction decides, and what is left unseen is steps, never an incommensurate phase.

aperiodic · Modulation The worst edge pair is the two most anomalous wavelengths. For each pair of the four wavelengths, and for the peak, inflection and high remote together, the median error of the phase difference between the whole structure and its anomalous atoms, recovered from both Friedel mates at each wavelength with a 1% error on every intensity. Beside each, the spread of f′ across the chosen wavelengths. The pairs that include a remote wavelength and one on the edge phase to seven or eight degrees; the peak and the inflection together, the two with the largest anomalous effect, to eighteen; the two remotes to fifty-five.

The two most anomalous wavelengths are the wrong pair

A MAD experiment measures both Friedel mates at two wavelengths near an absorption edge and solves for a phase. The obvious choice is the two wavelengths where the anomalous effects are largest, the peak of f″ and the dip of f′. Of all the pairs that use the edge, it is the worst. The sine of the phase is carried by f″, but the cosine is carried by how much f′ changes between the two wavelengths, and two points on the same edge barely change it.

diffraction · Friedel's law Three difference maps of one substructure, and the map they approximate. Patterson syntheses over 1,681 reflections of a structure with 120 light atoms and two anomalous scatterers, from squared Bijvoet differences at one wavelength, squared dispersive differences between two wavelengths, and the two normalised and summed; the fourth panel is the Patterson of the anomalous atoms' own intensities. Circles mark the two vectors between the anomalous atoms. Each map's correlation with the exact one is printed beneath it: Bijvoet 0.775, dispersive 0.838, combined 0.999.

A second wavelength completes the map

The Bijvoet difference carries the anomalous substructure's intensity times the sine squared of a phase; the dispersive difference between two wavelengths carries it times the cosine squared. Normalised and added, the two make the substructure's own Patterson, correlation 0.999. It needs no fewer reflections than either half, because the threshold was never the average. What the dispersive half brings is a way of failing the Bijvoet half cannot have: two datasets, and a scale between them.

diffraction · The Patterson function A scale error between two wavelengths has a sign. The contrast of the substructure's vectors in a dispersive difference Patterson, the weaker of its two peaks over the strongest other peak, as the second dataset is scaled wrongly, with no metal atom and with two metal atoms that do not scatter anomalously. An error that weakens the second dataset loses the map at 3.7% with no metal; one that strengthens it at 50%. The dashed line is where the substructure's peak is predicted to reach zero, a quarter of the fractional change in f′ between the wavelengths. The metals shorten the strong side and hardly move the weak one.

A scale error between two wavelengths has a sign

A dispersive difference compares two datasets, so a wrong scale factor between them adds the whole structure to it. On a structure whose own Patterson is a haze, the substructure's vectors survived a scale error of fifty per cent, and heavy atoms that do not scatter anomalously were expected to take over much sooner. They do, but only in one direction. Scale the second dataset the other way and the substructure is erased at four per cent whatever else is in the crystal, because the error's linear term cancels the signal. Scaling the datasets onto each other is itself an error of that sign.

diffraction · The Patterson function Fitting the outer shells makes it worse, and correcting the fit recovers it. The substructure's weaker peak as a share of its height with no scaling, on data with no error, with the scale error each scaling builds in, for five falloffs and three scalings. Fitting everything loses 42% with no falloff and less as the falloff grows. Fitting only the outer shells loses more, and at the steepest falloffs builds in a large error of the opposite sign. The fit corrected by the ratio the composition expects recovers the whole peak at every falloff, within a few per cent.

Fit the scale, then put the lean back

Scaling two wavelengths to each other builds in an error of the cancelling sign, because their means differ for a physical reason. The proposal was to fit against the part the edge does not touch, at high resolution. It does the opposite: f₀ falls with angle and f′ does not, so the outer shells lean furthest and fitting them costs more. Taking the scale from the composition alone reproduces the lean as well. What works is fitting as usual and then putting back the difference the composition expects, and that recovers the whole peak.

diffraction · The Patterson function Completeness against sweep, for four spindles. Completeness against rotation range for one tetragonal crystal, class 4/mmm, on four different spindles. The four-fold spindle rises fastest at first, then stops at 88.9 per cent at forty degrees and never improves. The two-fold spindles reach completion near 87 degrees. The general spindle is highest at forty-five degrees and reaches completion at the same 87.

The same measurements, and a hole that stays

How much of the sphere must be swept for a given redundancy? The question has two answers, because it has two parts. The mean number of observations depends only on the sweep, to within two per cent whichever axis the crystal turns on. Completeness depends entirely on the axis, and on one kind of axis no sweep ever finishes: when the Laue group keeps the spindle's line, every copy of a blind reflection is blind too, and fifteen of a tetragonal crystal's 135 reflections are never measured.

diffraction · Resolution

Named alongside it

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

Structure factorAnomalous scatteringLaue classLimiting sphereMeasurementAverage structureDisorderEnumerationFriedel lawHeavy atomLeast squaresOccupancy

All concepts