Crystallography & Symmetry
patterns derived from their groups
Concept

Space group determination — where it appears

Fixing which of the two hundred and thirty groups a crystal has, from its absences, its Laue class and its intensity statistics. Symmetry alone narrows the answer to a class and no further, so the absences carry most of the load and often still leave a pair.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

What pg scatters. The diffraction pattern computed from the atom positions alone. Where a glide plane is present, alternate reflections along a row cancel exactly — and those missing spots are how the glide is identified in an experiment, since the glide itself is never seen.

Systematic absences

The most informative part of a diffraction pattern is the part that is not there. A glide plane cancels alternate reflections along a row, exactly, and those missing spots are how a symmetry nobody can see is identified.

diffraction · Diffraction p4m, displaced and then measured. Every atom of a p4m pattern moved by up to 1.2 per cent of a cell edge, and the resulting positions examined at 26 tolerances. At zero tolerance only the identity survives, so the structure has no exact symmetry whatever. Between 0.021 and 0.094 the count sits at 8, which is the group that was displaced. Above that it climbs to 15, accepting operations no version of this pattern has. The climb is not even steady: at 4 of the 25 steps the count falls as the tolerance is loosened, because operations accepted separately at one threshold merge into one at the next. The correct answer is a step on a staircase and nothing in the coordinates says which step.

Near-symmetry, and the tolerance that is not here

Every claim on this site is decided by integer arithmetic, so no threshold is ever chosen. Measured coordinates do not arrive that way, and the moment a tolerance is introduced the answer stops being a fact about the structure and becomes a fact about the threshold.

diffraction · Accidental symmetry N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does.

Whether there is a centre is a statistic

Everything else on this site is decidable: a pattern has a symmetry or it does not, and the detector settles it in integers. Whether a structure has an inversion centre is not like that. No single reflection carries the answer — the distribution of all of them does.

diffraction · Statistics pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice.

Seventeen groups, seven vector sets

A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

diffraction · The Patterson function -2 0 -3 is forbidden and is reached 64 ways. A layer of the reciprocal lattice of Cc: pale spots are reflections the group extinguishes, solid ones are allowed. The path shows a detour — the beam diffracts once at -3 -3 -3 and again at 1 3 0, and the two together send it in exactly the direction a single reflection at -2 0 -3 would. That reflection is forbidden, so intensity arrives where the symmetry said none could. There are 64 such routes to this one spot inside this window alone.

The absence that fills itself in

A systematic absence is the strongest evidence this subject has: a whole zone of reflections cancelling exactly, for reasons of symmetry rather than of arithmetic accident. The exactness belongs to a model — that the beam scatters once. A beam that has already been diffracted can be diffracted again, and the two events together land where the group said nothing could.

diffraction · Diffraction

Named alongside it

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

Systematic absenceGlide planeRound tripStructure factorAccidental symmetryAutocorrelationCentral limit theoremCentringCentrosymmetricCentrosymmetryDetectionDouble diffraction

All concepts