Crystallography & Symmetry
patterns derived from their groups
Concept

Laue class — where it appears

The point group a diffraction experiment reports, being a crystal class with an inversion added, of which there are eleven. Friedel's law makes the addition unavoidable, so the thirty-two classes collapse to eleven before a single reflection is indexed.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

{100} offered to 5 classes: one form between them — the shape names none of the 5. The same face, {100}, handed to 5 crystal classes — m3̅m, m3̅, 432, 4̅3m, 23 — with the orbit each one returns drawn as a stereogram. Filled marks are poles in the upper hemisphere and open ones their partners below. The face counts are 6, 6, 6, 6, 6, taking 1 distinct value; the sets of faces take 1, which is the number that matters, since two classes can return the same count and different faces. Here every class returns the identical set, so a crystal bounded by this form alone has said nothing about which of them grew it.

Five classes grow the same cube

A crystal's shape is the most obvious thing about it and the least informative. Five of the thirty-two classes produce an identical cube, diffraction cannot see an inversion centre and so collapses the thirty-two to eleven, and the measurements that finally separate them are etch pits, optical rotation and a heated crystal attracting ash.

applied · Forms What p3 scatters, and what the scattering shows. The structure on the left has point group 3, of order 3. The intensities it scatters, on the right, have point group 6, of order 6 — more symmetric than the thing that produced them. Reversing the sign of both indices conjugates every term in the sum and leaves the modulus alone, so a diffraction pattern always acquires a centre of symmetry, and in the plane a centre is a half turn. Both numbers are measured: the left from the operations, the right by testing each candidate against the computed intensities.

The symmetry diffraction adds

A diffraction pattern is always more symmetric than the crystal that made it. The extra symmetry is not a mistake in the experiment and no care removes it — it is a property of what a detector records, and it collapses the seventeen groups onto six.

diffraction · Accidental symmetry The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.

The eleven a diffraction pattern reports

A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

point-groups · Diffraction p3, single and twinned. Left, the diffraction pattern of a single crystal of p3. Right, the same crystal twinned, with 50 per cent of it in one orientation. Not one spot has moved — the twin law is a symmetry of the lattice, so the two reciprocal lattices lie exactly on top of one another — and 72 of the 81 reflections drawn have changed intensity. At a fifty-fifty twin the pattern acquires the full symmetry of the lattice's point group and is indistinguishable from a crystal that genuinely has it.

A merohedral twin moves no spot at all

The twin law is a symmetry of the lattice, so the two individuals have reciprocal lattices lying exactly on top of one another. Nothing splits, nothing appears in a new place, and the only thing that changes is that pairs of intensities which were different have been averaged — which produces a diffraction pattern with a symmetry the crystal does not have and no sign that anything is wrong.

applied · Twinning What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

Where the experiment runs out

Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

diffraction · Diffraction Friedel's law, as an equality rather than a resemblance. Each pair of bars is a reflection and its opposite for a structure of three atoms in no particular arrangement. They are the same height, and not approximately: with real scattering factors, negating the indices conjugates the structure factor, and conjugation does not change a modulus. The test behind this figure requires the largest difference over 40 pairs to be below 10⁻⁹ and it comes back exactly zero. This is why a diffraction pattern is centrosymmetric whatever the crystal is, and why the thirty-two classes collapse to eleven before a structure is even proposed.

The law that hides handedness

With real scattering factors, negating the indices conjugates the structure factor and leaves the intensity exactly alone — so every diffraction pattern is centrosymmetric whatever the crystal is. The escape is an imaginary component that the negation does not touch, and it is how the handedness of a molecule is measured.

diffraction · Friedel's law 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 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 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 What each Laue class buys, in measurements per reflection. The eleven Laue classes, with how many distinct reflections a block of indices holds under each and how many times a data set measures the average one. The redundancy is always below the order of the class and the gap is the special reflections. This is the number an experiment is planned around: repeated measurements of what symmetry says must agree are the only estimate of precision that does not come from a model, so a triclinic crystal has to be turned through far more of the sphere than a cubic one to be measured as well.

Every reflection, several times over

A diffraction experiment does not measure each reflection once. Symmetry relates a reflection to the others of its orbit, and those are the same reflection seen from another direction — so a hundred thousand measurements may contain twelve thousand reflections, each observed eight times.

diffraction · Resolution 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 Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups

Named alongside it

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

Friedel lawStructure factorCrystal classLimiting sphereReciprocal latticeResolutionSystematic absenceAnomalous scatteringCentrosymmetricCentrosymmetryChiralityDiffraction

All concepts