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 23 essays across 6 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 Which sites a smooth error can mislead. For the eighteen pairs of disorder models at three site symmetries, under a smooth scale error of degree two (five molecule sizes) and of degree four (three sizes): how many verdicts of perfect counting go the wrong way, how many pairs no data can decide, the largest correlation between the error and the two models' difference, and how many harmonics of the error's degree the site's group leaves unchanged. At the cubic site the degree-two error correlates with nothing, to rounding, and decides no pair wrongly; the degree-four error, which the cubic site keeps one harmonic of, reaches it.

A cubic site is blind to a degree-two error

A smooth scale error left uncorrected made perfect counting choose the wrong disorder model for eight of ninety pairs and sizes, and the obvious way to predict which is to correlate the error with the two models' difference before refining anything. That correlation predicts none of them. What it does predict, exactly, is which pairs can never go wrong: at a cubic site a scale error of degree two correlates with nothing and is absorbed by nothing, because a cubic group keeps no harmonic of degree two. A degree-four error, which it keeps one of, reaches the cubic site at once.

restriction · Local symmetry The first-order prediction against the actual verdict, for every pair and size. The 85 decidable pairs of disorder models at five molecule sizes, each placed by the first-order prediction of the wrong model's misfit minus the right model's, made from the two error-free refinements and their Jacobians, and by the actual difference after refining against data with a 3% smooth error. Shaded quadrants are agreement. 77 agree. Of the 8 wrong verdicts, ringed, 4 are predicted; 4 right verdicts are predicted wrong.

The wrong model moves too far for first order

Which disorder models a smooth error will turn the wrong way was left as a calculation to make before any data exist: project the error onto the directions each refinement's parameters can move. Made, it gets the sign of the verdict right in seventy-seven of eighty-five cases, catches four of the eight wrong verdicts and raises four false alarms. Every case it gets wrong is either a near-tie or a refinement that absorbed more than twice, or less than half, of what its first-order directions allow.

restriction · Local symmetry Seven cubic antiferromagnets, and what a powder of each can say. Seven collinear antiferromagnets on the three cubic lattices, each with the size of its propagation vector's orbit, the number of the forty-eight cubic operations that keep the vector, the kind of every magnetic ring out to a fixed radius, summed from the actual magnetic reflections of one domain, and what a powder pattern can therefore read about the moments' direction. The two whose halving the cubic group keeps are blind in every ring. An orbit of three leaves every ring uniaxial about a cube axis, an orbit of four about a body diagonal, and the body-centred orbit of six leaves rings that read the direction. The cubic chemistry is the same in every row; the propagation vector decides.

Where a powder goes blind

A neutron sees only the part of a magnetic moment across its path, so each reflection knows which way the moments point. A powder adds up every reflection of a given length, and for a ring with cubic symmetry that sum comes out the same whichever way they point. But the ring's symmetry is not the chemistry's: it is the symmetry the propagation vector keeps, and most cubic antiferromagnets have lost enough of it to be read.

point-groups · Magnetic Two hundred and thirty groups on a hundred and twenty-two diffraction symbols. Every space group placed in a column for its diffraction symbol — Bravais lattice, Laue group and systematic absences — with the columns grouped by lattice. There are 122 symbols. 50 of them name one group; 44 are shared by two, 23 by three, 2 by four and 3 by five.

Every ambiguity in a diffraction pattern, sorted by its cause

A diffraction pattern shows a lattice, the symmetry of its intensities and the reflections that are missing, and the 230 space groups show 122 different combinations of the three. Fifty of those name a single group. For the rest, each pair of groups that share one can be sorted by why they do, and there are five reasons. Four of them are ambiguities some further measurement resolves. The fifth, two groups with the same operations in the same places except for where their axes cross, happens exactly twice, and nothing but a solved structure decides it.

classification · Cohomology Which Laue groups have a sampling grid that touches no special point. Every Laue group on every Bravais lattice it occurs on, twenty-four pairs, with the smallest grid of wavevectors that keeps the whole group and has no point fixed by any operation but the identity. Such a grid exists for 1̅, 2/m, mmm and 4/m, at exactly as many points as the group has operations. It does not exist, as far as the search reaches, for 4/mmm or for any group with a three-fold axis: 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m.

A three-fold axis puts a grid point on itself

A grid of wavevectors offset from the zone centre can keep a crystal's whole symmetry and touch no mirror, no axis and no point where time reversal folds the zone onto itself, and then its share of the zone is exactly one over the order of the group, with no boundary excess at all. Four Laue groups have such a grid, as small as the group itself. Every group with a three-fold axis has none, for a reason that fits in a line: the rotoinversion fixes one point of the plane across its axis, and above that point the three-fold rotation fixes the grid.

operations · Fundamental domain Photoelastic constants of the eleven Laue classes, split into four parts. The number of independent photoelastic constants each Laue class permits, from 36 for 1̅ to 3 for m3̅m, drawn as squares and split into the elastic tensor's part, which is symmetric under exchanging the two index pairs, and the extra fifteen-dimensional part, which divides into an axial vector, a second quadrupole and an octupole. The axial vector survives only in 1̅, 2/m, 4/m, 3̅ and 6/m; the octupole is what separates m3̅ from m3̅m; and in m3̅m nothing extra survives, so the two counts agree at three.

Fifteen constants an energy would have forbidden

A strain changes a crystal's refractive index through a fourth-rank tensor with the elastic tensor's index pairs and without its one remaining symmetry, because no energy is differentiated twice to make it. The missing symmetry is worth fifteen components. They split into an axial vector, a quadrupole and an octupole, and each does something the elastic count cannot: the octupole gives 23 and m3̅ a fourth constant, and the axial vector is a part of the tensor no turn of the axes removes.

point-groups · Neumann's principle Every square-symmetric grid of wavevectors has a point on a mirror. One cell of the square reciprocal lattice, centred on the zone centre, with the mirror lines of 4mm through it and three grids the four-fold rotation keeps with an offset off the axis. The quarter turn allows that offset at one place only, the second four-fold centre of the grid's own lattice, ringed. On an aligned grid of odd side the rotation fixes a point; on an aligned grid of even side the offset lies on a diagonal mirror; on a grid turned by 45 degrees it lies on an axial mirror. No grid escapes all three.

The offset a quarter turn allows lies on a mirror

A census of sampling grids found free grids for the tetragonal group 4/m and none for 4/mmm, to a hundred and twenty-eight points, with no reason. The reason is two lines long. A quarter turn lets a symmetric grid be offset to exactly one place off its axis, the second four-fold centre of the grid's own lattice, and a square lattice that a mirror keeps can only be aligned with its mirrors or turned by half a right angle. Either way that centre lies on a mirror line.

operations · Fundamental domain The constants a turn of the axes removes, class by class and property by property. The thirteen crystal classes whose operations all commute with some turn of the axes, against six properties: pyroelectric, dielectric, optical activity, piezoelectric, elastic and photoelastic, and all six together. Each cell gives the printed number of constants, and where a turn removes some, the number left. Triclinic classes have three turns and lose up to three; the others have one and lose at most one. Classes 6 and 6/m lose nothing in any property, and the last column loses the turns once, however many properties could spend them.

A six-fold crystal hides its frame until rank six

Thirteen crystal classes can be turned without moving any of their operations, and the tables print each property's constants in one frame among the many such a turn allows. Spending the turn removes a constant only where a property has a part that turns with the frame, and an n-fold axis keeps such a part only at rank n or above. So a monoclinic crystal's free angle shows in its dielectric tensor, a tetragonal one's first in its elastic constants, and a hexagonal one's in nothing anybody measures. And a crystal has one frame, not one per property: the angle is spent once.

point-groups · Neumann's principle Where the two-fold axis sits decides what the coloured optic axes keep. Interference-figure positions of the two optic axes from 400 to 700 nm for four model crystals seen down the acute bisectrix. Orthorhombic: both mirrors survive and only the angle between the axes changes. Two-fold axis along the optic normal: the axes stay on one line and slide unequally (inclined dispersion), one mirror along that line. Along the obtuse bisectrix: the axial plane stays parallel and shifts (horizontal), one mirror across it. Along the acute bisectrix: the axial plane turns about the centre (crossed), a half-turn and no mirror.

Three places for the two-fold axis

A monoclinic crystal fixes one direction of its indicatrix, the two-fold axis, and an ellipsoid has three principal directions for it to be. Which one it is decides what an interference figure keeps in white light: a single mirror along the plane of the optic axes, a single mirror across it, or a half-turn and no mirror at all. Mineralogy calls the three inclined, horizontal and crossed dispersion. They are one subgroup computed three times, and in light of a single colour all three disappear.

point-groups · Optical Down one optic axis, at three colours, for two crystals. Interference figures of two model monoclinic crystals, inclined above and crossed below, each a plate cut across the optic axis at the sodium line, at 450, 589 and 650 nm. In the inclined crystal the melatope slides along the horizontal mirror and the figure stays symmetric across it. In the crossed crystal the melatope leaves the line away from sodium, and the pattern loses the mirror.

The mirror a plate keeps in one colour

Cut a plate of a crystal across one optic axis, and in the light the plate was cut for its interference figure has a perfect mirror, whatever the crystal's class. Change the colour and the mirror survives only if the crystal really has one. Computed ring by ring, five model crystals give the prediction exactly, including the part the eye would miss: the mirror does not come back when the line is moved to follow the melatope.

point-groups · Optical 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 factorFriedel lawProperty tensorReciprocal latticeResolutionSystematic absenceCrystal classIndependent componentsLimiting sphereTime reversalWavevectorAnomalous scattering

All concepts