Crystallography & Symmetry
patterns derived from their groups
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The theme: Exactly this many — page 15

Seven friezes, seventeen wallpapers, fourteen lattices, two hundred and thirty space groups. Each number is a theorem with a finite proof, and the word that matters in every one of them is 'exactly'.
Diffuse scattering gathered under every Bragg peak. The one-phonon thermal diffuse scattering of a square lattice with springs to nearest and next-nearest neighbours, over the reciprocal plane from minus 2.5 to 2.5 in each index, with the Bragg reflections marked. The intensity rises as one over the squared distance to every reflection, because the acoustic phonons' frequencies vanish there, and it grows with the square of the scattering vector, so the lobes are strongest far from the origin. The lobes are not round: a small step from (1,0) along k gives 2.7 times the intensity of the same step along h, because it probes transverse phonons, which are softer, and that ratio is what an experiment reads the elastic constants from. How it is known

A warm plane crystal has no Bragg peaks

Thermal motion is disorder that moves, and in space it costs a crystal a little of each Bragg peak and gathers it into diffuse scattering underneath. In a plane it costs more than a little. Computed exactly from the phonons of a harmonic square lattice, an atom's mean-square wandering grows by the same amount every time the crystal doubles in size, so no Debye–Waller factor survives, and every Bragg peak fades as a power of the crystal's size — a power proportional to the temperature times the square of the reflection's distance from the origin.

 A looser fibre with more longitudinal response. The three piezoelectric coefficients of a polar fibre of grains with d33 = 90, d31 = −33 and d15 = 560 pC/N, as the tilt spread grows from nothing to a uniform one, with the degree-one and degree-three parts of d33 dashed. The degree-one part is 265 and the degree-three part -175; the negative octupole decays faster, so d33 rises from 90 to 168 at a spread of 30° before falling to nothing. What symmetry decides

A looser fibre, more piezoelectric

Spread the grains of a fibre about its axis and every property decays, but not at one rate: each spherical harmonic a property is built from is multiplied by its own average, and the higher the degree the faster it goes. A property made of two degrees therefore changes shape as the fibre loosens. For a polar grain whose shear coefficient dwarfs its longitudinal one, the fibre's longitudinal response nearly doubles before it falls; for another, a shear coefficient changes sign.

 Thirty-one band groups from three signs. Every operation of a ribbon keeps or reverses the direction along it, the direction across it and the sheet's two faces, so its point group is a subgroup of the eight sign triples. Each operation that keeps the along-direction can also carry half a period along the ribbon, a glide or a screw, and the choice must be consistent. The sixteen subgroups, weighted by their consistent choices, give 1 + 10 + 16 + 4 = 31, the number of band groups, and the enumeration finds all 31 as distinct groups. Into space

Every band group is a ribbon

A ribbon cut from a sheet has a symmetry group with translations along one direction only, one of 31 band groups. The 31 can be counted from three signs: whether an operation reverses the ribbon's direction, its width, or the sheet's faces, and whether it slides half a period. Cutting the eighty layers every way reaches all 31, seven of them as single edges and twenty-four only as ribbons cut symmetrically about a special line. Five of those twenty-four have no improper operation at all.

 One arm of a star is cheap, and the whole star costs the cube. For nine wavevectors of the primitive cubic zone, the number of cells in the smallest periodic box that holds one arm of the wavevector's star, and in the smallest that holds every arm, on a logarithmic axis found by enumerating every box of each size. The zone corner R is held whole by two cells. The face centre X needs two cells for one arm and eight for its three; the thirds along an axis or a body diagonal need three for one arm and twenty-seven for the star; the quarters need four and up to sixty-four. Every smallest box for a whole star has the full symmetry of the cube. The classification

A whole star needs a cubic box

A periodic box for a cubic crystal holds only the wavevectors its own lattice allows, and a box chosen to hold one wavevector usually holds one arm of its star and not the others. One arm of the face centre X fits in two cells; all three need eight. One arm of a third along an axis fits in three; the whole star needs twenty-seven. The smallest box that holds a whole star always keeps the cube's full symmetry, so a calculation either pays for the cube or reports a symmetry its box imposed.

 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. How it is known

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.

 Every crystal class permits the Faraday effect. The thirty-two crystal classes, each with two counts: the number of independent components of the tensor that turns a magnetic field into an optical rotation, and the number of components of the natural gyration tensor. The first is never zero: 5 classes have 1, 11 classes have 2, 11 classes have 3, 3 classes have 5, 2 classes have 9. The second is zero in 17 classes, every one with an inversion centre or a mirror arrangement that cancels it. The Faraday count is the mean of the squared trace over the class, which is never below one because the isotropic part always survives. What symmetry decides

A field lets every class rotate light

Seventeen of the thirty-two classes forbid a crystal to rotate the plane of polarisation on its own. Put the same crystals in a magnetic field along the ray and not one of the thirty-two forbids it, the centrosymmetric ones included. The field supplies the one thing natural rotation needs a handed structure for, and time reversal is what tells the two rotations apart.

 How few of their own operations the space groups need. The 212 space-group types that their own rotations, screws, mirrors, glides and inversions generate, by the fewest of those operations that suffice: 64 need 2, 93 need 3, 46 need 4, 8 need 5, 1 need 6. For 198 the number equals a lower bound read off the group's largest elementary abelian quotient; for the other 14 a finite quotient proves that one more is needed, and a set of that size is found. Into space

Two operations make rock salt's group, and Pmmm needs six

Every space group that its own rotations, screws, mirrors, glides and inversions can build is built by a handful of them, and how small a handful is now known for all 212. A quotient of each group says how many it cannot do without, a search finds that many for 198, and for the other fourteen a finite quotient proves that one more is needed. Rock salt's group needs two operations, Pmmm six, and P312 needs four where its abelian quotient allows two.

 Which plane groups can be coloured with five, seven, eleven or thirteen colours. For each of the seventeen plane groups and each of the primes 5, 7, 11 and 13, the number of its colourings with that many colours, as subgroups of that index, and the group that permutes the colours: cyclic (C), dihedral (D), or the affine group of the colours with a multiplier of order 3, 4 or 6 (written p:3, p:4, p:6). Totals 104, 172, 268 and 420. The square groups p4 appear at 5 and 13, the three-fold and six-fold groups p3 and p6 at 7 and 13, and p4m, p4g, p3m1, p31m and p6m never. The classification

Five colours, seven colours, and the primes that split

Four colours gave the seventeen plane groups 281 colourings, five different groups permuting the colours and ninety-six designs. Five colours give 104 colourings, three groups and fourteen designs, and from five upwards every prime behaves the same way. A prime number of colours that the point group cannot divide forces every colouring onto a sublattice the rotations keep, and whether the square and hexagonal patterns have one is decided by whether the prime splits in the Gaussian or the Eisenstein integers.

 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. How it is known

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.

 Every fibre texture is a point in one plane. The two numbers that decide every coefficient of a polar fibre, the mean first and third Legendre polynomials of the grains' tilt, as a plane. A cone of grains at a single tilt is a point on the curve; any other texture is an average of such points and lies in the curve's convex hull. Spreads about the axis trace the inner path. For a grain with a large shear coefficient the largest d33 of any texture is at a point of the curve, the cone at 51°, 223 pC/N; the best spread about the axis reaches 168. What symmetry decides

The best texture is a cone

Every fibre texture of a polar grain, however its orientations are distributed about the axis, reduces for piezoelectric purposes to two numbers, the averages of the first and third Legendre polynomials of the tilt. The textures fill a region of a plane bounded by one curve, and a coefficient is a straight line across that plane, so its best texture sits on the curve: a cone of grains at one angle. For a grain with a large shear coefficient the best cone lies at 51° and gives 223 pC/N, where the best spread about the axis gives 168.

 Whole stars of the hexagonal zone, and the boxes that hold them. For ten wavevectors of the primitive hexagonal zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it differs, in the smallest n by n grid of cells holding the star. The corner K is a single arm up to sign and is held whole by three cells, against nine for a grid; K raised to the top face, H, by six against eighteen; halfway to K, T, by twelve against thirty-six. M needs two cells for one arm and four for its three, and L two and eight. Every smallest box for a whole star keeps the full hexagonal symmetry. The classification

The corner of a hexagon fits in three cells

A periodic box for a hexagonal crystal holds a whole star of wavevectors only if it keeps the crystal's full symmetry, as a cubic box must. What the cube never offered is a box turned through thirty degrees. It holds the corner of the hexagonal zone in three cells where every grid needs nine, and the saving is always exactly a factor of three and always at the same stars. Turn a box through any other angle that the six-fold axis allows and it loses the mirrors, and comes with a mirror image.

 Along the rod above a reflection, a free sheet loses its peak. The Bragg intensity per site squared of a warm square-lattice sheet at the in-plane reflection (1, 0) and at heights 0.1, 0.2 and 0.3 along the rod above it, against the sheet's size, on logarithmic axes. In the plane the peak falls as a power of the size whatever the sheet does out of it. Above the plane a free sheet's peak collapses faster and faster, to 2.3e-2 at size 128 and height 0.3; under tension every height falls as a power, to 0.38. How it is known

A flapping sheet keeps its peaks in the plane

A warm sheet one atom thick wanders out of its plane far more than within it: a free sheet with only bending stiffness has a mean-square height that grows with its area. That wandering cannot touch a reflection whose scattering vector lies in the plane, which fades exactly as it would if the sheet were held flat. Every reflection with a component along the normal loses its peak faster than any power of the size. Tension puts it back on a power law.

 Every crystal class, as a reflection class and its relative invariants. The thirty-two crystal classes, each with the smallest reflection class containing it after a relabelling of the axes, the index, and the degrees of the secondary invariants that make its invariant ring a free module over the reflection class's polynomial ring. Twelve classes are reflection classes themselves; seventeen have index two and one secondary invariant besides 1; three — 1̄, 4̄ and 3̄ — have index four and three. What symmetry decides

What the twenty lack is a product of mirrors

Twelve crystal classes have invariant rings generated by three independent polynomials, and they are the twelve generated by their own mirrors. The other twenty were left as the classes whose counts need a correction. The correction has a simple shape. Each of the twenty sits inside one of the twelve, at index two or four, and its invariants are the larger class's invariants times one or three extra polynomials. Each extra polynomial is a product of the mirror planes the smaller class lacks. For the rotations of the cube it is a single polynomial of degree nine, the product of all nine mirrors of the full cubic class.

 Whole stars of the rhombohedral zone, and the boxes that hold them. For eight wavevectors of the rhombohedral zone, the number of cells in the smallest periodic box holding one arm of the star, in the smallest holding every arm, and, where it costs more, in the better of the two regular grids: the rhombohedral cell repeated n times along each edge, or the hexagonal triple cell repeated n by n by m. The axis end T is held whole in two cells against six, the face centre F in four against eight, and the in-plane third K₃ in nine against twenty-seven; L needs eight either way. Every smallest box for a whole star keeps all twelve operations of 3̅m. The classification

A rhombohedral box saves a factor of two the hexagon never offered

A rhombohedral crystal is drawn on two cells, the small rhombohedron and the hexagonal cell three times its size, and a periodic box is usually one of them repeated. The smallest box that holds a whole star of wavevectors is the star's own lattice instead, and across every star with denominators up to eight it beats the better of the two grids by one, two or three, never more. Three was the hexagon's number. Two is new, and it is the face centre's. And the boxes that keep the three-fold axis and lose the mirrors still come in mirror pairs, now at nine cells as well.

 Almost no homometric pair on the torus is a reversed factor. For ten plane groups at several numbers of atoms on a 6 × 6 torus, the structures, the pairs of structures sharing one Patterson, how many of those pairs reversing one factor of a sumset explains, and how many are a patchwork of Galois classes. Factor reversal explains 2 of 3522; the patchwork explains all 3522. How it is known

A homometric twin is the structure moved class by class

On a ring, every homometric pair a small search found came from one construction: factor the structure as a sumset and reverse one factor. Imposing a plane group on a torus made homometry more common, and the obvious guess was that symmetry makes structures easier to factor. Tested pair by pair, factor reversal explains two of 3,522 homometric pairs on a 6 × 6 torus, and none at all at six atoms. What explains all 3,522 is a different mechanism: the partner is the structure's own Fourier transform moved independently on each Galois class of characters, by a shift, a sign or an inversion.

 Which plane groups can be coloured with five, twenty-five, seven and forty-nine colours. For each of the seventeen plane groups, its colourings with 5, 25, 7 and 49 colours, as subgroups of that index. At p² colours the count splits into those whose translations meet the lattice in the whole plane pL and those that meet it in a cyclic sublattice. The five groups with a mirror across a rotation, shaded, have none at 5 or 7 and p² of them at 25 and 49, all on pL. In all, 104, 1449, 172 and 4197 colourings. The classification

Twenty-five colours, and the mirrors come back

With a prime number of colours from five up, five of the seventeen plane groups can never be coloured at all: the ones with a mirror across a rotation. With the square of a prime they all can. A colouring with twenty-five colours may meet the lattice in the whole plane modulo five, which every group keeps, and the colours then form a plane the point group acts on by its matrices. The designs come out as the prime's count again, with one more for each of the seventeen groups.

 Where the third wavelength should go. The median Cramér–Rao bound on the phase difference for two hundred reflections, measured at the inflection and the high remote and at a third wavelength whose energy runs across the edge, twice: with f′ known everywhere, and with f′ at the edge energies refined from the data. Known, the best third sits 1 eV above the edge at 6.26°, and a repeat of either wavelength is nearly as good. Refined, both repeats determine nothing, and the best third moves to +2 eV at 6.30°. How it is known

The third wavelength is there to measure the edge

A three-wavelength MAD experiment is usually justified as more anomalous signal: a third look at the edge, placed where the first two left the most out. Counted with the scattering factors known, a third wavelength buys nothing a longer look at the best pair would not buy better. Its real job appears only when f′ at the edge has to be found from the data, which it always does. Two wavelengths then carry one equation too few, and any third energy that is not a repeat supplies it, for two hundredths of a degree.

 Success of the flipping solver against how far off the subtracted lead is. The share of forty starts on the cell of nineteen carbons and one lead that the flipping solver solves, with the lead's density subtracted at a position displaced from the true one by up to sixteen grid steps, half the cell. The rate does not fall with the offset: the points scatter without a trend about the pooled rate of 40%, the band marking two binomial standard deviations, and the original solver, with nothing subtracted, solves none. Refining the position from the map does no better than holding it. How it is known

The lead is wherever it is taken out

Charge flipping solves a cell of nineteen carbons and one lead once the lead's own density is taken out before each flip and put back after. The obvious worry was precision: a lead subtracted a fraction of a grid step from where it really is leaves a residual larger than any carbon. The residual is there, larger than a carbon at a tenth of a step. It does not matter. Subtracted two steps away, or half the cell away, the lead still lets the solver succeed at the same rate, because the solver builds the whole structure round wherever it is put.

 The two forty-sixes over the ten plane point groups. The forty-six two-colour plane groups by the point group of the uncoloured group, and the forty-six rods that turn their axis over by the plane point group their operations show end on. The colourings pile onto the rectangular mirror groups — seventeen on 2mm, ten on m, none on 3 — and the rods onto the hexagonal and square groups, eleven on 6mm and nine on 4mm. Seventeen of the forty-six would have to change point group for the two lists to match. The classification

Two forty-sixes that do not match

The layer groups contain forty-six two-coloured plane patterns, and the rod groups contain forty-six threads that turn their axis over. Both lists are lists of a group with a half of index two, and both halves have a point group in the plane, so they can be laid side by side. They do not line up on any invariant tried: the plane spends its forty-six on rectangular mirrors, which halve a lattice easily, and the rods spend theirs on high rotations, which carry the most kinds of screw. The equal totals are two sums that happen to meet.

 The window in which A₄ and its dual are both minima. The exact second derivative of the Gaussian sum at A₄ and at A₄ along their five-dimensional family of changes of shape, for widths within six hundred-thousandths of one. A₄'s turns positive at t = 0.99998110 and A₄'s turns negative at 1.00001890, the reciprocal. Between them both are positive, so both lattices are local minima; at t = 1 itself both equal 8.59e-6. Lattices

The self-dual width is not flat

A search over four-dimensional lattices found that A₄ and its dual stop being traps for the Gaussian sum almost exactly at the self-dual width, and read that as a flat of the sum sitting on the width itself. Computed exactly rather than by finite differences, the curvature there is not zero. It is six parts in a million of the other curvature, positive, and the same for both lattices, because duality makes their Hessians one matrix at that width. Both are minima, in a window four parts in a hundred thousand wide, with basins a few millionths across.

 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. How it is known

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.

 Three kinds of three-fold axis in one coset. Two space groups of the primitive cubic lattice, each seen down a body diagonal, with one cell of the lattice projected and the three-fold axes parallel to that diagonal marked by type and coloured by class in the group. Every axis has the same linear part, and the lattice places a rotation, a 3₁ screw and a 3₂ screw in one coset of it. P23 keeps all three apart; Pm3̄, whose inversion reverses a screw's hand, joins the 3₁ and the 3₂. Operations

Where the plane had places, space has pitches

Conjugacy in a plane group came down to one lattice, and it counted four families of half-turn to a cell, three of three-fold rotation, two of quarter-turn and one of sixth. In space the same lattice gives the same four numbers for every one of the 219 space-group types. What changes is what the families are. In a centred lattice, and along some directions of a primitive one, they are not rotations in different places but screws of different pitch, and a right-hand screw and a left-hand one in the same coset are one symmetry exactly when the group contains a reflection.

 Every stacking reflection of a warm stack of sheets falls as a power of its size. The Bragg peak per site squared per layer squared of the stacking reflections (0, 0, l), l = 1 to 5, of a stack of flapping square-lattice sheets with bending stiffness 1 and coupling 0.1 between neighbouring layers, at kT/k₁ = 0.02, against the stack's width, with the number of layers growing as the width squared. Each order falls as a fixed power of the size, and the powers grow as the square of the order: 0.20, 0.80, 1.79 and 3.18 for the first four. The fifth falls as the fourth power, as fast as the crystal's volume grows, which is what a reflection with no peak at all does. How it is known

A stack lends each sheet a tension

A single warm sheet wanders so far out of its plane that every reflection with a component along its normal is lost. Stack the sheets and couple each to its neighbours, and inside the stack each sheet wanders only as a sheet under tension does, logarithmically. The stacking reflections survive as peaks that fall as powers of the crystal's size, and the power grows as the square of the order, so the fifth order of the stack computed here has no peak at all. A stack of finitely many sheets keeps even the first only out to a width growing as the square root of its thickness.

 Every four-dimensional lattice with a five-fold rotation, on one interval. The lattices with a five-fold rotation in four dimensions are one ring, ℤ[ζ₅], with the form Tr(a·x·x̄), and the number t = log(a/σ(a)) / 2 log φ, folded into the interval from 0 to 1, says which lattice a is. The dots are the members with a = u + v√5 for u up to 9, 8 distinct lattices. The two ends are A₄* and A₄, with 240 symmetries each; every point between has 20. The arcs join each member to the one a centring by fifths makes of it, its reflection t ↦ 1 − t. Lattices

A five-fold lattice has one free number

In four dimensions a lattice can have a five-fold rotation, and A₄ and its dual both do. Every other lattice with one turns out to be the same ring, the integers of the fifth roots of unity, with a single number deciding its shape. Folded onto an interval, the number runs from A₄* at one end to A₄ at the other, and every lattice strictly between has twenty symmetries. Centring by fifths does not make a new lattice. It reflects the interval end to end, exchanging A₄ with A₄* and pairing every other lattice with a partner, and no lattice sits at the middle to be its own.

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