Crystallography & Symmetry
patterns derived from their groups
Theme

The theme: Exactly this many — page 14

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'.
E₈ has the smallest sum at every width. For four eight-dimensional lattices of unit covolume — D₈, its dual D₈, the cubic lattice Z⁸ and two copies of D₄ — the ratio of their Gaussian sums, less the constant term, to E₈'s, at widths from 0.2 to 5, logarithmic. Every ratio exceeds one everywhere: E₈ wins at every width against every rival. The closest approach is at t = 1, where D₈ and its dual are equal, a factor of 1.31 above E₈. D₈ is the better of the pair at wide widths and D₈ at narrow ones, and neither comes close to E₈ at either end. Lattices

In eight dimensions one lattice wins at every width

In space no lattice has the smallest Gaussian sum at every width, because the densest lattice wins when the Gaussian is narrow and its dual wins when it is wide, and fcc is not bcc. The argument fails exactly when the densest lattice is its own dual. In eight dimensions it is: E₈, with 240 shortest vectors, beats D₈, its dual, the cubic lattice and a sum of two four-dimensional lattices at every width tried, by a factor of at least 1.31, and its sum is mirror-symmetric in the logarithm of the width.

 Every group a symmetric motif can lift a plane group to. The seventeen plane groups arranged by lattice, with an arrow from each group to every group a motif of higher symmetry, placed at some position, turns it into, on the generic lattice of the first. Dashed arrows mark the lifts that also halve the cell, two of which return the same type on the smaller cell. p1 lifts only to p2, the rectangular groups to pmm, pmg and cmm, the square ones to p4g and p4m, and p3 to four different groups. Three groups, outlined, cannot be lifted at all without a special metric: p2, pmm and p6m. The classification

Where a symmetric motif lifts the group

A motif with more symmetry than its site asks for can raise the group of the whole pattern — or not, and which one is decided by the site, not the motif. The criterion is one line: a symmetry of the motif survives into the pattern exactly when it normalises the group. Run over every Wyckoff position of the seventeen, it finds twenty-nine of seventy-two where a motif can lift the group, general positions only in the four groups with a direction to slide along, two sites where the same motif gives different groups depending on how it is turned, and seven lifts between groups that halve the cell.

 Almost every descent ends at D₄. At five widths of the Gaussian, twelve local minimisations of the lattice sum over all four-dimensional lattices of unit covolume, each from a random lattice, each marked by where it ends and labelled with the number of shortest vectors of the lattice it ends at. Fifty-nine of the sixty end at D₄, with twenty-four shortest vectors. One, at the widest Gaussian, ends at A₄*, with ten, a lattice whose sum is higher than D₄'s but which no small change lowers. Lattices

The two lattices where a four-dimensional descent stops short

In four dimensions the argument that rules out a single best lattice in space has nothing to act on, so whether one lattice wins at every width is left to be measured. A search over every four-dimensional lattice finds D₄ from almost anywhere. The exceptions are the finding: two lattices, A₄ and its dual, where a descent can stop short, each at one end of the widths and neither in the middle, for the same reason Voronoi found four dimensions has two perfect forms.

 The seven edges a sheet can have. The seven frieze groups drawn as the profile of a cut edge: along the strip is along the edge, up the strip is the sheet's normal, and the half-sheet itself stretches away from the viewer. Each is labelled with what its operations are in the sheet — a mirror standing across the edge, a mirror or glide in the sheet's own plane, a half-turn about the cut's in-plane normal — and with how many of the eighty layers have it as the edge in some direction. Only p1 and p2 have no mirror image among their operations, so only they are chiral edges; every one of the seven occurs. Into space

Cutting a sheet can give it a hand

Cut a periodic sheet along a line and the edge that is left has a symmetry of its own: a frieze group, drawn in the plane across the cut. A chiral sheet can only have chiral edges. An achiral one usually has chiral edges too, in all but a few directions — and eight of the eighty layer groups have no achiral edge at all, however the sheet is cut.

 Two cuts that give back the mirror a single cut takes away. A sheet whose only symmetry beyond translation is a centre of inversion: each flag has a copy turned half round and moved to the underside, drawn open, and the dots are the centres. Every single cut leaves a chiral edge, and the piece on the far side carries its mirror image. The upper ribbon is cut symmetrically about a row of centres, so each centre on its middle line carries the ribbon onto itself with its two edges exchanged, and the ribbon is achiral. The lower ribbon, cut at the same width about a line with no centre on it, keeps nothing but translations and is chiral. Into space

A ribbon can be achiral where its edges are not

Cut an achiral sheet once and the edge is usually chiral, in almost every direction. Cut it twice, into a ribbon, and the ribbon may be achiral even when both its edges are chiral, because an operation can carry the ribbon onto itself by exchanging the edges — but only if the ribbon's middle line sits on a centre of inversion or a mirror parallel to the cut. Seven of the eight sheets with no achiral edge give achiral ribbons, and one achiral sheet gives nothing but chiral strips however it is cut.

 Two hundred and twelve of the 219 are made of their own operations. The 219 affine space-group types divided by which of their own operations generate them, lattice translations included. 110 are generated both by their screws and glides alone and by their rotations, mirrors and inversions alone; 67 by their screws and glides only; 29 by their point operations only; 6 only by the two kinds together; and 7 by nothing they contain — P1, P2, Pm, Pmm2, P4, P3, P6 — which need a translation given outright. Into space

Seven space groups cannot make their own translations

Every space group contains a lattice of translations, and for all but seven of the 219 types those translations are products of the group's other operations. A screw done twice climbs its axis and a glide done twice slides along its plane, so the non-symmorphic operations make lattices by themselves. The seven that cannot are exactly the seven plane groups with no glide, repeated along a line they all fix, and the reason is a difference between a glide line and a glide plane.

 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.

 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.

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