Crystallography & Symmetry
patterns derived from their groups
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The theme: Symmetry is decidable — page 15

Unusually for a physical science, the central questions here have exact answers computable in integer arithmetic. There is no tolerance to choose and no residual to interpret.
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.

 Heavier atoms spoil the line, and then mend it. Above, the scatter of Sayre's ratio about its fitted line, averaged over four random cells, as a quarter of the atoms is made heavier, for cells of eight and of twelve atoms. Below, one minus the square of ρ, the correlation between the density's transform and its square's, computed from the composition alone. The spread of atomic numbers grows without limit along this axis, but ρ does not keep falling: it is lowest near a weight of three and climbs back towards one as the heavy atoms come to dominate, because a cell ruled by a few equal heavy atoms is nearly a cell of equal atoms. The scatter rises and falls with it. At equal weight what scatter remains comes from atoms that overlap, more of them in the larger cell. How it is known

Unequal atoms break the equality, and very unequal ones mend it

Sayre's identity is exact for a cell of equal atoms. Give a quarter of them more weight and the line it predicts scatters, reflection by reflection, by exactly the difference between weighting atoms by Z and by Z squared. Make them heavier still and the scatter shrinks again, because a cell ruled by a few heavy atoms is nearly a cell of equal ones. The scatter measures the cell, but only if the phases are already known.

 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.

 Ninety-six four-colour designs. For each of the seventeen plane groups, the number of its four-colourings, as subgroups of index four, and the number of distinct designs they fall into under the affine normaliser, sorted by the group that permutes the four colours. The 281 colourings make 96 designs: 13 ℤ₄, 39 Klein, 39 D₄, 2 A₄, 3 S₄. The classification

Ninety-six four-colour designs

The seventeen plane groups have 281 four-colourings, counted as subgroups of index four, and many of them are the same design described from a different origin, on different axes or from a neighbouring cell. Identified under every affine map that carries the pattern's group to itself, they make ninety-six designs. The same computation run at two and three colours gives forty-six and twenty-three, and all three numbers match the classical counts of coloured plane groups.

 Which way a fibre of achiral crystals rotates light. For three achiral classes that permit a gyration tensor, every crystal direction that could be aligned along a fibre texture's axis, drawn on a disc as seen from above the z axis, and shaded by the sign of the rotation the texture then shows along its axis. In each class the rotation takes both signs, in regions separated by a cone on which it vanishes. For 4̅2m and mm2 that cone is exactly the class's mirror planes, where the texture inherits a mirror and must be inactive; for 4̅, which has no mirrors, the cone is where the rotation passes through zero between a right-handed texture and a left-handed one. What symmetry decides

A fibre of achiral crystals can rotate light

Four achiral crystal classes permit optical rotation, of both signs in different directions. Grind such a crystal into a random powder and the rotation averages away completely. Align the grains along one direction and it comes back: the texture rotates light along its axis exactly as one grain does, half as much the other way across it — and the direction chosen for the alignment decides the texture's hand.

 Which cells the flipping solver finds. For four cells of twenty atoms — all carbon, a quarter sulphur, one sulphur, one lead — the share of random starting phase sets that charge flipping takes to the right structure within 250 cycles, against the threshold below which density has its sign reversed. Each point pools several random arrangements of atoms with several starts each, and the atoms fall off with angle as real atoms do. The cells with sulphur are solved more often than carbon alone, the quarter-sulphur cell most of all, each at a threshold near one standard deviation; the lead cell is never solved at any threshold. How it is known

The heavy atom the flipping solver cannot see past

Sayre's identity degrades as a cell's atoms become unequal, most of all for a quarter of heavy atoms three times the weight of the rest, and the obvious prediction is that a solver built on the same sparseness should struggle there too. It does not. Charge flipping solves a cell of carbon with sulphur in it more often than carbon alone, and never solves a cell with one lead atom — because a threshold stated in standard deviations belongs to whichever atom owns the map's variance.

 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.

 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. The classification

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.

 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.

 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.

 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.

 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.

 Where a gradient term is allowed at a special point. For each of the seventeen plane groups and each special point of its zone — the centre, the edge midpoints, the corner of a rectangular or square zone and the corner K of a hexagonal one — every physically irreducible order parameter, by its real dimension, marked where it admits a Lifshitz invariant. They occur at 10 points in 7 groups: the edges of the glide groups pg, pmg, pgg and p4g, and the corner K of p31m, p6 and p6m. Nowhere else. Into space

Where a gradient term survives at the edge of the zone

A transition that freezes a mode at the edge of the zone doubles the cell, but only if the energy has no term linear in the mode's gradient. That condition was stated here for the zone boundary and never computed. Computed for every order parameter at every special point of the seventeen zones, it fails in exactly ten places: the edge points of the four glide groups where one glide acts and no second glide crosses it, and the zone corner of three hexagonal groups that have no glide at all.

 Every stratum a primitive cubic cell lies on, and the steps between them. The ten lattice strata a primitive cubic cell lies on, from triclinic, with no condition, to cubic, with five, each with the number of linear conditions it imposes on the cell's six parameters, joined wherever one adds conditions to the other with nothing between. There are 14 steps and 10 paths from the bottom to the top: nine of four steps, through a monoclinic, the orthorhombic and a tetragonal stratum, and one of two, through the rhombohedral. Operations

A cubic cell has fourteen chances to fail

In the plane, deciding whether a measured cell is square meant two steps up a short diagram, and dividing the error rate between them fixed the test. In space, a primitive cubic cell sits on ten strata joined by fourteen steps on ten different paths, and a walk that tests every step at five per cent calls more than a quarter of true cubic cells something less. Dividing the level by the steps of one path is not enough and dividing it by all fourteen is too much. Measured directly, the right level is two-thirds of a per cent a step, and calibrated that way the walk notices a tetragonal distortion more often than the single test it replaces.

 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. Operations

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.

 Where the four-fold screws of one crystal cross a plane. A space group of the body-centred class I4/m that has a centre of symmetry, seen along the direction of its four-fold axes: the points where the special lines along that direction cross a plane, over four cells of the projection. A rod cut along the large filled lines is p4₃, a four-fold screw of one hand; along the other large lines, p4₁, its mirror image. The crystal's centre of symmetry carries each kind onto the other, so both are there and neither hand is favoured. Into space

A crystal with no hand holds rods of both hands

Rolling a flat pattern into a tube reaches fifty-four of the seventy-five rod groups and never a screw that comes in a pair of hands. Cutting a column out of a crystal reaches all seventy-five. Sixteen of them are handed screws, and half of those can be cut from a crystal that has a centre of symmetry, from a line that happens to be a screw of one hand, with its mirror image a short distance away. A handed thread does not need a handed crystal, only a handed place in one.

 Four flipping rules on four cells. The best success rate of the flipping solver, pooled over five arrangements and eight starts, on four cells: twenty carbons, a quarter sulphur, one sulphur among twenty and one lead among twenty. Four rules for what is flipped: below a number of standard deviations of the map, below a height set by the lightest atom's weight, the same height above the map's mean, and the original rule with the known heavy atom's density removed first. On the lead cell only the last solves anything, 55% against nothing; on both sulphur cells it raises the rate to near certainty; on carbon alone it is the original rule. How it is known

The lead taken out solves what no threshold could

Charge flipping never solved a cell of nineteen carbons and one lead, and two repairs were proposed: a threshold set by the composition rather than by the map's spread, and the lead's own density removed before each flip. The threshold does nothing for the lead cell at any setting, measured from zero or from the map's mean. Removing the lead takes it from nought to fifty-five per cent, and lifts both cells with sulphur in them to near certainty. The trouble was never where the threshold sat.

 Which two kinds of operation make each of the 64 two-generator space groups. For each of the fifteen pairs of kinds of operation, rotation, screw, mirror, glide and rotoinversion (the inversion included), the number of the 64 space-group types generated by two of their own operations that some pair of those kinds generates. Every one of the 960 cases is decided: by a generating pair found, by the pair's linear parts failing to generate the point group, or by a proof in a finite quotient of the group. A screw and a glide make 32 types; two mirrors and a rotation with a mirror make none. Into space

Two mirrors never make a crystal

Sixty-four space groups are generated by two of their own operations. Which two kinds can do it is a finite question, and it has been settled for all sixty-four and all fifteen pairs of kinds. A screw and a glide make thirty-two of them. Two mirrors make none, and neither does a rotation with a mirror, for reasons that need no computation: two operations that undo themselves generate a dihedral group, and an axis and a plane always share a point or a direction. Two plain rotations make six groups, every one of them cubic and without a mirror.

 What Kleinman's rule removes from the second harmonic, class by class. The 20 crystal classes that permit second-harmonic generation, each with its independent susceptibility components drawn as squares: those that survive when the tensor is made totally symmetric, and those Kleinman's rule removes, split into the traceless part of the gyration tensor and a polar vector. In all, 83 components become 48. Two classes, 422 and 622, lose everything; four, 6̅, 6̅2m, 23, 4̅3m, lose nothing. What symmetry decides

Two classes go dark far from absorption

The second-harmonic tensor of a crystal has as many components as its piezoelectric tensor, and far from any absorption it gains a symmetry of its own that removes eight of the eighteen. What it removes is not arbitrary: it transforms as a polar vector plus the traceless part of the tensor that carries optical activity. So the classes that lose the most are the polar ones and the handed ones, and two of them, 422 and 622, lose everything.

 How many resonances each crystal class needs for a second-order response. The 32 crystal classes sorted by the number of distinct oscillators a model needs before the class can have any second-order optical response, with the dimensions of the class's invariants among the cubic couplings: fully symmetric (one oscillator), mixed (two), alternating (three). The eleven centrosymmetric classes have none. Eighteen need one; four of those (23, 4̅3m, 6̅, 6̅2m) have no mixed invariant and obey Kleinman's rule at every frequency. 422 and 622 need two. 432 needs three. What symmetry decides

A crystal in 422 needs two resonances

Kleinman's rule silences classes 422 and 622 far from absorption, and the obvious guess is that dispersion brings them back near it. In the simplest model of dispersion, one charged oscillator with an anharmonic potential, it does not: they stay dark at every frequency. Two oscillators with different resonances light them, with a response growing as the square of the frequency. 432 needs three, and then only mixes two different frequencies, growing as the sixth power. The number of resonances a class needs is a count of invariants, and it sets how fast the forbidden part can grow.

 Which three kinds of operation make each of the 93 three-generator space groups. For each of the 35 triples of kinds of operation (rotation, screw, mirror, glide, rotoinversion with the inversion), how many of the 93 space-group types that need three of their own operations some triple of those kinds generates. Of the 3255 cases 978 are generating triples found, 2275 are refuted (by the point group or in a finite quotient) and 2 remain undecided. Three mirrors make none; a screw and two glides make the most, 53. Into space

Three mirrors never make a crystal either

Ninety-three space groups need three of their own operations to generate them, and which three kinds can do it is a finite question with 3,255 cases. All but two are now decided. Three mirrors make none of the ninety-three, as a theorem about reflection groups requires. A rotation with two mirrors makes exactly three, all cubic. Three plain rotations make thirteen, every one without a mirror. One group can be made thirty ways and three can be made only one.

 Every special position of the seventeen plane groups, with its regular subgroups. The 55 special Wyckoff positions of the seventeen plane groups, one chip each, labelled by site symmetry and points to a cell and carrying the number of subgroups that act on the position regularly. None is zero. 26 positions have one point to a cell and are reached by the lattice alone; 12 can be reached only by subgroups containing rotations; four, the positions of three points to a cell in p3m1, p31m, p6 and p6m, are reached by exactly one subgroup each. Operations

No Petersen graph in the plane

A finite graph can look the same from every vertex and still be the picture of no group; the Petersen graph is the smallest that does. A periodic net in the plane cannot. Whether one is a Cayley graph depends only on where its vertices sit, and they sit on one of the fifty-five special positions of the seventeen plane groups or on a general one. Every one of the fifty-five has a subgroup that carries any of its points to any other in exactly one way, and four of them have exactly one.

 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. Operations

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.

 The ninety magnetic classes that are not grey, sorted by their centre. The ninety magnetic point groups that are not grey, in three columns by whether they contain the inversion, the inversion combined with time reversal, or no centre at all, each tile shaded by what a stress can do: magnetise a crystal that is already magnetised (31 classes), magnetise one that has no moment at rest (35), or nothing. A dot marks the classes that also permit the magnetoelectric effect. No class with the primed inversion is piezomagnetic and no class with the plain one is magnetoelectric. What symmetry decides

Thirty-five classes a squeeze can magnetise

A magnetisation proportional to a stress is permitted in sixty-six of the hundred and twenty-two magnetic classes. Thirty-one are the classes that may be magnetised already; the other thirty-five may have no moment at rest and acquire one when squeezed, and in them a stress tells apart two domains that differ only by the direction of time. Among the ninety classes that are not grey, it is forbidden only by a primed centre and in the three ordinary cubic classes with four-fold operations.

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