Crystallography & Symmetry
patterns derived from their groups
Concept

Chirality — where it appears

The property of having no improper symmetry operation, so that a structure and its mirror image cannot be brought into coincidence. Eleven crystal classes have it, and a molecule with a hand can only crystallise in a group that does.

Named by 18 essays across 6 fields — each of them below, with the objects they name alongside it.

A glide. The motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies.

The four motions of the plane

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

operations · What symmetry is 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 2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 3₁, 3₂. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

Eleven groups that are their own reflection's rival

There are two hundred and thirty space groups, and there are two hundred and nineteen. Both numbers are correct and they answer different questions, and the eleven that separate them are the reason a crystal can be built one way round and not the other.

space-groups · Space groups The eleven screw axes, enumerated. An n-fold axis admits a screw for every m from 1 to n − 1, so the orders a lattice permits — 2, 3, 4, 6 — give 11 screws in all. Each row shows the fraction of a cell one turn advances, plotted between zero and one. 3 of them are their own mirror image, which happens exactly when m is half of n; the others pair off into left- and right-handed twins.

Eleven ways to turn while climbing

Five rotation orders survive the crystallographic restriction, and each of them admits a screw for every whole number of cells its turns can amount to. That is a sum with four terms, and it is why there are eleven screw axes rather than some other number.

space-groups · Screws and glides 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 May be piezoelectric, against may be optically active. Two questions asked of all thirty-two classes, and the classes where the answers part company. 14 classes are in both lists, 6 in only the first, 1 in only the second and 11 in neither. Both lists are computed from the same character sum with a different tensor, so a class appearing in one and not the other is a statement about which representation survives rather than about anything measured. Every entry is a permission: a class in a column is a class whose symmetry fails to forbid the effect, which is a weaker statement than it is usually read as.

Each permits what the other forbids

432 and 4̅3m are both cubic, both of order twenty-four, both without a centre. One of them can be piezoelectric and the other can be optically active, and it is not the same one — which is as clean a demonstration as the subject offers that "amount of symmetry" is not a quantity.

point-groups · Properties 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

aperiodic · Aperiodic 15 classes may rotate light, 11 of them chiral. A crystal is chiral when its point group contains no improper operation, and there are 11 such classes. A crystal may rotate the plane of polarisation when its class permits a non-zero gyration tensor, and there are 15. The four in the difference — 4̅, m, 4̅2m, mm2 — are achiral and may still rotate light, which is why the two words are not synonyms. In each of the four, symmetry forces the tensor to be traceless, so the rotation changes sign with direction and cancels in any average over directions.

Fifteen may rotate light, and eleven are chiral

Optical rotation and handedness are treated as the same thing and are not. Eleven crystal classes are chiral; fifteen permit a crystal to rotate the plane of polarisation; and the four in between are the reason quartz and sodium chlorate are the examples everybody uses.

point-groups · Optical p4m: 4 and 4. The standard motif — three points in no particular arrangement — repeated by p4m, with each copy coloured by the sign of the area of the triangle it makes. 4 copies have one sign and 4 the other, because the group contains an operation that reverses orientation. A structure built from one enantiomer cannot sit here: the group would put its mirror image in the same crystal. The colours were computed from the coordinates rather than assigned.

The groups a single hand may sit in

A protein is built from one enantiomer of every amino acid, and a crystal of it contains nothing else. That single fact deletes most of the classification at a stroke: any operation reversing orientation would put the other hand in the same crystal. The criterion is one line of arithmetic, and in the plane the enumeration is complete — five of the seventeen.

space-groups · Chirality Unreflected copies stop at 1 ring. Copies of the hat, all of the same handedness, covering a core of 1 ring of hexagons — 9 tiles, every cell covered once. At 2 rings the same search runs to exhaustion and returns nothing: there is no such covering, and the failure is a proof for that region rather than a search that gave up. The reflected copy is not a convenience of the drawing; the tiling cannot proceed without it.

The tile that needs no reflection

One shape tiles the plane and never repeats, and it does it with copies of both hands. Cut the tiles out of card and that is nothing; ask for it in a molecule, where handedness cannot be undone by turning something over, and it is the whole question.

aperiodic · Monotile A triangle sliding from one mirror to another. Two vertices fixed and the third slid along a line. At the left end the triangle is isoceles about the vertical, at the right end it is isoceles about a different line, and both ends have a mirror — so the measure is exactly zero at both, by cancellation rather than by a search running out. Between them the triangle has no mirror at all and the measure rises to 0.0422. The faint curve is the handedness over the same family, which is a different quantity: it has a sign, it is largest where the measure is not, and it does not vanish at either end.

How chiral, as a number

A group answers one bit: a set either has an improper symmetry or it does not. Two shapes can both be chiral and one of them be a mirror-symmetric thing with a substituent out of place while the other is a helix, and nothing in the classification says which is which. A distance does.

space-groups · Chirality One number, and it is the fraction. Data simulated from crystals that are nought, a quarter, a half, three quarters and wholly inverted, each fitted for the single parameter. The fitted values sit on the diagonal to better than five parts in a hundred, which is what makes the parameter a measurement of composition rather than a test of a hypothesis: a crystal is allowed to be part one hand and part the other, and a value near a half is a real answer about the specimen rather than a failure of the determination.

How much of it is the other hand

A crystal of one enantiomer is a hypothesis, not an observation. What the diffraction actually measures is a fraction — how much of the specimen is the inverted structure — and the useful part of that measurement is the uncertainty on it.

diffraction · Friedel's law One achiral motif in p4, chiral along one line and achiral along two. The same motif — three points with a mirror and no other symmetry — repeated by p4, the plane group of quarter-turns with no mirror, and placed with its mirror along three different lines of the square lattice. Along the first line the pattern has no operation that reverses orientation: it is chiral, although every piece of it is achiral. Along the second and third the motif's mirror is also a mirror of the whole pattern, and the detected groups are p4m and p4g; the mirror lines of each pattern are drawn. All three verdicts come from detecting the symmetry of the points and agree with whether the motif's mirror normalises p4.

A hand made of pieces that have none

Quartz is built from tetrahedra that have no handedness, and every quartz crystal is left-handed or right-handed anyway. Put a piece with a mirror into a pattern whose group has none, and the pattern keeps the piece's mirror only if that mirror lies on one of a few lines the group's normaliser draws. Anywhere else, the arrangement has a hand its parts do not.

space-groups · Chirality Seventeen plane groups, and one chiral sheet over each. The seventeen plane groups, whether each is chiral as a pattern in the plane, how many sheets can be built over it by giving each operation a sign on the sheet's normal — 63 in all — and which of those sheets is chiral in space. There is always exactly one. For the five groups chiral in the plane it is the sheet whose two faces differ and nothing turns it over. For the twelve achiral in the plane it is the sheet turned over by exactly the operations that reverse orientation in the plane, so that every mirror line becomes a half-turn axis lying in the sheet.

Chiral in the plane is not chiral in the room

A pattern with mirrors all over it can be a sheet with a hand, and a pattern with no mirror can be a sheet without one. Whether a layer is chiral depends on what each of its operations does to the side of the sheet, and over every one of the seventeen plane groups exactly one sheet is chiral in space.

space-groups · Chirality The rectangle a (4, 2) tube is rolled from. A patch of honeycomb turned so that the rolling vector C = 4a₁ + 2a₂ lies along the page. C has length √28 ≈ 5.292; the shortest lattice vector perpendicular to it, T, has length 4.583; and the rectangle on the two holds 28 hexagons and 56 atoms. Rolling the rectangle so that its left and right edges meet makes one repeat of the tube. The two lines through the corner are the sheet's mirror directions nearest C: the zigzag direction along a₁ and the armchair direction thirty degrees from it. C makes an angle of 19.11° with the first and lies on neither.

The tube has a screw no lattice allows

Roll a honeycomb along one of its lattice vectors and the tube turns and climbs with a screw of order 14, 98 or 794 — orders the flat sheet could never have. The rolling keeps the sheet's translations and spends them on turns, and it keeps the sheet's mirrors only along two directions, which is why almost every carbon nanotube comes in two hands.

restriction · Finite groups Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle.

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

restriction · Finite groups A thread's two signs, and the four kinds of operation. Every operation of a rod group carries the axis to itself, so it does two independent things: it keeps or reverses the direction along the thread, by a sign σ, and it keeps or reverses the handedness of the plane across the thread, by the determinant of a 2 × 2 matrix. The determinant in space is the product, so the shaded cells are the proper operations — a turn or screw about the axis, and a half-turn crossing it — and the unshaded ones are the improper. A rod group is chiral when all of its operations sit on the shaded diagonal, and polar along its axis when all of them sit on the top row. The two conditions pick out different diagonals of the same square, which is why neither implies the other.

A thread's hand is not a choice

A sheet's handedness in space depends on a sign that the plane pattern does not fix, so one plane group carries several sheets and exactly one of them is chiral. A thread has no such freedom: 32 of the 75 rod groups are chiral, they sit over 9 of the 27 axial classes, and which they are is settled before any structure is drawn. Only its direction depends on how the class lies along it.

space-groups · Chirality Fifty-four of the seventy-five rod groups are a rolled plane pattern. Every rod group, one dot each, grouped by its crystal class. A dot is filled when some plane pattern rolled along some lattice vector has exactly that group, and the 106 crystallographic rollings of the seventeen plane groups fill 54 of them. The eighteen improper classes are full: every one of the 43 achiral rod groups is reached. The nine proper classes are not, and the twenty-one groups named on the right are what is missing — the sixteen whose screw is one of a left- and right-handed pair, and the five bare axes with no climb at all, p1, p112, p3, p4 and p6.

A rolled sheet is never one of a pair

Rolled up along every lattice vector that gives a crystallographic tube, the seventeen plane groups reach fifty-four of the seventy-five rod groups: every achiral one and eleven of the chiral. Not one of the sixteen screws that come in left- and right-handed pairs is among them, and the reason is a single fact about how far a rolled lattice can climb.

restriction · Finite groups

Named alongside it

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

EnantiomorphScrew axisPoint groupSohncke groupCrystallographic restrictionEnumerationHandednessHelixPlane groupRod groupOrientationAbsolute configuration

All concepts