Crystallography & Symmetry
patterns derived from their groups
Concept

Optical activity — where it appears

The rotation of the plane of polarisation by a crystal, permitted in fifteen classes of which eleven are chiral. The four in between are why quartz and sodium chlorate are the examples everybody uses, and why handedness and optical rotation are not the same property.

Named by 8 essays across 2 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 The twin laws of class 32. The point group of the lattice of class 32 has 24 operations and the class has 6, so it splits into 4 cosets: the crystal itself, and 3 twin laws. Every operation in a block produces the identical second orientation, which is why the block and not the operation is the law.

Quartz has exactly three twin laws, and its lattice is why

Class 32 on a hexagonal lattice has index four, so three twin laws and no more. They turn out to be the three the mineralogists named — Dauphiné, Brazil and the combination of the two — and reading quartz's lattice off its class instead of measuring it would have produced one law where there are three.

applied · Twinning 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 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 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.

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.

point-groups · Curie 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.

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.

point-groups · Optical 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.

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.

point-groups · Properties 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.

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.

point-groups · Properties

Named alongside it

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

Gyration tensorAxial tensorCharacterChiralityNeumann principleBirefringenceCrystal classEnantiomorphismIrreducible representationPiezoelectricityPolar vectorProperty tensor

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