Each permits what the other forbids

Two cubic crystal classes, both of order twenty-four, both lacking a centre of symmetry, both with four three-fold axes along the body diagonals.

4̅3m permits piezoelectricity and forbids optical activity. 432 forbids piezoelectricity and permits optical activity.

Neither class is “more symmetric” than the other. They have the same order, the same axes, and the same crystal system. What differs is the kind of operation on two of their directions, and each property is sensitive to a different kind.

Optical activity is an axial property #

Shine plane-polarised light through quartz along its three-fold axis and the plane of polarisation rotates — clockwise in one crystal, anticlockwise in another, and the two crystals are mirror images. The effect is optical activity, and its tensor is a symmetric rank-two object called the gyration tensor.

Rank two is even, so the parity argument that kills piezoelectricity in centrosymmetric classes does not apply. Something else does: the gyration tensor is axial.

An axial tensor picks up an extra factor of det M under an operation, on top of the ordinary tensor factors. Physically this is what it means for a quantity to have a sense of rotation rather than a direction — a screw thread is right-handed or left-handed, and looking at it in a mirror swaps the two.

So the character of an axial symmetric rank-2 property is

$$\chi(M) = \det M \cdot \tfrac{1}{2}\left[(\operatorname{tr} M)^2 + \operatorname{tr} M^2\right]$$

and that single det factor changes the answer completely.

Fifteen, not eleven #

Eleven of the thirty-two classes are enantiomorphic: they contain no improper operation at all, so a crystal in one of them comes in two mirror-image forms that no rotation relates. Those eleven are the classes a single-handed molecule can crystallise in without its mirror image, and they are 1, 2, 222, 4, 422, 3, 32, 6, 622, 23 and 432.

The obvious guess is that those eleven are exactly the optically active ones. Handedness produces optical rotation; a class with no handedness should produce none.

The guess is wrong, and it is wrong in the informative direction: fifteen classes permit optical activity, not eleven.

The four extra are m, mm2, 4̅ and 4̅2m. Every one contains improper operations, so a crystal in one of them is superposable on its mirror image, and yet the gyration tensor does not vanish.

What the four extras are doing #

In m, mm2, 4̅ and 4̅2m the gyration tensor is non-zero, but every surviving component is off-diagonal. The consequence is that the rotation is zero along the principal directions and non-zero along others, with the sense of rotation reversing between them.

So a crystal in one of these classes rotates polarised light one way along one direction and the other way along another, and averages to nothing. It is optically active in the strict sense — the tensor is not the zero tensor — and it does not have a handedness, because the two senses are both present in the same crystal.

This is the distinction the rule of thumb elides. Chirality is a property of the whole object; optical activity along a direction is a property of that direction. The eleven chiral classes are the ones where the whole object has a hand, and the four extras are ones where individual directions do while the crystal does not.

Measuring it is genuinely difficult, because the directions along which the effect is non-zero are also directions along which the crystal is birefringent, and the birefringence is orders of magnitude larger. The classic case is silver gallium sulfide in 4̅2m, where the effect was predicted from symmetry long before anybody separated it from the birefringence.

Eleven chiral classes, and 230 against 219 #

The eleven enantiomorphic classes are the point-group half of a count this site has already met from the other side.

Two hundred and thirty or two hundred and nineteen is the question of whether a space group and its mirror image are one group or two. Eleven pairs of space groups are enantiomorphic — P3₁ and P3₂, P4₁ and P4₃, and nine more — and counting each pair once gives 219 where counting them twice gives 230.

Those eleven pairs live in the eleven enantiomorphic classes, and the coincidence of the two elevens is not a coincidence: a space group can come in two hands only if its point group has no improper operation, because an improper operation is exactly what would relate the two hands inside a single group.

Not every group in a chiral class is one of a pair, though, which is where the two elevens part company. P2₁2₁2₁ is in class 222, which is enantiomorphic, and it is its own mirror image — the mirror image is the same group in a relabelled setting. What produces a genuine pair is a screw axis of a handedness: 3₁ against 3₂, 4₁ against 4₃, 6₁ against 6₅ and so on, and only the classes containing three-, four- or six-fold axes can hold one.

So the chain runs: eleven classes with no improper operation, of which the ones with a high-order axis can carry handed screws, giving eleven enantiomorphic pairs among the 230.

Back to the cubic pair #

Now the opening claim can be settled by inspection of the two characters.

4̅3m contains six mirrors. A mirror has det = −1, and it contributes to the axial character with the opposite sign to the way it contributes to the polar one. Summed over the group, the axial contributions cancel exactly and the gyration tensor vanishes. Its piezoelectric character, which has no det factor, does not cancel and one modulus survives.

432 contains no improper operation at all, so det = +1 throughout and the axial character equals the ordinary one — which for a symmetric rank-2 property in a cubic class comes to one. Its piezoelectric character, as the previous essay shows, cancels to zero through a conspiracy of the whole group.

A cubic crystal permitting optical activity is a slightly startling object, because a cubic crystal is optically isotropic — its refractive index is one number, its birefringence is identically zero — and the usual reason a material rotates polarised light does not apply. Sodium chlorate is the standard example: cubic, class 23, optically isotropic, and it rotates polarised light. It crystallises in left and right forms from the same solution, and which one a given crystal is is decided by chance at nucleation.

The three lists, and how they nest #

Three properties, three lists, and the nesting is not what a reader expects.

• Eleven enantiomorphic classes: no improper operation. These can hold a single-handed structure.
• Fifteen optically active: the eleven plus m, mm2, 4̅ and 4̅2m.
• Twenty piezoelectric: everything non-centrosymmetric except 432.

Ten classes permit all three. 432 permits optical activity and chirality and not piezoelectricity. 4̅3m permits piezoelectricity alone among the three. And the four extras permit optical activity and piezoelectricity and not chirality.

No one of the three lists contains another, and no ordering of the thirty-two by “symmetry” produces them. That is the argument this essay exists to make: symmetry is a group, and different properties are sensitive to different parts of the group, so the only reliable way to know what a class permits is to compute it.

Why a det factor is such a large change #

It is worth pausing on how much work one factor of det M does, because it is the difference between two properties that otherwise have identical characters.

The dielectric tensor and the gyration tensor are both symmetric rank two. Their characters differ by det M and nothing else. And the resulting counts differ enormously: every one of the thirty-two classes permits at least one dielectric constant — a crystal always has a permittivity — while seventeen of them permit no optical activity at all.

The mechanism is that det M is −1 for exactly half the operations of any group containing an improper one. So in such a group the axial character has as many sign-flipped terms as unflipped ones, and cancellation is the default rather than the exception. In a group with no improper operations det is identically +1, the axial and polar characters coincide, and the count is whatever the ordinary one is.

That is why the eleven chiral classes are all optically active with no calculation needed, and why the four extras are the interesting cases: they contain improper operations whose contributions happen not to cancel everything.

Where the exactness stops #

Optical activity in a birefringent crystal is hard to observe and easy to mis-measure. The rotation is small and the birefringence swamps it in every direction but the optic axes. A measurement that has not separated the two is not a measurement of the gyration tensor.

The count is of permitted components. A class permitting optical activity does not make a material rotate light measurably, and the four non-chiral classes are precisely where the permitted effect is most likely to be unobservable.

And chirality of the crystal is not chirality of the molecule. A solution of a chiral molecule is optically active whatever it crystallises into; a crystal of an achiral molecule can be optically active if the packing is chiral, which is what happens in quartz — silicon dioxide has no handedness and the helical arrangement of its tetrahedra does. Sodium chlorate is the same story: the ion is not chiral and the crystal is.

This is the point where the site’s ruling with molecular-geometry applies. The handedness of a molecule in a finite point group is that site’s; the handedness of an infinite periodic arrangement, and the class-by-class enumeration of which arrangements can have one, is this one’s.

Pasteur, and the crystal that came first #

The chain from crystal handedness to molecular handedness runs the opposite way round from how it is usually told, and the crystals were first.

In 1848 Pasteur crystallised the sodium ammonium salt of racemic acid and noticed under a lens that the crystals came in two shapes, mirror images of each other, distinguishable by which of a pair of small faces was present. He sorted them by hand with tweezers, dissolved each pile separately, and found that one solution rotated polarised light to the left and the other to the right — while the unsorted mixture rotated it not at all.

The crystals were the instrument. Molecular handedness had no structural meaning in 1848 — the tetrahedral carbon atom is van 't Hoff and Le Bel in 1874 — and what Pasteur had was a mirror-image morphology, visible in the hemihedral faces, and a mirror-image optical rotation, measurable in solution. The inference that the molecules themselves came in two hands was the conclusion rather than the premise.

The symmetry statement behind the observation is the one this essay computes. Sodium ammonium tartrate crystallises in an enantiomorphic class, so it can hold a single-handed structure; the two hands cannot interconvert without breaking the crystal; and the faces that gave them away are the hemihedral ones — the faces present in a class smaller than its holohedry, which are exactly what a merohedral class shows and a holohedral one does not.

So the eleven enantiomorphic classes are also the eleven where Pasteur’s method could have worked, and the method needed both halves: a class that permits two hands, and a growth habit that shows which hand a given crystal is.

Two properties, one crystal, and an experiment that needs both #

Quartz is in 32, which permits both effects, and it is the material where the pair is most often met together — which makes it a good place to see that they are answering different questions about the same object.

Its optical activity is along the three-fold axis and is large enough to be a standard laboratory demonstration: a plate cut perpendicular to the axis rotates the plane of polarisation by about twenty-one degrees per millimetre at the sodium D line, and the sense of rotation tells left-handed quartz from right-handed.

Its piezoelectricity is along different directions entirely — the two-fold axes perpendicular to the three-fold — and it is what an oscillator plate is cut to use.

Both are permitted by the same group and neither implies the other, and the pair of cubic classes shows why: 432 permits the first and forbids the second, 4̅3m the reverse. In quartz they coexist because 32 happens to permit both, not because handedness and piezoelectricity are connected.

The one place they do connect is in telling the two hands apart. Both left and right quartz are piezoelectric with the same magnitudes, so a piezoelectric measurement cannot distinguish them; optical activity can, and so can anomalous X-ray scattering. Which is the third appearance in this field of the same asymmetry — a property that is present in both hands says nothing about hand, and one whose sign reverses says everything.

One more contrast is worth drawing, because it separates two things the word chiral is doing.

A chiral class is one with no improper operation, and there are eleven. A chiral structure is a particular arrangement of atoms that is not superposable on its mirror image, and it must sit in one of the eleven. A chiral molecule is the same property one level down, and it may crystallise in a chiral class or not — a racemic mixture usually crystallises centrosymmetrically, with the two hands paired in the cell.

So the eleven constrain the structure and say nothing about the molecule, and the molecule constrains what structures are available and says nothing about which one forms. Three levels, three statements, and the middle one is the only one this field computes.

The cubic case is the easiest measurement in the subject #

The essay above records that optical activity is hard to measure because birefringence swamps it. The cubic classes are where that difficulty disappears entirely, and it is worth following, because it turns the strangest entry in the table into the cleanest experiment.

A cubic crystal is optically isotropic: its dielectric tensor has one independent component, so its refractive index is a single number and its birefringence is identically zero in every direction. There is nothing to swamp anything. A rotation of the plane of polarisation measured through a cubic crystal is therefore the whole of the optical effect, measurable along any direction with no orientation to get right and no compensator to null out.

Sodium chlorate is the standard demonstration. It is in class 23, it grows as large clear crystals from water, its two enantiomorphous forms are distinguishable by eye once the habit is known, and it rotates polarised light by a few degrees per millimetre in opposite senses in the two forms. Pasteur’s method works on it, and it works with no polarising microscope, because there is no birefringence to confuse the observation.

That is a case where the classification does something better than permit. Both 23 and 432 permit optical activity while forbidding any birefringence at all, so the effect is not merely allowed but isolated — the only optical anisotropy the crystal has. A class that permits several effects at once usually makes each of them harder to measure; this pair does the opposite.

The general lesson is about how the table should be read. The number of permitted components says how much freedom a property has, not how observable it is. Class 1 permits every component of everything and is where measurements are hardest, because the effects overlap; the cubic classes permit almost nothing and are where the little they permit is easiest to see. Those are opposite orderings, and a table sorted by count is sorted by neither.

The pair also makes a point about how this table should be built. Neither entry could be predicted from the other, so both are computed by the same character sum run twice with different characters, and the agreement of the two against the thirty-two is the whole of the check.

Where this goes #

The field now has its three counts — ten polar, twenty piezoelectric, fifteen optically active, eleven chiral — and the observation that none of them can be predicted from the others. What remains is to put the whole table in one place and say what a reader should take from it, which is the last essay in the field: symmetry is a filter of great precision and no predictive power, and knowing which of those two it is at any moment is the whole skill.