Fifteen may rotate light, and eleven are chiral

A crystal that rotates the plane of polarisation of light passing through it is called optically active, and the property is routinely explained by saying the structure is handed. That explanation is nearly right and the gap is exactly four crystal classes wide.

Chirality is a property of the group. A class is chiral when it contains no mirror, no inversion and no rotoinversion, so that a structure built on it cannot be superposed on its mirror image. Eleven of the thirty-two are like that, and they are the eleven enantiomorphous classes that make two hundred and thirty space groups out of two hundred and nineteen types.

Optical rotation is a property of a tensor, and like every other property on this site it is decided by Neumann’s principle: the tensor must be invariant under every operation of the class, and what is permitted is whatever survives that. The tensor is the gyration tensor, symmetric and second rank, and the answer is fifteen.

One determinant does all of it #

The gyration tensor has exactly the symmetry of the dielectric tensor — symmetric, second rank, six components — and every crystal has a dielectric tensor. So the difference cannot be in the tensor’s shape. It is in how an operation acts on it.

An axial tensor transforms with the determinant of the operation. Under a proper rotation the determinant is one and the two laws agree; under any improper operation the determinant is minus one and they differ by a sign.

The consequence at an inversion centre is immediate. Under minus the identity, the matrix part returns the tensor unchanged — two minus signs cancel — and the determinant flips it, so the tensor must equal its own negative and therefore vanish. No centrosymmetric crystal rotates light, and that is eleven of the thirty-two classes gone in one line.

Dropping the determinant and running the same calculation gives all thirty-two, which is the dielectric tensor’s answer and is no symmetry statement at all. That comparison is the check that the calculation is about the right object.

What survives, class by class #

Two routes, and they agree. The character sum gives the number of independent components as the average of det(M) times the symmetric-square character over the class — one number per class, no tensors involved. The projector averages an actual generic tensor over the class and reads the rank of the result. The first is cheap and says only how many; the second says which, and their agreement on all thirty-two is what makes either believable.

Three classes are worth taking in turn, because the same six cells drawn the same way say three different things about them, and the differences are the whole subject of the rest of this essay. The first is the one every textbook uses.

What matters in that picture is not the two but the trace. A quadratic form with a non-zero trace can be positive in every direction, and the rotation a ray sees is that form evaluated along the ray — so a class-32 crystal can turn the plane of polarisation the same way whichever way the light goes through it. That is what makes the effect easy to demonstrate and easy to use: a quartz plate is an instrument because its rotation does not depend on getting the orientation exactly right.

The opposite extreme is the class with no symmetry at all, and it is worth drawing because it is what “unconstrained” looks like in this tensor. Nothing is struck through, nothing is tied to anything, and the six numbers a measurement would have to supply are the six a symmetric tensor starts with.

Reading that figure as “class 1 is the most optically active” would be the standard mistake of this whole field. Six is the dimension of a space of permitted tensors, and a triclinic crystal with six permitted components may have all six too small to measure. What the six does say is that symmetry has contributed nothing to the answer, which is the least useful thing a class can do for an experimenter.

The four in the difference #

The third case is the interesting one, and it is a class where symmetry has forced something that no count of independent components would reveal.

The four are 4̅, m, mm2 and 4̅2m, and what they have in common is visible in the computation rather than in the group symbol: in each of them the permitted tensor is forced traceless.

That is not a coincidence and the reason is one line. In Cartesian coordinates every class matrix is orthogonal, so taking the trace of the averaged tensor gives the original trace times the mean of det(M) over the class — and that mean is zero for every class containing an improper operation. A chiral class has none, so the trace survives; an achiral class has as many improper operations as proper ones, so the trace dies.

A traceless quadratic form takes both signs. The rotation a ray sees is the tensor’s quadratic form along its direction, so in these four classes the crystal rotates the plane one way along some directions and the other way along others, with surfaces of zero rotation in between. There is no direction-averaged effect at all: a powder of such a crystal shows nothing, and a measurement along a symmetry axis shows nothing.

Which is why they are hard cases rather than counterexamples. Optical activity in 4̅2m was predicted from exactly this argument long before it was measured, and the measurement — in silver gallium sulphide and in a handful of other crystals — required light along a general direction, careful subtraction of the ordinary birefringence, and a great deal of care about which sign one was looking at. The classic demonstrations use quartz (class 32) and sodium chlorate (class 23), both chiral, both with a trace, both rotating the same way whichever direction the light takes.

The other tensor, for comparison #

The two tensors sit on the same crystal and answer different questions, and a measurement sees both at once. The optical character decides birefringence: how the refractive index varies with direction and polarisation, which is much the larger effect. The gyration tensor decides the rotation, which is superposed on it.

Along an optic axis — a direction where the birefringence vanishes — the rotation is what is left, and that is where it is measured. Quartz’s rotation along its optic axis is about 22 degrees per millimetre for sodium light, a number large enough that a quartz plate is a standard instrument.

Quartz, and the twin that hides the effect #

Quartz is the standard example of an optically active crystal and it is also the standard example of one that can fail to be, which makes it the right place to see what the tensor does and does not decide.

Quartz comes in two hands. Its class is 32, chiral, and its space group is one of the eleven enantiomorphic pairs — P3₁21 and P3₂21, the same structure wound the opposite way. Left quartz rotates the plane one way and right quartz the other, by the same amount, and the two are distinguishable by nothing but that and the small faces on the crystal that Pasteur would have sorted them by.

And a twin puts both hands in one crystal. Quartz has exactly three twin laws, and the Brazil law relates the two hands: a Brazil-twinned crystal contains regions of left and right quartz sharing a lattice. The rotation from one region is undone by the next, so a heavily twinned crystal is optically inactive — while being, class by class and region by region, made entirely of a material that rotates light.

That is the sharpest possible illustration of permitted is not present, and it is worth stating in both directions. The class permits the effect and the specimen does not show it, because the specimen is not a single crystal. Nothing about the symmetry argument is wrong; the argument is about a crystal and the object on the bench is two crystals interleaved.

The practical consequence is that optical rotation is one of the standard tests for Brazil twinning: a quartz plate cut for an oscillator is checked for it, because a twinned plate has the wrong mechanical properties for the same reason it has the wrong optical ones. The two are the same defect seen with different instruments.

How the rotation is actually measured #

The measurement is worth a paragraph because it explains why the traceless classes were doubted for so long.

Along an optic axis the birefringence vanishes and the rotation is all that is left. A quartz plate cut perpendicular to its three-fold axis rotates sodium light by about 22 degrees per millimetre, which is a large and unambiguous signal, and the classical demonstrations all use that geometry.

Off an optic axis the birefringence is enormous by comparison. A general direction in a birefringent crystal splits light into two polarisations travelling at different speeds, and the phase difference between them dominates any rotation superposed on it. Extracting a rotation of a fraction of a degree from that requires modulating the polarisation and detecting at the modulation frequency — the technique is called high-accuracy universal polarimetry, and it is what made the achiral cases measurable.

Which is exactly where the four traceless classes live. Their rotation is zero along their symmetry axes by symmetry, so the only directions where there is anything to see are the general ones where the birefringence is largest. The prediction was a symmetry argument of the kind above; the measurement needed forty years of instrumentation.

Permitted is not present #

The site has an essay about this and it applies here with unusual force.

Symmetry permits; it does not produce. A crystal in class 32 may rotate light and the rotation may be immeasurably small. Neumann’s principle says which components of a tensor are forced to vanish; it says nothing whatever about the size of the ones that are not, which depends on the actual atoms and their arrangement.

And the converse failure is the more interesting one. A measured rotation of zero does not establish that a class is centrosymmetric, because the tensor may be permitted and small, or permitted and traceless with the light along the wrong direction. What a non-zero rotation establishes is definite: the class permits a gyration tensor, so it is one of the fifteen, so it is not centrosymmetric. Optical activity is a test for the absence of a centre, in the same way piezoelectricity is, and the two tests exclude different classes — which is why each permits what the other forbids is a separate essay.

The gyration column is worth setting beside the others once, in words, because the shape of the disagreement is the point. Run the same machinery over the pyroelectric vector, the piezoelectric moduli and the dielectric tensor and four columns come out; not one of them orders the thirty-two classes the way another does. The dielectric column is never zero, because every crystal has a permittivity. The pyroelectric column is zero for twenty-two classes and the piezoelectric for twelve, and the two lists of survivors are not nested. Add the gyration column and it agrees with none of them: it is zero seventeen times, and the classes it spares are neither the polar ones nor the piezoelectric ones.

That is the general shape of every result in this field. The crystal systems sort classes by which lattice they need; a tensor property sorts them by which components survive an averaging; and the two orderings agree only by accident. A reader who expects “more symmetric” to mean “fewer properties” will be right often enough to be surprised when it fails, and the failure is what the tables are for.

Where the exactness stops #

The counts are exact and the tensors are floating point. The character sum is an average of integers over a group and is checked to be an integer, which is an assertion the character machinery has carried since it was written. The projector’s rank is read from an elimination with a tolerance far below any real entry, and its agreement with the character sum on all thirty-two is what keeps the tolerance honest.

The traceless condition some authors impose is not part of this calculation. A gyration tensor is sometimes defined with its trace subtracted off, on the grounds that an isotropic part contributes nothing observable in certain geometries. That is a physical convention rather than a symmetry statement, and it is applied after the averaging rather than during it: the invariant subspace this page counts is the one the group’s action alone determines, and a constraint that is not a symmetry of the tensor’s indices cannot be imposed by a character. A count made with the trace removed differs from the count here by one in every class whose permitted tensor has a trace, and the difference is a difference of definition rather than of arithmetic.

And nothing here is a measurement of anything. Fifteen is a count of classes whose symmetry fails to forbid a tensor, and the number of independent components in each is a dimension. What a crystal of one of those classes actually does to a beam of light depends on its atoms, its wavelength, its temperature and whether the specimen is a single crystal at all — which the Brazil twin above settles in the most direct way available.

The averaging is done in Cartesian coordinates, and doing it anywhere else gives the wrong answer. The trigonal and hexagonal classes are built here in a hexagonal basis where the matrices are integral, and a tensor averaged in that basis is not the physical tensor — the previous rung reported thirteen classes as biaxial for exactly this reason before it was fixed. A tensor’s principal axes are lengths and angles in space.

Nothing here computes a rotation angle. The size of the effect depends on the structure, and the site’s machinery decides which components may be non-zero, not what they are. Every number quoted for an actual crystal in this essay is a measurement from the literature and is labelled as one.

The tensor is the leading term. Optical rotation is described more fully by a spatial-dispersion expansion of the dielectric response, of which the gyration tensor is the first term; there are higher-order effects, and in some geometries they are what is observed. The symmetry argument applies to the leading term.

Who found it, and when #

Arago found optical rotation in quartz in 1811, and Biot established within a few years that it occurs in liquids too — turpentine, sugar solutions — which was the first evidence that the effect can belong to a molecule rather than to a crystal arrangement.

Pasteur separated the two tartrate crystals by hand in 1848, sorting them under a lens by the small faces that betrayed their handedness, and found the two solutions rotated light in opposite senses. That is the experiment that made chirality a chemical idea rather than a crystallographic curiosity, and it is also the origin of the conflation this essay is about: in Pasteur’s case handedness and optical activity really were the same thing.

The tensor treatment is twentieth-century, and the four achiral classes fall out of it immediately once the gyration tensor is recognised as axial. The prediction that 4̅2m and mm2 could be optically active was made from symmetry and doubted for years; the measurements followed, and in each case the effect changed sign with direction exactly as the traceless tensor requires.

The vocabulary is still unsettled. Some authors reserve optical activity for the chiral case and call the other four gyrotropic; others use the words interchangeably. This site’s usage is the tensor’s: fifteen classes permit a gyration tensor, eleven of them are chiral, and saying which is meant costs one clause.

The absorbing half of the same tensor #

Optical rotation has a companion effect measured on the same crystals with the same symmetry rules, and setting the two together explains why the tensor is written as a complex quantity.

A medium that turns the plane of polarisation is one in which left- and right-circularly polarised light travel at different speeds. A medium in which they are absorbed by different amounts shows circular dichroism — linearly polarised light entering it comes out elliptical, because one circular component has been attenuated more than the other.

The two are the real and imaginary parts of one response. They are tied together by the Kramers–Kronig relations, so a material with rotation at one wavelength necessarily has dichroism somewhere — typically near an absorption band, where the rotation also changes sign and swings sharply. That swing is the Cotton effect, and it is the signature by which a chiral molecule’s absolute configuration is assigned in solution.

The symmetry statement covers both, and that is the point of raising it here. Both effects are governed by the gyration tensor, so the fifteen classes that permit one permit the other, and the seventeen that forbid one forbid the other. Nothing in the character sum distinguishes real from imaginary parts, and nothing needs to.

Why the effect is so much smaller than the birefringence #

The essay records that birefringence swamps rotation and that the four traceless classes are therefore hard cases. The ratio has an order of magnitude and a reason, and both are worth having.

Birefringence is a difference between two refractive indices and is a local effect: it depends on the polarisability at a point. Optical rotation is not local — it comes from the response varying across a wavelength, which is why it is called a spatial-dispersion effect, and its size is set by how much the field changes over the size of a cell.

That gives the scale immediately. The ratio of a lattice spacing to a wavelength of visible light is a few ångström against a few thousand, which is about one part in a thousand, and the gyration is smaller than the dielectric anisotropy by roughly that factor. A birefringent crystal’s two indices differ in the third decimal place; its gyration shifts them in the sixth.

That single ratio explains the experimental history. Quartz’s rotation is large enough to demonstrate in a lecture only because it is measured along the optic axis, where the birefringence is exactly zero and the small effect is all that remains. The traceless classes have no such direction available, their effect is smaller still, and the measurements took a century and a half longer.

Where this ladder goes #

This anchor now has two rungs and they are the same machinery on two tensors. The optical character is the dielectric tensor’s shape: five isotropic classes, nineteen uniaxial, eight biaxial, and the isotropic ones are exactly the cubic classes. This rung is the gyration tensor’s: fifteen classes, eleven chiral, four traceless.

What they share is the discipline that makes either believable. Both counts are computed twice by routes with nothing in common — a character sum and a projector — and both are checked against a refusal that must fail: averaging the identity instead of a generic tensor reports every class as isotropic, and dropping the determinant reports every class as optically active. A calculation that gives the right answer for the wrong reason gives it silently, and on a site whose whole proposition is that symmetry is decidable, the second route is not a luxury.