Three optical characters, and the arithmetic that assigns them

Neumann’s principle states the rule — a physical property of a crystal has at least the symmetry of its point group — and twenty-one, thirteen, nine, three counts what it leaves of the elastic constants. This essay asks the same question of the humblest property there is, and gets an answer a mineralogist uses every day at a microscope.

A symmetric rank-two property — the dielectric tensor, the thermal conductivity, the thermal expansion, all of them at once — is an ellipsoid. Averaging a generic one over a point group leaves an ellipsoid with the group’s symmetry, and there are exactly three things that can happen to its three semi-axes.

Three characters, and what each means at a microscope #

All three equal — a sphere. The crystal is isotropic: it has one refractive index in every direction, cannot be birefringent, and between crossed polars stays dark at every rotation of the stage.

Two equal — a spheroid. The crystal is uniaxial: there is one direction, the optic axis, along which light of any polarisation travels at one speed, and every other direction has two. The optic axis is the principal symmetry axis, which symmetry fixes exactly.

All three different — a general ellipsoid. The crystal is biaxial: there are two directions along which the two speeds coincide, and they lie in the plane of the largest and smallest axes.

Those three are the whole of what a petrographic microscope reports about the class of an unknown grain, and which one a crystal has is decided before any measurement, by its point group alone.

Five, nineteen, eight #

The census comes out at 5 isotropic, 19 uniaxial, 8 biaxial, and none of the three numbers was sorted by hand. For each class a generic symmetric tensor is averaged over the class’s own matrices, its eigenvalues are computed in closed form, and the number of distinct eigenvalues is the character. The classes then group themselves.

The five isotropic classes are exactly the cubic ones — 23, m3̅, 432, 4̅3m, m3̅m — and that is the statement worth stopping on. A cubic crystal cannot be birefringent. Not “is usually not”; cannot. Diamond, halite, garnet and fluorite are all optically indistinguishable from glass under crossed polars, and no chemistry can change it, because the averaging leaves a multiple of the identity and a multiple of the identity has one refractive index.

The nineteen uniaxial classes are the tetragonal, trigonal and hexagonal ones, which are exactly the classes with a unique axis of order greater than two. The optic axis is that axis.

The eight biaxial classes are the triclinic, monoclinic and orthorhombic ones — the classes with no axis of order greater than two, so nothing forces two of the three principal values together.

The second thing symmetry decides, and the one usually left out #

Symmetry fixes the shape of the ellipsoid in every class. It fixes the ellipsoid’s orientation in only some of them, and the difference is a number a table of characters does not carry.

The count that carries it is the number of independent components of a symmetric rank-two tensor, computed as the dimension of the invariant subspace by averaging the character over the group. Three of those numbers are always the principal values; anything beyond three is orientation.

The monoclinic case is the one with a visible consequence. One axis of the ellipsoid must lie along the unique two-fold; the other two are free to rotate in the plane across it. They do — by different amounts at different wavelengths — and a thin section of a monoclinic mineral under white light shows dispersion of the optic axes: the extinction position is not the same for red light as for blue, and the interference figure shows coloured fringes that are not symmetric.

That is a symmetry argument with an appearance. Nothing about an orthorhombic crystal can do it, because symmetry nails all three axes; a triclinic crystal does it in three directions at once.

How the assignment is computed #

The projector is the whole method and it has one trap in it.

Averaging MᵀTM over the group projects T onto the tensors the group leaves alone, and starting from a generic T — one with no accidental equalities among its entries — gives a generic member of that invariant subspace. Starting from the identity does not: the identity is invariant under every group, so its average is a sphere in all thirty-two classes and the census reads 32 isotropic, 0, 0.

That is the refusal this essay carries, and it is computed rather than described. A check that never fed the projector a generic tensor would pass every assertion it made about symmetry and would be measuring nothing at all.

The eigenvalues are taken in closed form rather than by iteration, because the whole classification is a count of how many are equal. A Jacobi sweep leaves 10⁻¹² between two axes that symmetry makes identical, and a threshold that tolerates that would also tolerate a genuinely small birefringence.

And the matrices have to be Cartesian. The trigonal and hexagonal classes are built here in a hexagonal basis, where the axes are at 120° and the matrices are integral. Averaging MᵀTM in that basis is not the same operation as averaging in a Cartesian one, and the result’s eigenvalues are not principal values. The first version of this calculation reported every trigonal and hexagonal class as biaxial — wrong for all thirteen — and the contradiction was on the same row of its own output, where the component count said two.

That is the whole lesson of it: a tensor’s principal axes are lengths and angles in space, so the averaging has to be done where lengths and angles are what they look like.

Why the counts fall where they do #

The three numbers 5, 19 and 8 are not arbitrary, and reading why says something about what an ellipsoid can do.

An ellipsoid has three axes and a point group acts on them by permutation and reflection. For two of its principal values to be forced equal, the group must contain an operation carrying one principal direction to another — which is a rotation of order greater than two about the third.

So the character is decided by the highest rotation order. A class with a unique axis of order three, four or six carries the other two directions into each other and forces two values equal: uniaxial. A class with several such axes — the cubic ones have four three-folds — forces all three equal: isotropic. A class whose highest order is two carries nothing into anything: biaxial.

That is a one-line rule and it reproduces the whole census. It also explains the asymmetry in the counts: there are nineteen uniaxial classes because three systems have unique high-order axes, and only five isotropic ones because only the cubic system has several.

And it says why rank matters. The same argument for a rank-four tensor produces the counts 21, 13, 9, 3 rather than 6, 4, 3, 2, 1 — a higher-rank tensor has more components for symmetry to constrain and the constraints bite differently. The optical case is the simplest one there is, which is why it was understood first.

Where the exactness stops #

The character is exact. It is a count of distinct eigenvalues of a matrix produced by an exact projector, and the equalities symmetry forces are equalities to machine precision rather than approximations.

Not one magnitude is computed anywhere. Symmetry says which components may differ from each other and says nothing whatever about by how much. A uniaxial crystal may have a birefringence of 0.172, as calcite does, or of 0.0001, as apatite nearly does; both are uniaxial and this arithmetic cannot tell them apart.

A crystal may be isotropic without being cubic. Nothing forbids a tetragonal crystal from having its two principal values accidentally equal at some wavelength and temperature — that is an isotropic point, real minerals have them, and symmetry has nothing to say about it. The implication runs one way: cubic implies isotropic; isotropic does not imply cubic.

And optical activity is a different tensor. A crystal can be optically isotropic and still rotate the plane of polarisation, because rotation is governed by the gyration tensor rather than by the dielectric one — optically active and not piezoelectric is that story, and the classes that permit it are decided by a different character sum.

What a microscope actually does with this #

The optical character is the first thing a petrographer establishes about an unknown grain, and the procedure is a symmetry argument run backwards.

Isotropic is settled by looking. Rotate the stage under crossed polars: a grain that stays dark through a full turn is cubic or amorphous. That single observation eliminates five of the thirty-two classes and all of the glass.

Uniaxial against biaxial is settled by an interference figure, which is the pattern produced in convergent light: a uniaxial crystal gives a cross that stays put as the stage rotates, and a biaxial one gives a cross that splits into two hyperbolae. That observation splits the remaining twenty-seven classes into nineteen and eight.

And the sign — positive or negative — is settled by a compensator, which reports whether the extraordinary index is larger or smaller than the ordinary one. That is a magnitude question and symmetry does not answer it; it is a measurement, and it is what identifies a mineral rather than a class.

So the microscope gives the system in two observations and the species in a third, and only the first two are consequences of the arithmetic here. The division between them is exactly the division this whole field keeps: symmetry supplies permissions, and measurement supplies numbers.

Everything on this page before that figure is the version of the calculation without the determinant; everything after it is the version with. It is worth being explicit about how much the two differ, because the two tensors are otherwise the same object and the essays that treat them apart can make the difference look larger than it is. The dielectric tensor is permitted in all thirty-two classes and only its shape is ever in question, so the census above is a partition of the thirty-two into three characters with nobody left out. The gyration tensor is forbidden outright in seventeen of them, so the same calculation asked about it returns a list rather than a census. Those are different kinds of answer, and knowing which kind a figure is giving is most of what it takes to read this part of the subject correctly.

The mechanism behind the difference is the one already met in the elastic case, arriving from the other side. In a group containing improper operations, exactly half the operations have det M = −1, so 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 at all the determinant is identically +1, the axial character is the ordinary one, and the count is whatever the dielectric count was. That is why every chiral class permits optical rotation with no calculation needed, and why the classes that permit it without being chiral are the ones worth looking at.

Who found it, and when #

The indicatrix is Fletcher’s, from 1892, though the ellipsoid itself is older — Fresnel had the optical ellipsoid in the 1820s, and the connection between crystal symmetry and optical behaviour was worked out through the middle of the nineteenth century by Biot, Brewster and Herschel, largely before anybody had a theory of what a crystal was.

Neumann’s principle is Franz Neumann’s, stated in lectures from the 1830s and published by his student Voigt in 1910 — and the optical case is where it was first used seriously, because optical anisotropy was the only tensor property anyone could measure conveniently before X-rays.

The order of discovery is worth noticing. The classification of crystals by their optics — isotropic, uniaxial, biaxial, with the systems attached — was established decades before the classification by symmetry that explains it. Mineralogists sorted crystals into these three groups because that is what the microscope showed, and the group theory arrived afterwards and said the sorting was forced.

One consequence for structure determination #

The optical character is a system determination made in a minute, and that is why it belongs to a site about symmetry rather than to one about optics.

A grain that stays dark under crossed polars is cubic. A grain with a fixed interference cross is tetragonal, trigonal or hexagonal. A grain whose cross splits is orthorhombic, monoclinic or triclinic. Three observations, three answers, and each of them is a statement about the point group obtained without measuring a single reflection.

That was the routine before diffraction and it remains the routine in petrography, where a thin section carries thousands of grains and no one is going to mount them all. It also remains the first check on a diffraction result: a crystal whose cell refines as tetragonal and whose optics are biaxial has something wrong with it — twinning, most likely, or a wrongly assigned cell — and the disagreement is between two independent determinations of the same quantity.

Two routes to one answer, disagreeing, is the most useful thing an experiment can produce. The optical route sees the point group through a rank-two property; the diffraction route sees it through the Laue class. Neither sees it directly, and where they differ, the difference is information.

The classes that are optically indistinguishable #

One consequence of the census is worth stating as a limitation, since it bounds what the microscope can do.

Nineteen classes are uniaxial and the microscope sees one thing. 4, 4̅, 4/m, 422, 4mm, 4̅2m, 4/mmm, 3, 3̅, 32, 3m, 3̅m, 6, 6̅, 6/m, 622, 6mm, 6̅2m and 6/mmm all give the same interference figure, and no optical observation separates them. The optical character reports the system and stops.

And two systems share a character. Trigonal and hexagonal classes are both uniaxial, and the dielectric tensor cannot tell them apart at all — a fact that follows from the counts here, since both have two independent components and the same eigenvalue pattern. Separating them needs a rank-three or rank-four property, or diffraction.

That is the honest ceiling on this method: it is a system determination and a fast one. Everything sharper requires either a higher-rank property, where the classes separate — twenty of twenty-one is where a rank-three tensor divides them — or an experiment that sees the lattice.

The birefringence that cubic crystals are not allowed to have #

The strongest statement on this page is that a cubic crystal is optically isotropic. It is exact, it follows from the character, and cubic minerals are observed to be birefringent often enough that the effect has its own name.

Anomalous birefringence in garnet is the standard case. A garnet grain between crossed polars ought to be dark at every rotation, and many are not: they show weak birefringence, often arranged in sectors corresponding to the crystal’s own growth faces. The effect is small — a fraction of a per cent of an ordinary mineral’s — and it is entirely reproducible.

Nothing here is wrong, and the resolution is the one this collection reaches for whenever a prohibition appears to fail: the crystal is not in the class it was assigned to. A garnet whose cations order onto the sites of one sublattice rather than being distributed at random has a structure of lower symmetry than cubic, whatever its shape and its diffraction pattern suggest. A crystal grown with different faces incorporating different compositions is inhomogeneous, and each sector is separately strained by its neighbours. Strain is a rank-two quantity and it destroys a cubic dielectric tensor’s degeneracy directly.

So the observation is evidence about the structure rather than about the theorem, and it is the sharpest kind of evidence available: a symmetry-forbidden effect measured at all says the symmetry is absent, without any need to say how far. That is worth setting against the caution the classification usually carries. Permitted is not present is about the direction in which the table says nothing; this is the other direction, where the table says something very strong and a microscope can check it in a minute.

What the effect does not permit is a numerical reading. How much birefringence a given amount of ordering or strain produces is not a symmetry question, and the anomalous case is precisely where the exact answer and the tolerant one come apart — the structure is near-cubic, the tensor is near-spherical, and the size of near is set by the material.

Where the ladder goes next #

This rung establishes the shape of one property in every class. Three rungs sit above it.

Second-order effects. The electro-optic and photoelastic tensors are third and fourth rank, so their symmetry-permitted forms are the character sums twenty of twenty-one and twenty-one, thirteen, nine, three already compute, and the optical consequence — a field or a stress making an isotropic crystal birefringent — is a permission this ladder can state.

Dispersion as a symmetry statement. The free angles in the monoclinic and triclinic cases are functions of wavelength, and how they may vary is itself constrained: symmetry fixes which components can disperse and which cannot, which is a sharper statement than the count of components.

Magnetic optics. Adding time reversal changes which tensors are permitted, and the Faraday effect is a permission of the magnetic classes rather than the ordinary ones — the same character sum with one more operation, which is the operation that reverses time applied to an optical property.