The axes a class pins down

Three optical characters computes the shape of a crystal’s indicatrix: average a random ellipsoid over the class and see how many of its three axes come out different. A cubic crystal gives a sphere and cannot be birefringent whatever it is made of; nineteen classes give a spheroid; eight give a general ellipsoid.

Neumann’s principle computes how many numbers the tensor needs — six in a triclinic crystal, four in a monoclinic one, three in an orthorhombic one, one in a cubic one.

Neither says where the ellipsoid points, and that is a third thing symmetry decides.

Shape, magnitude, orientation #

A rank-two property is an ellipsoid, and an ellipsoid is three things at once: how big it is, how far from spherical, and which way it points. Symmetry can constrain each independently, and reading a symmetry statement without knowing which of the three it is about is how the subject’s most useful rule gets misapplied.

Magnitude is never constrained. No symmetry argument has ever said how large a refractive index is.

Shape is constrained by degeneracy. Where a class forces two principal values to be equal, the ellipsoid is a spheroid; where it forces all three, a sphere. That is the character computation, and it is a statement about the eigenvalues.

Orientation is constrained by the eigenvectors, and it is measured here. A direction that is a principal axis of every tensor the class permits is a direction symmetry has pinned; a direction that varies from one permitted tensor to another is one the material chooses.

The measurement follows the definition exactly. Average several different random symmetric tensors over the class; each average is a permitted tensor; and take the directions common to all of them.

Three, one, none #

The answer sorts the biaxial systems in a way the component count almost, but not quite, predicts.

Orthorhombic pins all three. The three two-fold axes are mutually perpendicular and each must be an eigenvector — an operation of order two carries an eigenvector to itself or to its negative and nothing else — so the ellipsoid’s axes are the crystal’s axes, always. Three components, three fixed directions, and no orientation left to measure.

Monoclinic pins one. The single two-fold axis is an eigenvector for the same reason; the other two lie somewhere in the plane perpendicular to it and symmetry has nothing to say about where. The tensor needs four numbers: three principal values and one angle, which is the orientation of the pair in that plane.

Triclinic pins none. Six numbers: three values and three angles, and every one of them is a property of the material.

And cubic pins none for the opposite reason. Its tensor is a multiple of the identity, so every direction is a principal axis and the question has no content. A census that reported three fixed axes there would be reporting the directions the random sampling happened to produce, which is why the machinery reports the degeneracy alongside the count and treats a fully degenerate tensor as having no axes at all.

What the free angles do #

An angle that symmetry does not fix is a number the material supplies, and a number the material supplies can change when anything about the measurement changes. That is not a theoretical possibility. It is the first strange thing a mineralogist meets down a polarising microscope.

Dispersion of the optic axes. The refractive indices of a crystal depend on wavelength — that is ordinary dispersion, and every material has it. In a monoclinic crystal the orientation of the indicatrix depends on wavelength too, because the free angle is a property like any other. A thin section between crossed polars therefore extinguishes at one angle in red light and another in blue, and shows coloured fringes at extinction rather than going properly dark.

It cannot happen in an orthorhombic crystal. The axes are the crystal’s own axes at every wavelength, because there is nowhere for them to go. Extinction is sharp and colourless, which is a diagnostic a microscopist uses to separate the systems before any measurement is made.

And in a triclinic crystal all three axes disperse, independently, which is why triclinic minerals are the awkward ones.

So the count in the fourth column of the table above is not bookkeeping. It is the number of degrees of freedom the orientation has, and every one of them is a way for the crystal to behave differently at a different colour.

The same statement for other properties #

Nothing above mentioned light. The computation is about a symmetric rank-two tensor and every such property obeys it, which is Neumann’s principle doing what it always does.

Thermal expansion is a symmetric rank-two tensor, so a monoclinic crystal has one expansion axis fixed by its two-fold and two that can rotate — with temperature, this time, rather than with wavelength. A crystal whose expansion axes turn as it is heated is a monoclinic crystal, and it cannot be an orthorhombic one.

Electrical conductivity, magnetic susceptibility, the dielectric constant — all the same, all with the same table.

The one thing that changes between them is which classes are relevant, because a property may be forbidden entirely in some classes: the ten polar classes are where a vector property may exist at all, and a rank-two property exists in every class. The orientation arithmetic here applies wherever the property does.

What the count does not say, again #

Two warnings, and they are the standing ones for this whole field.

A pinned axis is a constraint, not a prediction. Symmetry says an orthorhombic crystal’s principal axes are its crystallographic axes; it does not say the three principal values differ, and if two of them happen to be equal the crystal is optically uniaxial while being crystallographically orthorhombic. That is an accidental degeneracy, it occurs at particular wavelengths in particular materials, and it is exactly the situation a coincidence the group did not ask for is about — testable by moving the numbers symmetry does not decide and seeing what survives.

A free axis is not a prediction either. Symmetry permits the monoclinic pair to rotate with wavelength; whether it does, and by how much, is a fact about the material. Permitted is not present applies to orientation exactly as it applies to magnitude.

Why an operation of order two fixes an axis #

The measurement above is a census; the reason behind it is a single argument, and it is short enough to give in full.

Let T be a tensor the class permits, so that every operation M of the class satisfies MᵀTM = T. Take M to be a two-fold rotation about a direction a. Then M fixes a and reverses every direction perpendicular to it.

Apply T to a. The vector Ta satisfies M(Ta) = T(Ma) = Ta, because T commutes with M and M fixes a — so Ta is itself fixed by M. But the only directions M fixes are multiples of a. Therefore Ta is a multiple of a, which is precisely the statement that a is an eigenvector of T.

Nothing in that argument mentioned which tensor T is, so it holds for every permitted tensor at once, which is exactly what “pinned” means. And it needs only that the operation has a unique fixed direction — so a four-fold or six-fold axis pins its own direction just as well, with the difference that those operations also force the perpendicular plane to be degenerate, which is why the tetragonal, trigonal and hexagonal systems come out uniaxial.

The orthorhombic case is then three copies of the argument, one per two-fold, and the three axes are mutually perpendicular because the operations are. The monoclinic case is one copy. The triclinic case is none, because inversion fixes every direction and constrains nothing at all — the identity and the inversion act the same way on a rank-two tensor, which is why a centre of symmetry never appears anywhere in this subject’s counting.

The eigenvector computation, and why it is done twice #

There is a methodological choice in the measurement that is worth exposing, because it is the difference between a result and an artefact.

A single invariant tensor has three principal axes, and every one of them is a direction. Nothing about one tensor says which of its axes were forced and which merely came out that way — a random tensor averaged over a triclinic class is a perfectly good ellipsoid pointing in a perfectly definite direction, and reading its axes as the class’s axes would be reading the seed of the random number generator.

Averaging several tensors from different starts is what turns that into a measurement. A pinned direction is common to all of them; an unpinned one is different in each. Six samples is enough that a coincidence is implausible, and the census then requires every class of a system to give the same count — which is a second, independent check, since the classes of a system have different orders and different operations and only their systems in common.

That is the same discipline as the refusal that ends every gate on this site: a measurement that cannot fail is not a measurement. Here the way it could fail is visible — one triclinic sample would report three pinned axes — and the way it does not is the point.

Where the exactness stops #

Computed here: for each of the thirty-two classes, six invariant tensors from six different random starts; the eigenvalues and eigenvectors of each; the number of principal directions common to all six; and the degeneracy, so that a class whose tensor is a multiple of the identity is reported as having no axes rather than three arbitrary ones. Every class of a system is required to give the same answer, which is the check that the measurement is measuring the symmetry and not the sample.

A measurement rather than a proof. Six samples agreeing is strong evidence that a direction is pinned and it is not a derivation; the derivation is the one-line argument that an eigenvector of an operation of order two must lie along its axis or perpendicular to it. The two agree everywhere, and the computation is what makes the census exhaustive rather than case-by-case.

And the tolerance is real. Directions are compared numerically, and two axes that differ by a hundredth of a degree would be reported as the same. That is a threshold, it is stated in the code, and it is the only one in this essay.

Reading a thin section #

The practical payoff of the table is a diagnostic procedure a microscopist runs without doing any arithmetic, and it is worth writing out because every entry in it is a line of the census.

Turn the stage between crossed polars. A grain that stays dark at every angle is cubic — the indicatrix is a sphere and there is nothing to extinguish against. That single observation eliminates five classes.

Find the extinction directions. A grain that goes dark when its own crystallographic edges are parallel to the polars has its principal axes along those edges: orthorhombic, tetragonal, trigonal or hexagonal. A grain that goes dark at some other angle has axes that are not the crystal’s own, which is monoclinic or triclinic — and the angle itself is the free parameter symmetry left open.

Change the colour. If the extinction angle moves with wavelength, at least one axis is free and dispersing. In a monoclinic crystal that is one angle moving; in a triclinic one it is all three, and the grain never extinguishes cleanly at any colour.

Every step of that is the same computation asked at the bench. What makes it work is that the number of free angles is a property of the system rather than of the substance, so an observation about how a grain behaves is an observation about which of seven systems it belongs to — before anything has been measured, and long before a diffraction pattern is available.

That is a good deal more than a symmetry argument usually gives, and the reason is worth noting: this is a case where symmetry constrains an orientation rather than a magnitude, and an orientation is something a microscope measures directly.

Who found it, and when #

The indicatrix is Fletcher’s, from the 1890s, and the constraint symmetry puts on it was understood as soon as Neumann’s principle was — Franz Neumann lectured on it in the 1830s and his student Woldemar Voigt wrote it down in the Lehrbuch der Kristallphysik of 1910, which is still the reference for exactly this table.

Dispersion of the axes was observed long before it was explained: Brewster described the coloured extinction of monoclinic crystals in the 1810s, and it was a puzzle for as long as crystals were classified by their shapes rather than by their groups. It became a diagnostic the moment the systems were understood, because the number of dispersing axes is the number of free angles, which is a number the class supplies.

What a fixed axis is worth to a measurement #

A pinned axis is not only a fact about the crystal; it is a saving in every experiment that measures the property.

Three numbers instead of six. An orthorhombic crystal’s dielectric tensor is measured by three measurements along three known directions. A triclinic crystal’s needs six, and — worse — three of them are angles that have to be found by searching rather than by measuring along a known direction. The difference in effort is the last column of the census.

And a check that costs nothing. Measure an orthorhombic crystal along its axes and off them, and the off-axis result is determined by the three on-axis ones. If it is not, the crystal is not orthorhombic, or the sample is twinned, or the axes were misassigned. That kind of internal consistency check exists exactly where symmetry has fixed something, and it is unavailable in the triclinic case where every direction is independent.

This is the same economy the count of independent components provides for magnitudes, extended to orientation, and the two together are what makes a symmetry classification worth doing before an experiment rather than after it.

Where the ladder goes next #

Back, to the shape of the same ellipsoid: three optical characters, where averaging over a cubic group turns any ellipsoid into a sphere.

Sideways, to the tensor with a different symmetry: fifteen may rotate light is the gyration tensor, which is not symmetric and whose census is a different one — and where the classes that permit the effect are not the classes anybody expects.

And to the counting this is the orientation half of: twenty-one, thirteen, nine, three does the same arithmetic for the elastic tensor, where the components run to twenty-one and the orientation question has the same three answers.

The uniaxial systems, where the freedom is of a different kind #

The sorting into three, one and none covers the biaxial systems and the cubic one, and it leaves out the three systems in between — where the count of pinned axes is also one and the free pair behaves nothing like the monoclinic case.

A tetragonal, trigonal or hexagonal class has a unique axis of order three, four or six, and that axis is pinned by the argument above. The other two principal values are equal, forced by the rotation carrying one perpendicular direction to another, so the ellipsoid is a spheroid.

That makes every direction perpendicular to the axis a principal direction, and the “free pair” is free in the sense that any perpendicular pair will serve rather than in the sense that the material chooses one. There is nothing to disperse: the two values are equal at every wavelength, by symmetry, so no rotation of anything can occur.

So one pinned axis means two quite different things. In a monoclinic crystal the other two are determined by the material and are free to move with wavelength or temperature. In a uniaxial one they are undetermined by anything, because there is no distinction to make, and they cannot move because there is nothing to move.

The distinction shows in what an experiment sees. A monoclinic crystal’s optic axial plane rotates with wavelength and a tetragonal one has no optic axial plane at all — it has an optic axis, singular, along its unique direction, and the whole apparatus of dispersion of the axes is empty for it.

Reading a table that reports “one pinned axis” for both is therefore reading two different situations under one number, and which is which is decided by whether the remaining two values are equal — that is, by the shape of the ellipsoid rather than by the count.