Five classes grow the same cube

Pyrite grows cubes. So does halite, so does fluorite, so does galena, and so — in a laboratory — does gold. The cubes are the same cube: six faces, all square, all meeting at right angles, indistinguishable by shape.

The five substances are not in the same crystal class. Halite, fluorite, galena and gold are m3̅m, the full cubic holohedry with forty-eight operations. Pyrite is m3̅, which has twenty-four, and is missing every one of the six diagonal mirrors and the four-fold rotations that go with them.

A cube shows none of that. The face (100) is fixed by so much symmetry that it cannot tell m3̅ from m3̅m — its orbit is the same six faces under either — and the same is true of the other three cubic classes. Five of the thirty-two produce a cube, and the cube says only that the class is one of five.

What a shape can and cannot decide #

The chain is short and each link loses information.

A habit is what a specimen looks like. It depends on growth conditions, so it is not even a property of the substance — the previous rung but one made that the whole content of Steno’s law.

A form is a property of the substance: an orbit of faces under the class, with a multiplicity that is the class order over a stabiliser. A crystal usually shows two or three of them.

A class is what the forms are orbits under. Going from a set of observed forms back to the class is the inference morphological crystallography lived on, and it is under-determined in general.

The place where it is determined is the general form. A face lying on no symmetry element has the identity for a stabiliser, so its orbit is as large as the class, and counting its faces gives the order of the class outright. m3̅m’s general form has forty-eight faces; m3̅’s has twenty-four; 23’s has twelve. Find a general face and the class is nearly named.

The difficulty is that general faces are exactly the faces that do not grow. The law of rational indices said why: a face survives if it is dense in lattice points, dense means small indices, and small indices mean lying on symmetry elements. The faces that would identify the class are the ones a crystal is least likely to show.

Pyrite, and the striations that gave it away #

Pyrite’s cubes are covered in fine parallel lines, and the lines on adjacent faces run at right angles to each other. That observation is worth more than the cube it is on.

A striation is a series of narrow steps — a face oscillating between two neighbouring orientations as it grew. The pattern of striations on a face therefore reports the symmetry of that face, and the symmetry of a face is its stabiliser.

In m3̅m the face (100) has a four-fold axis on it, so anything decorating it must have four-fold symmetry: striations one way would be forbidden, because the four-fold would demand the perpendicular set too. In m3̅ there is only a two-fold on (100). Parallel striations are permitted, and they must run one way on one face and the perpendicular way on the neighbouring one, because the three-fold along the body diagonal cycles the three cube directions.

That is exactly what pyrite shows. The class is being read off the decoration rather than off the shape, and the decoration reports the site symmetry of the face, which the shape cannot.

Pyrite also grows a second form that settles the matter outright: the pyritohedron, twelve pentagonal faces of the type {210}. That form has twelve faces in m3̅ and twenty-four in m3̅m — the higher class’s diagonal mirrors double the orbit — so a pyrite crystal showing twelve pentagons has demonstrated that those mirrors are absent.

Diffraction has the same problem, one level up #

The natural response is that morphology is a nineteenth-century method and diffraction settles everything. It does not, and its blind spot is sharper and better characterised.

A diffraction pattern’s intensities obey Friedel’s law: the intensity at (hkl) equals the intensity at (h̅k̅l̅), for every structure, whether or not it has an inversion centre. So the measured intensities always look centrosymmetric even when the crystal is not, and the symmetry that can be read off them is the crystal’s point group with an inversion centre added.

The thirty-two classes collapse under that operation to eleven Laue classes. Pyrite’s m3̅ and the holohedral m3̅m are in different Laue classes, so diffraction does separate those two. But 432, 4̅3m and m3̅m all become m3̅m, so diffraction cannot tell those three apart at all — and neither can the shape, since all three grow cubes and octahedra.

Two independent methods, blind in overlapping ways. The overlap is not a coincidence: both are blind to the inversion for related reasons, morphology because a form and its inverse are the same set of face orientations, diffraction because the phase information that would distinguish them is not recorded.

What actually resolves it #

The measurements that finish the job are not measurements of shape or of intensity. They are measurements of a physical property, and which properties a class permits is precisely what Neumann’s principle decides.

Etch figures. Dissolve a face very slightly and it develops pits. A pit is a small negative crystal, and its symmetry is the site symmetry of the face — so a face with a four-fold axis gets square pits and a face with only a two-fold gets rectangular ones. Baumhauer built a whole method on this in the 1870s, and it is still the cheapest way to separate two classes that share a form.

Optical activity. A crystal rotates the plane of polarised light only if its class permits it, and fifteen of the thirty-two do. Quartz is optically active; its class, 32, is one of the fifteen. That single observation excludes 3̅m, which is the class quartz’s morphology would otherwise be read as, and it excludes it by a property with nothing to do with shape.

Pyroelectricity. Warm a crystal and see whether it attracts ash. It does only if the class has a direction fixed by every operation, which ten of them have. That is the oldest of the three by two thousand years and still one of the sharpest.

Piezoelectricity. Squeeze a crystal and look for charge. Twenty of the twenty-one non-centrosymmetric classes give it, so a positive result excludes eleven classes at a stroke.

Quartz, and the faces that are almost never there #

The most consequential case of this ambiguity is the commonest mineral on the planet’s surface.

A quartz crystal is a hexagonal prism capped at each end by what looks like a hexagonal pyramid. That shape has a six-fold axis, six mirrors and a horizontal mirror — it is the shape of class 6/mmm, and quartz is not in class 6/mmm. It is in class 32: one three-fold axis, three two-folds perpendicular to it, no mirrors of any kind, and no inversion.

The reason the shape lies is that quartz’s cap is not one pyramid but two rhombohedra of nearly equal development, and its prism faces are six faces of a form that would be three in a lower class. Every face on an ordinary quartz crystal is a special face lying on some element of the lattice’s symmetry, and specials cannot separate a class from its holohedry.

What settles it is a form that appears on perhaps one quartz crystal in twenty, in tiny faces at the junction of a prism face and a rhombohedron: the trigonal trapezohedron, six faces of a general position. Six faces means an orbit of six, so the class has order six, and 6/mmm has order twenty-four.

Those faces do something else that no count can: they sit to the left on some crystals and to the right on others, and never on both. Quartz comes in two hands. The handedness is the same one its optical rotation reports, and it is the same one the enantiomorphic space groups record — P3₁21 and P3₂21, a pair that differ only in the sense of a screw.

Anomalous scattering. Friedel’s law is not a law of nature; it is a consequence of assuming that the atoms scatter without absorbing. Tune the wavelength near an absorption edge of one of the elements present and the assumption fails: the scattering factor acquires an imaginary part, the intensities at (hkl) and (h̅k̅l̅) become measurably different, and the collapse of thirty-two onto eleven is undone. Bijvoet did it in 1951 on sodium rubidium tartrate and settled the absolute configuration of a molecule for the first time. It is the only entry in this list that recovers the missing information rather than routing around it, and it is why the handedness of a structure is now a routine result instead of a convention.

The general shape of the difficulty #

Every one of these methods is a permission test, and permission is weaker than presence in a direction this site has now stated four times.

A negative result is strong: no pyroelectric response along a candidate axis means either the class is not polar or the effect is too small to see, and the first is settled by trying harder. A positive result is stronger still: an observed optical rotation means the class is one of the fifteen, with no room for argument.

What neither gives is the converse. A class permitting an effect does not produce it, and permitted is not present is the whole content of that. So the standard procedure is a sieve: morphology narrows to a handful, diffraction narrows to a Laue class, each physical test removes a subset, and the class is what survives.

It fails sometimes. The classic failure is a crystal in a class permitting everything the tests can see, whose structure nonetheless has less symmetry than the tests report — a crystal that is very nearly, but not exactly, centrosymmetric. That is near-symmetry arriving in a laboratory rather than in a figure, and every one of these methods has a threshold below which it reports the higher symmetry.

The under-determination, counted #

It is worth asking how bad the ambiguity actually is, since the answer is computable from the census.

Take the twenty-four-face forms as an example. Several classes have one: m3̅m at {hhl} and {hk0}, m3̅ at {hkl}, 432 at {hkl}, 4̅3m at {hk0}. A crystal showing twenty-four faces of one form has narrowed its class to those, and no count of faces will do better.

The general pattern is that the special forms — the ones with the small indices, the ones that grow — are shared widely, and the general forms are not shared at all. The information a shape carries is therefore concentrated in exactly the faces a crystal is least likely to grow, which is a fairly complete account of why morphological crystallography took a century and needed a great many specimens.

There is a cheerful way to say the same thing. The forms a crystal is likely to show are the ones shared by many classes, and the forms that would identify it are the ones almost nothing grows — so the information content of a shape is inversely related to the probability of seeing it. That is not a peculiarity of crystals; it is what happens whenever the most stable state of a system is also its most symmetric, and it is the reason the tie-breakers in this subject are all measurements of something other than shape.

Groth’s Chemische Krystallographie, five volumes finished in 1919, is what that century produced: axial ratios, forms and angles for around ten thousand substances, measured on a goniometer. It is a magnificent object and it was made almost obsolete within a decade by a method that measures something else entirely. The ratios in it are still right.

How bad the diffraction ambiguity is, counted #

The collapse from thirty-two classes to eleven is quoted as a loss and the size of the loss is not evenly spread. Counting it says which crystals are badly served and which are not.

Each Laue class covers the classes that give it when an inversion is added, and the counts are: four classes each for the tetragonal and hexagonal holohedral Laue classes 4/mmm and 6/mmm; three each for 2/m, mmm, 4/m, 6/m, 3̄m and m3̄m; and two each for 1̄, 3̄ and m3̄. Thirty-two in total, as it must be.

So the worst case is a four-way ambiguity and the best is a two-way one, and no crystal is ever left with more than three alternatives to eliminate. That is a much better position than the raw ratio suggests — thirty-two onto eleven sounds like a threefold loss on average and is, but the loss is bounded rather than concentrated, and every crystal’s shortlist is short.

It is also structured. In every case the ambiguity is between a class and its non-centrosymmetric subgroups, so the alternatives always differ in whether they contain the inversion and in which of the reversing operations they keep. That is why the resolving measurements are what they are: each of them tests for a property that an inversion forbids, so each of them eliminates the centrosymmetric candidate outright and then discriminates among the rest.

What morphology still does better #

Ending on the method’s limits would misrepresent it, because there is one respect in which the goniometer beats the diffractometer and it is worth stating.

Interfacial angles are measured extremely precisely. A reflecting goniometer measures the angle between two faces by autocollimation, to seconds of arc, on a specimen that needs no preparation beyond being clean — and the angles are the same for every specimen of a substance, which is the constancy that founded the subject. That precision was available in 1800 and it is not bettered by an ordinary cell determination even now.

What the precision buys is the axial ratios, and through them the cell’s shape. A century of mineralogy identified substances by measuring a few angles and comparing them against a table, and it worked: a substance’s angles are a fingerprint even when its class is ambiguous, because the fingerprint is metric and the ambiguity is about symmetry.

So the two methods fail in opposite places. Morphology measures the lattice’s shape well and its symmetry badly. Diffraction measures the symmetry well — up to the inversion — and needs the crystal in a beam. That they are complementary rather than one superseding the other is why field mineralogy still teaches habit and forms, and why a crystal on a museum shelf carries information a powder of it would not.

Where the ladder goes next #

This anchor has arrived at a limit: the outside of a crystal reports its class incompletely, and the incompleteness is structural rather than a matter of care.

The next anchor takes up a phenomenon that looks, at first, like the same difficulty made worse. A twin is two orientations of one structure grown together, and a twinned crystal can present a shape with more symmetry than the crystal has — a false holohedron, complete with a false set of faces and false angles that check out.

The surprise is that twinning is not a further source of ambiguity but a completely determined one. Which twin laws a crystal can have is decided exactly, by the same coset arithmetic that has produced everything else in this field, and the answer depends on the lattice rather than on the growth conditions. A twin is not something that happened to a crystal. It is one of a short list of things that could happen to it, and the list is computable before any crystal is grown.