A zone is a vanishing dot product

Pick up a well-formed crystal and the first thing the eye finds is not a face. It is a band of faces running round the specimen, whose edges are all parallel to one another. Quartz has one along the prism; a garnet has several; a well-grown pyrite cube has three at right angles.

Such a band is a zone, the common direction is the zone axis, and belonging to one is decided by an equation with no geometry in it.

The law #

A face with Miller indices (hkl) lies in the zone of axis [uvw] exactly when

$$h\,u + k\,v + l\,w = 0$$

That is the whole of the Weiss zone law, from the 1810s, and its content is that a face is parallel to a direction when the plane’s normal is perpendicular to it — with the perpendicularity expressed in the lattice’s own coordinates, where it costs nothing to check.

The arithmetic that goes with it is as short:

• the zone axis of two faces is their cross product, reduced by the common factor;
• the face common to two zones is the cross product of the axes, by the same operation;
• three faces are in one zone exactly when the determinant of their indices vanishes.

What is remarkable about it #

The point worth pausing on is what the law does not need.

No cell dimensions. The zone law holds whatever the lengths of the axes and whatever the angles between them. A triclinic crystal, with nothing orthogonal anywhere, obeys it in exactly the form written above.

No metric at all. The reason is that indices are coordinates in the reciprocal basis and directions are coordinates in the direct basis, and the pairing between the two is the identity by construction. The sum hu + kv + lw is that pairing, and it is metric-free.

So the whole calculus of faces and zones is integer arithmetic, and it was in use for a century before anybody could measure a lattice. Weiss, Naumann and Miller were doing crystallography with a goniometer and a notebook, and the arithmetic they used is unchanged. What arrived later was the interpretation: that a face is a plane of lattice points and its indices are how many cells it cuts off along each axis.

The cost of that freedom is the one thing the projections here cannot show. A stereogram plots directions, and to plot a direction from an index triple something has to decide what the triple means as a direction — which needs the metric the zone law does without. Every projection in this essay reads (hkl) as a direction in an orthonormal frame, so the angles between the dots are the angles of a cubic crystal whatever system is named above them. What survives that simplification is exactly what the essay is about: which faces are co-zonal, how many zones there are, and how many faces each carries, all of which are decided by vanishing integer sums and are the same numbers whatever the cell. What does not survive is any claim about an interfacial angle. Two systems whose forms share an index set therefore share a picture, and the picture is right about the incidences and wrong about the shape.

That is worth stating as a limit rather than buried, because the confusion it invites is a real one. A goniometrist’s measurement is an angle and their conclusion is a set of integers, and the step between them needs the cell. The zone law lives entirely on the far side of that step, among the integers, which is why it could be stated correctly in 1817 and needs no correction now.

From angles to indices #

A goniometer measures interfacial angles. Turning those into indices is where the arithmetic meets a measurement, and where a refusal has to happen.

The procedure is: choose three faces that are not in one zone and call them the axes; express every other measured normal in that frame; and require the coefficients to be small integers. That last requirement is the law of rational indices, used as a decision procedure rather than as a slogan.

The refusal is the part that makes it a procedure. A normal that does not land within the tolerance of any small-index face is refused an index rather than given the nearest one. Given a large enough index bound, every direction is within a tiny angle of some rational face, so a procedure without a refusal would index anything — including a scratch, a cleavage step, or a face of a second crystal grown on the first.

That is the same discipline as everything else in this collection: a machinery that cannot decline is a machinery whose acceptances mean nothing.

Why zones make the assignment possible at all #

The indexing above is a brute search, which works because the crystal is small and the indices are bounded. Historically the search was not available, and zones are what replaced it.

Zones propagate indices. Two faces with known indices determine their zone axis by a cross product. A third face known to be in that zone then has one linear condition on its three indices, which reduces the search from three dimensions to two. A face in two known zones has two conditions and its indices are determined outright, as the cross product of the two axes.

So a mineralogist would index a specimen by finding the zones — which are visible, since the edges are parallel — assigning indices to a few faces by choosing axes, and then propagating round the zones. The whole assignment is a graph traversal, and the graph is drawn on the specimen.

The names, and why they are not translations of each other #

The vocabulary of morphology grew up before the vocabulary of groups and the two do not correspond term for term. Three pairs are worth keeping apart.

A form and an orbit. A form is the set of faces the symmetry requires to appear together, which is an orbit of a face under the point group — the same object, and the older word carries the extra sense of what actually appears on a specimen, since a crystal shows some of its possible forms and not others.

A zone and a subgroup. A zone is not a group of anything. It is a set of faces satisfying a linear condition — a null space, in the language of the next century — and its axis is a lattice row rather than a symmetry element. A zone axis is often a symmetry axis and does not have to be.

A habit and a class. The habit is which forms a specimen shows and how large each is; the class is the point group. A specimen’s habit is decided by growth and can differ between two specimens of one mineral; the class cannot.

Getting these confused is the standard way of over-reading a specimen. A crystal showing only three faces of a form is showing an incomplete form, not a lower symmetry, and the classical literature is careful about it in a way that a modern reader coming from group theory usually is not.

Three faces that are not in one zone #

The indexing procedure begins by choosing three faces as axes, and it requires them not to be co-zonal. That requirement is worth a paragraph because it is the only place the procedure can fail before it starts.

Three faces in one zone have normals lying in one plane, so they span two dimensions rather than three. Expressed in a frame built on them, a general direction has no coordinates at all: the frame is degenerate. The determinant of the three index triples vanishes, and the vanishing determinant is exactly the co-zonality test given above.

So the same arithmetic that identifies zones also identifies which faces cannot be used as axes, and the procedure tests for it rather than assuming a good choice. For a form with many faces the test almost always passes at the first attempt; for a specimen showing a single zone and nothing else, it never passes, and no amount of angle measurement will produce indices. That is a real limitation of morphology: a crystal grown as a needle, showing only prism faces, cannot be indexed at all.

Where a zone is not just a band #

The word does more work than the picture suggests, and two of its other uses are worth naming.

In reciprocal space a zone is a plane. The faces of a zone have normals perpendicular to [uvw], so their reciprocal-lattice vectors lie in the plane through the origin normal to that direction. An electron diffraction pattern taken with the beam along [uvw] shows exactly that plane of reflections and is called a zone-axis pattern for the same reason. The zone law is what says which reflections appear in it: hu + kv + lw = 0, and the reflections satisfying it are the pattern.

A zone axis is a lattice row. The direction [uvw] with integer indices is a direction in which the lattice repeats, so a zone is a set of faces parallel to a repeat direction. That is the physical content of the law: a face is a plane of lattice points, and it can only be parallel to a lattice row if the row lies in the plane.

Underneath both readings is one pairing. A face’s indices (hkl) are coordinates in the reciprocal basis and a zone axis’s [uvw] are coordinates in the direct one, and hu + kv + lw is the pairing between the two — which is why the expression is a bare sum of products with no metric in it at all. Every angle in this subject needs the cell’s lengths and angles; the zone law needs none of them, because the pairing of a basis with its own dual is the identity matrix whatever the cell is. That is the whole reason the arithmetic below is integer arithmetic and the whole reason it survived the discovery that a crystal is a lattice.

The convention the answer depends on #

Every result here rests on a choice of axes, and it is worth naming rather than implying.

The indices depend on which three faces were chosen as axes. Choose a different three and every index changes. What does not change is the relations — which faces are in which zones, which are cross products of which — because those are statements about the lattice rather than about the coordinates.

A crystallographer resolves this by convention. The axes are chosen along symmetry directions where the symmetry supplies them, and by a rule about which face gets (111) where it does not. That is why two mineralogists indexing the same specimen agree, and it is a convention rather than a discovery — the same kind of choice as the setting of a space group, with the same consequence that a table has to say which one it used.

What the outside of a crystal can and cannot say #

The zone law is one half of a bargain and the other half is worth stating.

What morphology gives: the point group’s action on directions, since the set of faces is an orbit and the zones are the incidences among them. A careful goniometric study of a well-formed crystal determines the crystal class, and the classical crystallographers determined all thirty-two of them that way.

What it does not give: the lattice’s dimensions, the centring, the space group, or anything at all about where the atoms are. Faces are directions and directions have no scale, so a crystal’s outside is silent about every length. The constancy of angles is exactly this fact stated as an observation: two specimens of one mineral, of wholly different shapes, have identical interfacial angles because both are reading off the same set of directions.

What is owned here #

The zone law as a predicate, the cross products, the co-zonality determinant, the enumeration of zones for forms in several systems, and the indexing run with its stated perturbation and its refusals.

Two of those numbers are worth repeating because they are the essay in a line. A cubic {321} form has forty-eight faces, which make one thousand one hundred and twenty-eight pairs, and those pairs name only two hundred and twenty distinct axes. The collapse is the whole phenomenon: a band on a specimen is a coincidence among pairs, and the arithmetic counts the coincidences without ever drawing the crystal.

Not owned: any real specimen, any goniometric data, and any claim about which faces a particular mineral shows — which is a growth question and is decided by rates rather than by arithmetic.

A note on the third integer, which the older literature does not have. Weiss and Naumann worked with ratios of intercepts rather than with their reciprocals, so their symbols are the reciprocals of Miller’s and a face this essay calls (210) appears in the older tables as a different string. Miller’s convention won because the reciprocals are the integers that make the zone law a dot product; the intercept ratios make it something clumsier. That is a small thing and it is the kind of small thing that decides whether a notation survives a change of subject, which is the section above.

Why the arithmetic survived the change of subject #

Morphology was the whole of crystallography until 1912 and is now a small corner of it. The vocabulary survived the transition intact, and the reason is worth stating.

Indices were defined by the law of rational indices before anybody knew what a lattice was, as the small whole numbers relating the intercepts a face makes on three chosen axes. When von Laue’s experiment showed that a crystal is a lattice, those numbers turned out to be exactly the indices of a lattice plane — the same integers, arrived at from measurements of angles rather than of anything periodic.

So the change of subject was a change of explanation and not of notation. A mineralogist’s (111) and a diffractionist’s (111) are the same three integers, related by the same zone law, and a nineteenth-century goniometric table can be read straight into a modern indexing program.

Very little in science does that, and the reason it happened here is that the old vocabulary was built on a relation that is metric-free. A notation that had committed to lengths would have needed replacing; one committed only to integers and their ratios had nothing to give up.

The same law, at an electron microscope #

Morphology is a small corner of crystallography now, and the zone law is not — it is applied dozens of times a day in electron microscopy, where it does the same job on a different object.

A transmission electron microscope forms a diffraction pattern from a thin specimen, and the pattern it forms is very nearly a plane of the reciprocal lattice: the one perpendicular to the direction the beam travels. Which plane depends on how the specimen is tilted, and the direction the beam travels down is called the zone axis, in exactly the sense this page defines.

So the reflections visible in such a pattern are the (hkl) satisfying hu + kv + lw = 0 for the beam direction [uvw], which is the Weiss zone law with the same integers doing the same work. Indexing the pattern is assigning indices to two of its spots, taking their cross product to get the zone axis, and then checking that every other spot satisfies the law — the identical procedure a mineralogist ran on a goniometer, on data of a completely different kind.

The propagation argument transfers with it. A pattern with two indexed spots determines everything, because the rest of the spots are integer combinations; and a spot that will not fit is evidence of a second phase, a twin, or a wrong lattice, exactly as an unassignable face was. What changed between 1820 and now is which instrument produces the directions, and nothing at all about the arithmetic that consumes them.

Four indices, and why the hexagonal system asks for one more #

There is a variation of the notation that a reader will meet on any hexagonal material, and it exists for a reason the zone law makes easy to state.

In a hexagonal crystal the three symmetry-equivalent directions in the basal plane are a₁, a₂ and their negative sum. With three indices, faces related by the three-fold rotation get index triples that are not permutations of one another — (100), (010) and (1̄1̄0) — so the notation hides a symmetry the crystal has.

The Miller–Bravais convention adds a redundant third basal index: (hkil) with i = −h − k by definition. Now the three-fold rotation permutes the first three indices, the six equivalent prism faces are the six permutations with signs, and the symmetry is visible in the symbol. Nothing is added — i carries no information — and what is bought is that equivalence becomes readable.

The zone law survives the change with one index more: a face (hkil) lies in the zone of the direction [uvtw] when hu + kv + it + lw = 0, and the direction indices carry the same redundancy t = −u − v. The care needed is that a direction’s four indices are not got by simply inserting −u−v into a three-index direction; the first two change as well. That is the standard trap in the convention, and it is a bookkeeping cost paid to make the symmetry legible — which is the trade every notation on this site makes in one direction or the other.

Where the ladder goes next #

Inwards, to the same problem in reciprocal space. A diffraction experiment gives a bag of spot positions with no labels on them, and recovering the cell from them is the same kind of step — a small set of integers chosen so that everything else comes out small too — with an ambiguity that no quantity of data removes.

Outwards, to what the faces of a crystal are for. A form is an orbit, whether it closes is decided without lengths, and which of the possible forms actually appear is the question morphology was invented to answer.

What both sections share is that the arithmetic did not change and the object it is applied to did. A face normal, a beam direction and a reciprocal-lattice row are three different things measured by three different instruments, and the same vanishing dot product decides all of them — which is what a metric-free relation buys, and the reason a notation from 1820 is still the notation.