The angles belong to the substance, the shape to the specimen

Two quartz crystals from the same vein can be a stubby hexagonal barrel and a long thin needle. They do not resemble each other. Photographed side by side, nothing suggests they are the same substance.

Measure the angle between the prism face and the face above it on each, and both give the same number, to the accuracy of the instrument. Do it on a quartz crystal from another continent and it gives the same number again. Do it on a synthetic one grown last week and the number does not move.

That is Steno’s law, stated in 1669 and the first quantitative regularity anybody found in crystals. It says something oddly specific: not that crystals of a substance have the same shape — they plainly do not — but that a particular relation between their faces is invariant while everything else about their appearance is free.

The distinction the law rests on #

A face has two independent properties: which way it faces, and how far out it is.

The first is its orientation, given by its indices, and the previous rung established that those are integers fixed by the lattice. The second is the distance it has grown to, which depends on how fast material arrived there — on the solvent, the temperature, the concentration gradient and the accidents of the container.

The angles are computed from the first alone — from a set of integers and the shape of the cell, exactly as the previous rung left them. The outline is computed from both. So the angles carry only the part of the description that the lattice controls, and the shape carries the part the laboratory controls as well.

Crystallography names the two halves separately, and the distinction is worth keeping sharp because ordinary language collapses it:

• The form is the set of faces symmetry requires together — a matter of indices and the class, and nothing else. It is what the next rung is about.
• The habit is what a particular crystal looks like: which forms are present and how far each has grown. Two crystals of the same form can have entirely different habits.

Steno’s law, restated in these terms, is: the form is a property of the substance and the habit is not.

What the angle actually is #

An interfacial angle is a computation on two reciprocal lattice vectors, and it comes out of the metric.

Write G for the metric tensor of the cell — the matrix of dot products of a, b and c with each other — and G* for its inverse, which is the metric of the reciprocal cell. The cosine of the angle between the normals of faces (h₁k₁l₁) and (h₂k₂l₂) is

$$\cos\theta = \frac{\mathbf{h}_1^{\mathsf T} G^* \mathbf{h}_2}{\sqrt{(\mathbf{h}_1^{\mathsf T} G^* \mathbf{h}_1)(\mathbf{h}_2^{\mathsf T} G^* \mathbf{h}_2)}}$$

and that is the whole of it. Two integer triples, one matrix, one square root.

The important feature of that expression is what is not in it. There is no reference to the size of the crystal, to how far any face has grown, or to which faces are present. Change the habit as violently as growth conditions allow and the formula does not notice, because none of its inputs changed.

The second important feature is that the metric enters only through ratios. Multiply the whole cell by two — every edge doubled — and G is multiplied by four, G* by a quarter, and every cosine is unchanged. An angle measurement cannot see the absolute size of the cell. It sees the ratios of the edges and the angles between them, and that is exactly all it sees.

So a goniometer is a lattice-parameter instrument #

Put those two observations together and the consequence is startling in retrospect.

The angles between faces are determined by the axial ratios and the interaxial angles of the unit cell. There are at most six such numbers, and one of them is a scale factor angles cannot see, so at most five are measurable. A crystal presents many more than five distinct interfacial angles. So the system is over-determined, and the cell shape can be solved for.

That is what nineteenth-century morphological crystallography was doing. A reflecting goniometer — Wollaston’s, from 1809, which bounced a light source off each face in turn and read the rotation required to bring the next face into position — measured angles to about a minute of arc. From a table of such angles, and an assumption about which small integers to assign to which face, the axial ratios of the substance followed.

Those ratios were published, argued over and tabulated for a hundred years before von Laue, alongside the symbols of the thirty-two classes that were being sorted out at the same time and for the same reason. They are correct. What was missing was the scale: morphology says that quartz has c/a ≈ 1.1, and it cannot say that a is 4.91 Å, because no measurement of an angle can. The absolute lengths arrive with diffraction, and the reason is exactly the one this site keeps returning to — a diffraction pattern’s spacings are lengths in reciprocal space, and a wavelength supplies the unit.

What survives a change of cell #

The previous rung noted that indices depend on the choice of cell. Angles do not, and it is worth seeing why, because the reason is not that angles are somehow more fundamental.

A change of cell is an integer matrix P with determinant ±1 — the same object the shortest basis is found by searching over. It sends indices h to Pᵀh, and it sends the metric G to PᵀGP — and when both substitutions are made in the cosine formula, the P’s cancel. The angle between two physical faces is unchanged, as it must be, because nothing physical happened.

This is the clean version of a point the cell is a choice made about lattices generally. A quantity computed from indices and the metric together is a property of the crystal; a quantity computed from indices alone is a property of the description. Miller indices are a description. Interfacial angles are a crystal.

There is a practical consequence that catches people out. Two published index sets for the same mineral can disagree entirely and both be right, and the way to check is to compute the angles from each and compare those. If the angle tables agree, the two descriptions are the same crystal in different settings; if they disagree, at least one of them is wrong about the substance.

The counts are short, and that is the identification #

A form of twenty-four faces has 276 pairs of faces. It does not have 276 distinct interfacial angles — it has a handful, because the symmetry that generated the form also acts on the pairs.

If g is an operation of the class and (h₁, h₂) is a pair of faces, then (gh₁, gh₂) is another pair with the same angle between them, since g preserves the metric. So the angle table is a list of orbits of pairs, and its length is a property of the class.

That shortness is what makes a table of measured angles an identification rather than a description. A mineralogist reading a dozen angles off an unknown crystal is not recording twelve independent facts; they are recording twelve samples of a very short list, and the list is a fingerprint of the cell shape.

Where the exactness stops, twice #

The rest of this site’s applied field is integer arithmetic. This essay is not, and the two places it stops being exact are different from each other.

An angle is a real number. The cosine formula produces an irrational value for almost every pair of faces, and no amount of care makes it otherwise. That is not a limitation of the method; it is what an angle is. What the lattice constrains exactly is which faces can occur, and that question was settled in integers on the previous rung.

A measured angle has an error bar. Steno’s law is a statement about ideal crystals, and real ones have faces that are curved, striated, etched or grown against the wall of a container. A good goniometer reading has a spread of a few minutes; a bad face has no well-defined normal at all. The law is exact and the measurement is not, and the gap between them is where the whole practice of morphological crystallography lived.

There is a third case, and it is the one that returns later in this field. Two different substances can have angles that agree to within measurement error, because their cell shapes happen to be close. Morphology cannot separate them and diffraction can. That is the same shape of problem as near-symmetry and the tolerance: a claim that is exact in arithmetic becomes a claim about a threshold the moment a real measurement is put in front of it, and the threshold is not in the mathematics.

Two substances with the same angles #

Steno’s law is often stated as though the angles identified the substance. They constrain it very tightly and they do not identify it, and the exception was found early enough to have its own name.

Eilhard Mitscherlich noticed in 1819 that potassium dihydrogen phosphate and potassium dihydrogen arsenate crystallise with the same shape and very nearly the same angles, and that the two would grow on each other and form mixed crystals in any proportion. He called it isomorphism, and the modern reading is unsurprising: the two substances have the same structure with one atom swapped, so their cells have the same shape and differ slightly in size — and size is the one thing an angle cannot see.

The consequence is that a table of angles identifies a structure type, not a compound. Alums are the classic family: a dozen different substances of the general form of a double sulfate, all cubic, all growing the same octahedra, all with the same angles because every angle in a cubic crystal is fixed by the symmetry alone and has no free parameter left to differ in.

That last clause is worth pausing on, because it is the extreme case of this essay’s argument. A cubic cell has no adjustable ratio: the metric is a multiple of the identity, and the cosine formula has nothing in it but integers. Every cubic crystal of every substance has exactly the same interfacial angles. Morphology therefore identifies a cubic mineral not at all, and the whole discriminating power of the method lives in the systems that have free parameters — which is to say, in the six systems that are not cubic and in the axial ratios the conventional cell records.

It is the same shape of limitation as what diffraction cannot tell apart, one experiment earlier: a measurement that is blind to a quantity will report identical answers for things that differ only in it, and the correct response is to name the blindness rather than to distrust the instrument.

Steno, and what he was actually doing #

Nicolas Steno was a Danish anatomist working in Florence, and the paper the law appears in — De solido intra solidum naturaliter contento, 1669 — is mostly about fossils and geological strata. The crystal result occupies a few pages and a diagram of quartz sections.

His interest was in growth. He was arguing that crystals grow by addition from outside rather than by expansion from within, against a live alternative in which minerals were held to vegetate. The constancy of the angles was his evidence: if a crystal grew by internal expansion, its proportions would be preserved and its angles could change; if it grew by adding layers to faces, the angles would be fixed and the proportions free. The observation settles it in favour of the second, and he says so.

That is a very good argument, and it is worth noticing that it is the same argument the modern account gives, with the lattice in place of the layers. Nothing in Steno’s reasoning needed atoms. What it needed was the idea that a face is an orientation which growth preserves, and that is the whole content of the law.

The law waited a century for Romé de l’Isle to measure it systematically across many minerals in 1783, and another twenty-six years for Wollaston’s instrument to make the measurement precise enough to be useful. Haüy’s law of rational indices came in between, and the two laws together are morphological crystallography: the indices say which orientations are possible, and the angles say what the lattice shape is that makes them so.

How many angles it takes #

The claim that the angles determine the cell up to scale can be turned into a count, and the count says how much measuring a determination needs — which is the practical form of the whole argument.

A general cell has six parameters and the scale is one of them, so five numbers remain to be found. Each measured interfacial angle is one equation in those five, so five independent angles suffice in the triclinic case, and the word independent is doing the usual work: five angles among faces that happen to lie in one zone constrain fewer than five directions and settle nothing.

Higher symmetry costs fewer. A monoclinic cell has four free parameters and three after the scale; an orthorhombic one has two; a tetragonal or hexagonal one has one, the axial ratio, so a single well-chosen angle fixes it; and a cubic cell has none, so its angles are the same for every cubic substance and carry no information at all.

That last row is the sharp one and it explains a fact about mineral identification. The angles of a cubic mineral identify nothing — a cube is a cube, an octahedron is an octahedron, and pyrite, galena and rock salt agree exactly. Everything a nineteenth-century crystallographer could say about a cubic substance came from which forms it showed rather than from the angles between them, and that is a much weaker kind of evidence.

What the wavelength supplies that a goniometer cannot #

The missing scale is worth chasing to its resolution, because the instrument that supplies it does so by a route morphology has no access to.

A goniometer measures directions, and directions are dimensionless. So the whole of morphological crystallography could determine that quartz’s cell has c/a = 1.100 and could not determine that a is 4.913 ångström — because nothing in the measurement carries a length.

A diffraction experiment measures directions too. What it adds is a wavelength: Bragg’s law relates a scattering angle to a spacing through λ, and λ is a length known independently — from the emission line of the source, itself calibrated against a standard. The wavelength is the ruler, and it is the only thing in the experiment that is not a ratio.

That is why the cell parameters arrived in 1912 and not before, and why they arrived with an accuracy limited by how well the wavelength was known rather than by the diffraction. It is also why the earliest values were revised: the wavelength scale was itself refined for decades, and every cell parameter in the literature moved with it.

The ratios did not move. A hundred years of goniometry produced axial ratios that the diffraction measurements confirmed rather than corrected, which is an unusually clean example of two techniques agreeing across the gap between them — one measuring the shape of a lattice with no idea that there was one.

Where this rung goes next #

Two things have now been separated. Which faces can occur is decided by the lattice, in integers. What angles they make is decided by the metric, in real numbers, and measures the shape of the cell.

The question left over is the one Steno’s law implies and does not answer: given one face, which other faces must be there? That is the symmetry question, it is answered by a group acting on the indices, and the answer is called a form. It is an orbit — the same object an orbit under a wallpaper group has been on this site from the beginning, with face indices in place of points and a crystal class in place of a plane group.

The surprise waiting on that rung is that whether a form encloses a volume at all — whether a crystal can be bounded by it, or must combine it with another — is decided without any metric whatever. The angles do not come into it, which is exactly the opposite of what this essay’s subject would suggest.