Why a crystal face carries small whole numbers

A crystal grown from solution is bounded by flat faces, and that is the first thing about crystals anybody noticed. It is also the first thing about them that needs explaining, because nothing else that grows does it. A drop of water minimises its area and comes out round; a crystal of alum grows facets and corners and keeps them for as long as it is left alone.

The explanation is short and it is entirely about the lattice underneath. A crystal is a periodic array of points — one of five arrangements in the plane and fourteen in space — so it contains planes densely populated with those points and other planes that catch almost none. A face is a plane of the first kind. And a plane that passes through lattice points is not free to lie at any angle: its orientation is named by three integers.

The claim, exactly #

Every crystal face has a normal that is a reciprocal lattice vector, so its orientation is given by three integers (hkl).

That is the law of rational indices, and it is a consequence rather than an observation. The argument runs in one direction only, and it is worth separating the two halves that are usually said in one breath.

The first half is a theorem: if a face lies on a plane containing lattice points, its indices are integers. That is true of every plane of lattice points in every lattice, no exceptions, and it needs nothing but periodicity.

The second half is a claim about which planes crystals actually choose: the faces that appear are the ones with small indices. That is not a theorem. It is an empirical rule with a good mechanical reason behind it, and it fails often enough to be interesting.

Why the integers are there #

Take the three cell edges a, b and c. A plane cuts them at some distances from the origin — say at a/h, b/k and c/l. If the plane contains lattice points, those intercepts are rational multiples of the edges, and clearing the denominators makes h, k and l whole numbers with no common factor.

Those three numbers are the Miller indices, and the reason they are written as reciprocals rather than as the intercepts themselves is that reciprocals behave. A plane parallel to c never meets it, so its intercept is infinite and its index is zero; a plane parallel to two axes has two zeroes. Written as intercepts, the parallel case is a special one that has to be handled apart. Written as reciprocals it is the ordinary case with a zero in it.

The deeper reason is that (hkl) is not a triple of numbers describing a plane. It is a vector — the dual lattice made concrete — and it lives in the reciprocal lattice: the vector ha* + kb* + lc* is perpendicular to the plane, and its length is the reciprocal of the spacing between successive planes of that family. So the object naming a face is the same object naming a diffraction spot, and the two subjects are the same subject seen from opposite ends of the experiment. The reflections that vanish systematically are indexed with the very integers a goniometer reads off a face.

The staircase, counted #

Haüy’s argument is worth doing arithmetically, because it is short and because the smallness of the indices falls out of it directly.

Take a cubic lattice of spacing one and build a face as a staircase: advance p cells along a for every q cells risen along c. The resulting surface has a normal proportional to (q 0 p), so the indices are the two step counts read across. A staircase of one-along-one-up gives (101); two-along-one-up gives (102); and a staircase that advances thirteen cells for every five it climbs gives (5 0 13).

Now count the corners. The one-in-one staircase has a corner every cell, and its treads are single cells. The five-in-thirteen staircase has treads thirteen cells long, and the surface is thirteen-fourteenths flat terrace and one-fourteenth step. A high-index face is a low-index face with occasional steps in it, and as the indices grow it becomes a vicinal surface — nominally a distinct orientation, physically a terraced version of a nearby simple one.

That is the honest content of “the indices are small”. They are not small because large ones are forbidden; they are small because a face with large indices is barely a distinct face at all, and because the steps that make it up are exactly where growth is fast. The same terraces reappear in this field two rungs further on, where a boundary between two grains turns out to be built from them.

How the machinery here uses it #

The site’s point-group work is done with integer matrices in the lattice basis, and face indices join it without any change of method. An operation that sends a lattice vector x to Mx sends the indices to (M⁻¹)ᵀ(hkl), which is again integral because a symmetry of a lattice has determinant ±1 and so an integer inverse.

That single line is what makes the whole of the next essay possible. Applying one of the thirty-two crystal classes to a set of indices produces another set of indices, exactly, with no rounding and no tolerance — and the set of everything it produces is a crystal form, which is an orbit in precisely the sense this site has used the word since its first essay.

It also has a trap in it that is invisible in the system most people learn on. In the cubic system the operations are orthogonal in the lattice basis, so (M⁻¹)ᵀ = M and indices transform exactly as directions do. In every other system they do not, and code that transforms a face normal as though it were a direction produces a picture that is right for cubic crystals and wrong for the other six.

Why the indices are small #

Nothing so far explains why a mineral collection is full of (100) and (111) and (110) and almost empty of (973). The integers being integers is settled; their being small is a different question and has a different kind of answer.

The mechanism is growth. A face advances by adding material to its own surface, and how fast it advances depends on how much a new layer costs. A plane densely covered in lattice points is a plane where a newly arrived atom has many neighbours to bind to, so a layer is cheap and easily completed — and a face that grows quickly grows itself out of existence, because it recedes from the centre until it is squeezed out by its slower neighbours. The faces that survive are the slow ones, and the slow ones are the dense ones, and the dense ones are the ones with small indices.

The relation between density and index size is the one the reciprocal construction gives: a family of planes with small (hkl) has a short reciprocal vector, so its planes are widely spaced, so each of them carries more lattice points per unit area. Small index, wide spacing, dense plane, slow growth, surviving face. That chain is Bravais’s law, published in 1866, and it is the best available account of a phenomenon that nobody had explained for two centuries.

It is also, honestly, an approximation. It knows nothing about the solution the crystal is growing from, nothing about impurities that poison one face and not another, and nothing about the screw dislocations that let a face grow without ever having to nucleate a fresh layer. Real habits depart from it routinely, and the departures are what the growth literature is about.

What the indices depend on, which is not nothing #

An index triple is not a property of a face alone. It is a property of a face and a choice of cell, and this site has already spent an essay on the fact that the cell is a choice and the lattice is not.

Change the cell and every index changes. The same physical face of the same crystal is (111) on one setting and (101) on another, and both are correct. That is why a mineralogical description that quotes indices without naming the setting is incomplete, and why the same face of calcite appears in the literature as (10̅4) in the hexagonal setting and as (100) in the rhombohedral one — the second being the description in which calcite’s cleavage rhomb is simply the cube of its own lattice.

What does not change is the angle between two faces, which is why the next essay is about angles. And what also does not change is the rationality: a face with integer indices on one cell has integer indices on every other, because the change of cell is an integer matrix with determinant ±1 in each direction. Rationality is a property of the face; the particular integers are a property of the description.

Where the exactness stops #

The argument’s first line is a crystal is a periodic array of points. Everything after it depends on that, and there is now a large class of solids for which it is false.

A quasicrystal has sharp diffraction and no lattice, and its faces are flat and well formed — icosahedral alloys grow spectacular pentagonal dodecahedra. Those faces cannot be indexed with three integers, because there is no three-dimensional reciprocal lattice for them to sit in. They can be indexed with six, which is the dimension of the periodic lattice a cut-and-project construction slices them out of, and six integers is what the crystallographic literature on these alloys uses.

So the law generalises rather than fails: a face is still named by integers, and the number of integers required is the dimension of the periodic structure standing behind the aperiodic one. What is lost is the claim that three will do — and with it, the tidy correspondence between a face and a direction in ordinary space. That is the same enlargement where five-fold becomes legal describes for rotations, arriving here as extra indices rather than as extra axes.

Haüy, and the argument from a dropped crystal #

The law is older than the lattice it rests on, which is the usual order of events in this subject.

René-Just Haüy was, by his own account, examining a friend’s mineral collection in 1781 when he dropped a crystal of calcite. It broke into fragments, and every fragment was a rhombohedron with the same angles as every other — including angles that had not been on the outside of the original. He went home, broke more calcite deliberately, and found the same thing every time.

The conclusion he drew is the right one and he drew it before there was any language for it. If a crystal cleaves indefinitely into identical small blocks, then it is built of identical small blocks, and any face it presents must be a staircase of those blocks. A staircase that rises one block for every one it advances gives one face; one block in two gives another; one in three another. Every face is therefore described by a ratio of small whole numbers, because a staircase with a period of seventeen blocks is not something a crystal would build.

Haüy’s blocks were the molécules intégrantes, which he took to be actual particles of the substance. They are not, and it did not matter: what the argument needs is the periodicity, not the physical reality of the block. Bravais supplied the modern reading in the 1840s, replacing the solid brick with the lattice point and the stack with the array, and every consequence Haüy had drawn survived the substitution intact. The replacement is the same move that turns a symmetry into a map rather than a picture: what does the work is the relation, not the thing.

William Hallowes Miller gave the indices their modern form in 1839, and the choice of reciprocals rather than intercepts is his. It looks like a notational convenience and it is a great deal more than one: it is the step that made the face normal into the vector it is, and so made the connection to diffraction available seventy years before anybody could measure a diffraction pattern.

The identity behind “widely spaced means densely populated” #

The claim that low-index planes are the densely populated ones is stated above as a relation the reciprocal construction gives, and it is an exact identity rather than a tendency. Writing it down is worth doing, because it is what every ranking of faces in this collection ultimately rests on.

The spacing of the family (hkl) is d = 1/|H|, with H the reciprocal lattice vector. And the lattice points in one plane of that family have an areal density of

$$\sigma = \frac{d}{V},$$

with V the cell’s volume. The derivation is one line: the planes of the family are spaced d apart and together contain every lattice point, so the number per unit area times the number of planes per unit length must give the number per unit volume, which is 1/V.

So density and spacing are proportional, exactly, and the two ways of saying which faces a crystal prefers — the widely spaced ones, or the densely populated ones — are the same statement rather than two correlated ones. That is the identity the table near the top of this essay measures: the spacings there come from the reciprocal metric and the densities are counted in the planes themselves, and the two columns are the same numbers divided by the cell volume.

That identity is what makes the growth ranking computable from the metric alone. A face’s spacing is arithmetic on the reciprocal metric; its density needs no separate calculation; and the whole ranking follows from six numbers and a list of index triples.

The triple that is a reflection and not a face #

There is a systematic difference between how morphology and diffraction read the same three integers, and it catches people who move between the two.

A face’s indices are coprime. The plane (200) is the same plane as (100) — the same orientation, the same normal, the same face — so a common factor carries no information and morphology removes it. A crystal has a (100) face and never a (200) one.

A reflection’s indices are not. 200 and 100 are different reflections, at different angles, with different intensities, because a reflection is a reciprocal lattice point rather than a direction. 200 is twice as far from the origin, its spacing is half, and it exists whether or not 100 does — indeed a centring or a screw may extinguish one and not the other.

So the same triple names a direction in one subject and a vector in the other, and the common factor is meaningless in the first and decisive in the second. That is why a morphological description lists a handful of coprime triples while a data set lists tens of thousands of triples with every common factor represented, and why the two lists cannot be compared entry by entry.

The confusion has a specific and recognisable form: a reader meeting 222 in a diffraction context and asking which face it is. There is no such face. There is a (111) face, and 222 is its second order — a reflection from the same family of planes, at half the spacing, and a perfectly ordinary member of the data.

What this settles and what it does not #

Three things are now decided, and it is worth being precise about which is which.

A face’s orientation is a triple of integers. Theorem, from periodicity alone.

The observed faces have small indices. Empirical rule, with a good mechanism behind it, and with exceptions.

Which faces appear on a particular crystal. Not decided at all — that depends on the solvent, the temperature, the impurities and the history of the specimen, and symmetry has nothing to say about it.

The third is the one this site will keep returning to, because it is the standing shape of what a symmetry argument can and cannot buy. The classification says which faces are permitted to appear together; nothing says any of them will appear. That is the same relationship permitted is not present drew for physical properties, and it is about to become the difference between a form and a habit.

The next rung takes the one quantity that survives every choice made here — the angle between two faces — and asks what it measures. The answer turns out to be the metric of the lattice, which means that a goniometer built in 1780 was measuring cell parameters a century and a half before anybody knew what a cell was.