Two stackings, one density

Lay identical spheres on a table as tightly as they will go and the arrangement is forced: a triangular array, each sphere touching six others, with two sets of hollows between them. Put a second layer in one set of hollows — either set, they are mirror images and equally good — and that is forced too.

The third layer is not. It can sit directly over the first, or over the hollows the second layer left unused. Both choices touch the same number of spheres, both fill the same fraction of space, and every layer after them faces the same decision again.

So there is not one close packing but infinitely many, and the two regular ones — ABAB and ABCABC — are what mineralogy and metallurgy call hexagonal and cubic close packing. Nothing about density chooses between them. What separates them is symmetry, and that is the whole of the subject.

The same number, to every digit #

The fraction of space filled is π/√18 = 0.740480…, and it is the same for both regular packings — not approximately, exactly, and the equality is worth being convinced of rather than told.

The site’s habit applies: measure it. Take each structure’s atom positions and cell, compare every atom against every atom of the surrounding cells to find the nearest-neighbour distance, take the radius as half of that, and divide the spheres’ volume by the cell’s. The two answers agree to every digit the arithmetic carries.

Coordination twelve in both cases, and even the arrangement of the twelve is nearly the same: six in the sphere’s own layer, three above and three below. The difference is only in how the upper three sit relative to the lower three — staggered in the cubic case, eclipsed in the hexagonal — so the two coordination polyhedra are the cuboctahedron and its twisted relative, which have the same faces and the same edges and differ by a rotation of one half.

The groups, rediscovered rather than named #

Handing each structure to the three-dimensional detector gives the two symbols, and the round trip is the point: the space group is not typed into the caption, it is found.

Cubic close packing is Fm3̅m, order 192 including the centring translations. Its close-packed layers are perpendicular to a body diagonal — which is the fact that makes the cubic case surprising, since nothing about a cube suggests a stack of triangular layers until the ⟨111⟩ direction is looked along.

Hexagonal close packing is P6₃/mmc, order 24. And it is worth pausing on what it is not: hexagonal close packing is not a lattice. Its cell contains two atoms that no translation relates, so it is a lattice with a basis, and the distinction first drawn in the plane has its clearest physical example here. Cubic close packing is a lattice — the face-centred cubic one — with a single atom per primitive cell.

The 6₃ screw, and where it comes from #

The screw axis in P6₃/mmc has an origin the picture makes obvious, and it is the cleanest illustration of a screw this site has.

Take the hexagonal stack. A rotation of a sixth of a turn does not map it onto itself — that would carry an A layer onto an A layer, and the layers alternate. But a sixth of a turn combined with a shift of half the repeat carries A onto B and B onto A, and it maps the structure onto itself exactly. That is a 6₃ screw: rotate by 60°, advance by half.

So the operation exists precisely because the stacking alternates, and no choice of origin removes it — the intrinsic half-translation is a property of the operation. A structure with the cubic stacking has no such axis and has four three-fold axes instead, along the four body diagonals.

Everything in between is legal #

Any sequence of layers with no two adjacent alike is a close packing of exactly the same density, so between the two regular cases there are infinitely many irregular ones. The ones that repeat are polytypes, and counting them is a clean combinatorial exercise.

A stacking of period n is a cyclic word in {A, B, C} with no two adjacent letters equal, including at the wrap-around. Two words describe the same structure when one becomes the other by rotating the cycle, reversing it, or relabelling the three positions — so the count is the number of classes under a group of order 6n.

Silicon carbide is the standard example and it is extraordinary: more than two hundred polytypes have been reported, some with repeat periods of hundreds of layers, several of which grow reproducibly enough to be commercial materials with different band gaps. Zinc sulphide, cadmium iodide and the micas all do the same thing to a lesser degree.

The energy differences between polytypes are tiny — thousandths of an electron volt per atom — which is exactly why so many of them occur, and exactly why nothing about their occurrence can be predicted from symmetry. What symmetry supplies is the list.

The notation the polytypes are named in #

The naming worth knowing is Ramsdell’s, because it is compact and it says exactly what the count above counts.

A polytype is written as a number and a letter: the number is how many layers are in the repeat, and the letter is the crystal system — C for cubic, H for hexagonal, R for rhombohedral. So 2H is ABAB, 3C is ABCABC, 4H is ABAC, and 6H is one of the two period-six packings. Where a period has more than one packing the entries are distinguished by a subscript.

The notation is a compression of the sequence and it loses information: 6H does not say which of the two period-six sequences is meant. That is deliberate — the letter carries the symmetry, which is what a crystallographer wants, and the full sequence is available for anybody who needs it. It also explains the strange-looking fact that 3C is cubic while every other polytype in the list is hexagonal or rhombohedral: three is the only period whose stacking has the cubic arrangement, so every other number in the catalogue takes an H or an R.

What the arithmetic decides and what it does not #

The division here is unusually clean and worth stating explicitly, because this field is the one where the temptation to overreach is strongest.

Decided by the arithmetic. That the density is π/√18 for every close packing. That the coordination number is twelve. That ABCABC has space group Fm3̅m and ABAB has P6₃/mmc. That there are exactly sixteen distinct packings of period ten. Each of those is computed above and none is quoted.

Not decided by anything here. Which packing a substance takes. Magnesium is hexagonal, copper is cubic, and cobalt changes its mind at 695 K — three facts about electronic structure, all invisible to symmetry. Why silicon carbide has two hundred polytypes and copper has none. Why the ideal axial ratio √(8/3) = 1.633 is approached by real hexagonal metals and never met exactly.

That last one is worth a sentence, because it is a measurement rather than a decision. Zinc has c/a = 1.856 and cadmium 1.886, both far from ideal, so their “close-packed” layers are not in contact in the way the geometry assumes — the structure is P6₃/mmc regardless, since the symmetry does not depend on the ratio, but the packing fraction is lower and the twelve neighbours are no longer at equal distances. The group is exact and the packing is approximate, which is the reverse of how the two are usually presented.

Where else this decision shows up #

The stacking freedom is not confined to spheres, and naming two other places it occurs makes the pattern visible.

Twinning. A close-packed structure that stacks ABCABC and then, after a mistake, continues CBACBA has produced a twin: two orientations of one structure sharing a plane, with the boundary at the layer where the sequence reversed. In face-centred cubic metals this is the commonest planar defect there is, it costs almost nothing to make, and it is the annealing twin visible in any micrograph of brass — the same object the coincidence arithmetic reaches as a Σ3 boundary.

Stacking faults. A single wrong layer in an otherwise regular sequence — ABCABABC — is a fault rather than a twin, a slab of the other packing one layer thick. Its energy is the difference between the two packings’ energies, which is the small number mentioned above, and materials with a low stacking-fault energy deform in a completely different way from materials with a high one.

So the free choice this page opens with is not an abstraction: it is the mechanism behind a defect that decides how a metal deforms, and behind the boundary that decides how it corrodes. The geometry says the choice is free; the material’s energy says how often it is made wrongly; and everything interesting is in the second.

Why the density cannot depend on the sequence #

The figure measures the two fractions and finds them equal. The reason they are equal is worth stating separately, because it explains why the infinitely many irregular packings have the same density too, and no amount of measuring individual cases would establish that.

A close packing is a stack of identical layers. Each layer has the same number of spheres per unit area — that is what “as tightly as they will go” fixes, and it is the two-dimensional problem, settled by the triangular arrangement. And each layer sits at the same height above the one below it, because the spacing is decided by three touching spheres of the layer beneath and a fourth resting in their hollow, which is one tetrahedron of edge d whatever the hollow chosen.

So the density is spheres per unit area, divided by the spacing, and neither quantity knows the sequence. Choosing B or C for the next layer is choosing which of two hollows to use, and the two hollows are related by a translation of the layer — a rigid motion, which changes no volume and no separation.

That is the argument, and it says something the measurement cannot: every close packing has density π/√18, including sequences with no period at all. There are uncountably many such stackings, so no computation over polytypes could have reached the statement; it needs the layers to be identical and the spacing to be forced, and then it needs nothing else.

What the packing fraction is not #

There is a much stronger statement in the neighbourhood, it is famous, and it is not what has been established here.

What this page shows is that two particular arrangements achieve 0.740480…. Kepler’s conjecture is that no arrangement of equal spheres, of any kind whatever — periodic, aperiodic, or with no structure at all — exceeds it. Those are entirely different claims, and the gap between them held for nearly four hundred years.

Gauss closed part of it in 1831 by proving that no lattice packing does better, which is the case where the sphere centres form a lattice and the problem becomes one about quadratic forms. That leaves the packings that are not lattices, and among them the close packings themselves, since hexagonal close packing has two spheres per cell and is not a lattice packing at all. Thomas Hales proved the general statement in 1998, by an argument reducing the problem to a few thousand cases of nonlinear optimisation and settling each by computer. The referees reported after four years that they were ninety-nine per cent certain it was correct and could not verify the computations completely, which is an unusual sentence to find in a mathematical journal. Hales then led a decade-long project to produce a proof checked mechanically from the axioms, completed in 2014.

The episode is worth keeping beside the arithmetic on this page for a reason this collection meets repeatedly. Computing what an arrangement does is easy; proving that nothing does better is a different subject. Every number in this essay is of the first kind — a density measured on a structure that was built, a group recovered by a detector, a count of sequences. The one statement of the second kind is quoted, and it is quoted with the length of its proof attached.

Where the polytype count comes from #

The count in the figure has a closed form behind it, and the derivation is two lines of the kind this collection prefers.

A stacking of period n is a closed walk of length n on the graph whose vertices are A, B and C and whose edges join every pair — no two adjacent layers alike being exactly the condition that consecutive letters differ. The number of closed walks of length n on that graph is the trace of the n-th power of its adjacency matrix, whose eigenvalues are 2, −1 and −1, giving 2ⁿ + 2(−1)ⁿ sequences.

Everything after that is quotienting. Two sequences describing the same packing differ by a cyclic rotation of the word, by a relabelling of the three letters, or by reading the stack from the other end — so the distinct polytypes are the orbits of that action, and Burnside’s count over the orbits is what the figure plots. Period two gives one, period three gives one, period four gives one, and the numbers climb from there. The arithmetic is exact and integer throughout, which is the only reason a count of an infinite family is trustworthy at all.

Barlow, and a century of not being believed #

William Barlow, a businessman with no formal training, worked out the close packings and their symmetries in 1883 — before X-ray diffraction, before anybody had seen a crystal structure, and on the argument that crystals are built of spheres in contact.

He was substantially right, and he was not taken seriously for thirty years. The objection was reasonable at the time: nothing in the evidence required atoms to be spheres in contact, the theory made no prediction anybody could test, and the sphere-packing picture had been proposed and abandoned before. When Bragg determined the structure of rock salt in 1913, Barlow’s arrangements turned out to be the arrangements, and his diagrams were correct in detail.

The lesson the episode leaves is not about vindication but about what a geometrical argument can and cannot claim in advance. Barlow could say what the possible packings were; he could not say which substance took which, and he did not claim to. The list was right and the assignment had to wait for a measurement — which is the same division this page ends on, a hundred and forty years later.

Where the exactness stops #

Three limits.

Spheres are a model, and a strong one. Atoms are not spheres and do not have a well-defined radius; the “close-packed” description is a good one for metals, where the bonding is undirected, and a poor one for anything with directional bonds — silicon does not close-pack at all, and its structure fills 34% of space rather than 74%. The packing fraction computed above is a property of the geometric model, not a measurement of a solid.

The polytype count is a count of sequences. Two polytypes with the same sequence have the same symmetry, and two different sequences can still be very similar structures — the classification is by stacking, which is the right unit for symmetry and not necessarily for any physical property.

Polytypes are indistinguishable in a powder pattern at low resolution. Two long-period polytypes differ in reflections that are weak and closely spaced, so what a powder measurement loses is exactly what tells them apart — which is why the silicon carbide catalogue was built from single crystals and why the number in it keeps changing.

And the detector was given the crystal system. The round trip in space enumerates the operations a stated lattice type permits, so the verification confirms that the structure has the group named and does not independently discover the lattice type. That limit is the same one the space-groups phase recorded, unchanged, and it is the reason the packings above are described as verified rather than as identified.

Where the ladder goes next #

Close packing leaves holes, and the holes are where most of chemistry happens: two tetrahedral and one octahedral hole per sphere, in both stackings, and every structure from rock salt to spinel is a close packing with some of its holes filled. Counting the holes is a Wyckoff-position question with an answer that does not depend on the stacking, and that independence is what makes the structural taxonomy of minerals possible at all.