Close Packing: FCC, HCP, and Stacking

From wallpaper to a grocer's pile of oranges

The last rung left us with a clean slogan: crystal = lattice + motif, one motif stamped on every point of a lattice. For a pure metal like copper or aluminium the motif is the simplest thing imaginable — a single atom. That is what makes metals the easiest crystals to picture: strip away the chemistry and you are left with a bare geometry puzzle. How do you arrange identical balls in space so they take up as little room as possible? The answer to that one question hands you the structures of most of the metals in the periodic table.

Why does packing tightly even matter? Because the metallic bond is non-directional — it does not care which way a neighbour lies, only that neighbours are close. A cloud of shared electrons glues the positive cores together and pulls equally in every direction, so the atoms behave almost like hard, slightly sticky balls that want as many close neighbours as they can get. When bonding has no preferred direction, geometry alone takes over, and geometry says: pack like a grocer stacking oranges. This is the idea of close packing, and it is the backbone of this whole rung.

One close-packed layer, and the two sets of dimples

Start flat. Pour balls onto a table and shake them together, and they settle into the tightest possible single sheet: every ball touches six others around it, and the row of dimples between them runs in three directions at 60 degrees. This is the close-packed layer — the densest way to cover a plane with equal circles. Call it layer A. Notice there is no wasted room between neighbours in this sheet; each ball's six contacts are the most a flat layer can give. The whole trick of building the densest solid is now just: how do we stack these perfect sheets on top of one another?

The second layer, call it B, cannot sit ball-on-ball — that would waste space. Instead each ball of B nestles into a dimple of A, and the layer drops lower and locks in. Here is the subtle part that decides everything. Layer A actually offers two different kinds of dimple, twice as many hollows as there are balls, and a single layer B can only fill half of them. The half it fills we call the B positions; the other, untouched set we call the C positions, sitting directly over yet a third pattern of holes. When it comes time to lay the third layer, that leftover choice — B's own dimples sit over either the original A spots or the never-used C spots — is the whole story.

ABCABC is FCC; ABAB is HCP

So the third layer has exactly two honest choices. Put its balls back over the A positions and the pattern repeats every two layers: A, B, A, B, A, B... That two-layer beat is the hexagonal close-packed structure, HCP — the choice made by magnesium, zinc, titanium and cobalt. Or put the third layer over the fresh C positions, and only on the fourth layer do you return to A: A, B, C, A, B, C... That three-layer beat is cubic close packing, which turns out to be the same thing as the face-centred cubic (FCC) structure — the choice made by copper, aluminium, gold, nickel and silver.

  A CLOSE-PACKED LAYER (top view)      each ball touches 6 in-plane

       O   O   O
     O   O   O    <- 3 balls nestle in dimples ABOVE,
       O   O   O     3 more nestle in dimples BELOW  ->  12 neighbours


  STACKING (side view, each letter = one close-packed layer)

     HCP  ... A B A B A B ...   two-layer repeat  (hexagonal)
     FCC  ... A B C A B C ...   three-layer repeat (cubic close-packed)

     both: coordination = 12,  packing factor = 0.74
     BCC (for contrast): NOT built from close-packed layers,
                         coordination = 8,  packing factor = 0.68

The FCC surprise deserves a pause. Nothing about the ABCABC recipe looks cubic — we built it out of tilted hexagonal sheets. Yet if you rotate the stack and squint, those close-packed layers turn out to be the {111} planes of a cube, sliced across its body diagonal, and the cube's own face-centred cell falls out of the geometry. So the close-packed layers of FCC are its close-packed planes, the {111} family, and the directions along which balls touch nose-to-nose within a layer are the close-packed directions, the <110> family. In HCP the close-packed plane is simply the flat basal layer (0001). This is not idle labelling: those very planes and directions become the highways along which metals deform, which is why the next rung on dislocations keeps coming back to them.

The descriptors: coordination, packing factor, stacking sequence

To compare structures we need a few honest numbers. The coordination number is just how many nearest neighbours touch a given atom. In both FCC and HCP the count is the same: 6 in your own layer, 3 in the dimples above, 3 in the dimples below — 12 in all, the maximum any equal spheres can achieve. The atomic packing factor then asks what fraction of space the balls actually fill, and the answer for either close packing is 0.74 (more precisely pi divided by (3 times the square root of 2), about 0.7405). That 0.74 is not merely the best we have found — it is provably the densest any identical spheres can be packed, a fact conjectured by Kepler in 1611 and only rigorously proved in our own era.

Once you see structures as stacked layers, the stacking sequence becomes a compact shorthand for the whole crystal: HCP is ...AB..., FCC is ...ABC..., and you can invent longer beats. When the same close-packed layers repeat with an unusually long or mixed period — say ABCACB over and over — the material shows polytypism: one chemistry, many stacking variants. Silicon carbide is the champion, with over 200 known polytypes that differ only in how their identical layers are stacked. A one-off mistake in the sequence — an ...ABCABABC... hiccup where a single C is skipped — is a stacking fault, a thin slab of the 'wrong' stacking embedded in the right one, and it is one of the cheapest, most common defects in close-packed metals.

How eagerly a metal tolerates such faults is set by its stacking-fault energy — the energy cost per unit area of the mis-stacked slab. Low-energy metals like brass and austenitic steels are riddled with faults and deform in flat, planar ways; high-energy metals like aluminium heal them quickly. That single number quietly steers how a metal work-hardens and which slip paths its dislocations prefer, tying this rung of geometry straight to the mechanical slip systems you will meet next. And it explains why FCC and HCP are not rivals but neighbours: because they differ only in stacking, a metal can slip from one into the other, or leave thin lamellae of one inside the other, at very little cost.

Cashing it out: theoretical density from the cell

Here is the reward for all this counting. Once you know the structure and the atom, you can predict the material's density with nothing but arithmetic — no scale needed. This is the theoretical density, and it comes from a single honest idea: density is the mass inside one unit cell divided by that cell's volume. Recall from the last rung that an FCC cell contains 8 corners times 1/8 plus 6 faces times 1/2 = 4 atoms. Weigh those four atoms, divide by the cube's volume, and you have the density of the bulk metal.

1. Count the atoms per cell. FCC copper has n = 4 (corners and face-centres, shared correctly).
2. Get the mass of one cell: n times the atomic mass divided by Avogadro's number. Here (4 times 63.55 g/mol) / (6.022 times 10^23 per mol) = 4.22 times 10^-22 g.
3. Get the cell volume from the edge length a = 3.615 angstrom = 3.615 times 10^-8 cm, so V = a^3 = 4.72 times 10^-23 cm^3.
4. Divide mass by volume: 4.22 times 10^-22 g / 4.72 times 10^-23 cm^3 = 8.9 g/cm^3 — dead on the measured density of copper.

That the calculation lands right on copper's real 8.96 g/cm^3 is the whole point of structure science in one number: get the arrangement right and a bulk property falls out for free. The same recipe covers HCP and, with coordination 8 and a packing factor of only 0.68, the looser body-centred cubic metals like iron and tungsten — we save BCC's full workout for guide 2. And it is worth naming the flip side: because these are all just different stackings and packings of the same atoms, one element can adopt several of them at different temperatures. That is polymorphism, and its elemental form, allotropy — carbon as soft graphite or hard diamond, iron as FCC gamma-austenite when hot and BCC alpha-ferrite when cool. A polymorphic transition is nothing more exotic than the atoms choosing a new way to stack.