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Packing Factor, Coordination, and BCC

Guide 1 stacked spheres into FCC and HCP; now we put numbers on them. Two descriptors — coordination number and packing factor — quantify how tightly atoms sit, the less-dense BCC joins the cast, and from the cell's contents alone you can predict a metal's density to three figures.

Counting a sphere's neighbours: coordination number

In guide 1 we stacked close-packed layers of oranges into two grocer's pyramids: ABCABC gave the face-centred cubic (FCC) metal, and ABAB gave the hexagonal close-packed (HCP) one. That told us the arrangement; now we need words that measure it. The first descriptor is the simplest question you can ask of any atom: how many nearest neighbours actually touch it? That count is the coordination number, and it is the first number every materials person quotes about a structure.

For any close-packed sphere the answer is exactly 12: six neighbours ring it within its own layer, three nestle in the dimples of the layer above, and three more in the layer below — six plus three plus three. This holds for BOTH stackings. An FCC atom and a hexagonal close-packed atom each have coordination 12; the ABC-versus-AB difference lives only in where the third layer sits, which is a longer-range distinction that a sphere cannot feel from its immediate touching neighbours. Locally, FCC and HCP are twins.

How full is the box: the atomic packing factor

The second descriptor asks how much of space the atoms actually occupy. The atomic packing factor (APF) is just the volume of the spheres inside one unit cell divided by the volume of the cell itself. To compute it we lean on the hard-sphere model: we pretend the atoms are rigid billiard balls of radius r that simply touch, never overlapping and never squashing. It is an idealisation — real atoms are fuzzy clouds with no hard edge — but it captures the geometry astonishingly well.

Work FCC through as a template. In a face-centred cubic cell the atoms touch along the face diagonal, which threads three atoms across a square face, so 4r = sqrt(2) a and therefore a = 2 sqrt(2) r. The cell holds 4 atoms (eight corners times 1/8 plus six faces times 1/2). So APF = 4 times (4/3)pi r^3 / a^3 = 4 times (4/3)pi r^3 / (2 sqrt(2) r)^3 = pi / (3 sqrt(2)) = 0.7405. The radius r cancels completely — the answer is pure geometry, the same 0.74 for copper, aluminium, or gold. Even the tightest possible packing of equal spheres leaves 26 percent of space empty.

The odd one out: body-centred cubic

Not every metal close-packs. Iron at room temperature, chromium, tungsten, and all the alkali metals choose the body-centred cubic (BCC) structure instead: an atom at each of the 8 corners plus one lone atom sitting in the very centre of the cube. BCC is emphatically NOT close-packed — there is no set of close-packed layers hiding inside it — and its numbers tell the story. The central atom reaches out to the 8 corner atoms along the body diagonal, so its coordination number is 8, not 12.

For the packing factor, the touching condition changes. In BCC the spheres kiss along the body diagonal, whose length is sqrt(3) a and which spans three atoms (corner, centre, corner), so 4r = sqrt(3) a and a = 4r / sqrt(3). The cell holds only 2 atoms (eight corners times 1/8 plus one whole centre). So APF = 2 times (4/3)pi r^3 / a^3 = sqrt(3) pi / 8 = 0.6802. That is looser than 0.74 — a BCC metal wastes about 32 percent of its volume rather than 26 percent. Why would nature pick the roomier arrangement? Because the hard-sphere APF is only geometry, and which structure a metal truly adopts is decided by subtle electronic energy differences, not by packing alone. Metallic bonding is non-directional, so several arrangements are close in energy and small effects tip the balance.

 structure  coordination  APF    atoms/cell  spheres touch along
 ---------------------------------------------------------------
 FCC            12         0.74      4        face diagonal:  4r = sqrt(2) a
 HCP            12         0.74      2        in-plane edge:  a  = 2r
 BCC             8         0.68      2        body diagonal:  4r = sqrt(3) a
The three workhorse metallic structures on one card: the two close-packed cousins (FCC and HCP) tie at coordination 12 and APF 0.74, while BCC trades density for a looser, higher-energy-tolerant arrangement.

From cell contents to density

Here is where these bookkeeping numbers pay a dividend you can weigh on a scale. If you know how many atoms sit in a unit cell and how big the cell is, you can predict the material's theoretical density from first principles — no measurement required. The recipe is one clean formula: rho = n times M / (N_A times V_cell), where n is the atoms per cell, M is the molar mass, N_A is Avogadro's number (6.022 times 10^23 per mole), and V_cell is the cell volume.

  1. Count n, the atoms (or formula units) per cell — corners count 1/8, faces 1/2, edges 1/4, and a body-centre atom counts as a whole 1.
  2. Look up M (the molar mass in grams per mole) and compute V_cell — for a cubic cell that is just a^3, with the edge converted to centimetres.
  3. Plug into rho = n times M / (N_A times V_cell), keeping every length in centimetres so the answer lands in grams per cubic centimetre.
  4. Compare with the measured density — the theoretical value is always a touch higher, because a real crystal carries vacancies, dislocations, and grain boundaries that the perfect-crystal formula ignores.

Try it on room-temperature iron, which is BCC. Its cell edge is a = 2.866 angstrom = 2.866 times 10^-8 cm, so V_cell = a^3 = 2.354 times 10^-23 cm^3. With n = 2 atoms per cell and M = 55.85 g/mol, rho = 2 times 55.85 / (6.022 times 10^23 times 2.354 times 10^-23) = 111.7 / 14.18 = 7.88 g/cm^3. The handbook value for alpha-iron is 7.87 g/cm^3 — agreement to three figures, from geometry and a periodic table alone. (As a sanity check, 4r = sqrt(3) a gives an iron radius of about 1.24 angstrom, exactly its known metallic radius.)

Same atoms, different packing: polymorphism

One substance can crystallise in more than one structure, a phenomenon called polymorphism (or, for a pure element, allotropy). Carbon is the famous case — soft, layered graphite and hard, tetrahedral diamond are the very same atoms in two arrangements. Iron does it with temperature: it is BCC (called alpha, or ferrite) below 912 degrees, transforms to FCC (gamma, or austenite) on heating through that point, and this polymorphic transition is the structural switch that the whole steel industry is built on.

Here the packing factors give a delightfully counter-intuitive prediction. When iron transforms from BCC (APF 0.68) to FCC (APF 0.74) on heating through 912 degrees, the atoms pack MORE tightly, so the metal actually contracts by about 1 percent — a rare case where heating a solid makes it shrink. The denser close-packing wins even though we added heat. It is a memorable reminder that the packing factor is not idle arithmetic: it predicts real volume changes that steelmakers must design around.

So a small vocabulary now describes any structure: coordination number (how many touch), packing factor (how full), and stacking sequence (the ABCABC-versus-ABAB order beyond nearest neighbours). When the same layers stack in many different long-range sequences we call it polytypism — silicon carbide famously has hundreds of polytypes, all built from identical SiC bilayers merely re-ordered. Carry these descriptors forward: they are the shared language for the interstitial holes of guide 3 and the named ionic and covalent structure types of guides 4 and 5.