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Packing, Coordination, and Theoretical Density

Guide 2 told you where the atoms sit; this one asks how tightly they are packed and how many neighbors each one touches — then turns that pure geometry into a density you can predict on paper and check on a scale.

Two questions the unit cell already answers

In the last guide we pinned atoms onto the corners, faces, and centers of a repeating unit cell and met the three metal structures: FCC, BCC, and HCP. That told us where the atoms sit. But a structure is more than a seating chart — the same handful of atoms can be crammed together or left roomy, and each atom can be pressed against many neighbors or just a few. Those two facts, packing tightness and neighbor count, quietly control stiffness, ductility, and density. Happily, once you have the unit cell, both fall out of pure geometry — no experiment required.

We give each a name. The coordination number is simply how many nearest neighbors an atom is touching. The atomic packing factor (APF) is the fraction of the unit-cell volume that is actually filled by atoms, if we picture each atom as a hard sphere of radius R. Both are dimensionless numbers you can compute by hand, and both come straight from how the spheres are stacked. Let us build them up from the idea of stacking, because that picture explains why FCC and HCP tie for the tightest packing nature allows.

Coordination number: how many hands are you holding?

Picture a greengrocer stacking oranges. The tidiest way is to lay a first layer where each orange nestles into the dimple between three others, then drop the next layer into the hollows on top. In such a close-packed pile, every interior orange touches six in its own layer, three in the layer above, and three below — twelve neighbors in all. That is the maximum any equal spheres can achieve, and it is exactly the coordination number of both FCC and HCP metals: 12. Copper, aluminum, nickel, magnesium, and titanium all live at this crowded optimum.

So if FCC and HCP both reach 12, what is the difference? Only the stacking sequence of those close-packed layers. Call the three possible layer positions A, B, and C. HCP repeats A-B-A-B, dropping every third layer directly back over the first; FCC repeats A-B-C-A-B-C, using a third offset before it returns. Same neighbor count, same tightness, two different rhythms — which is why they behave alike in density yet differently under stress. BCC metals like alpha-iron, chromium, and tungsten are not close-packed: the central atom touches only the eight corner atoms, so their coordination number is 8. Simple cubic, rare in real metals, manages just 6.

Atomic packing factor: how much is atom, how much is air?

Even at coordination 12, hard spheres cannot fill all of space — there are always gaps between the balls. The atomic packing factor measures exactly how much: APF = (volume of atoms inside the cell) / (volume of the cell). Work it out for FCC. The cell holds n = 4 atoms (eight corners shared eight ways give 1, plus six faces shared two ways give 3). The atoms touch along the face diagonal, so 4R equals a times the square root of 2, which means the edge is a = 2 times sqrt(2) times R, about 2.83 R. Then APF = [4 times (4/3)pi R^3] / a^3 works out to pi / (3 times sqrt(2)) = 0.74. Nearly three-quarters solid, one-quarter empty — the densest packing possible.

The same recipe grades every structure. HCP also gives 0.74 — no surprise, it is close-packed too. BCC has n = 2 atoms and its spheres touch along the body diagonal (4R = a times sqrt(3)), which loosens the edge to a = 4R / sqrt(3); running the numbers gives APF = 0.68. Simple cubic manages only 0.52. So the ranking is FCC = HCP (0.74) > BCC (0.68) > simple cubic (0.52). Notice the payoff of counting neighbors first: higher coordination number goes hand in hand with higher packing factor, because pressing against more neighbors is just another way of saying you have squeezed out more empty space.

STRUCTURE  atoms/cell  spheres touch    edge a          coord. no.  APF    examples
---------  ----------  --------------   -------------   ----------  -----  ----------------
simple     1           cube edge        a = 2R          6           0.52   polonium (rare)
BCC        2           body diagonal    a = 4R/sqrt(3)  8           0.68   alpha-Fe, Cr, W
FCC        4           face diagonal    a = 2 sqrt(2) R  12          0.74   Cu, Al, Ni, gamma-Fe
HCP        6 (hex cell) close-packed     c/a ~= 1.633    12          0.74   Mg, Ti, Zn

  APF = (n atoms x volume of one sphere) / (unit-cell volume)
      = [ n x (4/3) pi R^3 ] / a^3
The four packings side by side. Coordination number and APF rise together; FCC and HCP tie at the close-packed ceiling of 0.74.

From geometry to grams: theoretical density

Here is the beautiful part. If you know the structure, you know the mass in one cell and the volume of one cell — so you can predict the material's theoretical density before ever weighing it. The formula is rho = (n times A) / (Vc times NA): n is atoms per cell, A is the atomic weight in grams per mole, Vc is the cell volume, and NA is Avogadro's number, 6.022 times 10^23 per mole. The numerator is 'how much stuff is in one cell'; the denominator is 'how much room one cell takes up'. Divide, and out drops density in g/cm^3.

  1. Copper is FCC, so count the atoms: n = 4 per cell.
  2. Get the edge from the radius. FCC touches along the face diagonal, so a = 2 times sqrt(2) times R; with R = 0.128 nm this gives a = 0.362 nm = 3.62 times 10^-8 cm.
  3. Cube it for the cell volume: Vc = a^3 = (3.62 times 10^-8 cm)^3 = 4.75 times 10^-23 cm^3.
  4. Plug in A = 63.5 g/mol and NA: rho = (4 times 63.5) / (4.75 times 10^-23 times 6.022 times 10^23) = 254 / 28.6 = 8.89 g/cm^3.
  5. Compare with reality: measured copper is 8.96 g/cm^3. Within about 1 percent — a genuinely predictive result from pure geometry.

Same atoms, different packing: polymorphism

Because density flows from structure, one element can have two densities if it can adopt two structures — that is polymorphism (called allotropy for a pure element). Iron is the headline case. Below 912 degrees C it is BCC alpha-iron (ferrite), APF 0.68; heat it past 912 and it flips to FCC gamma-iron (austenite), APF 0.74. Here is the counterintuitive kicker: on heating through that point the iron contracts by about 1 percent, because tighter packing wins over thermal expansion. A metal that shrinks when you heat it, purely because its atoms restacked more efficiently — and this restacking is the very transformation the whole heat-treating of steel later exploits.

Carbon makes the point even more dramatically. The same carbon atoms build diamond — a rigid three-dimensional covalent network, density 3.51 g/cm^3, the hardest natural material — or graphite, flat sheets weakly stacked, density only 2.25 g/cm^3, soft enough to smear onto paper as pencil lead. Identical atoms; the packing alone spans hard-and-dense to soft-and-open. That is the whole thesis of this rung in one comparison: structure, not just composition, writes the properties.

Two honest caveats before we move on. First, the hard-sphere APF is a model: real atoms are not rigid billiard balls, and a real bar always measures a hair less dense than the theoretical value because it carries vacancies, porosity, and defects — the 8.89-versus-8.96 gap for copper is partly this. Second, the tightest-packed planes we invoked for coordination and APF are also the planes on which metals most easily slip and deform; naming and indexing those planes is the job of the next guide on Miller indices, where this geometry starts paying off in mechanical behavior.