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Lesson 2 · 20 min

Close packing: ABAB and ABCABC

Stack equal spheres layer by layer like oranges: the two choices for the third layer give hexagonal and cubic close packing, both 74 % full, with tetrahedral and octahedral holes between the layers.

In this lesson

Build ABAB (hcp) and ABCABC (ccp) stackings layer by layer and say how they differ
Derive the close-packed filling π/(3√2) = 74 % and compare it with bcc and simple cubic
Give the number of tetrahedral and octahedral holes per sphere and the largest radius each can hold
Find the close-packed layers inside the face-centred cubic cell

Oranges on a market stall are always stacked into pyramids: each orange in an upper layer sits in a dimple between three below. In 1611 Kepler conjectured that this arrangement, filling about 74 % of space, is the densest possible for equal spheres. The “obvious” claim was proved only in 1998, by Thomas Hales with computer assistance, and fully machine-checked in 2017. The atoms of most metallic elements are stacked in just this way.

One layer, two layers, and the third

First pack equal spheres as tightly as possible on a plane: each sphere touches 6 others in a triangular net — call it layer A. Above A lie two sets of hollows, B and C. Either can cradle the second layer, but only one set at a time — neighbouring hollows are too close to hold two spheres.

With the second layer on B, the third has two choices:

straight above A, giving ABAB…, hexagonal close packing (hcp), as in Mg, Zn and Ti;
over C, giving ABCABC…, cubic close packing (ccp), as in Cu, Al and Au.

Both have a layer spacing of $\sqrt{2/3}\,D \approx 0.816D$ (with $D$ the sphere diameter), both fill 74.05 % of space, and in both every sphere has 12 touching neighbours: 6 in its own layer, 3 above and 3 below.

Predict first

Layers A and B are in place. If the third layer goes over C instead of back over A, what happens to the packing fraction?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

It rises: ABC is tighter It falls: ABC is looser No change: both are 74 % It depends on the sphere size

Equal spheres · one layer at a time

Close-packing builder

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The next layer can go over the B / C hollows; putting it on A stacks spheres straight on top of each other.

Show holes between the top two layers Shrink spheres to see inside

Stacking sequence

A ·

h / c under a letter: h when the layers above and below it match (hexagonal surroundings), c when they differ (cubic).

Stacking: Too few layers to tell yet
Space filled: —

Start from a single layer A and build ABAB and ABCABC; then try putting a layer directly on an identical one (B on B) and watch the filling drop. Turn on the holes and count the tetrahedral and octahedral holes between two layers.

\phi = \frac{\tfrac{\pi}{6}D^3}{\tfrac{\sqrt3}{2}D^2 \times \sqrt{\tfrac23}\,D} = \frac{\pi}{3\sqrt2} = 0.7405

Each sphere “owns” its area in the layer times the layer spacing; its own volume divided by that is the packing fraction.

$\phi$: packing fraction
$D$: sphere diameter
$\tfrac{\sqrt3}{2}D^2$: area per sphere in the triangular net
$\sqrt{2/3}\,D$: spacing between neighbouring layers

Cubic close packing is face-centred cubic

Why is cubic close packing also called face-centred cubic? Look along the body diagonal of the fcc cell (the [111] direction): the atomic layers perpendicular to it are exactly the triangular close-packed nets, repeating A, B, C. The cubic cell and the ABC stacking are two descriptions of one structure — the first is convenient for calculation, the second shows why it is so dense. hcp has no cubic symmetry; its cell is hexagonal, with an ideal axial ratio $c/a = \sqrt{8/3} = 1.633$ .

Lattice + basis · counting · coordination

Unit-cell explorer

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Cells

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Touching spheres Cell edges

Plane (hkl)

Copper Face-centred cubic (ccp) · Cubic a = 361.49 pm

How many atoms does one cell own?

Position	Centres	Share	Owned
$\ce{Cu}$ Corner	8	× 1/8	= 1
$\ce{Cu}$ Face	6	× 1/2	= 3
Total per cell			4
Formula units Z			4

This 2×2×2 block draws 63 atom centres but owns only 32 atoms — those on its surface are shared with cells outside.

Coordination number: 12 cuboctahedron
Nearest-neighbour distance: 255.6 pm then 6 more at 361.5 pm
Packing fraction: 74.0% touching hard spheres
Density (computed / measured): 8.935 / 8.933 g/cm³
Spacing d₁₁₁: 208.7 pm Cu Kα reflection at 2θ = 43.32°

The (111) planes are marked in a 2×2×2 copper block: the atoms on each yellow plane form a close-packed triangular net. Switch to Mg and choose (002) to see the A and B layers of the hexagonal cell.

Holes: room for smaller spheres

Even close packing leaves 26 % of space empty, in two kinds of hole:

tetrahedral holes, enclosed by 3 spheres below and 1 above (or the reverse), hold a sphere of radius $0.225r$ ; there are 2 per sphere;
octahedral holes, enclosed by 3 spheres below and 3 above, hold a sphere of radius $0.414r$ ; there is 1 per sphere.

Many ionic crystals are “large anions close-packed, small cations in the holes”: NaCl is $\ce{Cl-}$ in ccp with $\ce{Na+}$ in every octahedral hole; zinc blende ZnS is $\ce{S^2-}$ in ccp with $\ce{Zn^2+}$ in half the tetrahedral holes; swap the roles and fluorite $\ce{CaF2}$ is $\ce{Ca^2+}$ in ccp with $\ce{F-}$ in every tetrahedral hole. The numbers 0.225 and 0.414 return in the next lesson as radius ratios.

Worked example

The packing fraction of body-centred cubic

Sodium is body-centred cubic. Treating its atoms as touching hard spheres, find the packing fraction.

Packing fractions of common structures (touching hard spheres)
Structure	CN	Packing fraction	Examples
Simple cubic	6	$\pi/6 = 52.4\%$	α-Po
Body-centred cubic	8	$\sqrt3\,\pi/8 = 68.0\%$	Na, Fe, W
Face-centred cubic (ccp)	12	$\pi/(3\sqrt2) = 74.0\%$	Cu, Al, Au
Hexagonal close-packed	12	$74.0\%$	Mg, Ti
Diamond	4	$\sqrt3\,\pi/16 = 34.0\%$	C, Si, Ge

Source: Geometry of touching equal spheres; the same values are computed live in the crystal atlas.

Concept check

Which unit cell describes ABCABC… stacking?

Body-centred cubic Face-centred cubic Simple cubic Hexagonal

Try the calculation

In a close-packed array of $\ce{Cl-}$ ions (radius 181 pm), what is the largest cation radius that fits an octahedral hole?

Key takeaways

Equal spheres pack into triangular nets; the second layer sits on B or C, and the third layer’s choice gives hcp (ABAB) or ccp = fcc (ABCABC).
Both close packings fill π/(3√2) = 74 % of space with coordination number 12.
Each sphere has 2 tetrahedral holes (room for 0.225r) and 1 octahedral hole (0.414r); many ionic crystals are “anions packed, cations in the holes”.
Body-centred cubic fills 68 %, simple cubic 52 % and diamond 34 %: lower coordination, looser packing.

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Data and references

A. R. West, Basic Solid State Chemistry, 2nd ed. (Wiley, 1999), Ch. 1.
P. Atkins, T. Overton, J. Rourke, M. Weller, F. Armstrong, Shriver & Atkins’ Inorganic Chemistry, 5th ed. (OUP, 2010), Ch. 3.
T. C. Hales et al., “A formal proof of the Kepler conjecture”, Forum of Mathematics, Pi 5, e2 (2017).
W. M. Haynes (ed.), CRC Handbook of Chemistry and Physics, 97th ed. (CRC Press, 2016) — lattice parameters of Mg, Zn, Cd.