close packing of spheres
If you have ever seen oranges stacked at a market, or cannonballs in an old fort, you have seen close packing of spheres. Faced with a pile of identical balls and asked to use as little space as possible, you settle each new ball into the dimples left by the ones below. That stacking instinct is exactly how nature arranges atoms in most metals and in the anion frameworks of many ionic solids: as many neighbours as possible, with as little empty space as can be managed.
Start with one flat layer (call it A) where every sphere touches six others. The next layer (B) nestles into the dimples of A. The crucial choice comes with the third layer. If it sits directly above A again, the repeating pattern is ABAB and you get hexagonal close packing (hcp). If instead it sits over a brand-new set of dimples, the pattern is ABCABC and you get cubic close packing (ccp), which is the same arrangement as a face-centred cubic lattice. Either way each sphere ends up touching twelve neighbours (six in its own layer, three above, three below), and the spheres fill 74 percent of all space — the densest possible packing of equal spheres, a fact proved only recently as the Kepler conjecture.
Close packing matters because it gives a master template for the solid state. Most metals are simply close-packed atoms (copper, aluminium, gold are ccp; magnesium, zinc are hcp). And in ionic crystals the larger ions (usually the anions) often form a close-packed array, while the smaller ions tuck into the gaps left between them — the tetrahedral and octahedral holes. Knowing the packing and which holes are filled, and how completely, generates almost every common ionic structure type, from rock salt to spinel.
Copper metal is cubic close packing of copper atoms: each atom touches twelve others and the atoms occupy 74 percent of the space. View NaCl the same way and the chloride ions form a ccp array, with the much smaller sodium ions filling the octahedral holes between them.
The same close-packed template describes both metallic copper and the anion framework of rock salt.
hcp and ccp are equally dense (both 74 percent); the only difference is the stacking sequence (ABAB versus ABCABC). Not all solids are close-packed — body-centred cubic (e.g. CsCl, alkali metals) fills only about 68 percent.