Why Packing Efficiency and Coordination Number Are Linked

A fundamental question in solid state is: why do FCC and HCP have the highest packing efficiency (74%) and highest coordination number (12), while SC has the lowest of both (52.4%, CN = 6)?

The answer lies in geometry. In SC, each atom is surrounded by exactly 6 neighbours — one on each face of an imaginary cube centred on the atom. The maximum fraction of space that equal spheres can fill in this arrangement is only 52.4%, leaving nearly half the unit cell empty.

In BCC, the body-centre atom is in contact with 8 corner atoms (touching along body diagonal), giving CN = 8. This denser arrangement raises packing efficiency to 68%. However, it is still not close-packed because the corner atoms do not touch each other.

In FCC, every atom is in contact with 12 neighbours — 4 in the same layer, 4 in the layer above, and 4 in the layer below. The Kepler conjecture (proved in 1998) states that no arrangement of equal spheres can exceed 74.05% packing. Both FCC (ABCABC stacking) and HCP (ABAB stacking) achieve this theoretical maximum.

The local environment of each atom is identical in FCC and HCP — surrounded by 12 touching neighbours. They differ only in the stacking sequence of layers: the third close-packed layer in HCP is directly above the first (AB|AB), while in FCC/CCP it is offset from both (ABC|ABC). This difference creates cubic symmetry in CCP vs hexagonal symmetry in HCP, affecting macroscopic properties like slip planes, malleability, and thermal expansion — but not packing efficiency.

This explains why close-packed ionic structures (NaCl, ZnS) place smaller cations in voids within the FCC anion sublattice: the anions already pack at 74%, and the cations nestle in the remaining 26% void space without disturbing the packing framework.