Close Packing and Packing Efficiency

Packing efficiency is the percentage of a unit cell's total volume that is actually occupied by its

constituent atoms, treated as hard, touching spheres of equal size. It is a direct measure of how efficiently a

given arrangement fills space — a higher packing efficiency means the spheres are packed more tightly, with less

empty (void) space between them.

Simple cubic. In this arrangement, atoms at adjacent corners touch each other along the cube's edge, so the

edge length equals two atomic radii: $a = 2r$, i.e. $r = a/2$. The volume of one atom is $\tfrac{4}{3}\pi r^3$, and

since $Z = 1$, the volume occupied by atoms in the cell is $\tfrac{4}{3}\pi\left(\tfrac{a}{2}\right)^3 = \tfrac{\pi a^3}{6}$. Packing efficiency $= \dfrac{\pi a^3/6}{a^3} \times 100 = \dfrac{\pi}{6}\times100 \approx 52.4\%$. Simple cubic packing is the least efficient of the arrangements studied here, leaving nearly half the

cell's volume empty — which is why very few elements actually crystallise this way (polonium is the standard

exception).

Body-centred cubic (bcc). Here atoms touch along the cube's body diagonal, not the edge. For a cube of edge

$a$, the body diagonal has length $\sqrt{3}\,a$, and it spans one full atom's diameter at the centre plus a radius

from each corner atom it touches: $\sqrt{3}\,a = 4r$, so $r = \dfrac{\sqrt{3}\,a}{4}$. With $Z = 2$, the volume

occupied is $2 \times \tfrac{4}{3}\pi r^3 = \tfrac{8}{3}\pi\left(\dfrac{\sqrt{3}\,a}{4}\right)^3 = \dfrac{\sqrt{3}\,\pi a^3}{8}$. Packing efficiency $= \dfrac{\sqrt{3}\pi/8}{1}\times100 \approx 68.0\%$ — noticeably

better than simple cubic, and the structure adopted by iron, chromium, and the alkali metals at room temperature.

Face-centred cubic (fcc / ccp). Here atoms touch along a face diagonal. For a cube of edge $a$, a face

diagonal has length $\sqrt{2}\,a$, spanning a corner atom, a face-centre atom, and the opposite corner atom:

$\sqrt{2}\,a = 4r$, so $r = \dfrac{\sqrt{2}\,a}{4}$. With $Z = 4$, the volume occupied is

$4\times\tfrac{4}{3}\pi r^3 = \tfrac{16}{3}\pi\left(\dfrac{\sqrt{2}\,a}{4}\right)^3 = \dfrac{\sqrt{2}\,\pi a^3}{6}$.

Packing efficiency $= \dfrac{\sqrt{2}\pi/6}{1}\times100 \approx 74.0\%$ — the highest possible packing efficiency

for identical spheres, shared by both the face-centred cubic and the closely related hexagonal close-packed (hcp)

arrangement, which is why fcc/ccp and hcp are together called the close-packed structures. Copper, silver, …