Number of Atoms per Unit Cell

A unit cell's corners, faces, edges and body-centre are not private to that one cell — most of these positions are

shared with neighbouring unit cells in the stacked lattice, so an atom sitting there only "belongs" fractionally

to any single cell. Working out the true number of atoms per unit cell, usually given the symbol $Z$, requires

counting each atom's fractional contribution based on how many cells share that position.

Sharing rules. A corner atom is shared among $8$ unit cells that meet at that point, so it contributes

$\tfrac{1}{8}$ of an atom to any one cell. A face-centre atom sits on the boundary shared by exactly $2$ unit

cells, so it contributes $\tfrac{1}{2}$ of an atom. An edge-centre atom is shared among $4$ unit cells, so it

contributes $\tfrac{1}{4}$ of an atom. A body-centre atom lies wholly inside a single unit cell, shared with no

other cell, so it contributes the full atom, $1$.

Simple cubic: atoms sit only at the $8$ corners. $Z = 8 \times \tfrac{1}{8} = 1$ atom per unit cell.

Body-centred cubic (bcc): atoms sit at the $8$ corners plus $1$ atom at the body centre.

$Z = \left(8 \times \tfrac{1}{8}\right) + \left(1 \times 1\right) = 1 + 1 = 2$ atoms per unit cell.

Face-centred cubic (fcc): atoms sit at the $8$ corners plus $1$ atom at the centre of each of the $6$ faces.

$Z = \left(8 \times \tfrac{1}{8}\right) + \left(6 \times \tfrac{1}{2}\right) = 1 + 3 = 4$ atoms per unit cell.

These values of $Z$ — $1$ for simple cubic, $2$ for bcc, $4$ for fcc — are not just bookkeeping; they are the

starting point for two of the most important quantitative results in this chapter: the packing efficiency of each …