Number of Atoms per Unit Cell
Imagine you're stacking identical cubes to build a giant crystal. Each cube is a unit cell — the smallest repeating block of the crystal. The question is: how many whole atoms "belong" to one cube?
The trick is that atoms at the corners, edges, or faces of a cube are shared with neighbouring cubes. You can't count the same atom twice. So we need a fair way to divide each atom among the cubes that touch it.
The sharing rule
An atom's contribution to one unit cell depends on where it sits:
- Corner atom: Touched by 8 cubes (4 in the layer above, 4 below). Each cube gets 81 of that atom.
- Face-centre atom: Shared by exactly 2 cubes (one on each side of the face). Each cube gets 21.
- Edge-centre atom: Shared by 4 cubes (two along the edge direction, two stacked). Each cube gets 41.
- Body-centre atom: Lies entirely inside one cube — no sharing. Contribution = 1.
Visualise a corner atom as the meeting point of 8 cubes, like the corner of a room where 8 boxes meet. Only one-eighth of that atom's volume sits inside your cube.
Counting for the three basic cubic cells
Simple Cubic (SCC) — atoms only at the 8 corners.
8 corners×81=1 atom per unit cell
Body-Centred Cubic (BCC) — 8 corners + 1 atom at the body centre.
8×81+1×1=1+1=2 atoms
Face-Centred Cubic (FCC) — 8 corners + 6 face centres.
8×81+6×21=1+3=4 atoms
A common mistake is to count a face-centre atom as 41 because it's "on a face." But a face is shared by only two cells, not four. The edge-centre atom is the one shared by four cells. …