Lattices and unit cells: bookkeeping for atoms

Pour a pinch of table salt onto dark paper and look through a magnifier: nearly every grain is a crisp little cube. Nobody carved them — the shape comes from inside. The sodium and chloride ions are stacked layer upon layer along three perpendicular directions with exactly the same spacing; a grain 1 mm on a side contains about $5.6\times10^{18}$ identical cubic “bricks”. A crystal’s shape is that inner repetition magnified billions of times.

This lesson answers three questions: how can an endlessly repeating structure be described with the least information? How many atoms does one unit cell really contain? And how many neighbours surround each atom?

Two things describe any crystal. The lattice is a mathematical array of points: start from any point, move by whole multiples of three basis vectors $\mathbf a$, $\mathbf b$, $\mathbf c$, and every point you reach has identical surroundings. The basis is the small group of atoms placed at every lattice point. Lattice + basis = crystal structure.

Copper’s basis is a single Cu atom; sodium chloride’s is one $\ce{Na+}$ and one $\ce{Cl-}$; diamond’s is two carbon atoms. Yet all three share the same face-centred cubic lattice — the differences lie entirely in the basis.

The parallelepiped spanned by the three basis vectors is the unit cell, described by six lattice parameters: edge lengths $a$, $b$, $c$ and angles $\alpha$, $\beta$, $\gamma$. Copy the cell without gaps in three directions and you have the whole crystal.

Every crystal falls into one of these seven crystal systems. Allowing extra lattice points at the body centre or the face centres gives the 14 Bravais lattices. This unit works mostly with the cubic system, which has three of them: simple (P), body-centred (I) and face-centred (F).

The cell boundary slices through atoms, so counting needs bookkeeping by shares. Where an atom sits decides how many cells share it:

• corner: shared by 8 cells, 1/8 to each;
• edge: shared by 4 cells, 1/4 to each;
• face: shared by 2 cells, 1/2 to each;
• interior: owned outright, 1.

The rule holds for any parallelepiped cell, hexagonal cells included: the corner angles differ, but each corner lattice point is still shared by exactly 8 cells.

The number of atoms in the nearest shell around an atom is its coordination number (CN). The three cubic metals give very different answers:

• simple cubic: one neighbour each way along the three axes, CN = 6, at distance $a$;
• body-centred cubic: the body-centre atom is surrounded by the 8 corners, CN = 8, at half the body diagonal, $\frac{\sqrt3}{2}a$;
• face-centred cubic: every atom has 12 neighbours, at half the face diagonal, $\frac{\sqrt2}{2}a$.

The higher the coordination, the tighter the packing. Spheres fill only 52 % of a simple cubic crystal, which is why it is so rare — at ordinary pressure only polonium adopts it. Body-centred cubic fills 68 % and face-centred cubic 74 %; the next lesson shows that 74 % is the most equal spheres can manage.