the Bravais lattices
/ bruh-VAY /
Take the seven box shapes of the crystal systems and ask a further question: besides putting points at the corners of the box, can we also place identical points in the middle of the box, or on its faces, and still have every point see the same surroundings? Working through this systematically yields fourteen unique arrangements, called the Bravais lattices after the French physicist Auguste Bravais.
The extra points come from centring. A primitive lattice (P) has points only at the corners. Body-centred (I) adds one point at the very centre of the box. Face-centred (F) adds a point on each of the six faces. Base-centred (C) adds points on just one opposing pair of faces. Not every centring produces something new in every system — for instance, a face-centred tetragonal cell is really a smaller body-centred tetragonal cell in disguise — so only fourteen distinct lattices survive, not seven times four.
These fourteen exhaust every possible way to fill space with a lattice of identical points, so every crystal on Earth maps onto one of them. In particular, the face-centered cubic and body-centered cubic metals correspond to the F and I cubic Bravais lattices — the two workhorses of metallurgy.
In the cubic system, three Bravais lattices exist: simple (corners only), body-centred (BCC, a point at the cube's centre), and face-centred (FCC, a point on each face). A hypothetical 'base-centred cubic' turns out to equal a smaller tetragonal cell, so it is not counted.
The three cubic Bravais lattices: P (simple), I (BCC), and F (FCC).
Only 14 combinations are distinct because some centrings simply duplicate others in a different setting. A common error is to expect 7 systems times 4 centrings = 28.