Split a lump of rock salt with a knife and it does not crumble into random dust — it cleaves along flat, glinting faces meeting at clean right angles. Grow a quartz crystal and it comes out with smooth facets and sharp edges. Catch a snowflake and, however intricate, it is always six-fold symmetric. Cut a diamond and it splits along particular planes and no others. Why should metals, salts, gemstones and ice — utterly different materials — all share this taste for flat faces and exact angles?

Because underneath, they are all built the same way: atoms stacked in a repeating, orderly pattern that fills space. The outward geometry we can see with our eyes is a faithful echo of an invisible geometry at the atomic scale. That microscopic scaffolding is a crystal lattice, and pinning down exactly what it is — and how to name the planes and directions in it — is the foundation on which the rest of condensed matter physics is built. This page teaches that one idea: the geometry of the crystalline solid.

Crystal = lattice + basis

The single most important sentence on this page is deceptively short:

The \mathbf{a}_i are the primitive translation vectors. Starting from any lattice point and hopping by whole-number combinations of them lands you on another lattice point that looks exactly the same — same neighbours, same view in every direction. That "every point looks identical" property is the mathematical heart of a crystal, and it is what forces the flat faces and fixed angles we started with.

The basis can be a single atom (as in many pure metals), a pair (one sodium and one chlorine in table salt), or thousands of atoms (a protein crystal). But the lattice — the set of translation points — is a separate, simpler object. Keeping these two apart is the whole game, so we will draw the distinction explicitly.

The unit cell: primitive vs conventional

We never draw the whole infinite lattice. Instead we pick a small tile — a unit cell — that reproduces the entire crystal when repeated by the lattice translations. There are two flavours, and mixing them up is a classic source of factor-of-two errors:

• A primitive cell contains exactly one lattice point. It has the smallest possible volume that still tiles space. (The Wigner–Seitz cell is the tidy, symmetric choice of primitive cell.)
• A conventional cell is a larger cell chosen to show off the symmetry — usually a neat cube — even though it may contain several lattice points. We use it because a cube is far easier to think about than a squashed primitive rhombohedron.

How many lattice points sit inside a conventional cell? You cannot just count the corners, because a corner point is shared between the cells that meet there. This sharing bookkeeping is the key skill of the whole page:

The Bravais lattices — and the cubic family that matters most

How many genuinely different lattices are there? In three dimensions the answer, worked out by Auguste Bravais in 1848, is exactly 14. These 14 Bravais lattices are the only distinct ways to arrange points periodically in space so that every point has identical surroundings. They sort into seven crystal systems (cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, triclinic) with a few "centred" variants added on.

You do not need to memorise all fourteen. What you do need cold is the cubic family, because it contains the structures of most metals and many simple compounds. There are three cubic Bravais lattices, and they differ only in where the extra lattice points sit:

Notice the pattern: as we add lattice points, the coordination number (the number of nearest neighbours) climbs 6 \to 8 \to 12, and the fraction of space actually filled by hard spheres climbs 0.52 \to 0.68 \to 0.74. FCC (tied with the hexagonal close-packed structure) reaches 0.74, which Kepler conjectured in 1611 and Thomas Hales finally proved in 1998 to be the densest possible packing of identical spheres. Nature likes to pack tightly, which is why FCC and HCP metals are so common.

Worked example — atoms per cell

BCC. Eight corner atoms, each shared among 8 cells, contribute 8 \times \tfrac18 = 1 atom. The single body-centre atom sits entirely inside and contributes 1. Total:

FCC. The eight corners again give 1 atom. Each of the six face atoms is shared between 2 cells, contributing 6\times\tfrac12 = 3. Total:

These small integers — 1, 2, 4 for SC, BCC, FCC — feed straight into density calculations: multiply by the mass of one atom and divide by the cell volume a^3 to get the mass density, a favourite exam step.

Miller indices: naming planes and directions

Crystals cleave, diffract and grow along specific planes, so we need a compact way to name them. The convention is Miller indices (hkl). The recipe has one counter-intuitive step — you take reciprocals:

1. Find where the plane crosses the three axes, in units of the cell edges: intercepts at x a, y a, z a.
2. Take the reciprocals \tfrac1x, \tfrac1y, \tfrac1z.
3. Clear fractions to the smallest set of integers, and wrap in round brackets: (hkl). A negative index is written with a bar, \bar h.

Worked example. A plane cuts the axes at 2a,\ 3a,\ 1a. Reciprocals: \tfrac12, \tfrac13, \tfrac11. Put over a common denominator 6: \tfrac36, \tfrac26, \tfrac66. Drop the common denominator and read off the numerators:

The reciprocal step has a beautiful pay-off: a plane parallel to an axis never crosses it — the intercept is at infinity, and 1/\infty = 0, so a 0 in a Miller index simply means "parallel to that axis". That is why the cube faces are the clean (100), (010), (001) planes. (Directions, by contrast, are written in square brackets [hkl] and are not reciprocated — just the components of the direction vector cleared to integers.)