Crystal structure and lattices

Not every solid has the same kind of internal order. In a crystalline solid, atoms are arranged in a pattern that repeats regularly over long distances — metals, salts, quartz, and diamond are all crystalline. In an amorphous solid such as ordinary glass, atoms are still packed together tightly, but without that long-range repeating order; local bonding arrangements are similar from place to place, but there is no periodic pattern that extends indefinitely through the material. This distinction — long-range periodic order versus its absence — is what "crystalline" means physically.

Describing a crystal's structure precisely requires separating two ideas that are easy to conflate. The lattice is a purely mathematical object: an infinite, periodic array of points in space, generated by repeating a fixed set of translation vectors, with every lattice point having an environment identical to every other. The lattice by itself says nothing about atoms — it is a scaffold. The actual crystal is built by attaching a basis — one or more atoms, in a fixed arrangement — to every point of the lattice. A simple crystal like copper has a basis of a single atom per lattice point; other crystals, including many common minerals and semiconductor compounds, have a basis of two, several, or many atoms repeated at each lattice point, producing much more complex unit cells while the underlying lattice remains simple.

This lattice-plus-basis picture is not just bookkeeping — it is the foundation for essentially all of solid-state physics. The periodicity of the lattice is what allows electron wavefunctions in a crystal to be classified using Bloch's theorem, which in turn is what produces the allowed and forbidden energy bands (covered in the next concept) that determine whether a material is a conductor, semiconductor, or insulator. The same periodicity is also what makes X-ray and electron diffraction techniques work, since a periodic array of scattering centers produces the constructive interference patterns (Bragg diffraction) used to determine a crystal's structure experimentally in the first place.