basis
/ BAY-sis /
Picture a sticker album where every page has the same numbered slots, and into each slot you press the same little cluster of stickers — maybe a star, a moon, and a tiny rocket grouped just so. The slots are like the lattice points, marking where things go. The cluster of stickers you press into every slot is the basis: the actual contents, copied identically at each position.
The basis is the specific group of atoms — it might be a single atom, or two, or hundreds — that is attached to each point of the lattice. The lattice supplies the repeating addresses; the basis supplies what physically lives at each address, including how many atoms, which kinds, and their precise relative positions. The full crystal structure is then summed up by a simple recipe: structure equals lattice plus basis. Place the basis on every lattice point and the whole material appears.
The basis matters because it is where chemistry meets geometry: the same lattice with a one-atom basis might be a metal, while with a two-atom basis it becomes a compound like a salt. This is also a frequent source of confusion — a crystal with two atoms per lattice point is not the same as a lattice with twice as many points. The number of lattice points and the number of atoms are separate ideas, joined only through the basis.
In diamond, the lattice points form a face-centered cubic pattern, but the basis is two carbon atoms, not one. That two-atom basis is exactly what makes diamond's open, tetrahedral, super-strong network instead of a dense metal.
Same lattice, different basis — and a different material.
Basis here means the cluster of atoms at each lattice point, not the unrelated mathematical sense of basis as a set of coordinate vectors. The coordinate arrows that span the cell are the lattice vectors, a separate idea.