the reciprocal unit cell

The reciprocal lattice, like any lattice, has a smallest repeating box that tiles all of reciprocal space — its unit cell. This reciprocal unit cell is the parallelepiped (a squashed box shape) whose three edges are the reciprocal basis vectors a*, b*, c*. Stack copies of this box in every direction and you rebuild the whole reciprocal lattice, exactly as stacking the real unit cell rebuilds the real crystal lattice.

Its shape and size are locked to the real unit cell by the star definitions, and the tie is an inverting one. Reciprocal edge lengths are set by a* = 1/d_100, b* = 1/d_010, c* = 1/d_001, and the reciprocal ANGLES (alpha*, beta*, gamma*) are the angles between the star vectors, not the real ones. For a cubic real cell of edge a, the reciprocal cell is a cube of edge 1/a. For non-orthogonal cells the two boxes are not simply scaled copies — a monoclinic direct cell gives a monoclinic reciprocal cell whose unique angle is the supplement of the real one. There is one tidy exception: a hexagonal cell's reciprocal is also hexagonal, but rotated 30 degrees about the c-axis.

The reciprocal unit cell matters because everything about diffraction geometry — where the spots fall, how to index them, how to compute d-spacings — is read off from a*, b*, c* and the reciprocal angles. Software that solves crystal structures works almost entirely in the reciprocal cell. One honest note: the reciprocal cell is NOT the same as the Brillouin zone. Both are 'the cell of the reciprocal lattice', but the reciprocal unit cell here is the parallelepiped on a*, b*, c*, whereas the Brillouin zone is the Wigner-Seitz (symmetric) primitive cell of the same lattice — same lattice, different choice of repeating shape.