Periodicity as a measurable signal
Diffraction responds to a spatially varying scattering density. In one dimension, model an ideal infinite lattice of spacing a as a comb of identical scattering sites:
The function f describes one site's scattering distribution. In the ideal point limit it can be a Dirac delta; a real atom has a spatial electron density. Keeping f explicit separates the lattice geometry from the scattering strength of one site.
With Fourier convention ρ̃(q) = ∫ρ(x)e−iqx dx, translation by na contributes a phase e−iqna. Repetition reinforces only those spatial frequencies for which every translated site has the same phase:
These allowed values form the reciprocal lattice. The amplitudes are modulated by f̃(q), the Fourier transform of an individual site. A motif with several atoms adds its own phase weighted sum, the structure factor; some allowed reciprocal lattice points can therefore have zero intensity.
Why this matters experimentally
An incident wave scattered through momentum transfer q accumulates phase from each site. Peaks appear when q matches a reciprocal lattice vector, subject to the scattering geometry. A finite crystal broadens peaks because a finite number of sites cannot create infinitely sharp interference. Strain, disorder and defects alter peak positions, widths and diffuse intensity.
The reciprocal lattice is thus a map of the spatial frequencies permitted by translational order. It is not a second array of atoms in physical space.