the static structure factor
When you shine X-rays, neutrons, or electrons at a crystal, the scattered waves add up only at sharp special angles, giving bright spots — the Bragg peaks. Shine the same beam at a liquid or a glass, which has no lattice, and something different happens: instead of spots you get broad, blurry rings, called diffuse halos. The static structure factor, written S(Q), is simply the shape of that scattered intensity plotted against scattering angle (packaged as the scattering vector magnitude Q). It is what the detector actually records for a disordered material, and it is the raw fingerprint of liquid or glassy structure.
S(Q) is the frequency-space partner of the radial distribution function: the two are an exact Fourier-transform pair. A liquid's S(Q) oscillates gently around the value 1: it has a first, broad principal peak at a Q corresponding roughly to 2 times pi divided by the nearest-neighbour spacing (often around 2 to 3 inverse angstrom), then weaker humps that fade out to a flat S(Q) = 1 at high Q. Compare this to a crystal, whose S(Q) is a forest of infinitely sharp spikes at the reciprocal-lattice points: the broad humps of a glass are what those sharp spikes 'melt' into when long-range order is destroyed. Because S(Q) and g(r) carry the same information, you measure S(Q) in the experiment and then Fourier-transform it to get g(r) in real space — this whole procedure is called total scattering or PDF analysis.
The practical payoff is that S(Q) is directly observable and quantitative. The position of its main peak gives the dominant interatomic spacing; the peak's width tells you how far structural correlations persist (a narrower peak means order survives to larger distances); and a distinct feature at low Q — the first sharp diffraction peak — reveals medium-range order. To extract a reliable g(r) you must measure S(Q) out to a large maximum Q, because a Fourier transform truncated too early produces spurious ripples in g(r) — a genuine, well-known artefact. An honest caveat: for a material with more than one kind of atom, a single X-ray S(Q) is a weighted blend of all the pair correlations, so untangling which atoms pair with which usually needs several experiments (X-rays plus neutrons, or isotope substitution).
A diffraction pattern from window glass shows no sharp spots — just one or two broad, ghostly rings. Plotted as intensity versus Q, those rings become the humps of S(Q): a big first hump near 2 inverse angstrom set by the silicon-oxygen network spacing, plus a weaker bump at even lower Q that reports the medium-range ring structure. Fourier-transforming the whole S(Q) recovers g(r), from which the Si-O distance and the four-fold coordination of silicon fall straight out.
Sharp Bragg spots for a crystal; broad diffuse halos, captured as S(Q), for a glass or liquid.
The broad halos of a glass are not 'no diffraction' or blurry Bragg spots from tiny crystals — they are the genuine diffraction pattern of a disordered structure. Reading them requires transforming the whole S(Q), not just measuring where a ring sits, and truncating that transform too soon fakes ripples that are not really there.