pair-distribution-function analysis
Ordinary crystal diffraction is choosy: it listens only to the sharp Bragg peaks, which come from the perfectly repeating, long-range part of a structure — and it throws away the fuzzy 'diffuse' scattering in between the peaks as background noise. But that diffuse scattering is not noise; it carries the information about local, short-range arrangement — exactly the part that matters in a glass, a nanoparticle, a disordered or defective crystal, where there is little or no long-range order to make Bragg peaks. Pair-distribution-function analysis is the technique that keeps ALL the scattering — sharp peaks and diffuse alike — and turns it into a direct picture of the local atomic structure.
Here is how it works, in plain steps. You measure the total scattering out to high angles (which needs short-wavelength, high-energy X-rays or neutrons), then take a Fourier transform of the whole pattern. The result is a curve called the pair distribution function, written G(r): a graph whose horizontal axis is distance r between atoms and whose peaks tell you how likely you are to find a pair of atoms separated by that distance. A peak at r = 2.5 angstrom means 'lots of atom pairs sit 2.5 angstrom apart' — a nearest-neighbour bond length; the next peak is the second-neighbour distance, and so on. The height and width of each peak reveal how many neighbours there are and how much that distance varies. Crucially, the PDF works whether or not the material is crystalline: for a perfect crystal the peaks march out to large r; for a glass they fade after a few neighbour shells, quantifying exactly how far the local order extends.
PDF analysis matters because it is the way to study the structure of the 'unstructured' — materials that give few or no Bragg peaks and so defeat ordinary crystallography. It reveals the nearest-neighbour geometry in glasses and liquids, the real local structure inside nanoparticles (which is often not what their tiny, broad Bragg peaks suggest), the local distortions in disordered and defective crystals, and short-range order in complex alloys. It is honest about a subtlety many methods hide: the local structure of a material is frequently different from its average crystallographic structure — a crystal can look cubic on average yet be locally distorted — and the PDF is the tool that sees that difference, which is why it has become central to nanomaterials, batteries and disordered-materials research.
A 3-nanometre gold nanoparticle gives Bragg peaks too broad and few to solve by ordinary crystallography. Its PDF, by contrast, shows a sharp first peak at 2.88 angstrom (the gold-gold nearest-neighbour bond) and further peaks that fade out by about 3 nanometres — directly reading off both the bond length and the particle's size, from a material that looks nearly amorphous to a Bragg analysis.
PDF analysis: Fourier-transform ALL the scattering into G(r), a direct map of interatomic distances — works even without long-range order.
PDF keeps the diffuse scattering that ordinary crystallography discards, which is why it sees LOCAL structure — and the local structure often differs from the average crystallographic one (a crystal can be cubic on average yet locally distorted). Do not confuse the pair distribution function G(r) with a probability density function (also 'PDF').