Pair distribution function

The pair distribution function describes how particle density varies with distance from a reference particle. For a homogeneous material, the dimensionless function g(r) is the local pair density at separation r divided by the density expected for an ideal uniform distribution; values above one indicate enhanced separations and values below one indicate depleted separations.

In diffraction research, “PDF” often denotes a related dimensional real-space function rather than g(r) itself. Common conventions include G(r) = 4πrρ₀[g(r) − 1] and D(r) = 4πrρ₀g(r), where ρ₀ is the average number density. Because notation varies among fields and software packages, the definition and normalization must be checked before comparing results.¹

A PDF is usually obtained by Fourier transforming the suitably corrected total-scattering structure factor from reciprocal space into real space. X-ray and neutron experiments measure intensity over a range of scattering-vector magnitudes, commonly written as Q; background, absorption, multiple scattering, polarization, container, and inelastic-scattering effects must be treated before transformation.

The resulting function combines Bragg and diffuse scattering rather than retaining only sharp Bragg reflections. High maximum Q improves real-space resolution, while a wider continuous range reduces termination artifacts. Finite Q range produces ripples, and imperfect corrections can create spurious peaks or distort coordination numbers, so experimental normalization is part of the scientific interpretation rather than a merely cosmetic processing step.¹

Peak positions in a PDF identify favored interparticle distances, and their widths reflect disorder, thermal motion, static variation, instrumental resolution, and sometimes finite particle size. Integrating a properly normalized first-neighbor peak gives a coordination number, while successive peaks can reveal medium-range order that persists beyond the nearest-neighbor shell.

PDF analysis is therefore complementary to conventional crystallography. A crystal structure model can be tested against both long-range Bragg intensities and short-range PDF features, whereas an amorphous or nanocrystalline sample may be analyzed directly in real space without assuming translational periodicity. Fits can be performed over selected distance ranges: short ranges emphasize local bonding, and longer ranges test correlations and structural coherence at larger length scales.¹

The PDF is not restricted to ideal three-dimensional bulk crystals. It can characterize liquids, glasses, defects, surfaces, nanoparticles, battery materials, catalysts, and mixtures in which different local environments overlap. Neutrons can provide strong sensitivity to light elements and isotopic substitution, while X-rays generally weight pairs according to electronic structure; combining the two can separate chemically similar or structurally overlapping contributions.

Real-space fitting can also expose local structures that are invisible in an average unit-cell model, including correlated atomic displacements, disordered occupancy, nanoscale strain, and competing structural motifs. Yet a PDF peak is a distance distribution, not a unique bond assignment: different structural models may produce similar curves, and conclusions depend on density, composition, instrumental corrections, and the fitting range. Software implementations such as PDFgui helped make such comparisons accessible, but reproducible parameter definitions remain essential.

Pair distribution function

A function describing the statistical distribution of separations between pairs of particles in a material.

Total scattering

Scattering analysis that includes both Bragg and diffuse contributions.

Structure factor

A reciprocal-space quantity encoding interference from particles or atoms.

Coordination number

The average number of neighboring particles within a specified distance range.

Termination ripple

An artificial oscillation introduced when a Fourier transform is truncated at a finite maximum scattering vector.