A histogram of every interatomic distance

A crystallographic refinement uses the intensity under the Bragg peaks. That intensity reports the periodic average of the structure — what you would see if every unit cell were identical. Everything that makes cells differ from one another, which is to say the local structure, scatters between the peaks, as diffuse intensity that a conventional analysis subtracts as background.

Total scattering keeps it all. You measure the entire pattern — Bragg peaks and diffuse alike — correct it properly, put it on an absolute per-atom scale, and transform it. Nothing about that procedure requires the sample to be crystalline, which is why the same analysis works on a glass, a melt, a nanoparticle suspension and a well-ordered crystal.

The one thing it does demand is range. Real-space resolution is set by how far out in momentum transfer you measured, Δr ≈ π/Q_max, and Q = 4πsinθ/λ is capped by your wavelength. This single number decides whether a source is usable:

Time-of-flight neutron instruments have a structural advantage here: detector banks at different angles each cover their own Q range, and the whole white beam contributes, so a single measurement spans a very wide Q with good counting statistics at the top end — exactly where the sharpest real-space features come from.

Stand on an atom. Count how many atom centres lie in a thin shell between r and r+dr. Divide by the volume of that shell to get a local number density ρ(r), then repeat for every atom in the sample and average. Compare the result with the bulk number density ρ₀:

That is the whole definition. G(r) is a histogram of interatomic distances, weighted by how strongly each kind of atom scatters, with the average density subtracted off so that it oscillates about zero instead of growing without bound.

Three consequences are worth memorising.

• Below the shortest bond, G(r) is a straight line. No pairs exist there, so ρ(r) = 0 and G(r) = −4πrρ₀ exactly. You know ρ₀ independently from the density, so this region is a free, absolute check on your corrections and normalisation. If the low-r region is not that line, do not refine anything yet.
• Peak positions are distances. Not shifted, not phase-corrected, not model-dependent — unlike an EXAFS Fourier transform, this axis really is r.
• Peak areas are coordination numbers — weighted ones. Integrating R(r) = rG(r) + 4πr²ρ₀ across a peak gives the number of neighbours in that shell, multiplied by the scattering weight of that pair. Getting a bare coordination number out therefore requires knowing the weights, which is the subject of two sections from now.

Because the definition never mentions symmetry, translation or a unit cell, there is no structural prerequisite whatsoever. A liquid has a pair distribution function; so does a protein solution, a 2 nm nanocrystal and a piece of quartz.

The literature contains at least six closely related real-space functions and several conventions for the reciprocal-space ones. They differ by factors of r, by whether the average density has been subtracted, and by normalisation. Papers frequently fail to say which they used. The reliable move is to check the low-r behaviour of any published curve: it identifies the function immediately.

The transform that links the two halves is a sine transform:

Note that it is Q[S(Q)−1] that gets transformed, not S(Q) itself. The factor of Q does the same job as k-weighting in EXAFS: the oscillations in S(Q) decay with Q, and multiplying by Q gives the high-Q data — which carry the sharpest real-space information — a fair share of the transform. It also means that noise at high Q is amplified, which is why counting statistics up there matter so much.

A sample with more than one element does not have one pair distribution function; it has one for every pair of element types. What you measure is a single weighted sum of them:

with c the atomic fractions and b the scattering power. The weights always sum to one. Everything that distinguishes a neutron measurement from an X-ray measurement of the same material lives in that b.

For neutrons, b is the coherent nuclear scattering length: a property of the nucleus, tabulated in femtometres, and — because the nucleus is a point on the scale of the neutron wavelength — completely independent of Q. It varies erratically from element to element, it differs between isotopes of the same element, and it can be negative. Light atoms scatter as strongly as heavy ones.

For X-rays, b becomes the atomic form factor f(Q): the Fourier transform of the atom's electron density. It equals Z at Q = 0 and falls off as Q rises. Two things follow. Heavy elements dominate, quadratically — a Pb–Pb pair outweighs an O–O pair by a factor of a hundred. And the weighting is not constant across the pattern, so the effective partial mixture drifts as Q increases; the Faber–Ziman normalisation by ⟨f(Q)⟩² removes most, not all, of that drift.

Values are the standard tabulated coherent scattering lengths (Sears, Neutron News 3, 26, 1992). Check them against the current tables before using them in anger.

The transform integral runs from zero to infinity. Your data do not. Four instrumental facts leave their fingerprints on every G(r), and recognising them is most of what separates a careful practitioner from an enthusiastic one.

• Q_max → termination ripples. Cutting the integral off abruptly convolves G(r) with a sinc function. Every peak acquires symmetric side ripples of period 2π/Q_max, and the real-space resolution is Δr ≈ π/Q_max — 0.10 Å at 30 Å⁻¹, 0.31 Å at 10. A Lorch or similar modification function suppresses the ripples by smoothing the cutoff, at the cost of broadening every peak; use it for looking, not for refining.
• Q_min → a slowly varying baseline error. The smallest measured angle is typically 0.5–1 Å⁻¹, and the missing data contribute a smooth, low-frequency error that is hard to see and easy to absorb into a spurious refined parameter.
• Instrument Q-resolution → an envelope. Finite resolution damps G(r) as exp(−r²Q_damp²/2). With Q_damp around 0.03 Å⁻¹ the signal is still usable past 50 Å; at 0.10 it is gone by 20. You measure Q_damp once by refining a standard — nickel, silicon, LaB₆ — and then hold it fixed for everything else measured that way.
• Correlated motion → sharp near-neighbour peaks. Two atoms joined by a stiff bond move together, so their separation fluctuates far less than either atom's absolute displacement would suggest. Near-neighbour peaks are therefore much narrower than independent thermal motion predicts, and the effect dies away with distance. Refinement programs model it as a δ/r correction to the peak width.

The simulator below computes the whole chain for an ideal crystal — shell distances and multiplicities from the lattice, F(Q) from the Debye sum, G(r) by numerical sine transform — separately for neutrons and for X-rays.

Two quite different strategies exist, and choosing between them is really a choice about how many parameters your data can support.

Small-box modelling — often called real-space Rietveld — refines a unit cell against G(r). You adjust lattice parameters, atomic positions, anisotropic displacement parameters, a scale factor, the correlated-motion parameter δ₁ or δ₂, and, for a nanomaterial, a particle diameter. This is what PDFgui, Diffpy-CMI and TOPAS do. It is quantitative, the parameters mean something, and the goodness of fit R_w is comparable between models of the same data.

Big-box modelling — Reverse Monte Carlo, as implemented in RMCProfile — moves tens of thousands of atoms in a large supercell until the calculated G(r), S(Q) and Bragg profile all match simultaneously. The output is an atomic configuration rather than a parameter list, and you interrogate it afterwards for bond-angle distributions, polyhedral tilts, local ordering. It has vastly more degrees of freedom than the data constrain, so the honest use of RMC is to ask what configurations are consistent with the data, never to claim it has found the structure.

The single most useful trick in the whole method: refine the same model over different r ranges. A fit over 1–10 Å reports the local structure; a fit over 20–50 Å approaches the crystallographic average. If the two disagree — a larger local lattice parameter, a distortion that only appears at short r — that disagreement is the physics, and it is precisely what Bragg analysis alone cannot see.

Sanity checks that catch most mistakes: the low-r baseline must be the straight line; the refined scale factor should be near one if your normalisation was right; Q_damp must come from a standard rather than floating freely, since it correlates strongly with displacement parameters; and a fit that looks excellent over 1–5 Å but drifts systematically above 20 usually means a wrong Q_damp or an unmodelled particle-size effect, not new physics.