Measuring bond lengths with an interference pattern

An X-ray absorption experiment is a very simple experiment done very carefully. A monochromator — usually a pair of Si(111) or Si(220) crystals — picks a single photon energy out of a synchrotron beam, and you step that energy through the absorption edge of the element you care about while recording how much beam the sample removes.

In transmission, ion chambers before and after the sample give I₀ and I₁, and Beer–Lambert gives the absorption directly: μ(E)x = ln(I₀/I₁). For dilute samples the edge step disappears into the background, so instead you count the element's own fluorescence with an energy-discriminating detector and use μ(E) ∝ I_f/I₀. A third ion chamber and a reference foil sit downstream so that every single scan carries its own energy calibration.

The edge energy is a core-level binding energy, so it belongs to one element and one core level. That is the origin of the technique's defining virtue: you choose which atom you are standing on, and the rest of the sample is invisible.

Three regions get named, and the boundaries are conventions rather than physics. The pre-edge region is background. XANES (roughly the first 50 eV) is intense and dominated by multiple scattering and by the density of unoccupied states — it reports on oxidation state and site symmetry, and it is usually read by comparison with standards rather than fitted. EXAFS is everything above that, out to as far as the signal survives: weak, oscillatory, and quantitatively interpretable.

X-ray absorption is a transition from a core state to an unoccupied final state. Above the edge, that final state is a photoelectron travelling outward. For an isolated atom there is nothing else to say, and μ(E) decays smoothly.

Put neighbours around the absorber and the outgoing electron wave scatters off them. Part of it is backscattered and returns to the absorbing atom, where the final state is now the sum of the outgoing wave and everything that has come back. The transition matrix element depends on the amplitude of that final state at the absorber, so absorption is modulated by whether the returning wave arrives in phase or out of phase with the outgoing one.

The photoelectron's wavelength is set by its kinetic energy, so sweeping the X-ray energy sweeps the wavelength continuously. A neighbour at distance R imposes a round trip of 2R, so the interference cycles through constructive and destructive as sin(2kR + φ). That is the whole idea: a distance becomes a frequency.

Two consequences follow immediately. Because the interference is evaluated inside the tiny volume of the core orbital, EXAFS is a strictly local probe and needs no long-range order — glasses, liquids, solutions and nanoparticles are as tractable as crystals. And because the returning electron must survive the round trip, the useful range is only a few ångströms.

Since the physics is a wave phenomenon, the natural variable is not photon energy but photoelectron wavenumber. Take the photon energy above the edge as the electron's kinetic energy and

so 50 eV above the edge is k ≈ 3.6 Å⁻¹ and 1000 eV above it is k ≈ 16 Å⁻¹. Real data typically become useless somewhere between 12 and 18 Å⁻¹.

The fine structure itself is defined as the fractional modulation of the absorption:

The numerator subtracts the smooth atomic-like background; the denominator normalises by the measured edge step so that χ is per absorbing atom and independent of sample thickness and concentration.

μ₀ is not measured. It is a smooth spline drawn through the data, and choosing it is the most subjective step in the whole analysis. The standard algorithm (AUTOBK) constrains the spline so that it cannot absorb signal from above a chosen distance R_bkg, typically about 1 Å or half the first-shell distance. Set it too high and the spline eats your first shell; too low and low-frequency background survives into χ(k).

Finally, χ(k) is always plotted and fitted with a k-weight: k¹, k² or k³. The oscillations decay steeply with k, and multiplying by a power of k rebalances the plot so that the whole range contributes. It is not cosmetic: light backscatterers dominate at low k and heavy ones at high k, so fitting with several k-weights at once is a genuine discriminator between candidate structures.

A sum of sines in k, each with a frequency set by 2R, is exactly the thing a Fourier transform separates. Transforming kⁿχ(k) over a finite window gives a complex function of R whose magnitude shows a peak for each shell. This was the original insight of Sayers, Stern and Lytle in 1971, and it is what turned a curiosity into a structural technique.

Three things about that plot need to be said out loud, because every newcomer gets them wrong at least once.

• The peaks are not at the bond lengths. The phase shift φ(k) has a roughly linear part, and a linear term in the phase is a shift in the conjugate variable. Peaks appear 0.2–0.5 ångström short. The Cu–Cu bond at 2.55 Å shows up near 2.2 Å.
• It is not a radial distribution function. Peak heights are weighted by F(k), σ², λ(k) and by your choice of k-weight and window. A large peak can be four heavy atoms or twelve light disordered ones.
• Some features are artefacts of the transform. A finite k range produces truncation ripples and side lobes; that is why a window function (Hamming, Kaiser–Bessel) with a soft edge dk is applied first.

Fitting is normally done in R space, over a window that isolates the shells you intend to model, but the model itself is always built in k space and transformed identically to the data. Fitting the complex transform — real and imaginary parts, not just the magnitude — is what makes the distances precise, because the phase carries the information that the magnitude throws away.

You do not fit the EXAFS equation with free-floating F(k) and φ(k); those are computed ahead of time. You give FEFF a plausible atomic cluster, it builds self-consistent muffin-tin potentials, enumerates the scattering paths, and returns F_j(k), φ_j(k) and λ(k) for each one. The fit then adjusts a small number of physically meaningful parameters per path — typically ΔR, N (through S₀²N), σ² and a shared ΔE₀ — until the model matches the data over a chosen R window.

The single most important discipline is knowing how few parameters you are entitled to. A band-limited signal measured over a finite range contains a finite number of independent points:

For a good first-shell dataset — k from 3 to 14 Å⁻¹, R from 1 to 3 Å — that is about 14–16 independent points, depending on which additive constant you adopt. Four variables per path means you are already at the limit with three or four paths, and a fit with more free parameters than independent points is not a fit, whatever the software reports.

Two parameter pairs are strongly correlated and no amount of good data fully separates them: N with σ² (both set the amplitude, and S₀² joins the same fight), and R with ΔE₀ (both set the phase). Always look at the correlation matrix; a fit with a 0.95 correlation between N and σ² is reporting one number, not two.

The usual defences are constraints and restraints: one ΔE₀ for all paths, coordination numbers fixed by known stoichiometry, σ² tied across chemically similar paths or to an Einstein/correlated-Debye temperature, fits performed simultaneously with k¹, k² and k³ weighting, and where possible with data from two edges or several temperatures fitted together.