chapter I

What XAS measures

A core electron absorbs an X-ray. Everything else is inference.

Where a spectrum comes from

Before any analysis: a monochromator selects one energy at a time, ion chambers measure the beam before and after the sample, and is built up point by point as the energy sweeps across the edge.

open the XAS experiment demo

Drive a monochromator sweep across the Fe K-edge: ion chambers and a fluorescence detector record μ(E) live, and when the scan finishes, detector noise is propagated through a fixed background model into k³χ(k) and |χ(R)|, the quantities the rest of this course analyzes.

The edge is an element-specific ruler

Each element's core levels sit at characteristic energies, so an absorption edge identifies the element. Chemical shifts of that edge are sensitive to oxidation state and local bonding.

simulated Fe K-edge · normalized μ(E)

metal referencesample

oxidation state: +2

edge position

7114.9 eV

shift +2.9 eV vs. metal

The modeled edge shifts higher with formal oxidation. Measured chemical shifts also depend on element, ligand field, covalency, and the operational definition of E₀, so calibrate against relevant reference compounds measured in the same session. White-line intensity depends on material, edge, symmetry, and covalency; it is not a universal oxidation-state scale.

Fig. 1 | Simulated Fe K-edge XANES. Move the oxidation-state slider; the modeled edge shifts and the readout reports the chemical shift against the metal reference.

One spectrum, two regimes

Drag the cursor across the edge. Near it (XANES), the slow photoelectron scatters many times and encodes electronic structure and site symmetry. Past roughly 50 eV (EXAFS), single scattering often supplies the leading terms, while important multiple-scattering paths can remain. The natural variable becomes the wavenumber instead of energy.

normalized μ(E) · drag the cursor

energy →

122 eV

5.6 Å⁻¹

EXAFS

The faster photoelectron is often described by a path expansion led by single scattering, with multiple-scattering paths retained when important. The oscillations constrain local distances and coordination through a fitted model.

Fig. 2 | One spectrum, two regimes. Drag the energy cursor; the readouts convert E − E₀ to photoelectron wavenumber k and name the regime under the cursor.

chapter II

From measurement to χ(k)

The oscillations you fit are not measured directly. They are extracted from μ(E), and every extraction choice is a modeling choice.

Normalization and background are part of the model

Pre-edge and post-edge lines, the pick, and the background spline all shape before any fit begins. Record them like you record fit parameters, because that is what they are.

raw μ(E) · with normalization lines

normalized μ(E) · edge jump ≡ 1

The gap between the two dashed lines at E₀ becomes the normalization unit; the background spline through the EXAFS becomes the zero line of χ(k). Shift any of them and every downstream number shifts with them.

Fig. 3 | Normalization as modeling. The gap between the pre-edge and post-edge lines at E₀ becomes the unit that scales μ(E) to an edge jump of one.

The background spline decides what counts as signal

The spline through the post-edge region defines the zero line of . Its one knob, , controls the low-R minimization and therefore the spline's flexibility. Drag it too low and background leaks into the transform; too high and the spline can swallow first-shell signal.

|χ(R)| · after background removal

: 1.00 Å

background leak

30%

shell retained

100%

reasonable choice

Background is removed below Rbkg while the first shell is untouched. Rule of thumb: about half the first-shell distance.

Fig. 4 | The Rbkg compromise. Slide Rbkg to trade background leakage against eaten first-shell signal in |χ(R)|; the verdict updates with your choice.

chapter III

The EXAFS equation as an instrument

One shell, three physical parameters, three visible fingerprints.

N · neighbors → amplitudeR · distance → frequency, peak positionσ² · disorder → high-k dampingλ(k) · inelastic losses → path-length damping

Drive the equation

scales amplitude. sets the oscillation frequency and the FT peak position. kills the high-k signal. Now shrink k-max with the box window and watch ripples appear in the FT: truncation artifacts, not coordination shells. The hann window trades them for peak broadening.

k²·χ(k) · extracted signal you analyze

|FT of k²·χ(k)| · what you interpret

coordination N: 4.0distance R: 2.20 Ådisorder σ²: 0.006 Å²k-max: 12.0 Å⁻¹

window

FT peak at 1.91 Å · true R 2.20 Å

Fig. 5 | The EXAFS equation as an instrument. N, R, σ² and k-max each leave a distinct fingerprint in k³χ(k) and its Fourier transform; the window choice trades truncation ripples for peak broadening.

To keep every knob visible, this model sets , omits mean-free-path damping, and uses a linear phase . Real analysis takes , , and from a scattering calculation.

How many parameters can your data support

Using a common Stern convention, an EXAFS signal of range fit over a window contains only about independent numbers. Some software uses a +1 convention; either way, parameter counting is only a necessary first check, not a guarantee that correlations or systematics are controlled.

information budget · +2 convention

independent points in the data10.6

parameters your model asks for8

k-range: 3 to 12.0 Å⁻¹fit window ΔR: 1.5 Å

model

3 per shell (N, R, σ²) + 2 shared parameters (S₀², ΔE₀)

tight budget

Every parameter competes for the same information. Constrain something: calibrate ΔE₀, share justified parameters, or extend the usable k-range.

Fig. 6 | The information budget. N_idp = 2ΔkΔR/π + 2 counts the independent points in the data; the bars compare it with the parameters your model requests.

Read a two-shell spectrum

Everything at once: two coordination shells, resolved in -space. The scattering phase commonly shifts FT peaks below the corresponding path lengths. The display applies one uniform shift to show the direction of this effect; quantitative phase corrections are path- and species-dependent and come from the forward model.

|FT of k²·χ(k)| · simulated iron oxide

apply uniform display shift (+0.3 Å)

how to read it

1Two resolved peaks: a light first shell (Fe–O) and a heavier, more distant second shell (Fe–Fe, stronger backscatterer).
2Both peaks sit low because of the imposed linear phase. The toggle shifts the model trace; quantitative fits use path-specific phases.
3A two-shell fit asks for 8 parameters. Check that against the information budget above before believing any of them.

Fig. 7 | Reading a two-shell |χ(R)|. Peaks sit below the true path lengths; the toggle applies a uniform display shift to show the direction of the phase correction.

You can now read an EXAFS Fourier transform the way the equation does: which of its features are chemistry, which are artifacts, and how much information a fit can honestly claim.