X-ray absorption spectroscopy ($\mu (E)d=-\ln [I(E)/I_0(E)]$)

X-ray absorption spectroscopy measures a material's absorption response as the incident photon energy is scanned. Near a core-electron threshold, the X-ray can excite or remove a bound electron, creating an absorption edge. The edge position and surrounding spectral structure carry information about the absorbing element and its electronic and local atomic environment.

In a transmission experiment, upstream and downstream detectors measure $I_0(E)$ and $I(E)$. For a uniform sample of thickness $d$ obeying the simple attenuation model, $\mu (E)d=-\ln [I(E)/I_0(E)]$. A synchrotron radiation source and a tunable monochromator can supply the energy scan; energy calibration, sample thickness and detector response enter the measurement model. X-ray absorption fine structure studies the variations around and beyond the edge, rather than only its threshold position.

X-ray absorption fine structure is the structured variation of an X-ray absorption spectrum near and above a core-electron edge. The excited electron interacts with the absorbing atom's environment, so the spectrum is sensitive to local electronic states and nearby atoms even when the material lacks long-range crystalline order.

Near-edge X-ray absorption fine structure probes the complicated threshold region and available electronic states. Farther above the edge, extended X-ray absorption fine structure can be interpreted using the interference between outgoing and neighboring-atom-scattered photoelectron waves. Distances affect their phases; coordination and disorder affect their amplitudes. Extracting an interatomic distance therefore requires a scattering model and phase information, rather than simply reading a raw Fourier transform peak as a bond length.

Oscillatory absorption above a core-ionization edge caused by interference between outgoing and neighbor-scattered photoelectron waves. It probes local coordination, distances and disorder. The high-energy shell expansion is distinct from the more complicated near-edge regime; phase corrections are required before interpreting Fourier-transform peaks as bond lengths.

The bond-direction variance of relative atomic displacement. A Gaussian distance distribution damps the coherent round-trip phase by $e^{-2k^2{\sigma }^2}$. Correlated atomic motion and static bond disorder contribute, so this is not generally a single atom's mean-square displacement about its lattice site.

An absorption spectrum close to a core-level ionization threshold. Polarization-dependent core-to-unoccupied-state transitions probe orbital orientation and adsorbate geometry.

Variation of absorption with incident electric-field direction. For a localized $1s\to {\pi }^{\ast }$ transition the dipole matrix element is proportional to the component of the field along the pi-orbital normal.

For a diatomic molecular axis $\mathbf{n}$ and unit electric-field polarization $\mathbf{e}$, the axial and summed two transverse dipole transition directions have the displayed geometrical factors. Radial matrix elements differ between resonances. If axes have polar tilt $\beta$ and random azimuth, with electric-field angle $\alpha$ to the surface normal, the mean axial factor is ${\cos }^2\beta {\cos }^2\alpha +\frac{1}{2}{\sin }^2\beta {\sin }^2\alpha$. Polarization-dependent near-edge absorption therefore constrains an orientation distribution, subject to substrate hybridization and instrumental normalization.