X-ray absorption spectroscopy

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.