Past exam of the frontiers of experimental condensed matter physics course of the University of Cambridge / 2013 / Paper 1 / 2 / a / Solution

In near-edge X-ray absorption fine structure, a photon promotes a core electron into an unoccupied molecular or solid-state level. Near an oxygen K-edge this is principally an O $1s$ excitation. Bound or resonant final states, orbital symmetry, local bonding and multiple scattering shape the edge. A monochromatic tunable X-ray beam can measure transmission through a thin specimen, or fluorescence or electron yield as an absorption proxy. In scanning transmission X-ray microscopy, a focused spot is scanned while transmitted intensity is recorded; energy stacks across an edge distinguish regions with different chemistry even when their thicknesses are similar. A zone plate or another suitable X-ray optic sets spatial resolution, while energy resolution and reference spectra control chemical discrimination.

The dipole transition amplitude contains $\mathbf{e}\cdot \mathbf{r}$, so the incident electric-field direction selects final-orbital orientations. For an approximately free diatomic molecule with axis $\mathbf{n}$, the axial ${\sigma }^{\ast }$ transition has strength proportional to $|\mathbf{e}\cdot \mathbf{n}{|}^2$. Summing the two perpendicular ${\pi }^{\ast }$ directions gives strength proportional to $1-|\mathbf{e}\cdot \mathbf{n}{|}^2$, with separate radial matrix-element factors for the two resonance families. This NEXAFS orientation dependence of a diatomic molecule explains the complementary polarization behavior; the molecular axis is not itself the direction of a ${\pi }^{\ast }$ orbital.

In the supplied geometry, normal incidence has electric field parallel to the surface, whereas near-grazing incidence with the illustrated polarization has a large surface-normal field component. The single-layer curves show a stronger ${\pi }^{\ast }$ resonance and much weaker ${\sigma }^{\ast }$ structure in the latter geometry. This is consistent with oxygen axes lying largely parallel to the surface: a normal electric field is perpendicular to those axes and excites ${\pi }^{\ast }$ efficiently, while suppressing the axial ${\sigma }^{\ast }$ transition. For flat molecules with random in-plane azimuths, the ideal axial factor is $1/2$ for a tangential field and zero for a normal field.

The multilayer curves instead have a strong ${\pi }^{\ast }$ peak at normal incidence and substantially stronger relative ${\sigma }^{\ast }$ response at grazing incidence. Their axes therefore have a substantial surface-normal component, consistent with the proposed tilted, more upright packing. The distinction supports a different orientation distribution in the multilayer and monolayer, rather than one isotropically oriented molecular population in both.

For illustration, a molecular polar tilt $\beta$ and random azimuth give $⟨|\mathbf{e}\cdot \mathbf{n}{|}^2⟩={\cos }^2\beta {\cos }^2\alpha +\frac{1}{2}{\sin }^2\beta {\sin }^2\alpha$, where $\alpha$ is the electric-field angle to the surface normal. The original diagram plots these ideal factors for two chosen tilts; it is not a reconstruction or fit of the supplied spectra. Quantitative tilt extraction would additionally need intensity normalization, polarization purity, azimuthal distribution and substrate-induced changes of the resonances. The spectra alone do not determine unique bond lengths or every adsorption site.