Angle-resolved photoemission from crystal-field splitdshells of adsorbed atoms

A theory of the angular distribution (AD) of photoemitted electrons from filled $d$ shells of atoms adsorbed on solid surfaces is presented. The crystal field at the surface of the substrate splits the degenerate $d$ states of the adsorbate into at least $e_g$ and $t_{2g}$ components. The angular distribution is then calculated for photoemission from the $e_g$ group (since the distribution from the $t_{2g}$ group is easily related to this). The final state is written as a partial-wave sum. For photoelectron kinetic energies less than about 10 eV, transitions from $d$ to $p$ partial waves dominate the AD and these AD's are azimuthally symmetric (for unpolarized light at normal incidence). Above 10 eV, the delayed onset (due to passing over the centrifugal barrier) of $d$ to continuum $f$ partial-wave emission occurs and this channel then dominates. The $d_{x^2-y^2}$ initial state, composed of spherical harmonics $Y_{2\pm 2}$ is connected to $Y_{3\pm 3}$ and $Y_{3\pm 1}$ partial waves by the dipole operator. The calculated differential cross section, of the form $\frac{d\sigma }{d\Omega }=a\left(\theta \right)-b\left(\theta \right)cos4\phi$, is fourfold symmetric, as expected, owing to interference effects between the $m=\pm 1$ and $m=\pm 3$ partial waves. The anisotropy parameter $\alpha \left(\theta \right)\equiv \frac{b\left(\theta \right)}{a\left(\theta \right)}$ changes sign at $\theta =63.43\mathrm{}°$ and this manifests itself as a 45° azimuthal rotation of the fourfold pattern. Specific systems for studying this effect experimentally are discussed. The deposition of Cu, Ag, or Au on a wide-band-gap insulator such as LiF appears promising as LiF should provide a large crystal field and the noble-metal $d$ states should fall within the gap, thus remaining sharp and resolvable. The importance of the partial-wave interferences in other angle-resolved photoemission studies of oriented atoms, molecules, and surfaces is noted.