Quantitative modeling of electron spectroscopy intensities for supported nanoparticles: The hemispherical cap model for non-normal detection

Nanoparticles of one element or compound dispersed across the surface of another substrate element or compound form the basis for many materials of great technological importance, such as heterogeneous catalysts, fuel cells and other electrocatalysts, photocatalysts, chemical sensors and biomaterials. They also form during film growth by deposition in many fabrication processes. The average size and number density of such nanoparticles are often very important, and these can be estimated with electron microscopy or scanning tunneling microscopy. However, this is very time consuming and often unavailable with sufficient resolution when the particle size is ~ 1 nm. Because the probe depth of electron spectroscopies like X-Ray Photoelectron Spectroscopy (XPS) or Auger Electron Spectroscopy (AES) is ~ 1 nm, these provide quantitative information on both the total amount of adsorbed material when it is in the form of such small nanoparticles, and the particle thickness. For electron spectroscopy conducted with electron detection normal to the surface, Diebold et al. (1993) derived analytical relationships between the signal intensities for the adsorbate and substrate and the particles' average size and number density, under the assumption that all the particles have hemispherical shape and the same radius. In this paper, we report a simple angle- and particle-size-dependent correction factor that can be applied to these analytical expressions so that they can also be extended to measurements made at other detection angles away from the surface normal. This correction factor is computed using numerical integration and presented for use in future modeling. This correction factor is large (> 2) for angles beyond 60°, so comparing model predictions to measurements at both 0° and ≥ 60° will also provide a new means for testing the model's assumptions (hemispherical shape and fixed size particles). The ability to compare the hemispherical cap model at several angles simultaneously also should enable more accurate estimates of surface structural parameters when elastic diffraction effects cause strong peaks in the angular distributions of emitted electrons.