Mean escape depth of signal photoelectrons from amorphous and polycrystalline solids

The mean escape depth (MED) of the signal photoelectrons in x-ray photoelectron spectroscopy (XPS) is a useful measure of the surface sensitivity in WS measurements. The MED has previously been given by the product of the inelastic mean free path for the signal electrons and the cosine of the photoelectron emission angle, but, due to anisotropy in photoemission and to the effects of elastic electron scattering, evaluation of MED has become more complex. The theoretical formulation presented here is based on the solution of a kinetic equation within the transport approximation. This approximation allows us to solve the boundary-value problem with simplifying assumptions for the scattering properties of atoms constituting an amorphous or polycrystalline solid. To illustrate this approach, MED values have been computed for the 2s, 3s, and 4s subshells of aluminum, silver, and gold, respectively, for a large number of possible experimental configurations, and compared with corresponding MED values obtained from Monte Carlo simulations; satisfactory agreement has been obtained. Similar comparisons have also been made for the 3p and 3d subshells of silver, again with good agreement. It has been found that the MED is strongly affected by elastic scattering of electrons on their way out of the solid. The MED Values are up to about 30% less than the values expected from the oversimplified formalism (where elastic electron-scattering effects are neglected) for near-normal emission angles although MED Values can be much larger (by up to about a factor of 2) than those from the simple theory for near-grazing emission angles or for certain other experimental configurations. These calculations show that MED values can deviate substantially (typically by +/-30% for common measurement conditions) from those expected from the simple formalism due to combined effects of elastic electron scattering and of anisotropy in the photoionization process. MED values can be calculated much more rapidly, by orders of magnitude, with the new formalism than from Monte Carlo simulations.