Electron backscattering from surfaces: Azimuth-resolved distributions

The paper develops an invariant-embedding approach to calculate various distributions of particles reflected from a solid surface. The technique is most efficient for calculations resolving at once multiple variables; in particular, it represents a powerful tool for quantitative surface analysis utilizing reflection electron energy loss spectroscopy and elastic-peak electron spectroscopy data. Based upon deterministic equations, the approach involves no numerical simulations of particle transport and, though less generally applicable, offers significant advantages over the Monte Carlo method in computing speed, accuracy, and resolution. This work generalizes our recent treatment of azimuth-integrated distributions [Phys. Rev. B 68, 155409 (2003)] to those additionally resolved in azimuthal angles. It is a striking feature of the invariant-embedding approach that resolution in the extra variable, and thus much larger extent of the resulting data, require only a moderate (typically, less than a factor of 2) increase in the associated computing time. Within ~10.3em0exmin on a PC, the scheme produces an extremely detailed description of electron backscattering for arbitrarily sophisticated scattering cross sections: various distributions (in the numbers of elastic and/or inelastic collisions, path lengths, polar and azimuthal angles, etc.) are determined for all incidence and emission directions at once. The paper outlines the analytical formalism and discusses specific features of an appropriate calculation scheme. Some of the capabilities and advantages of the present approach are illustrated by calculations of various distributions of 300 and 2000 eV electrons backscattered from Al and Au. Compared to the earlier results, the additional azimuth angle resolution and the possibility to calculate for non-normal electron incidence yield a greater variation in the observed distributions.