Analytical and numerical methods for computing electron partial intensities in the case of multilayer systems

Abstract

We present two novel methods for computing energy spectra and angular distributions of electrons emitted from multi-layer solids. They are based on the Ambartsumian–Chandrasekhar (AC) equations obtained by using the invariant imbedding method. The first method is analytical and relies on a linearization of AC equations and the use of the small-angle approximation. The corresponding solution is in good agreement with that computed by using the Oswald–Kasper–Gaukler (OKG) model, which is extended to the case of layers of finite thickness. The second method is based on the discrete ordinate formalism and relies on a transformation of the AC equations to the algebraic Ricatti and Lyapunov equations, which are solved by using the backward differential formula. Unlike the previous approach, this method can handle both linear and nonlinear equations. We analyze the applicability of the proposed methods to practical problems of computing REELS spectra. To demonstrate the efficiency of the proposed methods, several computational examples are considered. Obtained numerical and analytical solutions show good agreement with the experimental data and Monte-Carlo simulations. In addition, the impact of nonlinear terms in the Ambartsumian–Chandrasekhar equations is analyzed.