A Comparison between Monte Carlo Method and the Numerical Solution of the Ambartsumian-Chandrasekhar Equations to Unravel the Dielectric Response of Metals

TL;DR

This paper compares Monte Carlo and numerical solutions of Ambartsumian-Chandrasekhar equations for modeling electron energy loss spectra in metals, highlighting similar accuracy but significantly lower computational cost for the numerical approach.

Contribution

It introduces and evaluates a numerical solution method for REEL spectra, demonstrating its efficiency and accuracy compared to the Monte Carlo method.

Findings

Numerical solution matches experimental spectra with high accuracy.

Numerical method is several orders of magnitude faster than Monte Carlo.

Both methods effectively describe bulk and surface plasmon contributions.

Abstract

In this work we describe two different models for interpreting and predicting Reflection Electron Energy Loss (REEL) spectra and we present results of a study on metallic systems comparing the computational cost and the accuracy of these techniques. These approaches are the Monte Carlo (MC) method and the Numerical Solution (NS) of the Ambartsumian-Chandrasekhr equations. The former is based on a statistical algorithm to sample the electron trajectories within the target material for describing the electron transport. The latter relies on the numerical solution of the Ambartsumian-Chandrasekhar equations using the invariant embedding method. Both methods receive the same input parameters to deal with the elastic and inelastic electron scattering. To test their respective capability to describe REEL experimental spectra, we use copper, silver, and gold as case studies. Our simulations include both bulk and surface plasmon contributions to the energy loss spectrum by using the effective electron energy loss functions and the relevant extensions to finite momenta. The agreement between MC and NS theoretical spectra with experimental data is remarkably good. Nevertheless, while we find that these approaches are comparable in accuracy, the computational cost of NS is several orders of magnitude lower than the widely used MC. Inputs, routines and data are enclosed with this manuscript via the Mendeley database.