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\section{Introduction}  The interaction of charged particles with matter has been a subject of extensive research over many decades; it its  findings provide precise information for many technological applications such as nuclear safety, applied material science, medical physics and fusion and fission applications \cite{Komarov_2013,Patel_2003,Caporaso_2009,Odette_2005}. When a fast ion moves through a solid, it loses kinetic energy due to the excitations of the target electrons and the path of their trajectory.   This energy-loss phenomenon plays an important role in many experimental studies involving radiation in solids, surfaces and nanostructures \cite{Chenakin_2006,Figueroa_2007,Markin_2008,Kaminsky_1965,Lehmann_1978,Sigmund_2014,Nastasi_1996}.   The complexity of describing the dynamic interaction between charged particles and solids has initiated a large amount of research both experimentally and theoretically in recent years; in the latter the theoretically, The  condensed matter community, however, have initiated sophisticated computer simulation techniques with considerable success. success\cite{Echenique_1989}.