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\section{Introduction}
The interaction of charged particles with matter has been a subject of extensive research over many decades;
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, The theoretically.
In particular, the condensed matter
community, however, have initiated community has introduced sophisticated computer simulation techniques with considerable success \cite{Echenique_1989}.