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\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 %The  complexity of describing the dynamic interaction between charged particles and solids has initiated a large amount of both theoretical and experimental research. In particular, the condensed matter community has introduced sophisticated computer simulation techniques with considerable success for this fundamentally non-adiabatic problem \cite{Echenique_1989}.