deletions | additions       diff --git a/section_Introduction_The_study_of__.tex b/section_Introduction_The_study_of__.tex  index 2e59fc4..ab81c6e 100644  --- a/section_Introduction_The_study_of__.tex  +++ b/section_Introduction_The_study_of__.tex 
...
\section{Introduction}  Thestudy of the interaction of  charged particles particle interaction  with matter has been a subject of extensive research over many decades; it 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_2005,2005}. 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 solids, surfaces and nanostructures. The complexity of describing the dynamic interaction between charged particles and solids has initiated a large amount of research both experimentally and theoretically; theoretically in recent years;  in the latter the condensed matter community community, however,  have initiated sophisticated computer simulation techniques with great considerable  success. Among the many measurable quantity the stopping power $\mathrm(S)$ \cite{Ferrell_1977} has received much attention; it provided detailed information regarding the energy transfer between the incoming projectile and the solid target.   The theoretical models employed to study stopping of elementary charged particles in solids \cite{Bloch_1933,Bethe_1930}, \cite{Bloch_1933,Bethe_1930}  has stimulated this kind of study. In the low energy regionfor metal  the energy loss in metal  is due to the excitation of a portion of electrons around the Fermi level to empty states in the conducting band. But at higher energies, a minimum momentum transfer of the projectile is possible due to its short duration near close to  the target. In this region the electronic curve has a maximum due to the limited response time of target bound electrons to the projectile ions. In recent times, years,  the development of time-dependent methods have enhanced the diverse study of many body problems involving the slowing down of charged particles either projectiles both  in matter or solids and  gases. The time dependent density functional theory (TDDFT) on the other hand has enjoyed much consideration owing to its electron dynamics both self-consistency and non-perturbative way. Most recently Correa {\emph et al} \cite{Correa_2012} have reported the role of radiation damage in ion-solid interactions. They have shown that the electronic excitations due to molecular dynamics (MD)  are quite different from the adiabatic outcome. The inclusion of non adiabatic effects in real calculations remains a challenging problem even today. Using the first principles descriptions  Schleife {\emph et al} \cite{Schleife_2015} have calculated the electronic stopping by $\mathrm{H}$ and $\mathrm{He}$ projectile including non-adiabatic interactions employing first principles descriptions. interactions.  It was observed that the  role of both off-channeling trajectories and consideration of semicore electrons enhances the stopping power and the yields better  agreement with the experimental results. Using a quantal method based on TDDFT, Quijada {\emph et al} \cite{Quijada_2007} have studied the energy loss of protons and anti-protons moving inside metalic Al and obtained good results for the projectile-target energy transfer over a wider energy range.   Recently Uddin {\emph et al.} \cite{Alfaz_Uddin_2013} have calculated stopping cross sections for various media with atomic number $Z=2$ to $100$ using realistic electron density with four fitted parameters and obtained close agreement ($\sim 15\%$) with the \textsc{Srim} data.   We report here an application of the TDDFT that embodies a plane-wave basis set that represents accurately the electron dynamics \cite{Correa_2012,Schleife_2012,Schleife_2014} for proton impact collision of $\mathrm{Cu}$ crystal. We have tested the strength The suitability  of this method to evaluate is tested by evaluating  the electronic stopping $\mathrm(S_\text{e})$. Our findings results  are compared with those of \textsc{Srim} as well as available experimental values.