deletions | additions       diff --git a/section_Introduction_The_interaction_of__.tex b/section_Introduction_The_interaction_of__.tex  index 47501aa..3973080 100644  --- a/section_Introduction_The_interaction_of__.tex  +++ b/section_Introduction_The_interaction_of__.tex 
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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 $\sim 15\%$ agreement with the \textsc{Srim} data \cite{Ziegler_2010}. Using a single formula with fewer parameters Haque {\emph et al.} \cite{Haque_2015} have reported proton impact $\mathrm{SCS}$ with encouraging results.   In the low energy region the energy loss The recent measurement \cite{Cantero_2009} by slow $\mathrm{H^+}$  in metal is $\mathrm{Cu}$ reveals the stopping  due to the conduction band electronic  excitation at lower velocities. The combined effects  of a portion of electrons around both  the Fermi level to empty states in free electrons and  the conducting band. But at higher energies, loosely bound $d$ electrons causes  a minimum momentum transfer change  of the projectile is possible due to its short duration close to the target. In slope. This study supports  this region the electronic curve has a maximum due to even upto $v = 0.01 ~\mathrm{a.u.}$ (see Figure \ref{fig:log_stopping_power}). The experimental results of Nomura and Kiyono \cite{Nomura_1975} on $\mathrm{H^+ + Cu}$ film show  the limited response time dependence  of target electrons to $\mathrm(S_\text{e})$ on incident velocity agrees with  the projectile ions. calculation of Lindhard {\emph et al} \cite{Lindhard_Scharff_Schiott}.  The recent measurement \cite{Cantero_2009} by slow $\mathrm{H^+}$ in $\mathrm{Cu}$ reveals In the low energy region  the stopping energy loss in metal is  due to conduction band electronic the  excitationat lower velocities. The combined effects  of both the free a portion of  electrons and around  the loosely bound $d$ electrons causes Fermi level to empty states in the conducting band. But at higher energies,  a change minimum momentum transfer  of the slope. This study supports projectile is possible due to its short duration close to the target. In  this even upto $v = 0.01 ~\mathrm{a.u.}$ (see Figure \ref{fig:log_stopping_power}). The experimental results of Nomura and Kiyono \cite{Nomura_1975} on $\mathrm{H^+ + Cu}$ film show region  the dependence of $\mathrm(S_\text{e})$ on incident velocity agrees with electronic curve has a maximum due to  the calculation limited response time  of Lindhard {\emph et al} \cite{Lindhard_Scharff_Schiott} target electrons to the projectile ions.  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. The suitability of this method is tested by evaluating the $\mathrm(S_\text{e})$.  Our results are compared with those of \textsc{Srim} as well as available experimental values.%