deletions | additions       diff --git a/section_Introduction_The_interaction_of__.tex b/section_Introduction_The_interaction_of__.tex  index 339acc2..ad466ef 100644  --- a/section_Introduction_The_interaction_of__.tex  +++ b/section_Introduction_The_interaction_of__.tex 
...
Among the measurable quantities associated to the interaction between ions and solid, 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.   Various models and theories have been proposed to calculate stopping cross sections; even today a unified \emph{ab initio} theoretical approach suitable for different projectiles and energies is not available in the literature. Employing the First Born Approximation (FBA), Bethe \cite{Bethe_1930_EN} has reported the calculation of inelastic and ionization cross section. The Bloch correction \cite{Bloch_1933} provides a convenient link between the Bohr and the Bethe scheme. Fermi and Teller \cite{Fermi_1947} using electron gas models had reported electronic stopping for various targets. The Bethe formula for stopping has been studied in details by Lindhard and Winther \cite{Lindhard_Winther} on the basis of the generalized linear-response theory.   All these models require ad-hoc assumptions for studying stopping processes. For calculating electronic stopping in metals there are few reviews (\cite{Race_2010} and references there in).The development of time dependent density functional theory (TDDFT) \cite{Runge_1984} enhanced the diverse study of many body problems and in particular the problem at hand.   It has enjoyed much consideration owing to its electron dynamics both self-consistency and non-perturbative way \cite{Kohn_1965}.  Recently, from a phenomenological point of view, Uddin {\emph et al.} \cite{Alfaz_Uddin_2013} have calculated $S$ 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.  The development of time dependent density functional theory (TDDFT) \cite{Runge_1984} enhanced the diverse study of many body problems and in particular the problem at hand.   It has enjoyed much consideration owing to its electron dynamics both self-consistency and non-perturbative way \cite{Kohn_1965} and allowed from an atomistic \emph{ab initio} perspective.  In studying the role of radiation damage in $\mathrm{H^+ + Al}$ interactions Correa {\emph et al} \cite{Correa_2012} have shown that the electronic excitations due to molecular dynamics (MD) are quite different from the adiabatic outcome. Even today the inclusion of non adiabatic effects in a real calculation poses a challenging problem. Recently Schleife {\emph et al} \cite{Schleife_2015} have calculated the electronic stopping $\mathrm(S_\text{e})$ by $\mathrm{H}$ and $\mathrm{He}$ projectile including non-adiabatic interactions and found that off-channeling trajectories along with the inclusion of semicore electrons enhance $S_\text{e}$ $\mathrm{S_\text{e}}$  resulting better agreement with the experiment.Recently Uddin {\emph et al.} \cite{Alfaz_Uddin_2013} have calculated $\mathrm{SCS}$ 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.  The recent measurement \cite{Cantero_2009} by slow $\mathrm{H^+}$ in $\mathrm{Cu}$ reveals the stopping due to conduction band electronic excitation at lower velocity. The combined effects of both the free electrons and the loosely bound $d$ electrons causes a change of the slope. This study supports this even upto $v = 0.01 ~\mathrm{a.u.}$ (see Figure \ref{fig:log_stopping_power}). The experimental results of Nomura and Kiyota \cite{Nomura_1975} on $\mathrm{H^+ + Cu}$ film show the dependence of $\mathrm{S_\text{e}}$ on incident velocity agrees with the calculation of Lindhard {\emph et al} \cite{Lindhard_Scharff_Schiott}. In the low energy region 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 close to the target. In this region the electronic curve has a maximum due to the limited response time of 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 used for tested by  evaluating the $S_\text{e}$. $\mathrm{S_\text{e}}$.  Our results are compared with those of \textsc{Srim} as well as available experimental values.%  %  %In recent years, the development of time-dependent methods have enhanced the diverse study of many body problems involving the slowing down of charged projectiles both in 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.