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Edwin E. Quashie edited section_Introduction_The_interaction_of__.tex
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Varies models and theories have been proposed to calculate stopping cross sections ($\mathrm{SCS}$); even today a unified 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 ref. there in\} show theoretical progress and we not repeating them here. The development of time dependent density functional theory (TDDFT) \cite{Runge_1984} enhanced the diverse study of many body problems. It has enjoyed much consideration owing to its electron dynamics both self-consistency and non-perturbative way \cite{Kohn_1965}.
In studying the role of radiation damage in
ion-solid $\mathrm{H^+ + Al}$ 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.
The Even today the inclusion of non adiabatic effects in
a real
calculations remains calculation poses a challenging
problem even today. Using the first principles descriptions 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.
It was observed interactions and found that
the role of both off-channeling trajectories
and consideration along with the inclusion of semicore electrons
enhances the stopping power and yields enhance $\mathrm{S_\text{e}}$ resulting 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. experiment.
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
($\sim 15\%$) ($better agreement with the experiment$) with the \textsc{Srim} data. Using a single formula with
less 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 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.
%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.
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 electronic stopping $\mathrm(S_\text{e})$.
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.
%
%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.