Edwin E. Quashie edited section_Introduction_The_interaction_of__.tex  over 8 years ago

Commit id: bdf23215b6c3241bacffc71833da0d3dab17ec25

deletions | additions      

       

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. %models employed to study stopping of elementary charged particles in solids \cite{Bloch_1933,Bethe_1930} has stimulated this kind of study.  All these models require ad-hoc assumptions for studying stopping processes. A recent review \cite{Race_2010} has full detail os theoretical developments for calculating electronic stopping in metals.  Most recently 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.  In 2012  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. It was observed that the role of both off-channeling trajectories and consideration of semicore electrons enhances the stopping power and 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.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.  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 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.%