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%\subsection{Optical and phenomenological potentials \editor{Gunter}}

The effect of collective nuclear elastic interaction upon primary and secondary  particles
is approximated by a nuclear potential. 

For projectile protons and neutrons this scalar potential
is given by the local Fermi momentum $p_{F}(r)$
\begin{equation}
\label{EQPotNucleon}
V(r) = \frac{p_{F}^2(r)}{2 m}
\end{equation}
where $m$ is the mass of the neutron $m_n$ or the mass of proton $m_p$.

For pions the potential is given by the lowest order optical potential
\cite{stricker79}

\begin{equation}
\label{EQPotPion}
V(r) =  \frac{-2 \pi (\hbar c)^2 A }{ \overline{m}_\pi } ( 1 + \frac{m_\pi}{M} ) b_0 \rho(r)
\end{equation}
where $A$ is the nuclear mass number,
$m_\pi$, $M$ are the pion and nucleon mass,
$\overline{m}_\pi$ is the reduced pion mass
  $\overline{m}_\pi  = (m_\pi m_N) / (m_\pi + m_N)$, with $m_N$ is the
mass of the nucleus, and $\rho(r)$ is the nucleon density distribution.
The parameter $b_0 $ is the effective $s-$wave scattering length and is
obtained from analysis to pion atomic data to be about  $-0.042 fm$.