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1 | %\subsection{Optical and phenomenological potentials \editor{Gunter}} |
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2 | |
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3 | The effect of collective nuclear elastic interaction upon primary and secondary particles |
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4 | is approximated by a nuclear potential. |
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5 | |
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6 | For projectile protons and neutrons this scalar potential |
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7 | is given by the local Fermi momentum $p_{F}(r)$ |
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8 | \begin{equation} |
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9 | \label{EQPotNucleon} |
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10 | V(r) = \frac{p_{F}^2(r)}{2 m} |
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11 | \end{equation} |
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12 | where $m$ is the mass of the neutron $m_n$ or the mass of proton $m_p$. |
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13 | |
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14 | For pions the potential is given by the lowest order optical potential |
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15 | \cite{stricker79} |
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16 | |
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17 | \begin{equation} |
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18 | \label{EQPotPion} |
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19 | V(r) = \frac{-2 \pi (\hbar c)^2 A }{ \overline{m}_\pi } ( 1 + \frac{m_\pi}{M} ) b_0 \rho(r) |
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20 | \end{equation} |
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21 | where $A$ is the nuclear mass number, |
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22 | $m_\pi$, $M$ are the pion and nucleon mass, |
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23 | $\overline{m}_\pi$ is the reduced pion mass |
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24 | $\overline{m}_\pi = (m_\pi m_N) / (m_\pi + m_N)$, with $m_N$ is the |
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25 | mass of the nucleus, and $\rho(r)$ is the nucleon density distribution. |
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26 | The parameter $b_0 $ is the effective $s-$wave scattering length and is |
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27 | obtained from analysis to pion atomic data to be about $-0.042 fm$. |
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28 | |
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29 | |
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