1 | %\subsection{The description of the target nucleus and fermi motion \editor{Gunter}} |
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2 | |
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3 | The nucleus is constructed from $A$ |
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4 | nucleons and $Z$ protons with nucleon coordinates $\mathbf{r}_i$ and momenta |
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5 | $\mathbf{p}_i$, with $i = 1,2,...,A$. |
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6 | We use a common initialization Monte Carlo procedure, which |
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7 | is realized in the most of the high energy nuclear interaction models: |
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8 | \begin{itemize} |
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9 | \item Nucleon radii $r_i$ are selected randomly in the nucleus rest frame |
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10 | according to nucleon density $\rho(r_i)$. |
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11 | For heavy nuclei with $A > 16$ \cite{GLMP91.BC} nucleon density is |
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12 | \begin{equation} |
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13 | \label{NIS1.BC}\rho(r_i) = |
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14 | \frac{\rho_0}{1 + \exp{[(r_i - R)/a]}} |
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15 | \end{equation} |
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16 | where |
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17 | \begin{equation} |
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18 | \label{NIS2.BC}\rho_0 \approx \frac{3}{4\pi R^3}(1+\frac{a^2\pi^2}{R^2})^{-1}. |
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19 | \end{equation} |
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20 | Here $R=r_0 A^{1/3}$ \ fm and $r_0=1.16(1-1.16A^{-2/3})$ \ fm and $a |
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21 | \approx 0.545$ fm. For light nuclei with $A < 17$ nucleon density is |
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22 | given by a harmonic oscillator shell model \cite{Elton61.BC}, e. g. |
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23 | \begin{equation} |
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24 | \label{4aap6.BC} \rho(r_i) = (\pi R^2)^{-3/2}\exp{(-r_i^2/R^2)}, |
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25 | \end{equation} |
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26 | where $R^2 = 2/3<r^2> = 0.8133 A^{2/3}$ \ fm$^2$. |
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27 | To take into account nucleon |
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28 | repulsive core it is assumed that internucleon distance $d > 0.8$ \ fm; |
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29 | |
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30 | \item The nucleus is assumed to be isotropic, i.e. we place each nucleon |
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31 | using a random direction and the previously determined radius $r_i$. |
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32 | |
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33 | \item The initial momenta of the nucleons $p_i$ are randomly choosen between $0$ |
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34 | and $p^{max}_F(r)$, where |
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35 | the maximal momenta of nucleons (in the local Thomas-Fermi |
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36 | approximation \cite{DF74.BC}) depends from |
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37 | the proton or neutron density $\rho$ according to |
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38 | \begin{equation} |
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39 | \label{NIS5.BC} p^{max}_F(r) = \hbar c(3\pi^2 \rho(r))^{1/3} |
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40 | \end{equation} |
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41 | |
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42 | |
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43 | \item To obtain momentum components, it is assumed that nucleons are distributed |
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44 | isotropic in momentum space; i.e. the momentum direction is chosen at random. |
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45 | |
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46 | \item The nucleus must be centered in momentum space around |
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47 | $\mathbf{0}$, \textit{i. e.} |
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48 | the nucleus must be at rest, i. e. $\sum_i {\bf p_i} = \bf 0$; |
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49 | To achieve this, we choose one nucleon to compensate the sum the remaining |
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50 | nucleon momenta $p_rest=\sum_{i=1}^{i=A-1}$. If this sum is larger than maximum |
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51 | momentum $p^{max}_F(r)$, we change the direction of the momentum of a few |
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52 | nucleons. If this does not lead to a possible momentum value, than we repeat the |
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53 | procedure with a different nucleon having a larger maximum momentum |
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54 | $p^{max}_F(r)$. In the rare case this fails as well, we choose new momenta for |
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55 | all nucleons. |
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56 | |
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57 | This procedure gives special for hydrogen $^1$H, where the proton has momentum |
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58 | $p=0$, and for deuterium $^2$H, where the momenta of proton and neutron are |
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59 | equal, and in opposite direction. |
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60 | |
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61 | |
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62 | \item We compute energy per nucleon $e = E/A = m_{N} + B(A,Z)/A$, |
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63 | where $m_N$ is nucleon mass and the nucleus binding energy $B(A,Z)$ is given |
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64 | by the tabulation of \cite{nucleus_binding}: |
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65 | and find the effective mass of each nucleon $m^{eff}_i = |
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66 | \sqrt{(E/A)^2 - p^{2\prime}_i}$. |
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67 | \end{itemize} |
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