1 | \section{Reaction initial state simulation.} |
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2 | |
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3 | \subsection[Allowed projectiles and bombarding energy range]{Allowed projectiles and bombarding energy range for interaction with nucleon and nuclear targets} |
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4 | \hspace{1.0em} |
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5 | The GEANT4 parton string models are capable to predict final states (produced |
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6 | hadrons |
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7 | which belong to the scalar and vector meson nonets and the baryon (antibaryon) |
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8 | octet and decuplet) of reactions on nucleon and nuclear targets |
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9 | with nucleon, pion and kaon |
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10 | projectiles. The allowed bombarding energy |
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11 | $\sqrt{s} > 5$ \ GeV is recommended. |
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12 | Two approaches, based on diffractive excitation or soft scattering with |
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13 | diffractive admixture according to cross-section, are considered. |
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14 | \hspace{1.0em}Hadron-nucleus collisions in the both |
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15 | approaches (diffractive and parton exchange) are considered |
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16 | as a set of the independent hadron-nucleon collisions. |
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17 | However, the string excitation procedures |
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18 | in these approaches are rather different. |
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19 | |
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20 | \subsection{ MC initialization procedure for nucleus.} |
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21 | \hspace{1.0em}The initialization of each nucleus, consisting from $A$ |
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22 | nucleons and $Z$ protons with coordinates $\mathbf{r}_i$ and momenta |
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23 | $\mathbf{p}_i$, where $i = 1,2,...,A$ is performed. |
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24 | We use the standard initialization Monte Carlo procedure, which |
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25 | is realized in the most of the high energy nuclear interaction models: |
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26 | \begin{itemize} |
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27 | \item Nucleon radii $r_i$ are selected randomly in the rest of nucleus according |
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28 | to proton or neutron density $\rho(r_i)$. |
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29 | For heavy nuclei with $A > 16$ \cite{GLMP91} nucleon density is |
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30 | \begin{equation} |
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31 | \label{NIS1}\rho(r_i) = |
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32 | \frac{\rho_0}{1 + \exp{[(r_i - R)/a]}} |
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33 | \end{equation} |
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34 | where |
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35 | \begin{equation} |
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36 | \label{NIS2}\rho_0 \approx \frac{3}{4\pi R^3}(1+\frac{a^2\pi^2}{R^2})^{-1}. |
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37 | \end{equation} |
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38 | Here $R=r_0 A^{1/3}$ \ fm and $r_0=1.16(1-1.16A^{-2/3})$ \ fm and $a |
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39 | \approx 0.545$ fm. For light nuclei with $A < 17$ nucleon density is |
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40 | given by a harmonic oscillator shell model \cite{Elton61}, e. g. |
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41 | \begin{equation} |
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42 | \label{4aap6} \rho(r_i) = (\pi R^2)^{-3/2}\exp{(-r_i^2/R^2)}, |
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43 | \end{equation} |
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44 | where $R^2 = 2/3<r^2> = 0.8133 A^{2/3}$ \ fm$^2$. |
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45 | To take into account nucleon |
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46 | repulsive core it is assumed that internucleon distance $d > 0.8$ \ fm; |
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47 | |
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48 | \item The initial momenta of the nucleons are randomly choosen between $0$ and |
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49 | $p^{max}_F$, where |
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50 | the maximal momenta of nucleons (in the local Thomas-Fermi |
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51 | approximation \cite{DF74}) depends from |
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52 | the proton or neutron density $\rho$ according to |
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53 | \begin{equation} |
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54 | \label{NIS5} p^{max}_F = \hbar c(3\pi^2 \rho)^{1/3} |
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55 | \end{equation} |
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56 | with $\hbar c = 0.197327$ GeV fm; |
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57 | |
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58 | \item To obtain coordinate and momentum components, it |
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59 | is assumed that nucleons are distributed isotropicaly in configuration |
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60 | and momentum spaces; |
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61 | |
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62 | \item Then perform shifts of nucleon coordinates ${\bf r_j^{\prime}} |
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63 | = {\bf r_j} - 1/A \sum_i {\bf r_i}$ and momenta ${\bf p_j^{\prime}} |
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64 | = {\bf p_j} - 1/A \sum_i {\bf p_i}$ |
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65 | of nucleon momenta. The nucleus must be centered in configuration space around |
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66 | $\mathbf{0}$, \textit{i. e.} $\sum_i {\mathbf{r}_i} = \mathbf{0}$ and |
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67 | the nucleus must be at rest, i. e. $\sum_i {\bf p_i} = \bf 0$; |
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68 | |
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69 | \item We compute energy per nucleon $e = E/A = m_{N} + B(A,Z)/A$, |
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70 | where $m_N$ is nucleon mass and the nucleus binding energy $B(A,Z)$ is given |
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71 | by the Bethe-Weizs\"acker formula\cite{BM69}: |
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72 | \begin{equation} |
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73 | \begin{array}{c} |
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74 | \label{NIS6} B(A,Z) = \\ |
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75 | = -0.01587A + 0.01834A^{2/3} + 0.09286(Z- \frac{A}{2})^2 + |
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76 | 0.00071 Z^2/A^{1/3}, |
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77 | \end{array} |
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78 | \end{equation} |
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79 | and find the effective mass of each nucleon $m^{eff}_i = |
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80 | \sqrt{(E/A)^2 - p^{2\prime}_i}$. |
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81 | \end{itemize} |
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82 | |
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83 | \subsection{Random choice of the impact parameter.} |
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84 | |
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85 | \hspace{1.0em}The impact parameter $0 \leq b \leq R_t$ is randomly |
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86 | selected according to the probability: |
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87 | \begin{equation} |
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88 | \label{NIS11}P({\bf b})d{\bf b} = b d{\bf b}, |
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89 | \end{equation} |
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90 | where $R_t$ is the target radius, |
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91 | respectively. In the case of nuclear projectile or target the nuclear radius is |
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92 | determined from condition: |
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93 | \begin{equation} |
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94 | \label{NIS12}\frac{\rho(R)}{\rho(0)} = 0.01. |
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95 | \end{equation} |
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