1 | \section{Fermi break-up simulation for light nuclei.} |
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2 | \hspace{1.0em} The GEANT4 Fermi break-up model is capable to predict |
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3 | final states as result of an excited nucleus with atomic number $A < 17$ |
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4 | statistical break-up. |
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5 | |
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6 | For light nuclei the values of excitation |
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7 | energy per nucleon are often comparable with nucleon binding |
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8 | energy. Thus a light excited nucleus breaks into two or more fragments |
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9 | with branching given by available phase space. To describe a process of |
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10 | nuclear disassembling the so-called Fermi break-up model is used |
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11 | \cite{Fermi50}, \cite{Kretz61}, \cite{EG67}. This statistical approach |
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12 | was first used by Fermi \cite{Fermi50} to describe the multiple |
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13 | production in high energy nucleon collision. |
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14 | |
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15 | |
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16 | |
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17 | \subsection{ Allowed channel.} |
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18 | |
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19 | \hspace{1.0em}The channel will be allowed for decay, if the total |
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20 | kinetic energy $E_{kin}$ of all fragments of the given channel at the |
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21 | moment of break-up is positive. This energy can be calculated according |
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22 | to equation: |
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23 | \begin{equation} |
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24 | \label{FBS1}E_{kin} = U+M(A,Z)-E_{Coulomb} - \sum_{b=1}^{n}(m_b+\epsilon_{b}), |
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25 | \end{equation} |
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26 | $m_{b}$ and $\epsilon_{b}$ are masses and excitation energies of fragments, |
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27 | respectively, $E_{Coulomb}$ is the Coulomb barrier for a given channel. It |
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28 | is approximated by |
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29 | \begin{equation} |
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30 | \label{FBS2}E_{Coulomb} = \frac{3}{5} \frac{e^2}{r_{0}}(1 + |
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31 | \frac{V}{V_{0}})^{-1/3} |
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32 | (\frac{Z^2}{A^{1/3}}-\sum_{b=1}^{n}\frac{Z^2}{A_b^{1/3}}), |
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33 | \end{equation} |
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34 | where $V_0$ is the volume of the system corresponding to the normal |
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35 | nuclear matter density and $\kappa = \frac{V}{V_0}$ is a parameter ( |
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36 | $\kappa = 1$ is used). |
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37 | |
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38 | \subsection{Break-up probability.} |
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39 | |
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40 | \hspace{1.0em}The total probability for nucleus to |
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41 | break-up into $n$ componets (nucleons, deutrons, tritons, alphas etc) |
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42 | in the final state is given by |
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43 | \begin{equation} |
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44 | \label{FBS3}W(E,n) = (V/\Omega)^{n-1}\rho_{n}(E), |
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45 | \end{equation} |
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46 | where $\rho_{n}(E)$ is the density of a number of final states, $V$ is |
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47 | the volume of decaying system and $\Omega = (2\pi \hbar)^{3}$ is the |
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48 | normalization volume. The density $\rho_{n}(E)$ can be defined as a |
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49 | product of three factors: |
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50 | \begin{equation} |
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51 | \label{FBS4}\rho_{n}(E)=M_{n}(E)S_nG_n. |
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52 | \end{equation} |
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53 | The first one is the phase space factor defined as |
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54 | \begin{equation} |
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55 | \label{FBS5}M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} |
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56 | \delta(\sum_{b=1}^{n} {\bf p_{b}}) \delta(E-\sum_{b=1}^{n}\sqrt{p^2+m^2_b}) |
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57 | \prod_{b=1}^{n} d^3p_b, |
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58 | \end{equation} |
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59 | where ${\bf p_b}$ is fragment $b$ momentum. The second one is the spin |
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60 | factor |
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61 | \begin{equation} |
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62 | \label{FBS6} S_n = \prod_{b=1}^{n}(2s_b+1), |
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63 | \end{equation} |
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64 | which gives the number of states with different spin orientations. The |
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65 | last one is the permutation factor |
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66 | \begin{equation} |
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67 | \label{FBS7}G_n = \prod_{j=1}^{k}\frac{1}{n_j !}, |
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68 | \end{equation} |
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69 | which takes into account identity of components in final state. $n_j$ is |
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70 | a number of components of $j$- type particles and $k$ is defined by $n = |
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71 | \sum_{j=1}^{k}n_{j}$). |
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72 | |
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73 | In non-relativistic case (Eq. ($\ref{FBS10}$) the integration in |
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74 | Eq. ($\ref{FBS5}$) can be evaluated analiticaly (see e. g. \cite{BBB58}). |
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75 | The probability for a nucleus with energy $E$ disassembling into $n$ |
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76 | fragments with masses $m_b$, where $b = 1,2,3,...,n$ equals |
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77 | \begin{equation} |
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78 | \label{FBS8} W(E_{kin},n) = |
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79 | S_nG_n (\frac{V}{\Omega})^{n-1}(\frac{1}{\sum_{b=1}^{n}m_b} |
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80 | \prod_{b=1}^{n} |
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81 | m_{b})^{3/2} |
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82 | \frac{(2\pi)^{3(n-1)/2}}{\Gamma(3(n-1)/2)}E_{kin}^{3n/2-5/2}, |
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83 | \end{equation} |
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84 | where $\Gamma(x)$ is the gamma function. |
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85 | |
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86 | \subsection{Fermi break-up model parameter.} |
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87 | |
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88 | \hspace{1.0em}Thus the Fermi break-up model has only one free parameter |
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89 | $V$ is the volume of decaying system, which can be calculated as |
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90 | follows: |
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91 | \begin{equation} |
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92 | \label{FBS9} V = 4\pi R^3/3 = 4\pi r_{0}^3 A/3, |
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93 | \end{equation} |
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94 | where $r_{0} = 1.4 $ fm is used. |
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95 | |
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96 | \subsection{ Fragment characteristics.} |
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97 | |
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98 | We take into account the formation of fragments in their ground and |
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99 | low-lying excited states, which are stable for nucleon |
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100 | emission. However, several unstable fragments with large lifetimes: |
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101 | $^{5}He$, $^{5}Li$, $^{8}Be$, $^{9}B$ etc are also considered. Fragment |
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102 | characteristics $A_b$, $Z_b$, $s_b$ and $\epsilon_b$ are taken from |
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103 | \cite{AS81}. |
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104 | |
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105 | |
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106 | \subsection{ MC procedure.} |
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107 | |
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108 | \hspace{1.0em}The nucleus break-up is described by the Monte Carlo (MC) |
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109 | procedure. We randomly (according to probability Eq. ($\ref{FBS8}$) and |
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110 | condition Eq. ($\ref{FBS1}$)) select decay channel. Then for given |
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111 | channel we calculate kinematical quantities of each fragment according |
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112 | to $n$-body phase space distribution: |
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113 | \begin{equation} |
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114 | \label{FBS10}M_{n} = \int_{-\infty}^{+\infty}...\int_{-\infty}^{+\infty} |
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115 | \delta(\sum_{b=1}^{n} {\bf p_{b}}) \delta(\sum_{b=1}^{n} |
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116 | \frac{p^2_b}{2m_b}-E_{kin}) |
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117 | \prod_{b=1}^{n} d^3p_b. |
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118 | \end{equation} |
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119 | The Kopylov's sampling procedure \cite{Kopylov70} is applied. The angular |
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120 | distributions for emitted fragments are considered to be isotropical. |
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