1 | \section{Simulation of pre-compound reaction} |
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2 | |
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3 | \hspace{1.0em}The precompound stage of |
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4 | nuclear reaction is considered until nuclear |
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5 | system is not an equilibrium state. |
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6 | Further emission of nuclear fragments or photons from excited |
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7 | nucleus is simulated using an equilibrium |
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8 | model. |
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9 | |
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10 | |
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11 | \subsection{Statistical equilibrium condition} |
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12 | |
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13 | \hspace{1.0em}In the state of statistical equilibrium, which is |
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14 | characterized by an eqilibrium number of excitons $n_{eq}$, all three |
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15 | type of transitions are equiprobable. Thus $n_{eq}$ is fixed by |
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16 | $\omega_{+2}(n_{eq},U) = \omega_{-2}(n_{eq},U)$. From this condition we |
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17 | can get |
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18 | \begin{equation} |
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19 | \label{PCS1}n_{eq} = \sqrt{2gU}. |
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20 | \end{equation} |
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21 | |
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22 | \subsection{Level density of excited (n-exciton) states} |
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23 | |
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24 | \hspace{1.0em}To obtain Eq. ($\ref{PCS1}$) it was assumed an equidistant |
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25 | scheme of single-particle levels with the density $g \approx 0.595 aA$, |
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26 | where $a$ is the level density parameter, when we have the level density |
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27 | of the $n$-exciton state as |
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28 | \begin{equation} |
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29 | \label{PCS2} \rho_{n}(U) = \frac{g(gU)^{n-1}}{p!h!(n-1)!}. |
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30 | \end{equation} |
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31 | |
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32 | \subsection{Transition probabilities} |
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33 | |
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34 | \hspace{1.0em}The partial transition probabilities changing the exciton |
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35 | number by $\Delta n$ is determined by the squared matrix element |
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36 | averaged over allowed transitions $<|M|^{2}>$ and the density of final |
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37 | states $\rho_{\Delta n}(n,U)$, which are really accessible in this |
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38 | transition. It can be defined as following: |
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39 | \begin{equation} |
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40 | \label{PCS3}\omega_{\Delta n}(n,U)=\frac{2\pi}{h}<|M|^{2}>\rho_{\Delta n}(n,U). |
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41 | \end{equation} |
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42 | The density of final states $\rho_{\Delta n}(n,U)$ were derived in paper |
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43 | \cite{Williams70} using the Eq. ($\ref{PCS2}$) for the level density of |
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44 | the $n$-exciton state and later corrected for the Pauli principle and |
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45 | indistinguishability of identical excitons in paper \cite{ROB73}: |
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46 | \begin{equation} |
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47 | \label{PCS4}\rho_{\Delta n = +2}(n,U)=\frac{1}{2}g\frac{[gU - F(p+1,h+1)]^2} |
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48 | {n+1} [\frac{gU - F(p+1,h+1)}{gU - F(p,h)}]^{n-1}, |
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49 | \end{equation} |
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50 | \begin{equation} |
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51 | \label{PCS5}\rho_{\Delta n = 0}(n,U)=\frac{1}{2}g\frac{[gU - F(p,h)]}{n} |
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52 | [p(p-1) + 4ph + h(h-1)] |
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53 | \end{equation} |
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54 | and |
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55 | \begin{equation} |
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56 | \label{PCS6}\rho_{\Delta n = -2}(n,U)=\frac{1}{2}gph(n-2), |
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57 | \end{equation} |
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58 | where $F(p,h)=(p^2 + h^2 + p -h)/4 - h/2$ and it was taken to be equal |
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59 | zero. To avoid calculation of the averaged squared matrix element |
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60 | $<|M|^2>$ it was assumed \cite{GMT83} that transition probability |
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61 | $\omega_{\Delta n = +2}(n,U)$ is the same as the probability for |
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62 | quasi-free scattering of a nucleon above the Fermi level on a nucleon of |
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63 | the target nucleus, i. e. |
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64 | \begin{equation} |
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65 | \label{PCS7}\omega_{\Delta n =+2}(n,U)=\frac{<\sigma(v_{rel})v_{rel}>}{V_{int}}. |
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66 | \end{equation} |
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67 | In Eq. ($\ref{PCS7}$) the interaction volume is estimated as |
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68 | $V_{int}=\frac{4}{3}\pi(2r_c + \lambda/2\pi)^3$, |
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69 | with the De Broglie wave length |
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70 | $\lambda/2\pi$ corresponding to the relative velocity |
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71 | $<v_{rel}>=\sqrt{2T_{rel}/ |
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72 | m}$, where $m$ is nucleon mass and $r_c = 0.6$ fm. |
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73 | |
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74 | The averaging in $<\sigma(v_{rel})v_{rel}>$ is further simplified by |
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75 | \begin{equation} |
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76 | \label{PCS8}<\sigma(v_{rel})v_{rel}> =<\sigma(v_{rel})><v_{rel}>. |
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77 | \end{equation} |
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78 | For $\sigma (v_{rel})$ we take approximation: |
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79 | \begin{equation} |
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80 | \label{PCS9}\sigma(v_{rel})=0.5[\sigma_{pp}(v_{rel})+\sigma_{pn}(v_{rel}]P(T_F/T_{rel}), |
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81 | \end{equation} |
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82 | where factor $P(T_F/T_{rel})$ was introduced to take into account the |
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83 | Pauli principle. It is given by |
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84 | \begin{equation} |
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85 | \label{PCS10} P(T_F/T_{rel})=1 - \frac{7}{5}\frac{T_F}{T_{rel}} |
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86 | \end{equation} |
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87 | for $\frac{T_F}{T_{rel}} \leq 0.5$ and |
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88 | \begin{equation} |
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89 | \label{PCS11} P(T_F/T_{rel})=1 - \frac{7}{5}\frac{T_F}{T_{rel}}+ \frac{2}{5} |
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90 | \frac{T_{F}}{T_{rel}}(2 - |
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91 | \frac{T_{rel}}{T_F})^{5/2} |
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92 | \end{equation} |
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93 | for $\frac{T_F}{T_{rel}} > 0.5$. |
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94 | |
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95 | The free-particle proton-proton $\sigma_{pp}(v_{rel})$ and |
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96 | proton-neutron $\sigma_{pn}(v_{rel})$ interaction cross sections are |
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97 | estimated using the equations \cite{Metro58}: |
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98 | \begin{equation} |
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99 | \label{PCS12}\sigma_{pp}(v_{rel}) = |
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100 | \frac{10.63}{v^2_{rel}}-\frac{29.93}{v_{rel}}+42.9 |
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101 | \end{equation} |
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102 | and |
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103 | \begin{equation} |
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104 | \label{PCS13}\sigma_{pn}(v_{rel}) = |
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105 | \frac{34.10}{v^2_{rel}}-\frac{82.2}{v_{rel}}+82.2, |
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106 | \end{equation} |
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107 | where cross sections are given in mbarn. |
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108 | |
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109 | The mean relative kinetic energy $T_{rel}$ is needed to calculate |
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110 | $<v_{rel}>$ and the factor $P(T_F/T_{rel})$ was computed as |
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111 | $T_{rel}=T_{p}+T_{n}$, where mean kinetic energies of projectile |
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112 | nucleons $T_p = T_F +U/n$ and target nucleons $T_N = 3T_F/5$, |
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113 | respecively. |
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114 | |
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115 | |
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116 | Combining Eqs. ($\ref{PCS3}$) - ($\ref{PCS7}$) and assuming that |
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117 | $<|M|^{2}>$ are the same for transitions with $\Delta n = 0$ and $\Delta |
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118 | n = \pm 2$ we obtain for another transition probabilities: |
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119 | \begin{equation} |
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120 | \begin{array}{c} |
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121 | \label{PCS14}\omega_{\Delta n =0}(n,U)= \\ |
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122 | =\frac{<\sigma(v_{rel})v_{rel}>}{V_{int}} |
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123 | \frac{n+1}{n}[\frac{gU - F(p,h)}{gU - F(p+1,h+1)}]^{n+1} |
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124 | \frac{p(p-1) + 4ph +h(h-1)}{gU - F(p,h)} |
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125 | \end{array} |
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126 | \end{equation} |
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127 | and |
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128 | \begin{equation} |
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129 | \begin{array}{c} |
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130 | \label{PCS15}\omega_{\Delta n |
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131 | = -2}(n,U)= \\ |
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132 | =\frac{<\sigma(v_{rel})v_{rel}>}{V_{int}} |
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133 | [\frac{gU - F(p,h)}{gU - F(p+1,h+1)}]^{n+1} |
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134 | \frac{ph(n+1)(n-2)}{[gU - F(p,h)]^2}. |
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135 | \end{array} |
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136 | \end{equation} |
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137 | |
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138 | \subsection{Emission probabilities for nucleons} |
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139 | |
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140 | \hspace{1.0em}Emission process probability has been choosen similar as |
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141 | in the classical equilibrium Weisskopf-Ewing model \cite{WE40.pre}. |
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142 | Probability to emit nucleon $b$ in the energy interval $(T_b, T_b+dT_b)$ |
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143 | is given |
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144 | \begin{equation} |
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145 | \label{PCS16}W_{b}(n,U,T_b) = \sigma_{b}(T_b)\frac{(2s_b+1)\mu_b}{\pi^2 h^3} |
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146 | R_b(p,h) |
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147 | \frac{\rho_{n-b}(E^{*})}{\rho_n(U)}T_b, |
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148 | \end{equation} |
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149 | where $\sigma_{b}(T_b)$ is the inverse (absorption of nucleon $b$) |
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150 | reaction cross section, $s_b$ and $m_b$ are nucleon spin and reduced |
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151 | mass, the factor $R_b(p,h)$ takes into account the condition for the |
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152 | exciton to be a proton or neutron, $\rho_{n-b}(E^{*})$ and $\rho_n(U)$ |
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153 | are level densities of nucleus after and before nucleon emission are |
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154 | defined in the evaporation model, respectively and $E^{*}=U-Q_b-T_b$ is the |
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155 | excitation energy of nucleus after fragment emission. |
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156 | |
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157 | \subsection{Emission probabilities for complex fragments} |
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158 | |
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159 | \hspace{1.0em}It was assumed \cite{GMT83} that nucleons inside excited |
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160 | nucleus are able to "condense" forming complex fragment. The |
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161 | "condensation" probability to create fragment consisting from $N_b$ |
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162 | nucleons inside nucleus with $A$ nucleons is given by |
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163 | \begin{equation} |
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164 | \label{PCS17} \gamma_{N_b}=N^3_b(V_b/V)^{N_b -1}=N^3_b(N_b/A)^{N_b -1}, |
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165 | \end{equation} |
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166 | where $V_b$ and $V$ are fragment $b$ and nucleus volumes, respectively. |
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167 | The last equation was estimated \cite{GMT83} as the overlap integral of |
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168 | (constant inside a volume) wave function of independent nucleons with |
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169 | that of the fragment. |
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170 | |
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171 | During the prequilibrium stage a "condense" fragment can be emitted. |
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172 | The probability to emit a fragment can be written as \cite{GMT83} |
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173 | \begin{equation} |
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174 | \label{PCS18}W_{b}(n,U,T_b) =\gamma_{N_b}R_b(p,h) |
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175 | \frac{\rho(N_b, 0, T_b + Q_b)}{g_b(T_b)} |
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176 | \sigma_{b}(T_b)\frac{(2s_b+1)\mu_b}{\pi^2 h^3} |
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177 | \frac{\rho_{n-b}(E^{*})}{\rho_n(U)}T_b, |
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178 | \end{equation} |
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179 | where |
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180 | \begin{equation} |
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181 | \label{PCS19}g_b(T_b)=\frac{V_b(2s_b+1)(2\mu_b)^{3/2}}{4\pi^2 h^3}(T_b+Q_b)^{1/2} |
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182 | \end{equation} |
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183 | is the single-particle density for complex fragment $b$, which is |
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184 | obtained by assuming that complex fragment moves inside volume $V_b$ in |
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185 | the uniform potential well whose depth is equal to be $Q_b$, and the |
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186 | factor $R_b(p,h)$ garantees correct isotopic composition of a fragment |
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187 | $b$. |
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188 | |
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189 | \subsection{The total probability} |
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190 | |
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191 | \hspace{1.0em}This probability is defined as |
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192 | \begin{equation} |
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193 | \label{PCS20} W_{tot}(n,U) =\sum_{\Delta n =+2,0,-2}\omega_{\Delta n }(n,U) + |
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194 | \sum_{b=1}^{6}W_b(n,U), |
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195 | \end{equation} |
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196 | where total emission $W_b(n,U)$ probabilities to emit fragment $b$ can |
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197 | be obtained from Eqs. ($\ref{PCS16}$) and ($\ref{PCS18}$) by |
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198 | integration over $T_b$: |
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199 | \begin{equation} |
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200 | \label{PCS21} |
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201 | W_{b}(n,U)=\int_{V_b}^{U-Q_b} W_b(n,U,T_b)dT_b. |
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202 | \end{equation} |
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203 | |
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204 | |
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205 | \subsection{Calculation of kinetic energies for emitted particle} |
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206 | |
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207 | \hspace{1.0em}The equations ($\ref{PCS16}$) and ($\ref{PCS18}$) |
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208 | are |
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209 | used to sample kinetic energies of emitted fragment. |
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210 | |
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211 | \subsection{Parameters of residual nucleus.} |
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212 | |
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213 | \hspace{1.0em}After fragment emission we update parameter |
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214 | of decaying nucleus: |
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215 | \begin{equation} |
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216 | \label{PCS24} |
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217 | \begin{array}{c} |
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218 | A_f=A-A_b; Z_f=Z-Z_b; P_f = P_0 - p_b; \\ |
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219 | E_f^{*}=\sqrt{E_f^2-\vec{P}^2_f} - M(A_f,Z_f). |
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220 | \end{array} |
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221 | \end{equation} |
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222 | Here $p_b$ is the evaporated fragment four momentum. |
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223 | |
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224 | |
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