1 | \section{Fission process simulation.} |
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2 | |
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3 | \subsection{Atomic number distribution of fission products.} |
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4 | |
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5 | \hspace{1.0em}As follows from experimental data \cite{VH73} mass |
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6 | distribution of fission products consists of the symmetric and the |
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7 | asymmetric components: |
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8 | \begin{equation} |
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9 | \label{FPS1} F(A_f) = F_{sym}(A_f) + \omega F_{asym}(A_f), |
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10 | \end{equation} |
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11 | where $\omega(U,A,Z)$ defines relative contribution of each component |
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12 | and it depends from excitation energy $U$ and $A,Z$ of fissioning |
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13 | nucleus. It was found in \cite{ABIM93} that experimental data can be |
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14 | approximated with a good accuracy, if one take |
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15 | \begin{equation} |
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16 | \label{FPS2} F_{sym}(A_f) = \exp{[-\frac{(A_f - A_{sym})^2}{2\sigma_{sym}^2}]} |
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17 | \end{equation} |
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18 | and |
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19 | \begin{equation} |
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20 | \begin{array}{c} |
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21 | \label{FPS3} F_{asym}(A_f) = \exp{[-\frac{(A_f - A_{2})^2}{2\sigma_{2}^2}]} + |
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22 | \exp{[-\frac{{A_f - (A - A_{2})}^2}{2\sigma_{2}^2}]} + \\ |
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23 | + C_{asym}\{\exp{[-\frac{(A_f - A_{1})^2}{2\sigma_{1}^2}]} + |
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24 | \exp{[-\frac{{A_f - (A - A_{1})}^2}{2\sigma_{2}^2}]}\}, |
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25 | \end{array} |
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26 | \end{equation} |
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27 | where $A_{sym} = A/2$, $A_1$ and $A_2$ are the mean values and |
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28 | $\sigma^2_{sim}$, $\sigma^2_1$ and $\sigma^2_2$ are dispertions of the |
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29 | Gaussians respectively. From an analysis of experimental data |
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30 | \cite{ABIM93} the parameter $C_{asym} \approx 0.5$ was defined and the |
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31 | next values for dispersions: |
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32 | \begin{equation} |
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33 | \label{FPS4} \sigma^2_{sym} = \exp{(0.00553U + 2.1386)}, |
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34 | \end{equation} |
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35 | where $U$ in MeV, |
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36 | \begin{equation} |
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37 | \label{FPS5} 2\sigma_1 = \sigma_2 = 5.6 \ MeV |
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38 | \end{equation} |
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39 | for $A \leq 235$ and |
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40 | \begin{equation} |
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41 | \label{FPS6} 2\sigma_1 = \sigma_2 = 5.6 + 0.096 (A - 235) \ MeV |
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42 | \end{equation} |
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43 | for $A > 235$ were found. |
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44 | |
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45 | The weight $\omega(U,A,Z)$ was approximated as follows |
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46 | \begin{equation} |
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47 | \label{FPS7} \omega = \frac{\omega_{a} - F_{asym}(A_{sym})} |
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48 | {1 - \omega_a F_{sym}((A_1 + A_2)/2)}. |
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49 | \end{equation} |
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50 | The values of $\omega_a$ for nuclei with $96 \geq Z \geq 90$ were |
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51 | approximated by |
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52 | \begin{equation} |
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53 | \label{FPS8} \omega_a(U) = \exp{(0.538U - 9.9564)} |
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54 | \end{equation} |
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55 | for $U \leq 16.25$ MeV, |
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56 | \begin{equation} |
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57 | \label{FPS9} \omega_a(U) = \exp{(0.09197U - 2.7003)} |
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58 | \end{equation} |
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59 | for $U > 16.25$ MeV and |
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60 | \begin{equation} |
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61 | \label{FPS10} \omega_a(U) = \exp{(0.09197U - 1.08808)} |
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62 | \end{equation} |
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63 | for $z = 89$. |
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64 | For nuclei with $Z \leq 88$ the authors of \cite{ABIM93} constracted |
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65 | the following approximation: |
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66 | \begin{equation} |
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67 | \label{FPS11}\omega_a(U) = |
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68 | \exp{[0.3(227 - a)]} \exp{ \{0.09197[U - (B_{fis} - 7.5)] |
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69 | - 1.08808 \}}, |
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70 | \end{equation} |
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71 | where for $A > 227$ and $U < B_{fis} - 7.5$ the corresponding factors occuring |
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72 | in exponential functions vanish. |
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73 | |
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74 | \subsection{Charge distribution of fission products.} |
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75 | |
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76 | \hspace{1.0em}At given mass of fragment $A_f$ the |
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77 | experimental data \cite{VH73} on the charge $Z_f$ distribution of |
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78 | fragments are well approximated by Gaussian with dispertion |
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79 | $\sigma^2_{z} = 0.36$ and the average $<Z_f>$ is described by |
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80 | expression: |
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81 | \begin{equation} |
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82 | \label{FPS12} <Z_f> = \frac{A_f}{A}Z + \Delta Z, |
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83 | \end{equation} |
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84 | when parameter $\Delta Z = -0.45$ for $A_f \geq 134$, $\Delta Z = - |
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85 | 0.45(A_f -A/2)/(134 - A/2)$ for $ A - 134 < A_f < 134$ and $\Delta Z = |
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86 | 0.45$ for $A \leq A - 134$. |
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87 | |
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88 | After sampling of fragment atomic masses numbers and fragment charges, |
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89 | we have to check that fragment ground state masses do not exceed initial |
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90 | energy and calculate the maximal fragment kinetic energy |
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91 | \begin{equation} |
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92 | \label{FPS13a}T^{max} < U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}), |
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93 | \end{equation} |
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94 | where $U$ and $M(A,Z)$ are the excitation energy and mass of initial |
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95 | nucleus, $M_1(A_{f1}, |
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96 | Z_{f1})$, and $M_2(A_{f2}, Z_{f2})$ are masses |
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97 | of the first and second fragment, respectively. |
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98 | |
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99 | |
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100 | \subsection{Kinetic energy distribution of fission products.} |
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101 | |
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102 | \hspace{1.0em}We use the empiricaly defined \cite{VKW85} dependence of |
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103 | the average kinetic energy $<T_{kin}>$ (in MeV) of fission fragments on |
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104 | the mass and the charge of a fissioning nucleus: |
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105 | \begin{equation} |
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106 | \label{FPS13}<T_{kin}> = 0.1178 Z^2/A^{1/3} + 5.8. |
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107 | \end{equation} |
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108 | This energy is distributed differently in cases of symmetric and |
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109 | asymmetric modes of fission. It follows from the analysis of data |
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110 | \cite{ABIM93} that in the asymmetric mode, the average kinetic energy of |
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111 | fragments is higher than that in the symmetric one by approximately |
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112 | $12.5$ MeV. To approximate the average numbers of kinetic energies |
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113 | $<T_{kin}^{sym}$ and $<T_{kin}^{asym}>$ for the symmetric and asymmetric |
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114 | modes of fission the authors of \cite{ABIM93} suggested empirical |
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115 | expressions: |
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116 | \begin{equation} |
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117 | \label{FPS14} <T_{kin}^{sym}> = <T_{kin}> - 12.5 W_{asim}, |
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118 | \end{equation} |
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119 | \begin{equation} |
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120 | \label{FPS15} <T_{kin}^{asym}> = <T_{kin}> + 12.5 W_{sim}, |
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121 | \end{equation} |
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122 | where |
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123 | \begin{equation} |
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124 | \label{FPS16} W_{sim} = \omega \int F_{sim}(A)dA/\int F(A)dA |
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125 | \end{equation} |
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126 | and |
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127 | \begin{equation} |
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128 | \label{FPS17} W_{asim} = \int F_{asim}(A)dA/\int F(A)dA, |
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129 | \end{equation} |
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130 | respectively. In the symmetric fission the experimental data for the |
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131 | ratio of the average kinetic energy of fission fragments |
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132 | $<T_{kin}(A_f)>$ to this maximum energy $<T^{max}_{kin}>$ as a function |
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133 | of the mass of a larger fragment $A_{max}$ can be approximated by |
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134 | expressions |
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135 | \begin{equation} |
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136 | \label{FPS18} <T_{kin}(A_f)>/<T^{max}_{kin}> = |
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137 | 1 - k [(A_f - A_{max})/A]^2 |
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138 | \end{equation} |
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139 | for $A_{sim} \leq A_f \leq A_{max} + 10$ and |
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140 | \begin{equation} |
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141 | \label{FPS19} <T_{kin}(A_f)>/<T^{max}_{kin}> = |
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142 | 1 - k(10/A)^2 - 2 (10/A)k(A_f - A_{max} - 10)/A |
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143 | \end{equation} |
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144 | for $A_f > A_{max} + 10$, where $A_{max} = A_{sim}$ and $k = 5.32$ and |
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145 | $A_{max} = 134$ and $k = 23.5$ for symmetric and asymmetric fission |
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146 | respectively. For both modes of fission the distribution over the |
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147 | kinetic energy of fragments $T_{kin}$ is choosen Gaussian with their own |
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148 | average values $<T_{kin}(A_f)>= <T_{kin}^{sym}(A_f)>$ or |
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149 | $<T_{kin}(A_f)>=<T_{kin}^{asym}(A_f)>$ and dispersions $\sigma^2_{kin}$ |
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150 | equal $8^2$ MeV or $10^2$ MeV$^2$ for symmetrical and asymmetrical |
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151 | modes, respectively. |
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152 | |
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153 | \subsection{Calculation of the excitation energy of fission products.} |
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154 | |
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155 | \hspace{1.0em}The total excitation energy of fragments $U_{frag}$ |
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156 | can be defined according to equation: |
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157 | \begin{equation} |
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158 | \label{FPS21} U_{frag} = U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}) - |
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159 | T_{kin}, |
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160 | \end{equation} |
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161 | where $U$ and $M(A,Z)$ are the excitation energy and mass of initial |
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162 | nucleus, $T_{kin}$ is the fragments kinetic energy, $M_1(A_{f1}, |
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163 | Z_{f1})$, and $M_2(A_{f2}, Z_{f2})$ are masses |
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164 | of the first and second fragment, respectively. |
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165 | |
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166 | The value of excitation energy of fragment $U_f$ determines the fragment |
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167 | temperature ($T = \sqrt{U_f/a_f}$, where $a_f \sim A_f$ is the parameter |
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168 | of fragment level density). Assuming that after disintegration |
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169 | fragments have the same temperature as initial nucleus than the total |
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170 | excitation energy will be distributed between fragments in proportion to |
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171 | their mass numbers one obtains |
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172 | \begin{equation} |
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173 | \label{FPS22} U_f = U_{frag} \frac{A_f}{A}. |
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174 | \end{equation} |
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175 | |
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176 | \subsection{Excited fragment momenta.} |
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177 | |
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178 | \hspace{1.0em}Assuming that fragment kinetic energy $T_f= |
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179 | P^2_f/(2(M(A_{f},Z_{f}+U_f)$ we are |
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180 | able to calculate the absolute value of fragment c.m. momentum |
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181 | \begin{equation} |
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182 | \label{FPS23} |
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183 | P_f=\frac{(M_1(A_{f1},Z_{f1}+U_{f1})(M_2(A_{f2},Z_{f2}+U_{f2})}{ |
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184 | M_1(A_{f1},Z_{f1})+U_{f1} + M_2(A_{f2},Z_{f2})+U_{f2}}T_{kin}. |
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185 | \end{equation} |
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186 | and its components, assuming fragment isotropical distribution. |
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