[1208] | 1 | |
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| 2 | \section{Parameterized Fission Model} |
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| 3 | |
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| 4 | \subsection{Fission Process Simulation} |
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| 5 | |
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| 6 | \subsubsection{Atomic number distribution of fission products} |
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| 7 | |
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| 8 | \hspace{1.0em}As seen in experimental data \cite{VH73} the mass |
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| 9 | distribution of fission products consists of symmetric and asymmetric |
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| 10 | components: |
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| 11 | \begin{equation} |
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| 12 | \label{FPS1} F(A_f) = F_{sym}(A_f) + \omega F_{asym}(A_f), |
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| 13 | \end{equation} |
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| 14 | where $\omega(U,A,Z)$ defines the relative contribution of each component |
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| 15 | and depends on the excitation energy $U$ and $A,Z$ of the fissioning |
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| 16 | nucleus. It was found \cite{ABIM93} that experimental data can be |
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| 17 | approximated with good accuracy, if one takes |
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| 18 | \begin{equation} |
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| 19 | \label{FPS2} F_{sym}(A_f) = \exp{[-\frac{(A_f - A_{sym})^2}{2\sigma_{sym}^2}]} |
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| 20 | \end{equation} |
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| 21 | and |
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| 22 | \begin{equation} |
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| 23 | \begin{array}{c} |
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| 24 | \label{FPS3} F_{asym}(A_f) = \exp{[-\frac{(A_f - A_{2})^2}{2\sigma_{2}^2}]} + |
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| 25 | \exp{[-\frac{{A_f - (A - A_{2})}^2}{2\sigma_{2}^2}]} + \\ |
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| 26 | + C_{asym}\{\exp{[-\frac{(A_f - A_{1})^2}{2\sigma_{1}^2}]} + |
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| 27 | \exp{[-\frac{{A_f - (A - A_{1})}^2}{2\sigma_{2}^2}]}\}, |
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| 28 | \end{array} |
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| 29 | \end{equation} |
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| 30 | where $A_{sym} = A/2$, $A_1$ and $A_2$ are the mean values and |
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| 31 | $\sigma^2_{sym}$, $\sigma^2_1$ and $\sigma^2_2$ are the dispersions of the |
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| 32 | Gaussians, respectively. From an analysis of experimental data |
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| 33 | \cite{ABIM93} the parameter $C_{asym} \approx 0.5$ was defined, followed by |
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| 34 | the dispersion values: |
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| 35 | \begin{equation} |
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| 36 | \label{FPS4} \sigma^2_{sym} = \exp{(0.00553U + 2.1386)}, |
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| 37 | \end{equation} |
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| 38 | where $U$ is in MeV, |
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| 39 | \begin{equation} |
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| 40 | \label{FPS5} 2\sigma_1 = \sigma_2 = 5.6 \ MeV |
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| 41 | \end{equation} |
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| 42 | for $A \leq 235$ and |
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| 43 | \begin{equation} |
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| 44 | \label{FPS6} 2\sigma_1 = \sigma_2 = 5.6 + 0.096 (A - 235) \ MeV |
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| 45 | \end{equation} |
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| 46 | for $A > 235$ were found. |
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| 47 | |
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| 48 | The weight $\omega(U,A,Z)$ was approximated as follows |
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| 49 | \begin{equation} |
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| 50 | \label{FPS7} \omega = \frac{\omega_{a} - F_{asym}(A_{sym})} |
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| 51 | {1 - \omega_a F_{sym}((A_1 + A_2)/2)}. |
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| 52 | \end{equation} |
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| 53 | The values of $\omega_a$ for nuclei with $96 \geq Z \geq 90$ were |
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| 54 | approximated by |
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| 55 | \begin{equation} |
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| 56 | \label{FPS8} \omega_a(U) = \exp{(0.538U - 9.9564)} |
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| 57 | \end{equation} |
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| 58 | for $U \leq 16.25$ MeV, |
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| 59 | \begin{equation} |
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| 60 | \label{FPS9} \omega_a(U) = \exp{(0.09197U - 2.7003)} |
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| 61 | \end{equation} |
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| 62 | for $U > 16.25$ MeV and |
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| 63 | \begin{equation} |
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| 64 | \label{FPS10} \omega_a(U) = \exp{(0.09197U - 1.08808)} |
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| 65 | \end{equation} |
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| 66 | for $z = 89$. |
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| 67 | An approximation for nuclei with $Z \leq 88$ \cite{ABIM93} is given by: |
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| 68 | \begin{equation} |
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| 69 | \label{FPS11}\omega_a(U) = |
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| 70 | \exp{[0.3(227 - a)]} \exp{ \{0.09197[U - (B_{fis} - 7.5)] |
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| 71 | - 1.08808 \}}, |
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| 72 | \end{equation} |
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| 73 | where for $A > 227$ and $U < B_{fis} - 7.5$ the corresponding factors occurring |
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| 74 | in the exponential functions vanish. |
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| 75 | |
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| 76 | \subsubsection{Charge distribution of fission products} |
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| 77 | |
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| 78 | \hspace{1.0em} For a given mass of the fragment $A_f$ the experimental |
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| 79 | data \cite{VH73} on the charge $Z_f$ distribution of fragments are well |
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| 80 | approximated by a Gaussian with dispersion $\sigma^2_{z} = 0.36$ and the |
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| 81 | average $<Z_f>$ is described by the expression: |
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| 82 | \begin{equation} |
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| 83 | \label{FPS12} <Z_f> = \frac{A_f}{A}Z + \Delta Z, |
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| 84 | \end{equation} |
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| 85 | when parameter $\Delta Z = -0.45$ for $A_f \geq 134$, $\Delta Z = - |
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| 86 | 0.45(A_f -A/2)/(134 - A/2)$ for $ A - 134 < A_f < 134$ and $\Delta Z = |
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| 87 | 0.45$ for $A \leq A - 134$. |
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| 88 | |
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| 89 | After the sampling of fragment atomic mass numbers and fragment charges, |
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| 90 | we must check that the fragment ground state masses do not exceed the |
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| 91 | initial energy. The maximal fragment kinetic energy is calculated as |
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| 92 | \begin{equation} |
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| 93 | \label{FPS13a}T^{max} < U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}), |
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| 94 | \end{equation} |
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| 95 | where $U$ and $M(A,Z)$ are the excitation energy and mass of the initial |
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| 96 | nucleus. $M_1(A_{f1}, Z_{f1})$ and $M_2(A_{f2}, Z_{f2})$ are the masses |
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| 97 | of the first and second fragment, respectively. |
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| 98 | |
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| 99 | \subsubsection{Kinetic energy distribution of fission products} |
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| 100 | |
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| 101 | \hspace{1.0em} The average kinetic energy $<T_{kin}>$ (in MeV) of fission |
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| 102 | fragments has an empiricaly defined \cite{VKW85} dependence on the mass and |
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| 103 | charge of a fissioning nucleus: |
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| 104 | \begin{equation} |
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| 105 | \label{FPS13}<T_{kin}> = 0.1178 Z^2/A^{1/3} + 5.8. |
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| 106 | \end{equation} |
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| 107 | This energy is distributed differently in cases of symmetric and |
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| 108 | asymmetric modes of fission. It follows from the analysis of data |
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| 109 | \cite{ABIM93} that in the asymmetric mode, the average kinetic energy of |
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| 110 | fragments is higher than that in the symmetric one by approximately |
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| 111 | $12.5$ MeV. Empirical expressions have been suggested \cite{ABIM93} |
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| 112 | to approximate the average values of the kinetic energies $<T_{kin}^{sym}$ |
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| 113 | and $<T_{kin}^{asym}>$ for the symmetric and asymmetric modes of fission, |
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| 114 | \begin{equation} |
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| 115 | \label{FPS14} <T_{kin}^{sym}> = <T_{kin}> - 12.5 W_{asim}, |
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| 116 | \end{equation} |
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| 117 | \begin{equation} |
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| 118 | \label{FPS15} <T_{kin}^{asym}> = <T_{kin}> + 12.5 W_{sim}, |
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| 119 | \end{equation} |
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| 120 | where |
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| 121 | \begin{equation} |
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| 122 | \label{FPS16} W_{sim} = \omega \int F_{sim}(A)dA/\int F(A)dA |
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| 123 | \end{equation} |
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| 124 | and |
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| 125 | \begin{equation} |
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| 126 | \label{FPS17} W_{asim} = \int F_{asim}(A)dA/\int F(A)dA, |
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| 127 | \end{equation} |
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| 128 | respectively. For symmetric fission the experimental data for the |
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| 129 | ratio of the average kinetic energy of fission fragments |
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| 130 | $<T_{kin}(A_f)>$ to this maximum energy $<T^{max}_{kin}>$ as a function |
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| 131 | of the mass of a larger fragment $A_{max}$, can be approximated by the |
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| 132 | expressions |
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| 133 | \begin{equation} |
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| 134 | \label{FPS18} <T_{kin}(A_f)>/<T^{max}_{kin}> = |
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| 135 | 1 - k [(A_f - A_{max})/A]^2 |
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| 136 | \end{equation} |
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| 137 | for $A_{sim} \leq A_f \leq A_{max} + 10$ and |
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| 138 | \begin{equation} |
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| 139 | \label{FPS19} <T_{kin}(A_f)>/<T^{max}_{kin}> = |
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| 140 | 1 - k(10/A)^2 - 2 (10/A)k(A_f - A_{max} - 10)/A . |
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| 141 | \end{equation} |
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| 142 | These are valid for $A_f > A_{max} + 10$, where $A_{max} = A_{sim}$ and |
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| 143 | $k = 5.32$ for symmetric fission, and $A_{max} = 134$ and $k = 23.5$ for |
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| 144 | asymmetric fission. For both modes of fission the distribution over the |
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| 145 | kinetic energy of fragments $T_{kin}$ is chosen to be Gaussian with their |
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| 146 | own average values $<T_{kin}(A_f)>= <T_{kin}^{sym}(A_f)>$ or |
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| 147 | $<T_{kin}(A_f)>=<T_{kin}^{asym}(A_f)>$, and dispersions $\sigma^2_{kin}$ |
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| 148 | equal $8^2$ MeV and $10^2$ MeV$^2$ for symmetrical and asymmetrical |
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| 149 | modes, respectively. |
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| 150 | |
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| 151 | \subsubsection{Calculation of the excitation energy of fission products} |
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| 152 | |
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| 153 | \hspace{1.0em} The total excitation energy of fragments $U_{frag}$ |
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| 154 | can be defined according to the equation |
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| 155 | \begin{equation} |
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| 156 | \label{FPS21} U_{frag} = U + M(A,Z) - M_1(A_{f1}, Z_{f1}) - M_2(A_{f2}, Z_{f2}) - |
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| 157 | T_{kin}, |
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| 158 | \end{equation} |
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| 159 | where $U$ and $M(A,Z)$ are the excitation energy and mass of the initial |
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| 160 | nucleus, $T_{kin}$ is the fragment kinetic energy, $M_1(A_{f1}, |
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| 161 | Z_{f1})$, and $M_2(A_{f2}, Z_{f2})$ are the masses of the first and second |
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| 162 | fragments, respectively. |
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| 163 | |
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| 164 | The value of the excitation energy of fragment $U_f$ determines the fragment |
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| 165 | temperature ($T = \sqrt{U_f/a_f}$, where $a_f \sim A_f$ is the parameter |
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| 166 | of fragment level density). Assuming that after disintegration |
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| 167 | fragments have the same temperature as the initial nucleus, then the total |
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| 168 | excitation energy will be distributed between fragments in proportion to |
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| 169 | their mass numbers. One then obtains |
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| 170 | \begin{equation} |
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| 171 | \label{FPS22} U_f = U_{frag} \frac{A_f}{A}. |
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| 172 | \end{equation} |
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| 173 | |
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| 174 | \subsubsection{Excited fragment momenta} |
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| 175 | |
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| 176 | \hspace{1.0em} Assuming that the fragment kinetic energy $T_f = |
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| 177 | P^2_f/(2(M(A_{f},Z_{f}+U_f)$, we are able to calculate the absolute value |
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| 178 | of fragment c.m. momentum |
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| 179 | \begin{equation} |
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| 180 | \label{FPS23} |
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| 181 | P_f=\frac{(M_1(A_{f1},Z_{f1}+U_{f1})(M_2(A_{f2},Z_{f2}+U_{f2})}{ |
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| 182 | M_1(A_{f1},Z_{f1})+U_{f1} + M_2(A_{f2},Z_{f2})+U_{f2}}T_{kin}, |
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| 183 | \end{equation} |
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| 184 | and its components, assuming the isotropic distribution of fragments. |
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