[1208] | 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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