1 | \section{Nuclear fission cross section.} |
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2 | |
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3 | \hspace{1.0em}The probability $P_{n}^{fis}$ that fission occurs at any |
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4 | step of evaporation chain with $n$ evaporated fragments can be defined |
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5 | as |
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6 | \begin{equation} |
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7 | \label{FCS1}P_{n}^{fis} = 1-P_{n}, |
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8 | \end{equation} |
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9 | where $P_{n}$ is the probability of a transition from an excited state |
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10 | to the ground state for the nucleus only by evaporation of $n$ |
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11 | fragments. The probability $P_{n}$ can be calculated using equation: |
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12 | \begin{equation} |
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13 | \label{FCS2}P_{n}=\prod_{i=1}^{n}[1-W_{fis}(E^{*}_i,A_i,Z_i)/W_{tot}(E^{*}_i,A_i,Z_i)], |
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14 | \end{equation} |
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15 | where $W_{fis}$ fission probability (per unit time) in the Bohr and |
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16 | Wheeler theory of fission \cite{BW39}. It is assumed to be proportional |
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17 | to the level density $\rho_{fis}(T)$ at the saddle point: |
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18 | \begin{equation} |
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19 | \label{FCS3}W_{fis}=\frac{1}{2\pi \hbar \rho_c(U_c)} |
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20 | \int_{0}^{U_f-B_{fis}} |
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21 | \rho_{fis}(U_f-B_{fis}-T)dT, |
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22 | \end{equation} |
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23 | where $U_f= E^{*} - \Delta_f$ and pairing energy |
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24 | \begin{equation} |
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25 | \label{FCS3a} \Delta_{f} = \kappa \frac{14}{\sqrt{A}} \ [MeV] |
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26 | \end{equation} |
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27 | In Eq. ($\ref{FCS3}$) $B_{fis}$ is the fission barrier height. |
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28 | $W_{tot}$ |
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29 | is total decay probability (per unit time) of a nucleus: |
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30 | \begin{equation} |
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31 | \label{FCS4} W_{tot}=W_{fis}+\sum_{b=1}^{6}W_{b} |
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32 | \end{equation} |
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33 | and $W_{b}$ is the probability to evaporate fragment of type $b$. In |
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34 | the Weisskopf and Ewing theory of particle evaporation \cite{WE40}: |
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35 | \begin{equation} |
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36 | \label{FCS5}W_{b}(T_b) = \sigma_{b}(T_b)\frac{(2s_b+1)m_b}{\pi^2 \hbar^3} |
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37 | \frac{\rho_b(U_b - Q_b-T_b)}{\rho_c(U_c)}T_b, |
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38 | \end{equation} |
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39 | where $\sigma_{b}(T_b)$ is the inverse (absorption of particle $b$) |
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40 | reaction cross section, $s_b$ and $m_b$ are particle spin and mass, |
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41 | $\rho_c$ and $\rho_b$ are level densities of compound nucleus and |
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42 | nucleus after particle evaporation, respectively. |
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43 | The energies $U_b$ and $U_c$ are defined as $U_b = E^{*} - \Delta_b$ |
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44 | and $U_c = E^{*} - \Delta_c$, where $\Delta_{b,c}$ are pairing energies |
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45 | $\Delta_{Pair}$ |
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46 | of the compound and residual nuclei, respectively. |
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47 | The pairing energy $\Delta_{Pair}$ is calculated according to |
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48 | \begin{equation} |
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49 | \label{FCS5a} \Delta_{Pair} = \kappa \frac{12}{\sqrt{A}} \ [MeV] |
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50 | \end{equation} |
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51 | with $\kappa = 0$, $1$, or $2$ for odd-odd, odd-even or even-even |
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52 | nuclei, respectively. |
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53 | |
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54 | The Eq. ($\ref{FCS1}$) gives us a possibility to calculate numericaly |
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55 | the so-called fissility of nucleus $P_{fis} = |
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56 | \sigma_{fis}/\sigma_{in}$ (see e.g. \cite{ICC80}), where $\sigma_{in}$ |
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57 | is the inelastic nuclear reaction cross section and hence the fission |
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58 | cross section $\sigma_{fis}$. E.g. |
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59 | \begin{equation} |
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60 | \label{FCS2} \sigma_{fis}=\sigma_{in}P_{fis}=\sigma_{in}\frac{1}{N_{ch}} |
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61 | \sum_{n=1}^{N_{ch}}P^{fis}_{n}, |
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62 | \end{equation} |
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63 | where $N_{ch}$ is the number of fragment evaporation chains, which is |
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64 | used for averaging. |
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65 | |
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66 | As one can see from Eq. ($\ref{FCS3}$) the fission barrier height |
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67 | $B_{fis}$ and the parameter of the level density of a nucleus $a_{fis}$ |
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68 | at saddle point are the basic ingredients of model, which are necessary |
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69 | for the calculation of fission cross section. |
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