1 | For neutron induced fission, we take first chance, second chance, third chance |
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2 | and forth chance fission into account. |
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3 | |
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4 | Neutron yields are tabulated |
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5 | as a function of both the incoming and outgoing neutron energy. |
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6 | The neutron angular distributions are either tabulated, or represented in terms |
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7 | of an expansion in legendre polynomials, similar to the angular distributions |
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8 | for neutron elastic scattering. In case no data are available on the angular |
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9 | distribution, isotropic emission in the centre of mass system of the collision |
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10 | is assumed. |
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11 | |
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12 | There are six different possibilities implemented to represent the neutron |
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13 | energy distributions. The energy distribution of the fission neutrons |
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14 | $f(E\rightarrow E')$ |
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15 | can be tabulated as a normalised |
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16 | function of the incoming and outgoing neutron energy, again using the ENDF/B-VI |
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17 | interpolation schemes to minimise data volume and maximise precision. |
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18 | |
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19 | The energy distribution can also be represented |
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20 | as a general evaporation spectrum, |
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21 | $$f(E\rightarrow E')~=~f\left(E'/\Theta(E)\right).$$ |
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22 | Here $E$ is the energy of the incoming neutron, $E'$ is the energy of a fission |
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23 | neutron, and $\Theta(E)$ is effective temperature used to characterise the |
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24 | secondary neutron energy distribution. Both the effective temperature and the |
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25 | functional behaviour of the energy distribution are taken from tabulations. |
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26 | |
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27 | Alternatively energy distribution can be represented |
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28 | as a Maxwell spectrum, $$f(E\rightarrow E')~\propto~\sqrt{E'}{\rm e}^{E'/\Theta(E)},$$ |
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29 | or a evaporation spectrum |
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30 | $$f(E\rightarrow E')~\propto~E'{\rm e}^{E'/\Theta(E)}.$$ |
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31 | In both these cases, the temperature is tabulated as a function of the incoming |
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32 | neutron energy. |
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33 | |
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34 | The last two options are the energy dependent Watt spectrum, and the Madland |
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35 | Nix spectrum. For the energy dependent Watt spectrum, the energy distribution |
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36 | is represented as |
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37 | $$f(E\rightarrow E')~\propto~{\rm e}^{-E'/a(E)}\sinh{\sqrt{b(E)E'}}.$$ |
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38 | Here both the parameters a, and b are used from tabulation as function of the |
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39 | incoming neutron energy. |
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40 | In the case of the Madland Nix spectrum, the energy distribution is described |
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41 | as |
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42 | $$f(E\rightarrow E')~=~{1\over 2}\left[g(E',<K_l>)~+~g(E',<K_h>)\right].$$ |
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43 | Here |
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44 | $$g(E',<K>)~=~ {1\over 3\sqrt{<K>\Theta}}\left[u_2^{3/2}E_1(u_2)-u_1^{3/2}E_1(u_1) |
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45 | +\gamma(3/2, u_2) - \gamma(3/2, u_1)\right],$$ |
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46 | $$ u_1(E',<K>) = {(\sqrt{E'}-\sqrt{<K>})^2 \over \Theta},~{\rm and}$$ |
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47 | $$ u_2(E',<K>) = {(\sqrt{E'}+\sqrt{<K>})^2 \over \Theta}.$$ |
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48 | Here $K_l$ is the kinetic energy of light fragments and $K_h$ the kinetic energy |
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49 | of heavy fragments, $E_1(x)$ is the exponential integral, and $\gamma(x)$ is the |
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50 | incomplete gamma function. The mean kinetic energies for light and heavy |
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51 | fragments are assumed to be energy independent. |
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52 | The temperature $\Theta$ is tabulated as a function of the kinetic |
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53 | energy of the incoming neutron. |
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54 | |
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55 | Fission photons are describes in analogy to capture photons, where evaluated |
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56 | data are available. The measured nuclear excitation levels and transition |
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57 | probabilities are used otherwise, if available. |
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58 | |
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59 | As an example of the results is shown in figure\ref{fission} the energy |
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60 | distribution of the fission neutrons in third chance fission |
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61 | of 15~MeV neutrons on Uranium ($^{238}$U). This distribution contains two |
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62 | evaporation spectra and one Watt spectrum. |
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63 | Similar comparisons for neutron yields, energy and angular distributions, and |
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64 | well as fission photon yields, energy and angular distributions have |
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65 | been performed for |
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66 | ${\rm^{238}U}$, |
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67 | ${\rm^{235}U}$, |
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68 | ${\rm^{234}U}$, and |
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69 | ${\rm^{241}Am}$ |
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70 | for a set of incoming neutron energies. |
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71 | In all cases the agreement between evaluated data and Monte Carlo is very good. |
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72 | |
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73 | \begin{figure}[b!] % fig 1 |
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74 | % \centerline{\epsfig{file=hadronic/lowEnergyNeutron/neutrons/plots/fissionu238.tc.15mev.energy.epsi,height=3.5in,width=3.5in}} |
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75 | \includegraphics[angle=0,scale=0.6]{hadronic/lowEnergyNeutron/neutrons/plots/fissionu238.tc.15mev.energy.epsi} |
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76 | \vspace{10pt} |
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77 | \caption{Comparison of data and Monte Carlo for fission neutron energy |
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78 | distributions for induced fission by 15~MeV neutrons on Uranium ($^{238}U$). |
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79 | The curve represents evaluated data and the histogram is |
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80 | the Monte Carlo prediction.} |
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81 | \label{fission} |
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82 | \end{figure} |
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83 | |
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84 | |
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