1 | |
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2 | \section{Fission with improved multiplicity sampling} |
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3 | |
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4 | As an alternative to the fission model described in the previous |
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5 | section there is a modified model that produces more accurate |
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6 | multiplicity distributions for the emitted neutrons and gamma rays |
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7 | from spontaneous and neutron-induced fission. This was motivated by |
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8 | detailed statistical studies of fission chains in multiplying |
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9 | media. This model is data-driven and incorporates all available |
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10 | multiplicity measurements found in the literature. Empirical models |
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11 | are employed whenever multiplicity data are not available. |
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12 | Essentially no data are available for the correlations between the |
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13 | neutrons and gammas, so this model samples these distributions |
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14 | independently. By default, this model effectively scales the |
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15 | multiplicity data to match the average multiplicity value |
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16 | ($\bar{\nu}$) found in the GEANT4 evaluated data library. Therefore, |
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17 | only isotopes that have a measured $\bar{\nu}$ in the data library |
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18 | will emit fission gammas or neutrons. At present the gammas and |
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19 | neutrons are emitted isotropically. The data and empirical models are |
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20 | described in detail in the following subsections. |
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21 | |
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22 | %\section{Neutrons emitted by fission\label{neutrons}} |
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23 | |
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24 | \subsection{Neutron number distribution} |
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25 | |
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26 | Based on reasonable assumptions about the distribution of excitation |
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27 | energy among fission fragments, Terrell~\cite{Terrell 1957} showed |
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28 | that the probability P$_\nu$ of observing $\nu$ neutrons from fission |
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29 | can be approximated by a Gaussian-like distribution |
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30 | \begin{equation} |
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31 | \sum_{n=0}^{\nu}P_n = \frac{1}{2\pi}\int_{-\infty}^{\frac{\nu-\bar{\nu} |
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32 | + \frac{1}{2}+b}{\sigma}}e^{-\frac{t^2}{2}dt} |
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33 | \end{equation} |
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34 | where $\bar{\nu}$ is the average number of neutrons, $\sigma$ (set to |
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35 | 1.079) is the width of the distribution, and $b$ is a small correction |
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36 | factor ($b<0.01$) that ensures that the discrete probability |
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37 | distribution has the correct average $\bar{\nu}$. This model is used |
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38 | when no explicit multiplicity data are available. |
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39 | |
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40 | \subsubsection*{Neutron-induced fission data} |
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41 | |
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42 | Zucker and Holden~\cite{Zucker and Holden 1986} measured the neutron |
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43 | multiplicity distributions for $^{235}$U, $^{238}$U, and $^{239}$Pu |
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44 | (see Tables~\ref{Neutron number distribution for induced fission in 235U} |
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45 | -\ref{Neutron number distribution for induced fission in 239Pu (continued)}), |
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46 | as a function of the incident neutron energy $E_n$ from |
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47 | zero through ten MeV in increments of one MeV. Fig.~\ref{235U induced |
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48 | fission 6MeV} shows the neutron number distribution for induced |
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49 | fission of $^{235}$U. Gwin, Spencer and Ingle~\cite{Gwin 1984} |
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50 | measured the distribution at thermal energies for $^{235}$U. In |
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51 | addition, there are many measurements of $\bar{\nu}$, the average |
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52 | number of emitted neutrons, for many isotopes. Since there are multiple |
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53 | methods for parameterizing the multiplicity data and |
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54 | renormalizing the overall distributions to agree with the specific measured |
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55 | values of $\bar{\nu}$, we provide four options for generating |
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56 | neutron multiplicity distributions. |
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57 | %% \notgeant{These options are selected by the internal variable {\tt nudist}, default=3.} |
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58 | |
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59 | \begin{table}[ht] |
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60 | \footnotesize |
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61 | \begin{center} |
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62 | \begin{tabular}{|c|ccccccc|} \hline |
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63 | $E_n$ & $\nu$=0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline |
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64 | 0 & .0317223 & .1717071 & .3361991 & .3039695 & .1269459 & .0266793 & .0026322 \\ |
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65 | 1 & .0237898 & .1555525 & .3216515 & .3150433 & .1444732 & .0356013 & .0034339 \\ |
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66 | 2 & .0183989 & .1384891 & .3062123 & .3217566 & .1628673 & .0455972 & .0055694 \\ |
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67 | 3 & .0141460 & .1194839 & .2883075 & .3266568 & .1836014 & .0569113 & .0089426 \\ |
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68 | 4 & .0115208 & .1032624 & .2716849 & .3283426 & .2021206 & .0674456 & .0128924 \\ |
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69 | 5 & .0078498 & .0802010 & .2456595 & .3308175 & .2291646 & .0836912 & .0187016 \\ |
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70 | 6 & .0046272 & .0563321 & .2132296 & .3290407 & .2599806 & .1045974 & .0265604 \\ |
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71 | 7 & .0024659 & .0360957 & .1788634 & .3210507 & .2892537 & .1282576 & .0360887 \\ |
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72 | 8 & .0012702 & .0216090 & .1472227 & .3083032 & .3123950 & .1522540 & .0462449 \\ |
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73 | 9 & .0007288 & .0134879 & .1231200 & .2949390 & .3258251 & .1731879 & .0551737 \\ |
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74 | 10& .0004373 & .0080115 & .1002329 & .2779283 & .3342611 & .1966100 & .0650090 \\ \hline |
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75 | \end{tabular} |
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76 | \end{center} |
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77 | \caption{Neutron number distribution for induced fission in $^{235}$U.} |
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78 | \label{Neutron number distribution for induced fission in 235U} |
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79 | \end{table} |
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80 | |
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81 | \begin{table}[ht] |
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82 | \footnotesize |
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83 | \begin{center} |
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84 | \begin{tabular}{|c|c|c|} \hline |
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85 | $E_n$ & $\nu$=7 & $\bar{\nu}$ \\ \hline |
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86 | 0 & .0001449 & 2.4140000 \\ |
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87 | 1 & .0004546 & 2.5236700 \\ |
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88 | 2 & .0011093 & 2.6368200 \\ |
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89 | 3 & .0019504 & 2.7623400 \\ |
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90 | 4 & .0027307 & 2.8738400 \\ |
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91 | 5 & .0039148 & 3.0386999 \\ |
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92 | 6 & .0056322 & 3.2316099 \\ |
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93 | 7 & .0079244 & 3.4272800 \\ |
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94 | 8 & .0107009 & 3.6041900 \\ |
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95 | 9 & .0135376 & 3.7395900 \\ |
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96 | 10& .0175099 & 3.8749800 \\ \hline |
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97 | \end{tabular} |
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98 | \end{center} |
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99 | \caption{Neutron number distribution for induced fission in $^{235}$U (continued).} |
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100 | \label{Neutron number distribution for induced fission in 235U (continued)} |
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101 | \end{table} |
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102 | |
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103 | \begin{table}[ht] |
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104 | \footnotesize |
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105 | \begin{center} |
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106 | \begin{tabular}{|c|ccccccc|} \hline |
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107 | $E_n$ & $\nu$=0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline |
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108 | 0 & .0396484 & .2529541 & .2939544 & .2644470 & .1111758 & .0312261 & .0059347 \\ |
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109 | 1 & .0299076 & .2043215 & .2995886 & .2914889 & .1301480 & .0363119 & .0073638 \\ |
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110 | 2 & .0226651 & .1624020 & .2957263 & .3119098 & .1528786 & .0434233 & .0097473 \\ |
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111 | 3 & .0170253 & .1272992 & .2840540 & .3260192 & .1779579 & .0526575 & .0130997 \\ |
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112 | 4 & .0124932 & .0984797 & .2661875 & .3344938 & .2040116 & .0640468 & .0173837 \\ |
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113 | 5 & .0088167 & .0751744 & .2436570 & .3379711 & .2297901 & .0775971 & .0225619 \\ |
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114 | 6 & .0058736 & .0565985 & .2179252 & .3368863 & .2541575 & .0933127 & .0286200 \\ |
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115 | 7 & .0035997 & .0420460 & .1904095 & .3314575 & .2760413 & .1112075 & .0355683 \\ |
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116 | 8 & .0019495 & .0309087 & .1625055 & .3217392 & .2943792 & .1313074 & .0434347 \\ |
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117 | 9 & .0008767 & .0226587 & .1356058 & .3076919 & .3080816 & .1536446 & .0522549 \\ |
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118 | 10& .0003271 & .0168184 & .1111114 & .2892434 & .3160166 & .1782484 & .0620617 \\ \hline |
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119 | \end{tabular} |
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120 | \end{center} |
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121 | \caption{Neutron number distribution for induced fission in $^{238}$U.} |
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122 | \label{Neutron number distribution for induced fission in 238U} |
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123 | \end{table} |
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124 | |
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125 | \begin{table}[ht] |
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126 | \footnotesize |
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127 | \begin{center} |
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128 | \begin{tabular}{|c|cc|c|} \hline |
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129 | $E_n$ & $\nu$=7 & 8 & $\bar{\nu}$ \\ \hline |
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130 | 0 & .0005436 & .0001158 & 2.2753781 \\ |
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131 | 1 & .0006947 & .0001751 & 2.4305631 \\ |
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132 | 2 & .0009318 & .0003159 & 2.5857481 \\ |
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133 | 3 & .0013467 & .0005405 & 2.7409331 \\ |
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134 | 4 & .0020308 & .0008730 & 2.8961181 \\ |
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135 | 5 & .0030689 & .0013626 & 3.0513031 \\ |
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136 | 6 & .0045431 & .0031316 & 3.2064881 \\ |
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137 | 7 & .0065387 & .0031316 & 3.3616731 \\ |
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138 | 8 & .0091474 & .0046284 & 3.5168581 \\ |
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139 | 9 & .0124682 & .0067176 & 3.6720432 \\ |
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140 | 10& .0166066 & .0095665 & 3.8272281 \\ \hline |
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141 | \end{tabular} |
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142 | \end{center} |
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143 | \caption{Neutron number distribution for induced fission in $^{238}$U (continued).} |
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144 | \label{Neutron number distribution for induced fission in 238U (continued)} |
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145 | \end{table} |
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146 | |
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147 | \begin{table}[ht] |
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148 | \footnotesize |
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149 | \begin{center} |
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150 | \begin{tabular}{|c|ccccccc|} \hline |
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151 | $E_n$ & $\nu$=0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline |
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152 | 0 & .0108826 & .0994916 & .2748898 & .3269196 & .2046061 & .0726834 & .0097282 \\ |
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153 | 1 & .0084842 & .0790030 & .2536175 & .3289870 & .2328111 & .0800161 & .0155581 \\ |
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154 | 2 & .0062555 & .0611921 & .2265608 & .3260637 & .2588354 & .0956070 & .0224705 \\ |
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155 | 3 & .0045860 & .0477879 & .1983002 & .3184667 & .2792811 & .1158950 & .0301128 \\ |
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156 | 4 & .0032908 & .0374390 & .1704196 & .3071862 & .2948565 & .1392594 & .0386738 \\ |
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157 | 5 & .0022750 & .0291416 & .1437645 & .2928006 & .3063902 & .1641647 & .0484343 \\ |
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158 | 6 & .0014893 & .0222369 & .1190439 & .2756297 & .3144908 & .1892897 & .0597353 \\ |
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159 | 7 & .0009061 & .0163528 & .0968110 & .2558524 & .3194566 & .2134888 & .0729739 \\ |
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160 | 8 & .0004647 & .0113283 & .0775201 & .2335926 & .3213289 & .2356614 & .0886183 \\ |
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161 | 9 & .0002800 & .0071460 & .0615577 & .2089810 & .3200121 & .2545846 & .1072344 \\ |
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162 | 10& .0002064 & .0038856 & .0492548 & .1822078 & .3154159 & .2687282 & .1295143 \\ \hline |
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163 | \end{tabular} |
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164 | \end{center} |
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165 | \caption{Neutron number distribution for induced fission in $^{239}$Pu.} |
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166 | \label{Neutron number distribution for induced fission in 239Pu} |
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167 | \end{table} |
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168 | |
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169 | \begin{table}[ht] |
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170 | \footnotesize |
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171 | \begin{center} |
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172 | \begin{tabular}{|c|cc|c|} \hline |
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173 | $E_n$ & $\nu$=7 & 8 & $\bar{\nu}$ \\ \hline |
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174 | 0 & .0006301 & .0001685 & 2.8760000 \\ |
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175 | 1 & .0011760 & .0003469 & 3.0088800 \\ |
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176 | 2 & .0025946 & .0005205 & 3.1628300 \\ |
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177 | 3 & .0048471 & .0007233 & 3.3167800 \\ |
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178 | 4 & .0078701 & .0010046 & 3.4707300 \\ |
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179 | 5 & .0116151 & .0014149 & 3.6246800 \\ |
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180 | 6 & .0160828 & .0029917 & 3.7786300 \\ |
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181 | 7 & .0213339 & .0020017 & 3.9325800 \\ |
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182 | 8 & .0274895 & .0039531 & 4.0865300 \\ |
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183 | 9 & .0347255 & .0054786 & 4.2404900 \\ |
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184 | 10& .0432654 & .0075217 & 4.3944400 \\ \hline |
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185 | \end{tabular} |
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186 | \end{center} |
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187 | \caption{Neutron number distribution for induced fission in $^{239}$Pu (continued).} |
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188 | \label{Neutron number distribution for induced fission in 239Pu (continued)} |
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189 | \end{table} |
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190 | |
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191 | \begin{figure}[ht] |
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192 | \begin{center} |
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193 | \includegraphics[scale=0.4, angle=-90]{hadronic/theory_driven/Fission/eps/U235_6MeV_nudist.ps} |
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194 | \end{center} |
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195 | \caption{Induced fission in $^{235}$U, incident neutron energy = 6MeV} |
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196 | \label{235U induced fission 6MeV} |
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197 | \end{figure} |
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198 | |
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199 | The first option |
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200 | %% \notgeant{ ({\tt nudist=0})} |
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201 | uses a fit to the Zucker |
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202 | and Holden data \cite{Zucker and Holden 1986} by |
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203 | Valentine~\cite{Valentine 1996}~\cite{Valentine 2000}. Valentine |
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204 | expressed the P$_{\nu}$'s (for $\nu=0$, ..., 8) as 5$^{th}$ order |
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205 | polynomials in $E_n$, the incident neutron energy. These functions |
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206 | P$_{\nu}(E_n)$ are used to sample the neutron multiplicity for |
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207 | $E_n$ in the range 0 to 10 MeV. When $E_n$ is greater than 10 MeV, |
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208 | $E_n$=10 MeV is used to generate P$_{\nu}$. |
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209 | |
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210 | In addition to using the Zucker and Holden data above for incident |
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211 | neutron energies $E_n$ above 1 MeV, the second |
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212 | option |
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213 | %% \notgeant{ ({\tt nudist=1})} |
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214 | also uses the Gwin, Spencer and |
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215 | Ingle data~\cite{Gwin 1984} for $^{235}$U at thermal energies (0 MeV) |
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216 | to generate P$_{\nu}(E_n)$ polynomials. As in the first option, when |
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217 | $E_n$ is greater than 10 MeV, $E_n$=10 MeV is used to generate |
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218 | P$_{\nu}$. |
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219 | |
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220 | The third option |
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221 | %% \notgeant{ ({\tt nudist=2})} |
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222 | implements an alternative |
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223 | polynomial fit from Valentine ~\cite{Valentine 2000} of P$_{\nu}$ |
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224 | as a function of $\bar{\nu}$ instead of $E_n$, following the suggestion |
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225 | of Frehaut~\cite{Frehaut 1988}. |
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226 | % |
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227 | %{\it"A unique |
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228 | %relationship P$_{\nu}(\bar{\nu})$ can sufficiently |
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229 | %well capture the multiplicity distributions of a number of major |
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230 | %isotopes. This distribution is expressed as a function of the average |
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231 | %number of neutrons emitted $\bar{\nu}$.}" |
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232 | % |
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233 | When a neutron induces a fission, the algorithm converts the incident |
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234 | neutron energy $E_n$ into $\bar{\nu}$ using conversion tables |
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235 | (typically ENDF/EDNL), generates the P$_{\nu}$ distributions for that |
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236 | value of $\bar{\nu}$, and then samples the P$_{\nu}$ distributions to |
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237 | determine $\nu$. The least-square fits to the $^{235}$U data are used |
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238 | for both $^{235}$U and $^{233}$U neutron induced fission, the fits to |
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239 | $^{238}$U are used for $^{232}$U, $^{234}$U, $^{236}$U and $^{238}$U, while the |
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240 | fits to $^{239}$Pu are used for $^{239}$Pu and $^{241}$Pu. Data comes from |
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241 | Zucker and Holden. For $^{235}$U, data comes from Zucker and Holden |
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242 | for $E_n$ greater than 1 MeV, and Gwin, Spencer and |
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243 | Ingle for 0 MeV. The fits are only used when $\bar{\nu}$ is in the |
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244 | range of the $\bar{\nu}$'s for the tabulated data. Otherwise, |
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245 | Terrell's approximation is used. |
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246 | |
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247 | The fourth option, which is the default |
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248 | %% \notgeant{ ({\tt nudist=3})} |
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249 | , is similar to the third option except that the P$_{\nu}$ distributions |
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250 | are not functions of $\bar{\nu}$, but are left intact as multiplicity |
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251 | distributions for the data listed in Gwin, Spencer and |
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252 | Ingle, and for the data listed in Zucker and |
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253 | Holden. The multiplicity distribution P$_{\nu}$ from which the number |
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254 | of neutrons will be sampled is selected based on the value of |
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255 | $\bar{\nu}$ for a given induced fission event. For instance, if |
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256 | P$_{\nu}(1 MeV)$ has $\bar{\nu}=2.4$, P$_{\nu}(2 MeV)$ has |
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257 | $\bar{\nu}=2.6$, and $\bar{\nu}$ is 2.45 at the energy of the incident |
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258 | fission-inducing neutron (this value $\bar{\nu}$ comes typically from |
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259 | cross-section data libraries such and ENDF/ENDL), the probability of |
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260 | sampling the number of neutrons ${\nu}$ from P$_{\nu}(1 MeV)$ and |
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261 | P$_{\nu}(2 MeV)$ will be 25\% and 75\%, respectively. This technique |
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262 | is only used when $\bar{\nu}$ is in the range of the $\bar{\nu}$'s for |
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263 | the tabulated data. Otherwise, Terrell's approximation is used. This |
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264 | last way of computing ${\nu}$ has several advantages: first, the data |
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265 | as listed in the original papers is used exactly, as opposed to |
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266 | approximated by low-ordered polynomials least-square fitting the |
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267 | original data. Second, the data from the Gwin, Spencer and |
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268 | Ingle paper, and the data from the Zucker and Holden paper is |
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269 | entered as-is as a table in the code, and can easily be checked and |
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270 | maintained if necessary by the application developer. Third the method |
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271 | provides a simple and statistically correct mechanism of sampling the |
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272 | data tables. |
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273 | % \notgeant{The fission module behaves in this |
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274 | % manner when the 'nudist' option is set to 3, which is also the default |
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275 | % behavior.} |
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276 | |
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277 | \subsubsection*{Spontaneous fission data} |
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278 | |
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279 | For $^{252}$Cf, the fission module can be set to use either the |
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280 | measurements by Spencer~\cite{Spencer 1982} |
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281 | %% \notgeant{ ({\tt ndist=0})} |
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282 | , which is the default, |
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283 | or Boldeman~\cite{Boldeman 1985} |
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284 | %% \notgeant{ ({\tt ndist=1})} |
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285 | . |
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286 | |
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287 | For $^{238}$U, $^{238}$Pu, $^{240}$Pu, $^{242}$Pu, $^{242}$Cm, |
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288 | $^{244}$Cm, the probability distribution data comes from Holden and |
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289 | Zucker~\cite{Holden and Zucker BNL}. The measured data is summarized |
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290 | in Tables~\ref{Neutron number distribution for spontaneous fission} |
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291 | and \ref{Neutron number distribution for spontaneous fission (continued)}. |
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292 | |
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293 | \begin{table}[ht] |
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294 | \footnotesize |
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295 | \begin{center} |
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296 | \begin{tabular}{|c|ccccccc|} \hline |
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297 | isotope & $\nu$=0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline |
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298 | $^{238}$U & .0481677 & .2485215 & .4253044 & .2284094 & .0423438 & .0072533 & 0 \\ |
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299 | $^{238}$Pu & .0540647 & .2053880 & .3802279 & .2248483 & .1078646 & .0276366 & 0 \\ |
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300 | $^{242}$Pu & .0679423 & .2293159 & .3341228 & .2475507 & .0996922 & .0182398 & .0031364 \\ |
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301 | $^{242}$Cm & .0212550 & .1467407 & .3267531 & .3268277 & .1375090 & .0373815 & .0025912 \\ |
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302 | $^{244}$Cm & .0150050 & .1161725 & .2998427 & .3331614 & .1837748 & .0429780 & .0087914 \\ |
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303 | $^{252}$Cf~\cite{Spencer 1982} & .00211 & .02467 & .12290 & .27144 & .30763 & .18770 & .06770 \\ |
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304 | $^{252}$Cf~\cite{Boldeman 1985} & .00209 & .02621 & .12620 & .27520 & .30180 & .18460 & .06680 \\ \hline |
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305 | \end{tabular} |
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306 | \end{center} |
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307 | \caption{Neutron number distribution for spontaneous fission.} |
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308 | \label{Neutron number distribution for spontaneous fission} |
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309 | \end{table} |
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310 | |
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311 | \begin{table}[ht] |
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312 | \footnotesize |
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313 | \begin{center} |
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314 | \begin{tabular}{|c|ccc|} \hline |
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315 | isotope & $\nu$=7 & 8 & 9 \\ \hline |
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316 | $^{238}$U & 0 & 0 & 0 \\ |
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317 | $^{238}$Pu & 0 & 0 & 0 \\ |
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318 | $^{242}$Pu & 0 & 0 & 0 \\ |
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319 | $^{242}$Cm & .0007551 & .0001867 & 0 \\ |
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320 | $^{244}$Cm & .0002744 & 0 & 0 \\ |
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321 | $^{252}$Cf~\cite{Spencer 1982} & .01406 & .00167 & .0001 \\ |
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322 | $^{252}$Cf~\cite{Boldeman 1985} & .01500 & .00210 & 0 \\ \hline |
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323 | \end{tabular} |
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324 | \end{center} |
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325 | \caption{Neutron number distribution for spontaneous fission (continued).} |
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326 | \label{Neutron number distribution for spontaneous fission (continued)} |
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327 | \end{table} |
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328 | |
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329 | If no full multiplicity distribution data exists, the fission module |
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330 | uses Terrell~\cite{Terrell 1957}'s approximation with $\bar{\nu}$ from |
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331 | Ensslin~\cite{Ensslin 1998}. Ensslin has $\bar{\nu}$ data for the |
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332 | isotopes in table~\ref{Nubar for spontaneous fission}. |
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333 | |
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334 | \begin{table}[ht] |
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335 | \footnotesize |
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336 | \begin{center} |
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337 | \begin{tabular}{|c|c|} \hline |
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338 | isotope & $\bar{\nu}$ \\ \hline |
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339 | $^{232}$Th & 2.14 \\ |
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340 | $^{232}$U & 1.71\\ |
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341 | $^{233}$U & 1.76\\ |
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342 | $^{234}$U & 1.81\\ |
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343 | $^{235}$U & 1.86\\ |
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344 | $^{236}$U & 1.91\\ |
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345 | $^{238}$U & 2.01\\ |
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346 | $^{237}$Np & 2.05\\ |
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347 | $^{238}$Pu & 2.21\\ |
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348 | $^{239}$Pu & 2.16\\ |
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349 | $^{240}$Pu & 2.156\\ |
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350 | $^{241}$Pu & 2.25\\ |
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351 | $^{242}$Pu & 2.145\\ |
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352 | $^{241}$Am & 3.22\\ |
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353 | $^{242}$Cm & 2.54\\ |
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354 | $^{244}$Cm & 2.72\\ |
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355 | $^{249}$Bk & 3.40\\ |
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356 | $^{252}$Cf & 3.757\\ \hline |
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357 | \end{tabular} |
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358 | \end{center} |
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359 | \caption{Average number of neutrons per spontaneous fission.} |
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360 | \label{Nubar for spontaneous fission} |
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361 | \end{table} |
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362 | |
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363 | \subsection{Neutron energy distribution} |
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364 | |
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365 | All of the fission spectra in the Evaluated Nuclear Data Library, |
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366 | ENDL~\cite{ENDL 1975} are defined by a simple analytical function, a |
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367 | Watt spectrum defined as |
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368 | |
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369 | \begin{equation} |
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370 | W(a,b,E') = Ce^{-aE'}sinh(\sqrt{bE'}) |
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371 | \end{equation} |
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372 | |
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373 | where $C=\sqrt{\pi\frac{b}{4a}}\frac{e^{\frac{b}{4a}}}{a}$, and E' is the secondary neutron energy. |
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374 | The Watt spectrum for $^{235}$U and an incident neutron energy of 6 MeV is shown in Fig.~\ref{Watt spectrum for U235}. |
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375 | |
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376 | \begin{figure}[ht] |
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377 | \begin{center} |
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378 | \includegraphics[scale=0.4, angle=-90]{hadronic/theory_driven/Fission/eps/Wattspectrum_U235_6MeV.ps} |
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379 | \end{center} |
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380 | \caption{Watt spectrum for $^{235}$U and an incident neutron energy of 6 MeV.} |
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381 | \label{Watt spectrum for U235} |
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382 | \end{figure} |
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383 | |
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384 | The coefficients a and b vary weakly from one isotope to another and |
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385 | also vary weakly with the incident neutron energy. In the fission |
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386 | module, b is set identical to 1.0, and a is parametrized as a simple |
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387 | function of the incident neutron energy, as implemented in |
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388 | TART~\cite{TART 2003, Cullen 2004}. The fissioning isotope and |
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389 | incident neutron energy determine the value of a, and the energy E' of |
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390 | the secondary neutron emitted is sampled using the Los Alamos' Monte |
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391 | Carlo sampler attributed to Mal Kalos~\cite{Everett 1983}. |
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392 | |
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393 | The Watt spectrum is used for all isotopes except $^{252}$Cf, for which a |
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394 | special treatment summarized by Valentine~\cite{Valentine 2000} is |
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395 | applied. The neutron spectrum for $^{252}$Cf is sampled from |
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396 | the Mannhart~\cite{Mannhart 1987} corrected Maxwellian distribution, |
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397 | the Madland and Nix~\cite{Madland 1984} |
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398 | or the Watt fission spectra from |
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399 | Froehner~\cite{Froehner 1990}. |
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400 | %% \notgeant{These options are selected by the internal variable {\tt neng=0(default),1,2} respectively.} |
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401 | The Mannhart distribution is used by default. |
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402 | |
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403 | %\section{Gammas emitted by fission\label{gammas}} |
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404 | |
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405 | \subsection{Gamma-ray number distribution} |
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406 | |
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407 | The fission module uses Brunson~\cite{Brunson 1982}'s double Poisson |
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408 | model for the spontaneous fission gamma ray multiplicity of $^{252}$Cf |
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409 | (see Fig.~\ref{Fission gamma-ray multiplicity for 252Cf}). |
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410 | |
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411 | \begin{equation} |
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412 | \Pi(G)=0.682\frac{7.20^Ge^{-7.20}}{G!}+0.318\frac{10.71^Ge^{-10.72}}{G!} |
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413 | \end{equation} |
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414 | |
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415 | where $G$ is the gamma ray multiplicity. |
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416 | \begin{figure}[ht] |
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417 | \begin{center} |
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418 | \includegraphics[scale=0.4, angle=-90]{hadronic/theory_driven/Fission/eps/Cf252_nugdist.ps} |
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419 | \end{center} |
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420 | \caption{Fission gamma-ray multiplicity for $^{252}$Cf.} |
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421 | \label{Fission gamma-ray multiplicity for 252Cf} |
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422 | \end{figure} |
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423 | |
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424 | The prompt gamma ray multiplicity ranges from 0 to 20 gama rays per |
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425 | fission with an average of 8.32 gamma rays per fission. This model is |
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426 | a fit to experimental data measured by Brunson himself. |
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427 | |
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428 | For other isotopes, there is no data available for the multiplicity of |
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429 | prompt gamma rays. Valentine~\cite{Valentine 2001} used an |
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430 | approximation that was adopted by the fission module. The probability |
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431 | of emitting $G$ fission gamma rays obeys the negative binomial |
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432 | distribution: |
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433 | |
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434 | \begin{equation} |
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435 | \Pi(G)=\left(\begin{array}{c} \alpha+G-1 \\ G \end{array} \right) p^G(1-p)^G |
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436 | \end{equation} |
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437 | |
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438 | where the parameter $p$ can be written as |
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439 | $p=\frac{\alpha}{\alpha+\bar{G}}$, $\alpha$ is approximately 26 and |
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440 | $\bar{G}$ is the average number of gamma rays per fission. $\bar{G}$ |
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441 | is approximated by |
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442 | |
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443 | \begin{equation} |
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444 | \bar{G} = \frac{E_t(\bar{\nu}, Z, A)}{\bar{E}} |
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445 | \end{equation} |
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446 | |
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447 | where $E_t(\bar{\nu}, Z, |
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448 | A)=(2.51(\pm0.01)-1.13\cdot10^{-5}(\pm7.2\cdot10^{-8})Z^2\sqrt{A})\nu+4.0$ |
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449 | is the total prompt gamma ray energy, and $\bar{E} = |
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450 | -1.33(\pm0.05)+119.6(\pm2.5)\frac{Z^{\frac{1}{3}}}{A}$ is the average |
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451 | prompt gamma ray energy. The multiplicity distribution for the |
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452 | spontaneous fission of $^{238}$U is shown in Fig.~\ref{Fission |
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453 | gamma-ray multiplicity for 238U}. |
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454 | |
---|
455 | \begin{figure}[ht] |
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456 | \begin{center} |
---|
457 | \includegraphics[scale=0.4, angle=-90]{hadronic/theory_driven/Fission/eps/U238_nugdist.ps} |
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458 | \end{center} |
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459 | \caption{Fission gamma-ray multiplicity for spontaneous fission of $^{238}$U.} |
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460 | \label{Fission gamma-ray multiplicity for 238U} |
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461 | \end{figure} |
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462 | |
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463 | These multiplicity distributions are only estimates and are not |
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464 | measured data. The fission module uses this model for estimating the |
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465 | number of gamma rays from both spontaneous and induced fission. |
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466 | |
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467 | \subsection{Gamma-ray energy distribution} |
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468 | |
---|
469 | The fission module implements Valentine's~\cite{Valentine 2000} model |
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470 | for the energy spectra of fission gamma-rays. The only measured |
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471 | energy spectra for fission gamma-rays are for the spontaneous fission |
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472 | of $^{252}$Cf and for the thermal-neutron-induced fission of $^{235}$U. |
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473 | Both spectra are similar~\cite{Wagemans 1991}. Because the $^{235}$U |
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474 | measurements are more precise, this data will be used for the fission |
---|
475 | gamma-ray spectrum. The energy spectrum of the prompt fission gamma |
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476 | rays is obtained from Maienschein's measurements~\cite{Maienschein |
---|
477 | 1958}~\cite{Goldstein 1959}: |
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478 | |
---|
479 | \begin{equation} |
---|
480 | N(E) = \left\{ |
---|
481 | \begin{array}{ll} |
---|
482 | 38.13 (E-0.085)e^{1.648E}& E<0.3\ \mathrm{MeV} \\ |
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483 | |
---|
484 | 26.8 e^{-2.30E} & 0.3<E<1.0\ \mathrm{MeV}\\ |
---|
485 | |
---|
486 | 8.0 e^{-1.10E} & 1.0<E<8.0\ \mathrm{MeV} |
---|
487 | \end{array} |
---|
488 | \right. |
---|
489 | \end{equation} |
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490 | |
---|
491 | This probability function is shown in Fig.~\ref{Fission gamma-ray |
---|
492 | spectrum for 235U}. Because gamma ray energy spectra are not |
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493 | available, the spectrum above is used for all isotopes, both for |
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494 | spontaneous and induced fission. |
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495 | |
---|
496 | \begin{figure}[ht] |
---|
497 | \begin{center} |
---|
498 | \includegraphics[scale=0.4, angle=-90]{hadronic/theory_driven/Fission/eps/U235_gspectrum.ps} |
---|
499 | \end{center} |
---|
500 | \caption{Fission gamma-ray spectrum for $^{235}$U.} |
---|
501 | \label{Fission gamma-ray spectrum for 235U} |
---|
502 | \end{figure} |
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503 | |
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504 | |
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505 | \subsection{Implementation} |
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506 | |
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507 | For neutron induced fission, this model is intended to be used with |
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508 | the low energy neutron interaction data libraries with class |
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509 | \textit{G4Fisslib} specified in the physics list as the |
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510 | \textit{G4HadronFissionProccess} instead of class |
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511 | \textit{G4NeutronHPFission}. |
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512 | %% \notgeant{ |
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513 | %% Here is an example code snippet for registering this model in the physics |
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514 | %% list: \input{snippet}} |
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515 | |
---|
516 | The constructor of \textit{G4FissLib} |
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517 | does two things. First it reads the necessary fission cross-section |
---|
518 | data in the file located in the directory specified by the environment |
---|
519 | variable \textit{NeutronHPCrossSections}. It does this by initializing |
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520 | one object of class \textit{G4NeutronHPChannel} per isotope present in |
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521 | the geometry. Second, it registers an instance of |
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522 | \textit{G4FissionLibrary} for each isotope as the model for that |
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523 | reaction/channel. When Geant4 tracks a neutron to a reaction site and |
---|
524 | the fission library process is selected among all other process for |
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525 | neutron reactions, the method \textit{G4FissLib::ApplyYourself} is |
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526 | called, and one of the fissionable isotopes present at the reaction |
---|
527 | site is selected. This method in turn calls |
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528 | \textit{G4NeutronHPChannel::ApplyYourself} which calls |
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529 | \textit{G4FissionLibrary::ApplyYourself}, where the induced neutrons |
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530 | and gamma-rays are emitted by sampling the fission library. |
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531 | |
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532 | For spontaneous fission the user must provide classes {\it |
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533 | PrimaryGeneratorAction}, {\it MultipleSource}, {\it |
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534 | MultipleSourceMessenger}, {\it SingleSource}, {\it SponFissIsotope} to |
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535 | generate spontaneous fission neutrons and gammas. Examples of these |
---|
536 | classes can be downloaded from {\tt |
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537 | http://nuclear.llnl.gov/CNP/simulation}. Spontaneous fissions are |
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538 | generated in the {\it PrimaryGeneratorAction} class. |
---|
539 | The spontaneous fission |
---|
540 | source needs to be described in terms of geometry, isotopic |
---|
541 | composition and fission strength. Once this information is given, the |
---|
542 | constructor creates as many spontaneous fission isotopes of class {\it |
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543 | SponFissIsotope} as specified, and adds them to the source of class |
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544 | {\it MultipleSource}. When Geant needs to generate particles, it calls |
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545 | the method {\it PrimaryGeneratorAction::GeneratePrimaries}, which |
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546 | first sets the time of the next fission based on the fission rates |
---|
547 | entered in the constructor, and then calls the method {\it |
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548 | MultipleSource::GeneratePrimaryVertex} which determines which one of |
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549 | the spontaneous fission isotopes will fission. This method in turn |
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550 | calls the method {\it SponFissIsotope::GeneratePrimaryVertex} for the |
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551 | chosen isotope. It is in this method that the neutrons and photons |
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552 | sampled from the fission library are added to the stack of secondary |
---|
553 | particles. Sources other than spontaneous fission isotopes can be |
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554 | added to the source of class {\it MultipleSource}. For instance, a |
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555 | background term emitting a large number of background gamma-rays can |
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556 | be added, as long as it derives from the class {\it SingleSource}. The |
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557 | intensity of that source would be set the same way as for the |
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558 | spontaneous fission isotope sources. |
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559 | |
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560 | Different sampling methods can be selected by calling the following functions. |
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561 | \subsection*{void setnudist\_(int *nudist) |
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562 | \label{setnudist}} |
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563 | |
---|
564 | This selects the data to be sampled for the neutron number |
---|
565 | distributions for neutron-induced fission. If there is no data |
---|
566 | available, then in all cases the Terrell approximation is used. |
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567 | The argument \textit{nudist} can take 3 values: |
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568 | |
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569 | \begin{tabbing} |
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570 | |
---|
571 | \indent 0 \hspace*{.6in} |
---|
572 | \= \parbox[t]{4in}{ Use the fit |
---|
573 | to the Zucker and Holden tabulated P$_\nu$ distributions as a function |
---|
574 | of energy for $^{235}$U, $^{238}$U and $^{239}$Pu.}\\ |
---|
575 | |
---|
576 | \indent 1 |
---|
577 | \> \parbox[t]{4in} |
---|
578 | {Use fits to the Zucker and Holden tabulated |
---|
579 | P$_\nu$ distribution as a function of energy for $^{238}$U and |
---|
580 | $^{239}$Pu, and a fit to the Zucker and Holden data as well as |
---|
581 | the Gwin, Spencer and Ingle data (at thermal |
---|
582 | energies) as a function of energy for $^{235}$U.}\\ |
---|
583 | |
---|
584 | \indent 2 |
---|
585 | \> \parbox[t]{4in} |
---|
586 | {Use the fit to the Zucker and Holden |
---|
587 | tabulated P$_\nu$ distributions as a function of $\bar{\nu}$. The $^{238}$U |
---|
588 | fit is used for the $^{232}$U, $^{234}$U, |
---|
589 | $^{236}$U and $^{238}$U isotopes, the $^{235}$U fit for $^{233}$U |
---|
590 | and $^{235}$U, the $^{239}$Pu fit for |
---|
591 | $^{239}$Pu and $^{241}$Pu.}\\ |
---|
592 | |
---|
593 | \indent 3 |
---|
594 | (default) \> \parbox[t]{4in} |
---|
595 | {Use the discrete Zucker and Holden |
---|
596 | tabulated P$_\nu$ distributions and corresponding $\bar{\nu}$s. |
---|
597 | Sampling based on the incident neutron $\bar{\nu}$. The $^{238}$U data tables |
---|
598 | are used for the $^{232}$U, $^{234}$U, $^{236}$U |
---|
599 | and $^{238}$U isotopes, the $^{235}$U data for $^{233}$U and |
---|
600 | $^{235}$U, the $^{239}$Pu data for $^{239}$Pu and $^{241}$Pu.} |
---|
601 | |
---|
602 | \end{tabbing} |
---|
603 | |
---|
604 | \subsection*{void setcf252\_(int *ndist, int *neng)} |
---|
605 | |
---|
606 | This function is specific to the spontaneous fission of $^{252}$Cf. It |
---|
607 | selects the data to be sampled for the neutron number and energy |
---|
608 | distributions and takes the following arguments: |
---|
609 | |
---|
610 | \begin{tabbing} |
---|
611 | \indent ndist: \= Sample the number of neutrons \\ |
---|
612 | \indent \> 0 (default) \= |
---|
613 | from the tabulated data measured by Spencer \\ |
---|
614 | \indent \> 1 \> from |
---|
615 | Boldeman's data \\ |
---|
616 | \\ |
---|
617 | \indent neng: Sample the spontaneous fission |
---|
618 | neutron energy \\ |
---|
619 | \indent \> 0 (default)\> from Mannhart corrected Maxwellian spectrum \\ |
---|
620 | \indent \> 1 \> from Madland-Nix theoretical spectrum \\ |
---|
621 | \indent \> 2 \> from the Froehner Watt spectrum \\ |
---|
622 | \end{tabbing} |
---|