1 | \chapter{INCL 4.2 Cascade and ABLA V3 Evaporation with Fission} |
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2 | |
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3 | \section{Introduction} |
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4 | |
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5 | There is a renewed interest in the study of spallation reactions. This |
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6 | is largely due to new technological applications, such as Accelerator |
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7 | Driven Systems, consisting of sub-critical nuclear reactor and |
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8 | particle accelerator. These applications require optimized spallation |
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9 | targets or spallation sources. This type of problem has typically a large number |
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10 | of parameters and thus it cannot be solved by trial and error |
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11 | method. One has to rely on simulations, which implies that very |
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12 | accurate simulation tools need to be developed and their validity and |
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13 | accuracy needs also to be assessed. |
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14 | |
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15 | Above the energy 200 MeV it is necessary to use reliable models due to |
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16 | the prohibitive number of open channels. The most appropriate modeling |
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17 | technique in this energy region is intranuclear cascade (INC) combined |
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18 | with evaporation model. One such pair of models is the Li\`ege cascade |
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19 | model INCL4.2 coupled with ABLA evaporation model. The strategy adopted |
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20 | by the INCL4.2 cascade is to improve the quasi-classical treatment of |
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21 | physics without relying on too many free parameters. |
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22 | |
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23 | This chapter introduces the physics provided by INCL4.2 and ABLA V3 codes as implemented in Geant4. |
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24 | Tables \ref{tbl:inclsummary} and \ref{tbl:ablasummary} will summarize the key features |
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25 | and provides references describing in detail the physics. |
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26 | |
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27 | |
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28 | \section{INCL4.2 cascade} |
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29 | \label{sec:inclmodel} |
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30 | |
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31 | INCL4.2 is a Monte Carlo simulation incorporating aforementioned cascade |
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32 | physics principles. INCL4.2 cascade algorithm consists of an |
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33 | initialization stage and the actual data processing stage. |
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34 | |
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35 | |
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36 | The INCL4.2 cascade can be used to simulate the collisions between |
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37 | bullet particles and nuclei. The supported bullet particles and the |
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38 | interface classes supporting them are presented in table |
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39 | \ref{tbl:inclsummary}. |
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40 | |
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41 | |
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42 | \begin{table}[ht] |
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43 | \caption{INCL 4.2 (located in the Geant4 |
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44 | directory {\tt source/\-processes/\-hadronic/\-models/\-incl}) |
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45 | feature summary.} |
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46 | \label{tbl:inclsummary} |
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47 | \vskip1cm |
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48 | \begin{center} |
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49 | \begin{tabular}{l|l} |
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50 | \hline |
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51 | {\bf Requirements} & \\ |
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52 | External data file & G4ABLA3.0 available at Geant4 site \\ |
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53 | Environment variable & {\tt G4ABLADATA} \\ |
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54 | for external data & \\ |
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55 | \hline |
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56 | {\bf Usage} & \\ |
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57 | Physics list & Not yet implemented, \\ |
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58 | & instead use the interfaces directly. \\ |
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59 | \hline |
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60 | {\bf Interfaces} & \\ |
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61 | {\tt G4InclCascadeInterface} & h--A \\ |
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62 | {\tt G4InclLightIonInterface} & A--A \\ |
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63 | \hline |
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64 | {\bf Projectile particles} & proton, neutron \\ |
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65 | & pions ($\pi^+$, $\pi^0$, $\pi^-$) \\ |
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66 | & deuteron, triton \\ |
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67 | & $\alpha$, $^3$He \\ |
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68 | \hline |
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69 | {\bf Energy range} & 200 MeV - 3 GeV \\ |
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70 | \hline |
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71 | {\bf Target nuclei} & \\ |
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72 | Lightest applicable & Carbon, C \\ |
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73 | Heaviest & Uranium, U \\ |
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74 | \hline |
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75 | {\bf Features} & No ad-hoc parameters \\ |
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76 | & Woods-Saxon nuclear potential \\ |
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77 | & Coulomb barrier \\ |
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78 | & Non-uniform time-step \\ |
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79 | & Pion and delta production cross sections \\ |
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80 | & Delta decay \\ |
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81 | & Pauli blocking \\ |
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82 | \hline |
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83 | {\bf Misc.} & 5 classes (see fig. \ref{fig:uml}), 8k lines \\ |
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84 | & 0.9 $<$ speed C++/F77 $<$ 1.1 \\ |
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85 | \hline |
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86 | {\bf References} & Key reference \cite{Boudard02a}, see also \cite{Cugnon97a, Cugnon81a, Cugnon87a, Cugnon89a} \\ |
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87 | \hline |
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88 | \end{tabular} |
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89 | \end{center} |
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90 | \end{table} |
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91 | |
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92 | The momenta and positions of the nucleons inside the nuclei are |
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93 | determined at the beginning of the simulation run by modeling the |
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94 | nucleus as a free fermi gas in a static potential well. The cascade is |
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95 | modeled by tracking the nucleons and their collisions. The collisions |
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96 | are assumed to be well separated in space and time. |
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97 | |
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98 | The possible reactions inside the nucleus are |
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99 | \begin{itemize} |
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100 | \item $NN \rightarrow N \Delta$ and $N \Delta \rightarrow NN$ |
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101 | \item $\Delta \rightarrow \pi N$ and $\pi N \rightarrow \Delta$ |
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102 | \end{itemize} |
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103 | |
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104 | %\begin{figure}[ht] |
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105 | %\begin{center} |
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106 | %\includegraphics[scale=0.6]{Pb208Proton1GeV.eps} |
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107 | %\includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/Incl/Pb208Proton1GeV.eps} |
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108 | %\end{center} |
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109 | %\caption{Colliding 1 GeV proton to Pb208 target. Here is presented the |
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110 | %mass number of the outcoming particles.} |
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111 | %\end{figure} |
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112 | |
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113 | \subsection{Model limits} |
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114 | |
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115 | |
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116 | |
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117 | The INCL4.2 model has certain limitations with respect to the bullet |
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118 | particle energy and target nucleus type. The supported energy range |
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119 | for bullets is: 200 MeV - 3 GeV. Acceptable target nuclei range from |
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120 | Carbon to Uranium. |
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121 | |
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122 | % Maybe too basic stuff... |
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123 | % \chapter{INCL and ABLA models} |
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124 | |
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125 | % Hadronic interactions can be modeled with cascade |
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126 | % models when we deal with so called medium energy range between 100 MeV |
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127 | % and 10 GeV \cite{thebook}. |
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128 | % %The medium energy range lies between 100 MeV and 10 GeV. |
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129 | % In this energy range we can simplify the problem with certain |
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130 | % assumptions and approximations. In this chapter we outline the basic |
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131 | % features of medium energy hadronic physics and how INCL and ABLA |
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132 | % models implement them. |
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133 | |
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134 | % %\section{Physics in intermediate energy range} |
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135 | |
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136 | % \section{Intra-nuclear cascade (INC)} |
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137 | % \index{intranuclear cascade} |
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138 | % \label{sec:inc} |
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139 | |
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140 | % Atomic nuclei can, in principle, be described in terms of quantum |
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141 | % mechanics. However, since the nucleus is fairly complex multi-particle |
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142 | % system purely quantum mechanical approach is usually not feasible for |
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143 | % describing nuclear reactions in the intermediate energy range. The INC |
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144 | % model of these processes is a semiclassical description of collision |
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145 | % between a particle and a nucleus. |
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146 | |
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147 | % The basic problem can be divided into two steps. First is the actual |
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148 | % \emph{cascade} stage. At this stage each nucleon collides few times |
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149 | % in the nucleus. The number of collisions is usually between 1 and 4 |
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150 | % depending on the weight of the nucleus (number of nucleons). As the |
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151 | % result of the collisions fast nucleons and pions exit the nucleus. |
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152 | % This stage lasts usually $10^{-23}$ - $10^{-22}$ seconds. |
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153 | |
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154 | % After cascade comes pre-equilibrium and evaporation cometing with |
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155 | % fission stages. At this stage more nucleons are ejected from the |
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156 | % nucleus. This process is slower than the \emph{cascade} taking |
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157 | % $10^{-18}$-$10^{-16}$ seconds. |
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158 | |
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159 | % \subsection{Basic assumptions} |
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160 | % \index{cascadeassumptions} |
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161 | |
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162 | % The basic assumptions of INC-model can be summarized as follows: |
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163 | % \begin{enumerate} |
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164 | % \item Motion of particles obeys \emph{classical mechanics}. |
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165 | % \item In collisions relativistic kinematics is used. |
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166 | % \item Collisions between pairs of particles are well separated in |
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167 | % space and time. This means that the spatial dimensions and time |
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168 | % scale of collisions are smaller than those of particle |
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169 | % transportation in the nuclear matter. |
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170 | % \end{enumerate} |
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171 | |
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172 | \subsection{Model principles} |
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173 | |
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174 | The INCL model uses only two external user defined parameters: the nuclear |
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175 | potential depth and scaling factor for time-step. All other parameters |
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176 | used during the calculation are obtained either from theory or |
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177 | experiments. In the actual simulation code good default values for |
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178 | these two parameters are predefined. |
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179 | |
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180 | During the initialization the necessary Woods-Saxon potential |
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181 | calculations are made. The results of these calculations are used at |
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182 | the beginning of a cascade to determine the positions and momenta of |
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183 | the nucleons inside the nucleus. The nucleons are modeled as free |
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184 | Fermi gas in a static potential well. The principle for doing this is |
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185 | presented in the following equations \ref{eqn:density} and |
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186 | \ref{eqn:rpcorrelations}. |
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187 | |
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188 | \index{nucleon density} |
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189 | The nucleon density in r-space is given by formula: |
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190 | \begin{equation} |
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191 | \rho(r) = \left\{ \begin{array}{ll} |
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192 | \frac{\rho_0}{1 + exp(\frac{r - R_0}{a})}& \textrm{for $r |
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193 | < R_{max}$}\\ |
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194 | 0 & \textrm{for $r > R_{max}$}\\ |
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195 | \end{array} \right. |
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196 | \label{eqn:density} |
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197 | \end{equation} |
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198 | The nucleon positions $r$ and momenta $p$ are correlated in the following way: |
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199 | \begin{equation} |
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200 | \rho(p)p^2dp = -\frac{d \rho(p)}{dr} \frac{r^3}{3}dr. |
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201 | \label{eqn:rpcorrelations} |
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202 | \end{equation} |
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203 | |
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204 | % \begin{figure}[ht] |
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205 | % \begin{center} |
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206 | % \includegraphics{rpcorrelations.eps} |
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207 | % \end{center} |
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208 | % \caption[Momentum-position correlations in INCL]{Nucleon |
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209 | % momentum-position correlations of the initial state in INCL are |
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210 | % determined using the Woods-Saxon potential. (Picture courtesy of |
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211 | % Alain |
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212 | % Boudard)} |
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213 | % \label{fig:rpcorr} |
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214 | % \end{figure} |
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215 | |
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216 | After the initialization a projectile particle, bullet, is shot |
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217 | towards the target nucleus. The impact parameter, i.e. the distance |
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218 | between the projectile particle and the center point of the projected |
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219 | nucleus surface is chosen at random. The value of the impact parameter |
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220 | determines the point where the bullet particle will hit the |
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221 | nucleus. After this the algorithm tracks the nucleons by determining |
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222 | the times at which an event will happen. The possible events are: |
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223 | \begin{itemize} |
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224 | \item collision |
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225 | \item decay of a delta resonance |
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226 | \item reflection from nuclear potential wall (only for nucleons) |
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227 | \end{itemize} |
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228 | |
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229 | The particles are assumed to propagate along straight line |
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230 | trajectories. The algorithm calculates the time at which events will |
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231 | happen and propagates the particles directly to their positions at |
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232 | that particular point in time. Practically this means that the length |
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233 | of the time step in simulation is not constant. |
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234 | |
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235 | \index{participant} |
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236 | \index{spectator} |
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237 | All particles in the model are labeled either as \emph{participants} |
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238 | or \emph{spectators}. Only collisions where participants are involved |
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239 | are taken into account. |
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240 | |
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241 | \index{delta resonance} |
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242 | The processes undergone by the nucleons, pions and delta resonances |
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243 | are as follows: |
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244 | \begin{itemize} |
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245 | \item $NN \rightarrow N \Delta$ and $N \Delta \rightarrow NN$ |
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246 | \item $\Delta \rightarrow \pi N$ and $\pi N \rightarrow \Delta$ |
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247 | \end{itemize} |
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248 | %Figure \ref{img:inc} demonstrates these prosesses during INC. |
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249 | |
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250 | |
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251 | \subsection{Conservation laws} |
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252 | |
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253 | During cascade INCL model conserves several quantities: |
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254 | energy, baryon number, charge number, energy, momentum and angular momentum. |
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255 | The conservation relations are: |
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256 | \begin{eqnarray} |
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257 | A_{projectile} + A_{target} = A_{ejectiles} + A_{remnant} \\ |
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258 | Z_{projectile} + Z_{target} = Z_{ejectiles} + Z_{remnant} \\ |
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259 | T_{laboratory} = K_{ejectiles} + W_{\pi} + E_{recoil} + E_{excitation} + S \\ %///FIXED-AH systematically use full word indexing |
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260 | \vec{p}_{laboratory} = \vec{p}_{ejectiles} + \vec{p}_{\pi} + |
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261 | \vec{p}_{remnant} \\ |
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262 | \vec{l} = \vec{l}_{ejectiles} + \vec{l}_{\pi} + \vec{l}_{remnant} + |
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263 | \vec{l}_{excitation}, |
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264 | \end{eqnarray} |
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265 | where |
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266 | $A$ is baryon number, $Z$ charge, $T$ energy, $K$ kinetic energy, |
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267 | $E_{recoil}$ remnant recoil energy, $E_{excitation}$ remnant nucleus |
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268 | excitation energy, $S$ separation energy, $\vec{p}$ momentum and |
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269 | $\vec{l}$ angular momentum. |
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270 | |
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271 | \subsection{Cascade stopping time} |
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272 | |
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273 | Stopping time is defined as the point in time when the cascade phase is finished |
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274 | and the excited nucleus remnant is given to evaporation model. |
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275 | In INCL model the stopping time, $t_{stop}$, is determined as: |
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276 | \begin{equation} |
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277 | t_{stop} = f_{stop}t_0 (A_{target}/208) ^{0.16}. |
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278 | \end{equation} |
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279 | Here $A_{target}$ is the target mass number and $t_0$ is a constant |
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280 | with default value 70 fm/c. Factor $f_{stop}$ is the second of |
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281 | the two free parameters in the model. It is a scaling factor for the |
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282 | stopping time. Good default value for this factor is |
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283 | $f_{stop}$ = 1.0. |
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284 | |
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285 | |
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286 | \subsection{Light ions} |
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287 | |
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288 | In addition to protons, neutrons and pions INCL4.2 supports also |
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289 | light ion projectiles: deuterons, tritons, He3 and alpha. |
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290 | In light of INCL physics modeling principles presented |
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291 | above, the extension to light ions is quite natural. Light ions are |
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292 | modeled in similar way to nucleus, except that ion potential is |
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293 | considered to be Gaussian instead of real Woods-Saxon shape. In table |
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294 | \ref{tbl:gaussianformslightions} the parameters for Gaussian forms |
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295 | used for distance and momentum distributions in light ions are |
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296 | presented. |
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297 | |
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298 | \begin{table}[h] |
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299 | \caption{The parameters for nucleon position and momentum |
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300 | distributions in light ions.} % \cite{incl4}.} |
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301 | \begin{center} |
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302 | \begin{tabular}{c|c c} |
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303 | \hline |
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304 | Particle & $\sqrt{(r_{mean})^2}$ [fm] & $\sqrt{(p_{mean})^2}$ [MeV/c]\\ |
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305 | \hline |
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306 | Deuteron & 1.91 & 77 \\ |
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307 | Triton & 1.8 & 110 \\ |
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308 | He3 & 1.8 & 110 \\ |
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309 | Alpha & 1.63 & 153 \\ |
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310 | \hline |
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311 | \end{tabular} |
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312 | \end{center} |
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313 | \label{tbl:gaussianformslightions} |
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314 | \end{table} |
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315 | |
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316 | |
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317 | \section{ABLA V3 evaporation} |
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318 | \index{evaporation} |
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319 | |
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320 | The ABLA V3 evaporation model takes excited nucleus parameters, |
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321 | excitation energy, mass number, charge number and nucleus spin, as |
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322 | input. It calculates the probabilities for emitting proton, neutron or |
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323 | alpha particle and also probability for fission to occur. |
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324 | The summary of Geant4 ABLA V3 implementation is represented in Table \ref{tbl:ablasummary}. |
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325 | |
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326 | |
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327 | The probabilities for emission of particle type $j$ are calculated using |
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328 | formula: |
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329 | \begin{equation} |
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330 | W_j(N,Z,E) = \frac{\Gamma_j(N,Z,E)}{\sum_k\Gamma_k(N,Z,E)}, |
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331 | \label{eqn:probabilities} |
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332 | \end{equation} |
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333 | where $\Gamma_j$ is emission width for particle $j$, $N$ is neutron |
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334 | number, $Z$ charge number and $E$ excitation energy. Possible emitted |
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335 | particles are \emph{protons}, \emph{neutrons} and \emph{alphas}. |
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336 | Emission widths are calculated using the following formula: |
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337 | \index{emission width} |
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338 | \begin{equation} |
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339 | \Gamma_j = \frac{1}{2 \pi \rho_c(E)} \frac{4 m_j R^2}{\hbar^2} T_j^2 \rho_j(E - S_j - B_j), |
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340 | \label{eqn:emissionwidth} |
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341 | \end{equation} |
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342 | where $\rho_c(E)$ and $\rho_j(E - S_j - B_j)$ are the level densities |
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343 | of the compound nucleus and the exit channel, respectively. $B_j$ is |
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344 | the height of the Coulomb barrier, $S_j$ the separation energy, $R$ is |
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345 | the radius and $T_j$ the temperature of the remnant nucleus after |
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346 | emission and $m_j$ the mass of the emitted particle. |
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347 | |
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348 | The fission width is calculated from: |
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349 | \index{fission width} |
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350 | \begin{equation} |
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351 | \Gamma_i = \frac{1}{2 \pi \rho_c(E)}T_f \rho_f(E - B_f), |
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352 | \label{eqn:fissionwidth} |
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353 | \end{equation} |
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354 | where $\rho_f(E)$ is the level density of transition states in the |
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355 | fissioning nucleus, $B_f$ the height of the fission barrier and $T_f$ |
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356 | the temperature of the nucleus. |
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357 | |
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358 | |
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359 | \begin{table}[ht] |
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360 | \caption{ABLA V3 (located in the Geant4 directory |
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361 | {\tt source/\-processes/\-hadronic/\-models/\-incl}) feature summary.} |
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362 | \label{tbl:ablasummary} |
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363 | \vskip1cm |
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364 | \begin{center} |
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365 | \begin{tabular}{l|l} |
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366 | \hline |
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367 | {\bf Requirements} & \\ |
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368 | External data file & G4ABLA3.0 available at Geant4 site \\ |
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369 | Environment variable & {\tt G4ABLADATA} \\ |
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370 | for external data & \\ |
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371 | \hline |
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372 | {\bf Usage} & \\ |
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373 | Physics list & Not yet implemented, \\ |
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374 | & instead use the interfaces directly. \\ |
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375 | \hline |
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376 | {\bf Interfaces} & \\ |
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377 | {\tt G4InclAblaCascadeInterface} & h--A \\ |
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378 | {\tt G4InclAblaLightIonInterface} & A--A \\ |
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379 | \hline |
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380 | {\bf Supported input} & Excited nuclei \\ |
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381 | \hline |
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382 | {\bf Output particles} & proton, neutron \\ |
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383 | & $\alpha$ \\ |
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384 | & fission products \\ |
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385 | & residual nuclei \\ |
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386 | \hline |
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387 | {\bf Features} & evaporation of proton, neutron and $\alpha$ \\ |
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388 | & fission \\ |
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389 | \hline |
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390 | {\bf Misc.} & 5 classes, 5k lines \\ |
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391 | & 0.9 $<$ speed C++/F77 $<$ 1.1 \\ |
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392 | \hline |
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393 | {\bf References} & Key reference: \cite{Junghans98a}, see also \cite{Benlliure98a} \\ |
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394 | \hline |
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395 | \end{tabular} |
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396 | \end{center} |
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397 | \end{table} |
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398 | |
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399 | \subsection{Level densities} |
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400 | \index{level density} |
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401 | |
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402 | Nuclear level densities are calculated using the following formula: |
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403 | \begin{equation} |
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404 | a = 0.073 A [MeV^{-1}] + 0.095 B_s A^{2/3} [MeV^{-2}], |
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405 | \label{eqn:leveldensity} |
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406 | \end{equation} |
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407 | where $A$ the nucleus mass number and $B_s$ dimensionless surface area |
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408 | of the nucleus. |
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409 | |
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410 | %The level density calculation is implemented in the |
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411 | %code as follows: |
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412 | %\begin{verbatim} |
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413 | %pa = (ald->av)*a + (ald->as)*pow(a,(2.0/3.0)) |
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414 | % + (ald->ak)*pow(a,(1.0/3.0)); |
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415 | %\end{verbatim} |
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416 | %where {\tt ald->av} is 0.073, {\tt ald->as} is 0.095 and {\tt ald->ak} |
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417 | %is 0. |
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418 | |
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419 | \subsection{Fission} |
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420 | \index{fission} |
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421 | |
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422 | Fission barrier, used to calculate fission width |
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423 | \ref{eqn:fissionwidth}, is calculated using a semi-empirical model |
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424 | fitting to data obtained from nuclear physics experiments. |
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425 | |
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426 | |
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427 | \section{External data file required} |
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428 | |
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429 | Both INCL4.2 and ABLA V3 need data files. These files contain ABLA V3 shell corrections and remnant nucleus masses for INCL4.2. |
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430 | To enable this data set, environment variable {\tt G4ABLADATA} needs to be set, |
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431 | and the data downloaded from Geant4 web page. For Geant4 9.1 release use data file G4ABLA3.0 |
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432 | |
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433 | \section{Implementation details} |
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434 | INCL4.2 and ABLA V3 are provided as alpha release for Geant4 9.1. |
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435 | In this first release design follows as closely as possibly the original codes, |
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436 | and class re-design is left for future Geant4 releases. |
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437 | Current simple design is shown in \ref{fig:uml} |
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438 | |
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439 | \begin{figure} |
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440 | \begin{center} |
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441 | %\includegraphics[angle=0,scale=1.0]{InclAblaUml.eps} |
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442 | \includegraphics[angle=0,scale=1.0]{hadronic/theory_driven/Incl/InclAblaUml.eps} |
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443 | \end{center} |
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444 | \caption[INCL4 and ABLA class diagram]{Simplified UML class diagram of |
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445 | INCL and ABLA implementations in Geant4.} |
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446 | \label{fig:uml} |
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447 | \end{figure} |
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448 | |
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449 | Testing of INCL and ABLA models is based on ROOT \cite{Brun97a} scripting. |
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450 | |
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451 | \section{Physics Performance} |
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452 | INCL4.2 together with ABLA V3 provides an up to date modeling tool particularly for spallation |
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453 | studies for hadron projectile energy range 200 MeV - 3 GeV. |
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454 | It provides an detailed description of double differential |
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455 | energy spectrum of cascading particles (see \ref{fig:pPbDoubleDifferential}) and remnants. |
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456 | |
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457 | Models are validated against recent data and continually updated. |
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458 | |
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459 | \begin{figure} |
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460 | \begin{center} |
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461 | %\includegraphics[angle=0,scale=0.65]{pPbDoubleDifferential.eps} |
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462 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/Incl/pPbDoubleDifferential.eps} |
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463 | \end{center} |
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464 | \caption{Geant4 implementation of INCL4.2 together with ABLA V3. Neutron double differential cross sections of reaction p(1GeV) + Pb |
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465 | For more detailed discussion on this subject |
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466 | % double differential cross sections for proton-induced reactions on |
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467 | % Pb targets with comparison to experimental data |
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468 | see reference \cite{Boudard02a}.} |
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469 | \label{fig:pPbDoubleDifferential} |
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470 | \end{figure} |
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471 | |
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472 | \section{Status of this document} |
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473 | |
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474 | {\bf 06.12.2007} Documentation for alpha release added. Pekka |
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475 | Kaitaniemi, HIP (translation); Alain Boudard, CEA (contact person |
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476 | INCL/ABLA); Joseph Cugnon, University of Li\`ege (INCL physics |
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477 | modelling); Karl-Heintz Schmidt, GSI (ABLA); Christelle Schmidt, IPNL |
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478 | (fission code); Aatos Heikkinen, HIP (project coordination) |
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479 | |
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480 | % \begin{thebibliography}{99} |
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481 | % \bibitem{incl1} J. Cugnon et al \emph{Nuc. Phys. A352} (1981) 505 |
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482 | % \bibitem{incl2} J. Cugnon et al \emph{Nuc. Phys. A462} (1987) 751 |
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483 | % \bibitem{incl3} J. Cugnon et al \emph{Nuc. Phys. A500} (1989) 701 |
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484 | % \bibitem{incl4} A. Boudard et al \emph{Phys. Rev. C66} (2002) 044615 |
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485 | % \bibitem{liegeuniversity} Li\`ege University |
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486 | % \href{http://www.ulg.ac.be/foreign/}{{\tt http://www.ulg.ac.be/foreign/}} |
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487 | % \bibitem{cea} CEA website |
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488 | % \href{http://www.cea.fr/gb/index.asp}{{\tt http://www.cea.fr/gb/index.asp}} |
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489 | % \end{thebibliography} |
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490 | |
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491 | \begin{latexonly} |
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492 | \begin{thebibliography}{99} |
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493 | |
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494 | % \bibitem{alsmiller90} |
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495 | % R.G. Alsmiller and F.S. Alsmiller and O.W. Hermann, |
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496 | % The high-energy transport code HETC88 and comparisons with experimental data, |
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497 | % Nuclear Instruments and Methods in Physics Research A 295, |
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498 | % (1990), 337--343, |
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499 | |
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500 | % [1] Barashenkov V.S., Toneev V.D. High Energy interactions of particles and nuclei with nuclei. Moscow, 1972 |
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501 | %(in Russian, but there is an English translation)) |
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502 | |
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503 | \bibitem{Boudard02a} A. Boudard et al \emph{Phys. Rev. C66} (2002) 044615 |
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504 | \bibitem{Cugnon81a} J. Cugnon et al \emph{Nuc. Phys. A352} (1981) 505 |
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505 | \bibitem{Cugnon87a} J. Cugnon et al \emph{Nuc. Phys. A462} (1987) 751 |
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506 | \bibitem{Cugnon89a} J. Cugnon et al \emph{Nuc. Phys. A500} (1989) 701 |
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507 | \bibitem{Cugnon97a} J. Cugnon et al \emph{Nuc. Phys. A620} (1997) 745 |
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508 | \bibitem{Junghans98a} A.R. Junghans et al \emph{Nuc. Phys. A629} (1998) 635 |
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509 | \bibitem{Benlliure98a} J. Benlliure et al \emph{Nuc. Phys. A628} (1998) 458 |
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510 | \bibitem{Kaitaniemi07a} P. Kaitaniemi et al. \emph{Implementation of |
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511 | INCL4 cascade and ABLA evaporation codes in Geant4} (To be |
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512 | published in the proceedings of CHEP 2007, September 2-6, |
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513 | Victoria, BC, Canada.) |
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514 | \bibitem{Brun97a} R. Brun, F. Rademakers \emph{Nucl. Inst \& |
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515 | Meth. in Phys. Res. A 389} (1997) 81 |
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516 | |
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517 | \end{thebibliography} |
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518 | \end{latexonly} |
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519 | |
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520 | |
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521 | |
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522 | \begin{htmlonly} |
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523 | \section{Bibliography} |
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524 | |
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525 | \begin{enumerate} |
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526 | \item{Boudard02a} A. Boudard et al \emph{Phys. Rev. C66} (2002) 044615 |
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527 | \item{Cugnon81a} J. Cugnon et al \emph{Nuc. Phys. A352} (1981) 505 |
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528 | \item{Cugnon87a} J. Cugnon et al \emph{Nuc. Phys. A462} (1987) 751 |
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529 | \item{Cugnon89a} J. Cugnon et al \emph{Nuc. Phys. A500} (1989) 701 |
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530 | \item{Cugnon97a} J. Cugnon et al \emph{Nuc. Phys. A620} (1997) 745 |
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531 | \item{Junghans98a} A.R. Junghans et al \emph{Nuc. Phys. A629} (1998) 635 |
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532 | \item{Benlliure98a} J. Benlliure et al \emph{Nuc. Phys. A628} (1998) 458 |
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533 | \item{Kaitaniemi07a} P. Kaitaniemi et al. \emph{Implementation of |
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534 | INCL4 cascade and ABLA evaporation codes in Geant4} (To be |
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535 | published in the proceedings of CHEP 2007, September 2-6, |
---|
536 | Victoria, BC, Canada.) |
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537 | \item{Brun97a} R. Brun, F. Rademakers \emph{Nucl. Inst \& |
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538 | Meth. in Phys. Res. A 389} (1997) 81 |
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539 | |
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540 | \end{enumerate} |
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541 | \end{htmlonly} |
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542 | |
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