1 | \chapter{Bertini Intranuclear Cascade Model in {\sc Geant4} } |
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2 | |
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3 | \section{Introduction} |
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4 | This model is based on a re-engineering of the INUCL code and |
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5 | includes the Bertini intra-nuclear cascade model with excitons, a |
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6 | pre-equilibrium model, a nucleus explosion model, a fission model, and an |
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7 | evaporation model. Intermediate energy nuclear reactions from 100~MeV to |
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8 | 5~GeV are treated for protons, neutrons, pions, photons and nuclear |
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9 | isotopes. We present an overview of the models, review results achieved |
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10 | from simulations and make comparisons with experimental data. |
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11 | |
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12 | The intranuclear cascade model (INC) was was first proposed by Serber in |
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13 | 1947 \cite{serber47}. He noticed that in particle-nuclear collisions the |
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14 | deBroglie wavelength of the incident particle is comparable (or shorter) than |
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15 | the average intra-nucleon distance. Hence, a description of interactions |
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16 | in terms of particle-particle collisions is justified. |
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17 | |
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18 | The INC has been used succesfully in Monte Carlo simulations at intermediate |
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19 | energies since Goldberger made the first hand-calculations in |
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20 | 1947 \cite{goldberger48}. The first computer simulations were done by |
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21 | Metropolis et al. in 1958 \cite{metropolis58}. Standard methods in INC |
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22 | implementations were developed when Bertini published his results in |
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23 | 1968 \cite{bertini68}. An important addition to INC was the exciton model |
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24 | introduced by Griffin in 1966 \cite{griffin66}. |
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25 | |
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26 | The current presentation describes the implementation of the Bertini INC |
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27 | model within the {\sc Geant4} hadronic physics |
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28 | framework \cite{geant4collaboration03}. This framework is flexible and |
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29 | allows for the modular implementation of various kinds of hadronic |
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30 | interactions. It is based on the concepts of physics processes and models. |
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31 | While the process is a general concept, models may be restricted according |
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32 | to process type, material, element and energy range. Several models can be |
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33 | utilized by one process class; for instance, a process class for inelastic |
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34 | collisions can use a different model for each energy range. |
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35 | |
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36 | The process classes use model classes to determine the secondaries produced |
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37 | in the interaction and to calculate the momenta of the particles. Here we |
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38 | present a collection of such models which describe a medium-energy |
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39 | intranuclear cascade. |
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40 | |
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41 | \section{The Geant4 Cascade Model} |
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42 | |
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43 | Inelastic particle-nucleus collisions are characterized by both fast and slow |
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44 | components. The fast ($10^{-23} - 10^{-22} s$) intra-nuclear cascade |
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45 | results in a highly excited nucleus which may decay by fission or |
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46 | pre-equilibrium emission. The slower ($10^{-18} - 10^{-16} s$) compound |
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47 | nucleus phase follows with evaporation. A Boltzmann equation must be solved |
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48 | to treat the collision process in detail. |
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49 | |
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50 | The intranuclear cascade (INC) model developed by |
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51 | Bertini \cite{bertini68, bertini71} solves the Boltzmann equation on |
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52 | average. This model has been implemented in several codes such as |
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53 | HETC \cite{alsmiller90}. Our model, which is based on a re-engineering of |
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54 | the INUCL code \cite{titarenko99a}, includes the Bertini intranuclear cascade |
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55 | model with excitons, a pre-equilibrium model, a simple nucleus explosion |
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56 | model, a fission model, and an evaporation model. |
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57 | |
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58 | The target nucleus is modeled as a three-region approximation to the |
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59 | continuously changing density distribution of nuclear matter within nuclei. |
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60 | The cascade begins when the incident particle strikes a nucleon in the |
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61 | target nucleus and produces secondaries. The secondaries may in turn |
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62 | interact with other nucleons or be absorbed. The cascade ends when all |
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63 | particles, which are kinematically able to do so, escape the nucleus. |
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64 | At that point energy conservation is checked. Relativistic kinematics is |
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65 | applied throughout the cascade. |
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66 | |
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67 | \subsection{Model Limits} |
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68 | |
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69 | The model is valid for incident protons, neutrons and pions. Particles |
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70 | treated in the model include protons, neutrons, pions, photons and nuclear |
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71 | isotopes. All types of targets are allowed. |
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72 | |
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73 | The necessary condition of validity of the INC model is |
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74 | $\lambda_{B} / v << \tau_{c} << \Delta t$, where $\delta_{B}$ is the deBroglie |
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75 | wavelenth of the nucleons, $v$ is the average relative velocity between two |
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76 | nucleons and $\Delta t$ is the time interval between collisions. |
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77 | At energies below $200 MeV$, this condition is no longer strictly valid, |
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78 | and a pre-quilibrium model must be invoked. At energies greater than |
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79 | $\approx$ 10 GeV) the INC picture breaks down. This model has been tested |
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80 | against experimental data at incident kinetic energies between 100~MeV and |
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81 | 5~GeV. |
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82 | |
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83 | \subsection{Intranuclear Cascade Model} |
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84 | |
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85 | The basic steps of the INC model are summarized as follows: |
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86 | |
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87 | \begin{enumerate} |
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88 | \item the space point at which the incident particle enters the nucleus is |
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89 | selected uniformly over the projected area of the nucleus, |
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90 | \item the total particle-particle cross sections and region-depenent nucleon |
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91 | densities are used to select a path length for the projectile, |
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92 | \item the momentum of the struck nucleon, the type of reaction and the |
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93 | four-momenta of the reaction products are determined, and |
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94 | \item the exciton model is updated as the cascade proceeds. |
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95 | \item If the Pauli exclusion principle allows and |
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96 | $E_{particle} > E_{cutoff}$ = 2~MeV, step (2) is performed to transport the |
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97 | products. |
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98 | \end{enumerate} |
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99 | |
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100 | After the intra-nuclear cascade, the residual excitation energy of the |
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101 | resulting nucleus is used as input for non-equilibrium model. |
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102 | |
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103 | \subsection{Nuclear Model} |
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104 | |
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105 | Some of the basic features of the nuclear model are: |
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106 | |
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107 | \begin{itemize} |
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108 | \item the nucleons are assumed to have a Fermi gas momentum distribution. |
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109 | The Fermi energy is calculated in a local density approximation i.e. the |
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110 | Fermi energy is made radius-dependent with Fermi momentum |
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111 | $p_{F}(r) = (\frac{3 \pi^2 \rho(r)}{2})^\frac{1}{3}$. |
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112 | %\item Nuclear density effects are re-calculated after each step, |
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113 | \item Nucleon binding energies (BE) are calculated using the mass formula. |
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114 | A parameterization of the nuclear binding energy uses a combination of the |
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115 | Kummel mass formula and experimental data. Also, the asymptotic high |
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116 | temperature mass formula is used if it is impossible to use experimental data. |
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117 | \end{itemize} |
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118 | |
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119 | \subsubsection{Initialization} |
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120 | The initialization phase fixes the nuclear radius and momentum according to |
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121 | the Fermi gas model. |
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122 | |
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123 | If the target is hydrogen (A = 1) a direct particle-particle collision is |
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124 | performed, and no nuclear modeling is required. |
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125 | |
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126 | If $1 < A < 4$, a nuclear model consisting of one layer with a radius of |
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127 | 8.0 fm is created. |
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128 | |
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129 | For $4 < A < 11$, the nuclear model is composed of three concentric spheres |
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130 | $i = \{1, 2, 3\}$ with radius |
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131 | $$r_{i}(\alpha_{i}) = \sqrt{C_{1}^{2} (1 - \frac{1}{A}) + 6.4} \sqrt{-log( \alpha_{i})}$$. |
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132 | |
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133 | Here $\alpha_{i} = \{0.01, 0.3, 0.7\}$ and $C_{1} = 3.3836 A^{1/3}$. |
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134 | |
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135 | If $A > 11$, a nuclear model with three concentric spheres is also used. The |
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136 | sphere radius is now defined as |
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137 | \begin{equation} |
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138 | r_{i}(\alpha_{i}) = C_{2} \log({\frac{1 + e^{- \frac{C_{1}}{C_{2}}}}{\alpha_{i}} - 1}) + C_{1} , |
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139 | \end{equation} |
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140 | where $C_{2} = 1.7234$. |
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141 | |
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142 | The potential energy $V$ for nucleon $N$ is |
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143 | \begin{equation} |
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144 | V_{N} = \frac{p_{F}^2}{2 m_{N}} + BE_{N}(A, Z) , |
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145 | \end{equation} |
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146 | where $p_f$ is the Fermi momentum and $BE$ is the binding energy. |
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147 | |
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148 | The momentum distribution in each region follows the Fermi distribution with |
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149 | zero temperature. |
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150 | |
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151 | \begin{equation} |
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152 | f(p) = c p ^2 |
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153 | \end{equation} |
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154 | where |
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155 | |
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156 | \begin{equation} |
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157 | \int_0^{p_F} f(p) dp = n_{p} \rm{ or } n_{n} |
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158 | \end{equation} |
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159 | where $n_p$ and $n_n$ are the number of protons or neutrons in the region. |
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160 | $P_f$ is the momentum corresponding to the Fermi energy |
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161 | |
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162 | \begin{equation} |
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163 | E_f = \frac{p_F^2}{2 m_N} = \frac{\hbar^2}{2 m_N}(\frac{3 \pi^{2}}{v})^\frac{2}{3} , |
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164 | \end{equation} |
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165 | which depends on the density $n/v$ of particles, and which is different for |
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166 | each particle and each region. |
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167 | |
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168 | \subsubsection{Pauli Exclusion Principle} |
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169 | The Pauli exclusion principle forbids interactions where the products would be |
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170 | in occupied states. Following the assumption of a completely degenerate Fermi |
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171 | gas, the levels are filled from the lowest level. |
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172 | The minimum energy allowed for the products of a collision correspond to the |
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173 | lowest unfilled level of the system, which is the Fermi energy in the region. |
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174 | So in practice, the Pauli exclusion principle is taken into account by |
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175 | accepting only secondary nucleons which have $E_N > E_f$. |
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176 | |
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177 | |
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178 | \subsubsection{Cross Sections and Kinematics} |
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179 | |
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180 | Path lengths of nucleons in the nucleus are sampled according to the local |
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181 | density and the free $N-N$ cross sections. Angles after the collision are |
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182 | sampled from experimental differential cross sections. |
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183 | %{\sc Geant4} cascade model uses tabulated cross-sections. |
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184 | Tabulated total reaction cross sections are calculated by Letaw's |
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185 | formulation \cite{letaw83, letaw93, pearlstein89}. |
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186 | %:::$45 A^0.7 (1+0.016 sin(5.3-2.63 log10(A)))^(1-0.62 exp(-E / 200) sin(10.9 E^(-0.28))) |
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187 | For $N-N$ cross sections the parameterizations are based on the experimental |
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188 | energy and isospin dependent data. |
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189 | The parameterization described in \cite{barashenkov72} is used. |
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190 | |
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191 | For pions the intra-nuclear cross sections are provided to treat elastic |
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192 | collisions and the following inelastic channels: |
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193 | $\pi^{-}$p $\rightarrow$ $\pi^{0}$n, |
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194 | $\pi^{0}$p $\rightarrow$ $\pi^{+}$n, |
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195 | $\pi^{0}$n $\rightarrow$ $\pi^{-}$p, and |
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196 | $\pi^+$n $\rightarrow$ $\pi^0$p. |
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197 | Multiple particle production is also implemented. |
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198 | |
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199 | The pion absorption channels are |
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200 | $\pi^{+}$nn $\rightarrow$ pn, $\pi^{+}$pn $\rightarrow$ pp, |
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201 | $\pi^{0}$nn $\rightarrow$ nn, $\pi^{0}$pn $\rightarrow$ pn, |
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202 | $\pi^{0}$pp $\rightarrow$ pp, $\pi^{-}$pn $\rightarrow$ nn , and |
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203 | $\pi^{-}$pp $\rightarrow$ pn. |
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204 | |
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205 | \subsection{Pre-equilibrium Model} |
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206 | |
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207 | The {\sc Geant4} cascade model implements the exciton model proposed by |
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208 | Griffin \cite{griffin66, griffin67}. In this model, nucleon states are |
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209 | characterized by the number of excited particles and holes (the excitons). |
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210 | Intra-nuclear cascade collisions give rise to a sequence of states |
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211 | characterized by increasing exciton number, eventually leading to an |
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212 | equilibrated nucleus. For a practical implementation of the exciton model |
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213 | we use parameters from \cite{ribansky73}, (level densities) |
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214 | and \cite{kalbach78} (matrix elements). |
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215 | |
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216 | In the exciton model the possible selection rules for particle-hole |
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217 | configurations in the source of the cascade are: |
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218 | $\Delta p = 0, \pm 1$ $\Delta h = 0, \pm 1$ $\Delta n = 0, \pm 2$, |
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219 | where $p$ is the number of particles, $h$ is number of holes and $n = p + h$ |
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220 | is the number of excitons. |
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221 | |
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222 | The cascade pre-equilibrium model uses target excitation data and the |
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223 | exciton configurations for neutrons and protons to produce non-equilibrium |
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224 | evaporation. The angular distribution is isotropic in the rest frame of the |
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225 | exciton system. |
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226 | |
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227 | Parameterizations of the level density are tabulated as functions of $A$ and |
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228 | $Z$, and with high temperature behavior (the nuclear binding energy using |
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229 | the smooth liquid high energy formula). |
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230 | |
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231 | \subsection{Break-up models} |
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232 | |
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233 | Fermi break-up is allowed only in some extreme cases, i.e. for light nuclei |
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234 | ($A < 12$ and $3 (A - Z) < Z < 6$ ) and $E_{excitation} > 3 E_{binding}$. |
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235 | A simple explosion model decays the nucleus into neutrons and protons and |
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236 | decreases exotic evaporation processes. |
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237 | |
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238 | The fission model is phenomenological, using potential minimization. A binding |
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239 | energy paramerization is used and |
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240 | some features of the fission statistical model are incorporated \cite{fong69}. |
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241 | |
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242 | \subsection{Evaporation Model} |
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243 | |
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244 | A statistical theory for particle emission of the excited nucleus remaining |
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245 | after the intra-nuclear cascade was originally developed by |
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246 | Weisskopf \cite{weisskopf37}. This model assumes complete energy |
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247 | equilibration before particle emission, and re-equilibration of excitation |
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248 | energies between successive evaporations. |
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249 | As a result the angular distribution of emitted particles is isotropic. |
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250 | |
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251 | The {\sc Geant4} evaporation model for the cascade implementation adapts the |
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252 | often-used computational method developed by |
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253 | Dostrowski \cite{dostrovsky59, dostrovsky60}. The emission of particles is |
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254 | computed until the excitation energy falls below some specific cutoff. |
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255 | If a light nucleus is highly excited, the Fermi break-up model is executed. |
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256 | Also, fission is performed if that channel is open. The main chain of |
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257 | evaporation is followed until $E_{excitation}$ falls below |
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258 | E$_{cutoff}$ = 0.1 MeV. The evaporation model ends with an emission chain |
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259 | which is followed until $E_{excitation} < E^{\gamma}_{cutoff} = 10^{-15}$ MeV. |
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260 | |
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261 | \section{Implementation} |
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262 | |
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263 | The cascade model is implemented in the {\sc Geant4} hadronic physics |
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264 | framework. All the models are used collectively through the interface |
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265 | method {\it Apply\-Yourself} defined in the class {\it G4Cascade\-Interface}. |
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266 | A {\sc Geant4} track ({\it G4Track}) and a nucleus ({\it G4Nucleus}) are given |
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267 | as parameters. |
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268 | |
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269 | The cascade models were first tested in release {\sc Geant4 5.0} for energies |
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270 | 100~MeV -- 5~GeV. Detailed comparisons with experimental data have been made |
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271 | in the energy range 160 -- 800 MeV. |
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272 | |
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273 | \section{Status of this document} |
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274 | |
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275 | 01.12.02 created by Aatos Heikkinen, Nikita Stepanov and Hans-Peter Wellisch \\ |
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276 | 14.06.05 grammar, spelling check and list of pion absorption channels |
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277 | corrected by D.H. Wright \\ |
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278 | |
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279 | \begin{latexonly} |
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280 | |
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281 | \begin{thebibliography}{99} |
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282 | |
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283 | \bibitem{alsmiller90} |
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284 | R.G. Alsmiller and F.S. Alsmiller and O.W. Hermann, |
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285 | The high-energy transport code HETC88 and comparisons with experimental data, |
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286 | Nuclear Instruments and Methods in Physics Research A 295, |
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287 | (1990), 337--343, |
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288 | |
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289 | % [1] Barashenkov V.S., Toneev V.D. High Energy interactions of particles and nuclei with nuclei. Moscow, 1972 |
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290 | %(in Russian, but there is an English translation)) |
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291 | |
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292 | \bibitem{barashenkov72} |
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293 | V.S. Barashenkov and V.D. Toneev, |
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294 | High Energy interactions of particles and nuclei with nuclei (In russian), |
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295 | (1972) |
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296 | |
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297 | \bibitem{bertini68} |
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298 | M. P. Guthrie, R. G. Alsmiller and H. W. Bertini, |
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299 | Nucl. Instr. Meth, |
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300 | 66, |
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301 | 1968, |
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302 | 29. |
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303 | % \bibitem{bertini69} |
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304 | % H.W.Bertini, |
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305 | % Intranuclear-Cascade Calculation of the Secondary Nucleon Spectra from Nucleon-Nucleus |
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306 | % Interactions in the Energy Range 340 to 2900 MeV and Comparisons with Experiment, |
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307 | % Phys. Rev., |
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308 | % 188, |
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309 | % 1969, |
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310 | % 1711 |
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311 | % |
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312 | \bibitem{bertini71} |
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313 | H. W. Bertini and P. Guthrie, |
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314 | Results from Medium-Energy Intranuclear-Cascade Calculation, |
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315 | Nucl. Phys.A169, |
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316 | (1971). |
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317 | |
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318 | \bibitem{dostrovsky59} |
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319 | I. Dostrovsky, Z. Zraenkel and G. Friedlander, |
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320 | Monte carlo calculations of high-energy nuclear interactions. III. Application to low-lnergy calculations, |
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321 | Physical Review, |
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322 | 1959, |
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323 | 116, |
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324 | 3, |
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325 | 683-702. |
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326 | |
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327 | \bibitem{dostrovsky60} |
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328 | I. Dostrovsky and Z. Fraenkel and P. Rabinowitz, |
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330 | Physical Review, |
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331 | 1960. |
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332 | |
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333 | \bibitem{fong69} |
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334 | P. Fong, |
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335 | Statistical Theory of Fission, |
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336 | 1969, Gordon and Breach, New York. |
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337 | |
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338 | \bibitem{geant4collaboration03} |
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339 | Geant4 collaboration, |
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340 | Geant4 general paper (to be published), |
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341 | Nuclear Instruments and Methods A, |
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342 | (2003). |
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343 | |
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344 | \bibitem{goldberger48} |
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345 | M. Goldberger, |
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346 | The Interaction of High Energy Neutrons and Hevy Nuclei, |
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347 | Phys. Rev. 74, |
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348 | (1948), |
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349 | 1269. |
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350 | |
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351 | \bibitem{griffin66} |
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352 | J. J. Griffin, |
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353 | Statistical Model of Intermediate Structure, |
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354 | Physical Review Letters 17, (1966), 478-481. |
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355 | |
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356 | \bibitem{griffin67} |
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357 | J. J. Griffin, |
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358 | Statistical Model of Intermediate Structure, |
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359 | Physics Letters 24B, 1 (1967), 5-7. |
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360 | |
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361 | \bibitem{iljinov94} |
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362 | A. S. Iljonov et al., |
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363 | Intermediate-Energy Nuclear Physics, |
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364 | CRC Press 1994. |
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365 | |
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366 | \bibitem{kalbach78} |
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367 | C. Kalbach, |
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368 | Exciton Number Dependence of the Griffin Model Two-Body Matrix Element, |
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369 | Z. Physik A 287, (1978), 319-322. |
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370 | |
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371 | \bibitem{letaw83} |
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372 | J. R. Letaw et al., |
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373 | The Astrophysical Journal Supplements 51, |
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374 | (1983), |
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375 | 271f. |
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376 | |
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377 | \bibitem{letaw93} |
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378 | J. R. Letaw et al., |
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379 | The Astrophysical Journal 414, |
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380 | 1993, |
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381 | 601. |
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382 | |
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383 | \bibitem{metropolis58} |
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384 | N. Metropolis, R. Bibins, M. Storm, |
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385 | Monte Carlo Calculations on Intranuclear Cascades. I. Low-Energy Studies, |
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386 | Physical Review 110, |
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387 | (1958), |
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388 | 185ff. |
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389 | |
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390 | \bibitem{pearlstein89} |
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391 | S. Pearlstein, |
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392 | Medium-energy nuclear data libraries: a case study, neutron- and |
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393 | proton-induced reactions in $^56$Fe, |
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394 | The Astrophysical Journal 346, |
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395 | (1989), |
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396 | 1049-1060. |
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397 | |
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398 | \bibitem{ribansky73} |
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399 | I. Ribansky et al., |
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400 | Pre-equilibrium decay and the exciton model, |
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401 | Nucl. Phys. A 205, |
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402 | (1973), |
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403 | 545-560. |
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404 | |
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405 | \bibitem{serber47} |
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406 | R. Serber, |
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407 | Nuclear Reactions at High Energies, |
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408 | Phys. Rev. 72, |
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409 | (1947), |
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410 | 1114. |
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411 | |
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412 | \bibitem{titarenko99a} |
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413 | Experimental and Computer Simulations Study of |
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414 | Radionuclide Production in Heavy Materials |
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415 | Irradiated by Intermediate Energy Protons, |
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416 | Yu. E. Titarenko et al., |
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417 | nucl-ex/9908012, |
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418 | (1999). |
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419 | |
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420 | \bibitem{weisskopf37} |
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421 | V. Weisskopf, |
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422 | Statistics and Nuclear Reactions, |
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423 | Physical Review 52, |
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424 | (1937), |
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425 | 295--302. |
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426 | |
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427 | \end{thebibliography} |
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428 | |
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429 | \end{latexonly} |
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430 | |
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431 | \begin{htmlonly} |
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432 | |
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433 | \section{Bibliography} |
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434 | |
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435 | \begin{enumerate} |
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436 | \item |
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437 | R.G. Alsmiller and F.S. Alsmiller and O.W. Hermann, |
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438 | The high-energy transport code HETC88 and comparisons with experimental data, |
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439 | Nuclear Instruments and Methods in Physics Research A 295, |
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440 | (1990), 337--343, |
---|
441 | |
---|
442 | % [1] Barashenkov V.S., Toneev V.D. High Energy interactions of particles and nuclei with nuclei. Moscow, 1972 |
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443 | %(in Russian, but there is an English translation)) |
---|
444 | |
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445 | \item |
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446 | V.S. Barashenkov and V.D. Toneev, |
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447 | High Energy interactions of particles and nuclei with nuclei (In russian), |
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448 | (1972) |
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449 | |
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450 | \item |
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451 | M. P. Guthrie, R. G. Alsmiller and H. W. Bertini, |
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452 | Nucl. Instr. Meth, |
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453 | 66, |
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454 | 1968, |
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455 | 29. |
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456 | % \bibitem{bertini69} |
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457 | % H.W.Bertini, |
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458 | % Intranuclear-Cascade Calculation of the Secondary Nucleon Spectra from Nucleon-Nucleus |
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459 | % Interactions in the Energy Range 340 to 2900 MeV and Comparisons with Experiment, |
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460 | % Phys. Rev., |
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461 | % 188, |
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462 | % 1969, |
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463 | % 1711 |
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464 | % |
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465 | \item |
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466 | H. W. Bertini and P. Guthrie, |
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467 | Results from Medium-Energy Intranuclear-Cascade Calculation, |
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468 | Nucl. Phys.A169, |
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469 | (1971). |
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470 | |
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471 | \item |
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472 | I. Dostrovsky, Z. Zraenkel and G. Friedlander, |
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473 | Monte carlo calculations of high-energy nuclear interactions. III. Application to low-lnergy calculations, |
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474 | Physical Review, |
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475 | 1959, |
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476 | 116, |
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477 | 3, |
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478 | 683-702. |
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479 | |
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480 | \item |
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481 | I. Dostrovsky and Z. Fraenkel and P. Rabinowitz, |
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482 | Monte Carlo Calculations of Nuclear Evaporation Processes. V. Emission of Particles Heavier Than $^4He$, |
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483 | Physical Review, |
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484 | 1960. |
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485 | |
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486 | \item |
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487 | P. Fong, |
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488 | Statistical Theory of Fission, |
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489 | 1969, Gordon and Breach, New York. |
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490 | |
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491 | \item |
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492 | Geant4 collaboration, |
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493 | Geant4 general paper (to be published), |
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494 | Nuclear Instruments and Methods A, |
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495 | (2003). |
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496 | |
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497 | \item |
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498 | M. Goldberger, |
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499 | The Interaction of High Energy Neutrons and Hevy Nuclei, |
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500 | Phys. Rev. 74, |
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502 | 1269. |
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503 | |
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504 | \item |
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505 | J. J. Griffin, |
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506 | Statistical Model of Intermediate Structure, |
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507 | Physical Review Letters 17, (1966), 478-481. |
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508 | |
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509 | \item |
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510 | J. J. Griffin, |
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511 | Statistical Model of Intermediate Structure, |
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512 | Physics Letters 24B, 1 (1967), 5-7. |
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513 | |
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514 | \item |
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515 | A. S. Iljonov et al., |
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516 | Intermediate-Energy Nuclear Physics, |
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517 | CRC Press 1994. |
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518 | |
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519 | \item |
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520 | C. Kalbach, |
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521 | Exciton Number Dependence of the Griffin Model Two-Body Matrix Element, |
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522 | Z. Physik A 287, (1978), 319-322. |
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523 | |
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524 | \item |
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525 | J. R. Letaw et al., |
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526 | The Astrophysical Journal Supplements 51, |
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527 | (1983), |
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528 | 271f. |
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529 | |
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530 | \item |
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531 | J. R. Letaw et al., |
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532 | The Astrophysical Journal 414, |
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533 | 1993, |
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534 | 601. |
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535 | |
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536 | \item |
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537 | N. Metropolis, R. Bibins, M. Storm, |
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538 | Monte Carlo Calculations on Intranuclear Cascades. I. Low-Energy Studies, |
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539 | Physical Review 110, |
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540 | (1958), |
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541 | 185ff. |
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542 | |
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543 | \item |
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544 | S. Pearlstein, |
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545 | Medium-energy nuclear data libraries: a case study, neutron- and |
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546 | proton-induced reactions in $^56$Fe, |
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547 | The Astrophysical Journal 346, |
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548 | (1989), |
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549 | 1049-1060. |
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550 | |
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551 | \item |
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552 | I. Ribansky et al., |
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553 | Pre-equilibrium decay and the exciton model, |
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554 | Nucl. Phys. A 205, |
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555 | (1973), |
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556 | 545-560. |
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557 | |
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558 | \item |
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559 | R. Serber, |
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560 | Nuclear Reactions at High Energies, |
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561 | Phys. Rev. 72, |
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563 | 1114. |
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564 | |
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565 | \item |
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566 | Experimental and Computer Simulations Study of |
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567 | Radionuclide Production in Heavy Materials |
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568 | Irradiated by Intermediate Energy Protons, |
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569 | Yu. E. Titarenko et al., |
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570 | nucl-ex/9908012, |
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571 | (1999). |
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572 | |
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573 | \item |
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574 | V. Weisskopf, |
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575 | Statistics and Nuclear Reactions, |
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576 | Physical Review 52, |
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577 | (1937), |
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578 | 295--302. |
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579 | |
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580 | \end{enumerate} |
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581 | |
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582 | \end{htmlonly} |
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583 | |
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584 | |
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