[1211] | 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 | An example of Bertini evaporation model in action is shown in Fig. \ref{models}. |
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| 262 | |
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| 263 | \begin{figure} |
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| 264 | |
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| 265 | %\includegraphics[angle=0,scale=1.0]{nFromSubModels.eps} |
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| 266 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/Cascade/nFromSubModels.eps} |
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| 267 | \caption{Secondary neutrons generated by Bertini INC with exitons and evaporation model.} |
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| 268 | \label{models} |
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| 269 | \end{figure} |
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| 270 | |
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| 271 | \section{Interfacing Bertini implementation} |
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| 272 | Typically Bertini models are used through physics lists, with 'BERT' in their name. |
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| 273 | User should consult these validated physics model collection to understand the inclusion mechanisms |
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| 274 | before using directly the actual Bertini cascade interfaces: |
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| 275 | |
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| 276 | \begin{description} |
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| 277 | \item[G4CascadeInterface] {\em All the Bertini cascade submodels} in integrated fashion, can be used collectively through this interface |
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| 278 | using method {\it Apply\-Yourself}. A {\sc Geant4} track ({\it G4Track}) and a nucleus ({\it G4Nucleus}) are given |
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| 279 | as parameters. |
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| 280 | \item[G4ElasticCascadeInterface] provides an access to elastic hadronic scattering. Particle treated are the same as in case for {\em G4CascadeInterface} but only elastic scattering is modeled. |
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| 281 | \item[G4PreCompoundCascadeInterface] provides an interface to INUCL intra nuclear cascade with exitons. |
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| 282 | Subsequent evaporation phase is {\em not} modeled. |
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| 283 | \item[G4InuclEvaporation] provides an interface to INUCL evaporation model. |
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| 284 | This interface with method {\em BreakItUp} inputs an exited nuclei {\em G4Fragment} to model evaporation phase. |
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| 285 | |
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| 286 | \end{description} |
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| 287 | |
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| 288 | %The cascade models were first tested in release {\sc Geant4 5.0} for energies |
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| 289 | %100~MeV -- 5~GeV. Detailed comparisons with experimental data have been made |
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| 290 | %in the energy range 160 -- 800 MeV. |
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| 291 | |
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| 292 | \section{Status of this document} |
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| 293 | |
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| 294 | 01.12.02 created by Aatos Heikkinen, Nikita Stepanov and Hans-Peter Wellisch \\ |
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| 295 | 14.06.05 grammar, spelling check and list of pion absorption channels |
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| 296 | corrected by D.H. Wright \\ |
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| 297 | 30.05.07 New interfaces documented Aatos Heikkinen |
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| 298 | \begin{latexonly} |
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| 299 | |
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| 300 | \begin{thebibliography}{99} |
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| 301 | |
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| 323 | % H.W.Bertini, |
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| 425 | R. Serber, |
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| 426 | Nuclear Reactions at High Energies, |
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| 427 | Phys. Rev. 72, |
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| 429 | 1114. |
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| 430 | |
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| 431 | \bibitem{titarenko99a} |
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| 432 | Experimental and Computer Simulations Study of |
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| 433 | Radionuclide Production in Heavy Materials |
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| 434 | Irradiated by Intermediate Energy Protons, |
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| 435 | Yu. E. Titarenko et al., |
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| 436 | nucl-ex/9908012, |
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| 437 | (1999). |
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| 438 | |
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| 439 | \bibitem{weisskopf37} |
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| 440 | V. Weisskopf, |
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| 441 | Statistics and Nuclear Reactions, |
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| 442 | Physical Review 52, |
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| 443 | (1937), |
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| 444 | 295--302. |
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| 445 | |
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| 446 | \end{thebibliography} |
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| 447 | |
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| 448 | \end{latexonly} |
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| 449 | |
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| 450 | \begin{htmlonly} |
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| 451 | |
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| 452 | \section{Bibliography} |
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| 453 | |
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| 454 | \begin{enumerate} |
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| 455 | \item |
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| 456 | R.G. Alsmiller and F.S. Alsmiller and O.W. Hermann, |
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| 457 | The high-energy transport code HETC88 and comparisons with experimental data, |
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| 458 | Nuclear Instruments and Methods in Physics Research A 295, |
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| 459 | (1990), 337--343, |
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| 460 | |
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| 461 | % [1] Barashenkov V.S., Toneev V.D. High Energy interactions of particles and nuclei with nuclei. Moscow, 1972 |
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| 462 | %(in Russian, but there is an English translation)) |
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| 463 | |
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| 464 | \item |
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| 465 | V.S. Barashenkov and V.D. Toneev, |
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| 466 | High Energy interactions of particles and nuclei with nuclei (In russian), |
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| 467 | (1972) |
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| 468 | |
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| 469 | \item |
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| 470 | M. P. Guthrie, R. G. Alsmiller and H. W. Bertini, |
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| 471 | Nucl. Instr. Meth, |
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| 472 | 66, |
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| 473 | 1968, |
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| 474 | 29. |
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| 475 | % \bibitem{bertini69} |
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| 476 | % H.W.Bertini, |
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| 477 | % Intranuclear-Cascade Calculation of the Secondary Nucleon Spectra from Nucleon-Nucleus |
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| 478 | % Interactions in the Energy Range 340 to 2900 MeV and Comparisons with Experiment, |
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| 479 | % Phys. Rev., |
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| 480 | % 188, |
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| 481 | % 1969, |
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| 482 | % 1711 |
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| 483 | % |
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| 484 | \item |
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| 485 | H. W. Bertini and P. Guthrie, |
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| 486 | Results from Medium-Energy Intranuclear-Cascade Calculation, |
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| 487 | Nucl. Phys.A169, |
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| 488 | (1971). |
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| 489 | |
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| 490 | \item |
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| 491 | I. Dostrovsky, Z. Zraenkel and G. Friedlander, |
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| 492 | Monte carlo calculations of high-energy nuclear interactions. III. Application to low-lnergy calculations, |
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| 493 | Physical Review, |
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| 494 | 1959, |
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| 495 | 116, |
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| 496 | 3, |
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| 497 | 683-702. |
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| 498 | |
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| 499 | \item |
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| 500 | I. Dostrovsky and Z. Fraenkel and P. Rabinowitz, |
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| 501 | Monte Carlo Calculations of Nuclear Evaporation Processes. V. Emission of Particles Heavier Than $^4He$, |
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| 502 | Physical Review, |
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| 503 | 1960. |
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| 504 | |
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| 505 | \item |
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| 506 | P. Fong, |
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| 507 | Statistical Theory of Fission, |
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| 508 | 1969, Gordon and Breach, New York. |
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| 509 | |
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| 510 | \item |
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| 511 | Geant4 collaboration, |
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| 512 | Geant4 general paper (to be published), |
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| 513 | Nuclear Instruments and Methods A, |
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| 514 | (2003). |
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| 515 | |
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| 516 | \item |
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| 517 | M. Goldberger, |
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| 518 | The Interaction of High Energy Neutrons and Hevy Nuclei, |
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| 519 | Phys. Rev. 74, |
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| 520 | (1948), |
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| 521 | 1269. |
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| 522 | |
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| 523 | \item |
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| 524 | J. J. Griffin, |
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| 525 | Statistical Model of Intermediate Structure, |
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| 526 | Physical Review Letters 17, (1966), 478-481. |
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| 527 | |
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| 528 | \item |
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| 529 | J. J. Griffin, |
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| 530 | Statistical Model of Intermediate Structure, |
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| 531 | Physics Letters 24B, 1 (1967), 5-7. |
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| 532 | |
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| 533 | \item |
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| 534 | A. S. Iljonov et al., |
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| 535 | Intermediate-Energy Nuclear Physics, |
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| 536 | CRC Press 1994. |
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| 537 | |
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| 538 | \item |
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| 539 | C. Kalbach, |
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| 540 | Exciton Number Dependence of the Griffin Model Two-Body Matrix Element, |
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| 541 | Z. Physik A 287, (1978), 319-322. |
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| 542 | |
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| 543 | \item |
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| 544 | J. R. Letaw et al., |
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| 545 | The Astrophysical Journal Supplements 51, |
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| 546 | (1983), |
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| 547 | 271f. |
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| 548 | |
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| 549 | \item |
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| 550 | J. R. Letaw et al., |
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| 551 | The Astrophysical Journal 414, |
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| 552 | 1993, |
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| 553 | 601. |
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| 554 | |
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| 555 | \item |
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| 556 | N. Metropolis, R. Bibins, M. Storm, |
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| 557 | Monte Carlo Calculations on Intranuclear Cascades. I. Low-Energy Studies, |
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| 558 | Physical Review 110, |
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| 559 | (1958), |
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| 560 | 185ff. |
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| 561 | |
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| 562 | \item |
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| 563 | S. Pearlstein, |
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| 564 | Medium-energy nuclear data libraries: a case study, neutron- and |
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| 565 | proton-induced reactions in $^56$Fe, |
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| 566 | The Astrophysical Journal 346, |
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| 567 | (1989), |
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| 568 | 1049-1060. |
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| 569 | |
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| 570 | \item |
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| 571 | I. Ribansky et al., |
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| 572 | Pre-equilibrium decay and the exciton model, |
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| 573 | Nucl. Phys. A 205, |
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| 574 | (1973), |
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| 575 | 545-560. |
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| 576 | |
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| 577 | \item |
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| 578 | R. Serber, |
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| 579 | Nuclear Reactions at High Energies, |
---|
| 580 | Phys. Rev. 72, |
---|
| 581 | (1947), |
---|
| 582 | 1114. |
---|
| 583 | |
---|
| 584 | \item |
---|
| 585 | Experimental and Computer Simulations Study of |
---|
| 586 | Radionuclide Production in Heavy Materials |
---|
| 587 | Irradiated by Intermediate Energy Protons, |
---|
| 588 | Yu. E. Titarenko et al., |
---|
| 589 | nucl-ex/9908012, |
---|
| 590 | (1999). |
---|
| 591 | |
---|
| 592 | \item |
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| 593 | V. Weisskopf, |
---|
| 594 | Statistics and Nuclear Reactions, |
---|
| 595 | Physical Review 52, |
---|
| 596 | (1937), |
---|
| 597 | 295--302. |
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| 598 | |
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| 599 | \end{enumerate} |
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| 600 | |
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| 601 | \end{htmlonly} |
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| 602 | |
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| 603 | |
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