1 | \chapter{Parametrization Driven Models} |
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2 | |
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3 | \section{Introduction} |
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4 | |
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5 | Two sets of parameterized models are provided for the simulation of high |
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6 | energy hadron-nucleus interactions. The so-called ``low energy model'' is |
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7 | intended for hadronic projectiles with incident energies between 1 GeV and |
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8 | 25 GeV, while the ``high energy model'' is valid for projectiles between 25 |
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9 | GeV and 10 TeV. Both are based on the well-known GHEISHA package of GEANT3. |
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10 | The physics underlying these models comes from an old-fashioned multi-chain |
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11 | model in which the incident particle collides with a nucleon inside the |
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12 | nucleus. The final state of this interaction consists of a recoil nucleon, |
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13 | the scattered incident particle, and possibly many hadronic secondaries. |
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14 | Hadron production is approximated by the formation zone concept, in which the |
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15 | interacting quark-partons require some time and therefore some range to |
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16 | hadronize into real particles. All of these particles are able to re-interact |
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17 | within the nucleus, thus developing an intra-nuclear cascade. \\ |
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18 | |
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19 | \noindent |
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20 | In these models only the first hadron-nucleon collision is simulated in |
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21 | detail. The remaining interactions within the nucleus are simulated by |
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22 | generating additional hadrons and treating them as secondaries from the |
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23 | initial collision. The numbers, types and distributions of the extra |
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24 | hadrons are determined by functions which were fitted to experimental data |
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25 | or which reproduce general trends in hadron-nucleus collisions. Numerous |
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26 | tunable parameters are used throughout these models to obtain reasonable |
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27 | physical behavior. This restricts the use of these models as generators for |
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28 | hadron-nucleus interactions because it is not always clear how the parameters |
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29 | relate to physical quantities. On the other hand a precise simulation of |
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30 | minimum bias events is possible, with significant predictive power for |
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31 | calorimetry. |
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32 | |
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33 | \section{Low Energy Model} |
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34 | |
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35 | In the low energy parameterized model the mean number of hadrons produced in |
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36 | a hadron-nucleus collision is given by |
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37 | \begin{equation} |
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38 | \label{pm.eq1} |
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39 | N_m = C(s) A^{1/3} N_{ic} |
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40 | \end{equation} |
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41 | where $A$ is the atomic mass, $C(s)$ is a function only of the center of |
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42 | mass energy $s$, and $N_{ic}$ is approximately the number of hadrons generated |
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43 | in the initial collision. Assuming that the collision occurs at the center of |
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44 | the nucleus, each of these hadrons must traverse a distance roughly equal to |
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45 | the nuclear radius. They may therefore potentially interact with a number of |
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46 | nucleons proportional to $A^{1/3}$. If the energy-dependent cross section |
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47 | for interaction in the nuclear medium is included in $C$ then Eq. \ref{pm.eq1} |
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48 | can be interpreted as the number of target nucleons excited by the initial |
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49 | collision. Some of these nucleons are added to the intra-nuclear cascade. |
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50 | The rest, especially at higher momenta where nucleon production is suppressed, |
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51 | are replaced by pions and kaons. \\ |
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52 | |
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53 | \noindent |
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54 | Once the mean number of hadrons, $N_m$ is calculated, the total number of |
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55 | hadrons in the intra-nuclear cascade is sampled from a Poisson distribution |
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56 | about the mean. Sampling from additional distribution functions provides |
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57 | \begin{itemize} |
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58 | \item the combined multiplicity $w(\vec{a},n_{i})$ for all particles |
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59 | $i$, $i = \pi^{+}, \pi^{0}, \pi^{-}, p, n, .....$, including the |
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60 | correlations between them, |
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61 | \item the additive quantum numbers $E$ (energy), $Q$ (charge), $S$ |
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62 | (strangeness) and $B$ (baryon number) in the entire phase space region, and |
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63 | \item the reaction products from nuclear fission and evaporation. |
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64 | \end{itemize} |
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65 | A universal function $f(\vec{b},x/p_{T},m_{T})$ is used for the distribution |
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66 | of the additive quantum numbers, where $x$ is the Feynman variable, $p_T$ is |
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67 | the transverse momentum and $m_T$ is the transverse mass. $\vec{a}$ and |
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68 | $\vec{b}$ are parameter vectors, which depend on the particle type of the |
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69 | incoming beam and the atomic number $A$ of the target. Any dependence on |
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70 | the beam energy is completely restricted to the multiplicity distribution |
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71 | and the available phase space. \\ |
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72 | |
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73 | \noindent |
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74 | The low energy model can be applied to the $\pi^+$, $\pi^-$, $K^+$, $K^-$, |
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75 | $K^0$ and $\overline{K^0}$ mesons. It can also be applied to the baryons $p$, |
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76 | $n$, $\Lambda$, $\Sigma^+$, $\Sigma^-$, $\Xi^0$, $\Xi^-$, $\Omega^-$, and |
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77 | their anti-particles, as well as the light nuclei, $d$, $t$ and $\alpha$. The |
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78 | model can in principal be applied down to zero projectile energy, but the |
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79 | assumptions used to develop it begin to break down in the sub-GeV region. |
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80 | |
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81 | |
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82 | \section{High Energy Model} |
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83 | |
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84 | The high energy model is valid for incident particle energies from 10-20 GeV |
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85 | up to 10-20 TeV. Individual implementations of the model exist for $\pi^+$, |
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86 | $\pi^-$, $K^+$, $K^-$, $K^0_S$ and $K^0_L$ mesons, and for $p$, $n$, |
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87 | $\Lambda$, $\Sigma^+$, $\Sigma^-$, $\Xi^0$, $\Xi^-$, and $\Omega^-$ baryons |
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88 | and their anti-particles. |
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89 | |
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90 | \subsection{Initial Interaction} |
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91 | In a given implementation, the generation of the final state begins with the |
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92 | selection of a nucleon from the target nucleus. The pion multiplicities |
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93 | resulting from the initial interaction of the incident particle and the target |
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94 | nucleon are then calculated. The total pion multiplicity is taken to be a |
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95 | function of the log of the available energy in the center of mass of the |
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96 | incident particle and target nucleon, and the $\pi^+$, $\pi^-$ and $\pi^0$ |
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97 | multiplicities are given by the KNO distribution. |
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98 | |
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99 | From this initial set of particles, two are chosen at random to be replaced |
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100 | with either a kaon-anti-kaon pair, a nucleon-anti-nucleon pair, or a kaon and |
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101 | a hyperon. The relative probabilities of these options are chosen according |
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102 | to a logarithmically interpolated table of strange-pair and |
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103 | nucleon-anti-nucleon pair cross sections. The particle types of the |
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104 | pair are chosen according to averaged, parameterized cross sections typical at |
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105 | energies of a few GeV. If the increased mass of the new pair causes the total |
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106 | available energy to be exceeded, particles are removed from the initial set as |
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107 | necessary. |
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108 | |
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109 | \subsection{Intra-nuclear Cascade} |
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110 | The cascade of these particles through the nucleus, and the additional |
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111 | particles generated by the cascade are simulated by several models. These |
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112 | include high energy cascading, high energy cluster production, medium energy |
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113 | cascading and medium energy cluster production. For each event, high energy |
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114 | cascading is attempted first. If the available energy is sufficient, this |
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115 | method will likely succeed in producing the final state and the interaction |
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116 | will have been completely simulated. If it fails due to lack of energy or |
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117 | other reasons, the remaining models are called in succession until the final |
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118 | state is produced. If none of these methods succeeds, quasi-elastic |
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119 | scattering is attempted and finally, as a last resort, elastic scattering is |
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120 | performed. These models are responsible for assigning final state momenta to |
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121 | all generated particles, and for checking that, on average, energy and |
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122 | momentum are conserved. |
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123 | |
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124 | \subsection{High Energy Cascading} |
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125 | |
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126 | As particles from the initial collision cascade through the nucleus more |
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127 | particles will be generated. The number and type of these particles are |
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128 | parameterized in terms of the CM energy of the initial particle-nucleon |
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129 | collision. The number of particles produced from the cascade is given |
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130 | roughly by |
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131 | \begin{equation} |
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132 | \label{he.eq1} |
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133 | N_m = C(s) [A^{1/3} - 1] N_{ic} |
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134 | \end{equation} |
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135 | where $A$ is the atomic mass, $C(s)$ is a function only of $s$, the square of |
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136 | the center of mass energy, and $N_{ic}$ is approximately the number of hadrons |
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137 | generated in the initial collision. This can be understood qualitatively |
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138 | by assuming that the collision occurs, on average, at the center of the |
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139 | nucleus. Then each of the $N_{ic}$ hadrons must traverse a distance |
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140 | roughly equal to the nuclear radius. They may therefore potentially interact |
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141 | with a number of nucleons proportional to $A^{1/3}$. If the energy-dependent |
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142 | cross section for interaction in the nuclear medium is included in $C(s)$ then |
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143 | Eq. \ref{he.eq1} can be interpreted as the number of target nucleons excited |
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144 | by the initial collision and its secondaries. |
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145 | |
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146 | Some of these nucleons are added to the intra-nuclear cascade. The rest, |
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147 | especially at higher momenta where nucleon production is suppressed, are |
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148 | replaced by pions, kaons and hyperons. The mean of the total number of |
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149 | hadrons generated in the cascade is partitioned into the mean number of |
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150 | nucleons, $N_n$, pions, $N_\pi$ and strange particles, $N_s$. Each of these |
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151 | is used as the mean of a Poisson distribution which produces the randomized |
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152 | number of each type of particle. |
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153 | |
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154 | The momenta of these particles are generated by first dividing the final state |
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155 | phase space into forward and backward hemispheres, where forward is in the |
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156 | direction of the original projectile. Each particle is assigned to one |
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157 | hemisphere or the other according to the particle type and origin: |
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158 | |
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159 | \begin{itemize} |
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160 | |
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161 | \item the original projectile, or its substitute if charge or strangeness |
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162 | exchange occurs, is assigned to the forward hemisphere and the target nucleon |
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163 | is assigned to the backward hemisphere; |
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164 | |
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165 | \item the remainder of the particles from the initial collision are assigned |
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166 | at random to either hemisphere; |
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167 | |
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168 | \item pions and strange particles generated in the intra-nuclear cascade are |
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169 | assigned 80\% to the backward hemisphere and 20\% to the forward hemisphere; |
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170 | |
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171 | \item nucleons generated in the intra-nuclear cascade are all assigned |
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172 | to the backward hemisphere. |
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173 | |
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174 | \end{itemize} |
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175 | |
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176 | It is assumed that energy is separately conserved for each hemisphere. If |
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177 | too many particles have been added to a given hemisphere, randomly chosen |
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178 | particles are deleted until the energy budget is met. The final state |
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179 | momenta are then generated according to two different algorithms, a cluster |
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180 | model for the backward nucleons from the intra-nuclear cascade, and a |
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181 | fragmentation model for all other particles. Several corrections are then |
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182 | applied to the final state particles, including momentum re-scaling, effects |
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183 | due to Fermi motion, and binding energy subtraction. Finally the |
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184 | de-excitation of the residual nucleus is treated by adding lower energy |
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185 | protons, neutrons and light ions to the final state particle list. \\ |
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186 | |
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187 | \noindent |
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188 | {\bf Fragmentation Model.} This model simulates the fragmentation of the |
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189 | highly excited hadrons formed in the initial projectile-nucleon collision. |
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190 | Particle momenta are generated by first sampling the average transverse |
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191 | momentum $p_T$ from an exponential distribution: |
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192 | |
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193 | \begin{equation} |
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194 | \label{he.eq2} |
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195 | exp [-a {p_T}^b ] |
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196 | \end{equation} |
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197 | where |
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198 | \begin{eqnarray} |
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199 | 1.70 \le a \le 4.00 ; \ |
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200 | 1.18 \le b \le 1.67 . |
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201 | \label{he.eq3} |
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202 | \end{eqnarray} |
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203 | |
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204 | The values of $a$ and $b$ depend on particle type and result from a |
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205 | parameterization of experimental data. The value selected for $p_T$ is |
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206 | then used to set the scale for the determination of $x$, the fraction of |
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207 | the projectile's momentum carried by the fragment. The sampling of $x$ |
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208 | assumes that the invariant cross section for the production of fragments |
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209 | can be given by |
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210 | |
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211 | \begin{equation} |
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212 | \label{he.eq4} |
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213 | E \frac{d^3 \sigma}{dp^3} = \frac{K}{(M^2 x^2 + {p_T}^2)^{3/2}} |
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214 | \end{equation} |
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215 | where $E$ and $p$ are the energy and momentum, respectively, of the produced |
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216 | fragment, and $K$ is a proportionality constant. $M$ is the average |
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217 | transverse mass which is parameterized from data and varies from 0.75 GeV to |
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218 | 0.10 GeV, depending on particle type. Taking $m$ to be the mass of the |
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219 | fragment and noting that |
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220 | \begin{equation} |
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221 | p_z \simeq x E_{proj} |
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222 | \label{he.eq5} |
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223 | \end{equation} |
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224 | in the forward hemisphere and |
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225 | \begin{equation} |
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226 | p_z \simeq x E_{targ} |
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227 | \label{he.eq6} |
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228 | \end{equation} |
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229 | in the backward hemisphere, Eq. \ref{he.eq4} can be re-written to give the |
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230 | sampling function for $x$: |
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231 | \begin{equation} |
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232 | \frac{d^3 \sigma}{dp^3} = \frac{K}{(M^2 x^2 + {p_T}^2)^{3/2}} |
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233 | \frac{1}{ \sqrt{m^2 + {p_T}^2 + x^2 E_i^2} } , |
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234 | \label{he.eq7} |
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235 | \end{equation} |
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236 | where $i = proj$ or $targ$. |
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237 | |
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238 | $x$-sampling is performed for each fragment in the final-state candidate list. |
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239 | Once a fragment's momentum is assigned, its total energy is checked to see |
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240 | if it exceeds the energy budget in its hemisphere. If so, the momentum of |
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241 | the particle is reduced by 10\%, as is $p_T$ and the integral of the |
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242 | $x$-sampling function, and the momentum selection process is repeated. If |
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243 | the offending particle starts out in the forward hemisphere, it is moved to |
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244 | the backward hemisphere, provided the budget for the backward hemisphere is |
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245 | not exceeded. If, after six iterations, the particle still does not fit, it |
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246 | is removed from the candidate list and the kinetic energies of the particles |
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247 | selected up to this point are reduced by 5\%. The entire procedure is |
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248 | repeated up to three times for each fragment. |
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249 | |
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250 | The incident and target particles, or their substitutes in the case of charge- |
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251 | or strangeness-exchange, are guaranteed to be part of the final state. |
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252 | They are the last particles to be selected and the remaining energy in their |
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253 | respective hemispheres is used to set the $p_z$ components of their momenta. |
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254 | The $p_T$ components selected by $x$-sampling are retained. \\ |
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255 | |
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256 | \noindent |
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257 | {\bf Cluster Model.} This model groups the nucleons produced in the |
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258 | intra-nuclear cascade together with the target nucleon or hyperon, and |
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259 | treats them as a cluster moving forward in the center of mass frame. The |
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260 | cluster disintegrates in such a way that each of its nucleons is given a |
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261 | kinetic energy |
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262 | \begin{equation} |
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263 | 40 < T_{nuc} < 600 \rm{MeV} |
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264 | \label{he.eq8} |
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265 | \end{equation} |
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266 | if the kinetic energy of the original projectile, $T_{inc}$, is 5 GeV or more. |
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267 | If $T_{inc}$ is less than 5 GeV, |
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268 | \begin{equation} |
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269 | 40 ( T_{inc}/5 \rm{GeV} )^2 < T_{nuc} < 600 ( T_{inc}/5 \rm{GeV} )^2 . |
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270 | \label{he.eq9} |
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271 | \end{equation} |
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272 | In each range the energy is sampled from a distribution which is skewed |
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273 | strongly toward the high energy limit. In addition, the angular distribution |
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274 | of the nucleons is skewed forward in order to simulate the forward motion |
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275 | of the cluster. \\ |
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276 | |
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277 | \noindent |
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278 | {\bf Momentum Re-scaling.} Up to this point, all final state momenta have |
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279 | been generated in the center of mass of the incident projectile and the |
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280 | target nucleon. However, the interaction involves more than one nucleon |
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281 | as evidenced by the intra-nuclear cascade. A more correct center of mass |
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282 | should then be defined by the incident projectile and all of the baryons |
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283 | generated by the cascade, and the final state momenta already calculated |
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284 | must be re-scaled to reflect the new center of mass. |
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285 | |
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286 | This is accomplished by correcting the momentum of each particle in the |
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287 | final state candidate list by the factor $T_1 / T_2$. $T_2$ is the total |
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288 | kinetic energy in the lab frame of all the final state candidates generated |
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289 | assuming a projectile-nucleon center of mass. $T_1$ is the total kinetic |
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290 | energy in the lab frame of the same final state candidates, but whose momenta |
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291 | have been calculated by the phase space decay of an imaginary particle. |
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292 | This particle has the total CM energy of the original projectile and a |
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293 | cluster consisting of all the baryons generated from the intra-nuclear |
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294 | cascade. \\ |
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295 | |
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296 | \noindent |
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297 | {\bf Corrections.} Part of the Fermi motion of the target nucleons is taken |
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298 | into account by smearing the transverse momentum components of the final |
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299 | state particles. The Fermi momentum is first sampled from an average |
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300 | distribution and a random direction for its transverse component is chosen. |
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301 | This component, which is proportional to the number of baryons produced in |
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302 | the cascade, is then included in the final state momenta. |
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303 | |
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304 | Each final state particle must escape the nucleus, and in the process reduce |
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305 | its kinetic energy by the nuclear binding energy. The binding energy is |
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306 | parameterized as a function of $A$: |
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307 | \begin{equation} |
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308 | E_B = 25\rm{MeV} \left( \frac{A-1}{120} \right) e^{-(A-1)/120)} . |
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309 | \label{he.eq10} |
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310 | \end{equation} |
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311 | |
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312 | Another correction reduces the kinetic energy of final state $\pi^0$s when |
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313 | the incident particle is a $\pi^+$ or $\pi^-$. This reduction increases |
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314 | as the log of the incident pion energy, and is done to reproduce |
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315 | experimental data. In order to conserve energy on average, the energy |
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316 | removed from the $\pi^0$s is re-distributed among the final state $\pi^+$s, |
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317 | $\pi^-$s and $\pi^0$s. \\ |
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318 | |
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319 | \noindent |
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320 | {\bf Nuclear De-excitation.} After the generation of initial interaction |
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321 | and cascade particles, the target nucleus is left in an excited state. |
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322 | De-excitation is accomplished by evaporating protons, neutrons, deuterons, |
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323 | tritons and alphas from the nucleus according to a parameterized model. |
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324 | The total kinetic energy given to these particles is a function of the |
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325 | incident particle kinetic energy: |
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326 | \begin{equation} |
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327 | T_{evap} = 7.716 \rm{GeV} \left( \frac{A-1}{120} \right) F(T) e^{-F(T) - (A-1)/120} , |
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328 | \end{equation} |
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329 | where |
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330 | \begin{equation} |
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331 | F(T) = \rm{max} [ 0.35 + 0.1304 ln(T) , 0.15 ] , |
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332 | \end{equation} |
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333 | and |
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334 | \begin{eqnarray} |
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335 | T = 0.1 \rm{GeV} \quad \rm{for} \quad T_{inc} < 0.1 \rm{GeV} \\ |
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336 | T = T_{inc} \quad \rm{for} \quad 0.1 \rm{GeV} \le T_{inc} \le 4 \rm{GeV} \\ |
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337 | T = 4 \rm{GeV} \quad \rm{for} \quad T_{inc} > 4 \rm{GeV} . |
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338 | \end{eqnarray} |
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339 | The mean energy allocated for proton and neutron emission is |
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340 | $\overline{T_{pn}}$ and that for deuteron, triton and alpha emission is |
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341 | $\overline{T_{dta}}$. These are determined by partitioning $T_{evap}$ : |
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342 | \begin{eqnarray} |
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343 | \overline{T_{pn}} = T_{evap} R \quad , \quad \overline{T_{dta}} = T_{evap} (1-R) \quad \rm{with} \nonumber |
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344 | \end{eqnarray} |
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345 | \begin{eqnarray} |
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346 | R = \rm{max}[ 1 - (T/4\rm{GeV})^2 , 0.5 ] . |
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347 | \end{eqnarray} |
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348 | The simulated values of $T_{pn}$ and $T_{dta}$ are sampled from normal |
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349 | distributions about $\overline{T_{pn}}$ and $\overline{T_{dta}}$ and their |
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350 | sum is constrained not to exceed the incident particle's kinetic energy, |
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351 | $T_{inc}$. |
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352 | |
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353 | The number of proton and neutrons emitted, $N_{pn}$, is sampled from a |
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354 | Poisson distribution about a mean which depends on $R$ and the number of |
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355 | baryons produced in the intranuclear cascade. The average kinetic energy |
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356 | per emitted particle is then $T_{av} = T_{pn}/N_{pn}$. $T_{av}$ is used |
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357 | to parameterize an exponential which qualitatively describes the nuclear |
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358 | level density as a function of energy. The simulated kinetic energy of |
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359 | each evaporated proton or neutron is sampled from this exponential. Next, |
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360 | the nuclear binding energy is subtracted and the final momentum is |
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361 | calculated assuming an isotropic angular distribution. The number of |
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362 | protons and neutrons emitted is $(Z/A)N_{pn}$ and $(N/A)N_{pn}$, |
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363 | respectively. |
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364 | |
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365 | A similar procedure is followed for the deuterons, tritons and |
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366 | alphas. The number of each species emitted is $0.6 N_{dta}$, $0.3 N_{dta}$ |
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367 | and $0.1 N_{dta}$, respectively. \\ |
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368 | |
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369 | \noindent |
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370 | {\bf Tuning of the High Energy Cascade} The final stage of the high |
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371 | energy cascade method involves adjusting the momenta of the produced |
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372 | particles so that they agree better with data. Currently, five such |
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373 | adjustments are performed, the first three of which apply only to |
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374 | charged particles incident upon light and medium nuclei at incident |
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375 | energies above $\simeq$ 65 GeV. |
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376 | |
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377 | \begin{itemize} |
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378 | |
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379 | \item If the final state particle is a nucleon or light ion with a |
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380 | momentum of less than 1.5 GeV/c, its momentum will be set to zero |
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381 | some fraction of the time. This fraction increases with the logarithm |
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382 | of the kinetic energy of the incident particle and decreases with |
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383 | $log_{10}(A)$. |
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384 | |
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385 | \item If the final state particle with the largest momentum happens |
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386 | to be a $\pi^0$, its momentum is exchanged with either the $\pi^+$ |
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387 | or $\pi^-$ having the largest momentum, depending on whether the |
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388 | incident particle charge is positive or negative. |
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389 | |
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390 | \item If the number of baryons produced in the cascade is a significant |
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391 | fraction ($ > 0.3 $) of $A$, about 25\% of the nucleons and light ions |
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392 | already produced will be removed from the final particle list, provided |
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393 | their momenta are each less than 1.2 GeV/c. |
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394 | |
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395 | \item The final state of the interaction is of course heavily |
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396 | influenced by the quantum numbers of the incident particle, particularly |
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397 | in the forward direction. This influence is enforced by compiling, for |
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398 | each forward-going final state particle, the sum |
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399 | \begin{equation} |
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400 | S_{forward} = \Delta_M + \Delta_Q + \Delta_S + \Delta_B, |
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401 | \end{equation} |
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402 | where each $\Delta$ corresponds to the absolute value of the |
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403 | difference of the quantum number between the incident particle and |
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404 | the final state particle. $M$, $Q$, $S$, and $B$ refer to mass, |
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405 | charge, strangeness and baryon number, respectively. For final |
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406 | state particles whose character is significantly different from the |
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407 | incident particle ($S$ is large), the momentum component parallel to |
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408 | the incident particle momentum is reduced. The transverse component |
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409 | is unchanged. As a result, large-$S$ particles are driven away from |
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410 | the axis of the hadronic shower. For backward-going particles, a |
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411 | similar procedure is followed based on the calculation of $S_{backward}$. |
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412 | |
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413 | \item Conservation of energy is imposed on the particles in the |
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414 | final state list in one of two ways, depending on whether or not |
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415 | a leading particle has been chosen from the list. If all the |
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416 | particles differ significantly from the incident particle in |
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417 | momentum, mass and other quantum numbers, no leading particle is |
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418 | chosen and the kinetic energy of each particle is scaled by the |
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419 | same correction factor. If a leading particle is chosen, its |
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420 | kinetic energy is altered to balance the total energy, while all |
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421 | the remaining particles are unaltered. |
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422 | |
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423 | \end{itemize} |
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424 | |
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425 | \subsection{High Energy Cluster Production} |
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426 | |
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427 | As in the high energy cascade model, the high energy cluster model |
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428 | randomly assigns particles from the initial collision to either a |
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429 | forward- or backward-going cluster. Instead of performing the |
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430 | fragmentation process, however, the two clusters are treated |
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431 | kinematically as the two-body final state of the hadron-nucleon |
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432 | collision. Each cluster is assigned a kinetic energy $T$ which is |
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433 | sampled from a distribution |
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434 | \begin{equation} |
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435 | exp[-aT^{1/b}] |
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436 | \end{equation} |
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437 | where both $a$ and $b$ decrease with the number of particles in a |
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438 | cluster. If the combined total energy of the two clusters is |
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439 | larger than the center of mass energy, the energy of each cluster |
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440 | is reduced accordingly. The center of mass momentum of each |
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441 | cluster can then be found by sampling the 4-momentum transfer |
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442 | squared, $t$, from the distribution |
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443 | \begin{equation} |
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444 | exp [t (4.0 + 1.6ln(p_{inc}) ) ] |
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445 | \end{equation} |
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446 | where $t < 0$ and $p_{inc}$ is the incident particle momentum. |
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447 | Then, |
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448 | \begin{equation} |
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449 | cos\theta = 1 + \frac{t - (E_c - E_i)^2 + (p_c - p_i)^2}{2p_c p_i}, |
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450 | \end{equation} |
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451 | where the subscripts $c$ and $i$ refer to the cluster and incident |
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452 | particle, respectively. Once the momentum of each cluster is |
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453 | calculated, the cluster is decomposed into its constituents. The |
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454 | momenta of the constituents are determined using a phase space decay |
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455 | algorithm. |
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456 | |
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457 | The particles produced in the intra-nuclear cascade are grouped into |
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458 | a third cluster. They are treated almost exactly as in the high |
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459 | energy cascade model, where Eq. \ref{he.eq1} is used to estimate the |
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460 | number of particles produced. The main difference is that the cluster |
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461 | model does not generate strange particles from the cascade. Nucleon |
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462 | suppression is also slightly stronger, leading to relatively higher |
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463 | pion production at large incident momenta. Kinetic energy and |
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464 | direction are assigned to the cluster as described in the cluster |
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465 | model paragraph in the previous section. |
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466 | |
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467 | The remaining steps to produce the final state particle list are |
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468 | the same as those in high energy cascading: |
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469 | \begin{itemize} |
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470 | \item re-scaling of the momenta to reflect a center of mass which |
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471 | involves the cascade baryons, |
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472 | \item corrections due to Fermi motion and binding energy, |
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473 | \item reduction of final state $\pi^0$ energies, |
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474 | \item nuclear de-excitation and |
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475 | \item high energy tuning. |
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476 | \end{itemize} |
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477 | |
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478 | \subsection{Medium Energy Cascading} |
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479 | The medium energy cascade algorithm is very similar to the high |
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480 | energy cascade algorithm, but it may be invoked for lower incident |
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481 | energies (down to 1 GeV). The primary difference between the two |
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482 | codes is the parameterization of the fragmentation process. The |
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483 | medium energy cascade samples larger transverse momenta for pions |
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484 | and smaller transverse momenta for kaons and baryons. |
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485 | |
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486 | A second difference is in the treatment of the cluster consisting |
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487 | of particles generated in the cascade. Instead of parameterizing |
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488 | the kinetic energies and angles of the outgoing particles, the |
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489 | phase space decay approach is used. |
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490 | |
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491 | Another difference is that the high energy tuning of the final |
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492 | state distribution is not performed. |
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493 | |
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494 | \subsection{Medium Energy Cluster Production} |
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495 | The medium energy cluster algorithm is nearly identical to the |
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496 | high energy cluster algorithm, but it may be invoked for incident |
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497 | energies down to 10 MeV. There are three significant differences |
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498 | at medium energy: less nucleon suppression, fewer particles |
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499 | generated in the intra-nuclear cascade, and no high energy tuning |
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500 | of the final state particle distributions. |
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501 | |
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502 | \subsection{Elastic and Quasi-elastic Scattering} |
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503 | When no additional particles are produced in the initial |
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504 | interaction, either elastic or quasi-elastic scattering is |
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505 | performed. If there is insufficient energy to induce an |
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506 | intra-nuclear cascade, but enough to excite the target nucleus, |
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507 | quasi-elastic scattering is performed. The final state is |
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508 | calculated using two-body scattering of the incident particle |
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509 | and the target nucleon, with the scattering angle in the center |
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510 | of mass sampled from an exponential: |
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511 | \begin{equation} |
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512 | exp [ - 2 b p_{in} p_{cm} (1 - cos\theta) ] . |
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513 | \end{equation} |
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514 | Here $p_{in}$ is the incident particle momentum, $p_{cm}$ is the |
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515 | momentum in the center of mass, and $b$ is a logarithmic function |
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516 | of the incident momentum in the lab frame as parameterized from |
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517 | data. As in the cascade and cluster production models, the |
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518 | residual nucleus is then de-excited by evaporating nucleons and |
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519 | light ions. |
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520 | |
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521 | If the incident energy is too small to excite the nucleus, |
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522 | elastic scattering is performed. The angular distribution of |
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523 | the scattered particle is sampled from the sum of two |
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524 | exponentials whose parameters depend on $A$. |
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525 | |
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526 | |
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527 | \section{Status of this document} |
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528 | |
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529 | 7.10.02 re-written by D.H. Wright \\ |
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530 | 1.11.04 new section on high energy model by D.H. Wright \\ |
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