1 | \chapter{Leading Particle Bias} |
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2 | |
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3 | \section{Overview} |
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4 | {\it G4Mars5GeV} is an inclusive event generator for hadron\,(photon) |
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5 | interactions with nuclei, and translated from the MARS code system\,(MARS13\,(98)). |
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6 | To construct a cascade tree, only a fixed |
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7 | number of particles are generated at each vertex. |
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8 | A corresponding statistical weight is assigned to each secondary particle |
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9 | according to its type and phase-space. Rarely-produced particles |
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10 | or interesting phase-space region can be enhanced. \\ [2mm] |
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11 | %% |
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12 | N.B. This inclusive simulation is implemented in Geant4 partially |
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13 | for the moment, not completed yet. \\ [3mm] |
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14 | %% |
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15 | {\bf MARS Code System} \\ |
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16 | MARS is a set of Monte Carlo programs for inclusive simulation |
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17 | of particle interactions, and high multiplicity or rare events |
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18 | can be simulated fast with its sophisticated biasing techniques. |
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19 | For the details on the MARS code system, see \cite{MARS98, MARSWWW}. \\ |
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20 | |
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21 | \section{Method} |
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22 | In {\it G4Mars5GeV}, three secondary hadrons are generated |
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23 | in the final state of an hadron(photon)-nucleus inelastic interaction, |
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24 | and a statistical weight is assigned to each particle |
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25 | according to its type, energy and emission angle. |
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26 | %% what really affects on weights? |
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27 | %% |
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28 | In this code, energies, momenta and weights of the secondaries are sampled, |
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29 | and the primary particle is simply terminated at the vertex. |
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30 | The allowed projectile kinetic energy is $E_0 \leq $~5~GeV, and |
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31 | following particles can be simulated; |
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32 | $$ |
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33 | p,~n,~\pi^+,~\pi^-,~K^+,~K^-,~\gamma,~\bar{p}~. |
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34 | $$ |
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35 | |
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36 | Prior to a particle generation, a Coulomb barrier is considered |
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37 | for projectile charged hadrons\,($p$, $\pi^+$, $K^+$ and $\bar{p}$) |
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38 | with kinetic energy of less than 200 MeV. The coulomb potential |
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39 | $V_{\rm coulmb}$ is given by |
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40 | \begin{equation} |
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41 | V_{\rm columb} = 1.11\times10^{-3} \times Z/A^{1/3}~~~~{\rm (GeV)}, |
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42 | \end{equation} |
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43 | where $Z$~and~$A$ are atomic and mass number, respectivelly. \\ [2mm] |
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44 | %% |
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45 | %% secondary particle generation |
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46 | %% |
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47 | \subsection{Inclusive hadron production} |
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48 | %% |
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49 | The following three steps are carried out in a sequence |
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50 | to produce secondary particles: |
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51 | \begin{itemize} |
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52 | \item nucleon production, |
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53 | \item charged pion/kaon production and |
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54 | \item neutral pion production. |
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55 | \end{itemize} |
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56 | %% |
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57 | These processes are performed independently, i.e. |
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58 | the energy and momentum conservation law is broken at each event, however, |
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59 | fulfilled on the average over a number of events simulated. \\ [5mm] |
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60 | %% |
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61 | %% -- nucleon production |
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62 | {\bf nucleon production} \\ [1mm] |
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63 | Projectiles $K^\pm$ and $\bar{p}$ are |
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64 | replaced with $\pi^\pm$ and $p$, individually to generate |
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65 | the secondary nucleon. |
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66 | Either of neutron or proton is selected randomly as the secondary |
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67 | except for the case of gamma projectiling. |
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68 | The gamma is handled as a pion. \\ [5mm] |
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69 | %% reason?? |
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70 | %% |
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71 | %% -- nucleon production |
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72 | {\bf charged pion/kaon production} \\ [1mm] |
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73 | If the incident nucleon does not have enough energy to produce |
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74 | the pion\,($> 280$ MeV), charged and neutral pions are not produced. |
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75 | A charged pion is selected with the equal probability, and a bias is eliminated |
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76 | with the appropriate weight which is assigned taking into account the difference between |
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77 | $\pi^+$ and $\pi^-$ both for production probability and for inclusive spectra. |
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78 | It is replaced with a charged kaon a certain fraction |
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79 | of the time, that depends on the projectile energy if $E_0 > 2.1$~GeV. |
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80 | The ratio of kaon replacement is given by |
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81 | \begin{equation} |
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82 | R_{\rm kaon} = 1.3 \times \biggl\{ C_{\rm min} + ( C - C_{\rm min} ) |
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83 | \frac{ \log(E_0/2) }{ \log(100/2) } \biggr\}~~~~~ |
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84 | ( 2.1 \leq E_0 \leq 5.2~{\rm GeV} ), |
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85 | \end{equation} |
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86 | where $C_{\rm min}$ is 0.03\,(0.08) for nucleon\,(others) projectiling, and \\ |
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87 | {\it ~\hspace*{35mm}Produced particle~\hspace*{5mm} |
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88 | Projectile particle} |
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89 | \begin{equation} |
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90 | C = \left\{ |
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91 | \begin{array}{ll} |
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92 | 0.071 & (~\pi^+~) \\ |
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93 | 0.083 & (~\pi^-~) |
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94 | \end{array} \right\} |
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95 | \times |
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96 | \left\{ |
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97 | \begin{array}{ll} |
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98 | 1.3 & (~\pi^\pm~) \\ |
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99 | 2.0 & (~K^\pm~) \\ |
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100 | 1.0 & (~{\rm others}~) |
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101 | \end{array} \right\} |
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102 | \end{equation} |
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103 | A similar strangeness replacement is not considered for nucleon production. |
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104 | \\ [3mm] |
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105 | %% |
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106 | %% -- nucleon production |
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107 | %% |
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108 | %% |
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109 | |
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110 | \subsection{Sampling of energy and emission angle of the secondary} |
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111 | % |
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112 | The energy and emission angle of the secondary particle depends on |
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113 | projectile energy. There are formulae depending on whether or not |
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114 | the interaction particle\,(IP) is identical to the secondary\,(JP). \\ [2mm] |
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115 | %% |
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116 | %% |
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117 | For IP $\neq$ JP, the secondary energy $E_2$ is simply given by |
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118 | \begin{equation} |
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119 | E_2 = E_{\rm th} \times \biggl(\frac{E_{\rm max}} |
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120 | {E_{\rm th}}\biggr)^\epsilon~~~~~~~~~~~~~~{\rm (MeV)}~, |
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121 | \end{equation} |
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122 | where $E_{\rm max} = max\,(E_0,0.5\,{\rm MeV})$, $E_{\rm th} = 1$ MeV, |
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123 | and $\epsilon$ is a uniform random between 0 and 1. \\ [2mm] |
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124 | %% |
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125 | %% |
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126 | For IP = JP, |
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127 | \begin{equation}\label{eq:energy} |
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128 | E_2 = \left\{ |
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129 | \begin{array}{l l l} |
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130 | E_{\rm th} + \epsilon\,(E_{\rm max} - E_{\rm th}) & |
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131 | & E_0 < 100\,E_{\rm th}~{\rm MeV} \\ |
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132 | E_{\rm th}\times\,e^{\epsilon\,(\beta+99)} & {\rm (MeV)} |
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133 | & E_0 \geq 100\,E_{\rm th}~{\rm and}~\epsilon < \eta \\ |
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134 | E_0\times\bigl( \beta\,(\epsilon - 1) + 1 + 99\epsilon )/100 & |
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135 | & E_0 \geq 100\,E_{\rm th}~{\rm and}~\epsilon \geq \eta |
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136 | \end{array} \right. |
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137 | \end{equation} |
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138 | Here, $\beta = \log(E_0/100\,E_{\rm th})$ and |
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139 | $\eta = \beta/(99 + \beta)$. |
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140 | If resulting $E_2$ is less than 0.5 MeV, nothing is generated. \\ [5mm] |
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141 | % |
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142 | % |
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143 | {\bf Angular distribution} \\ [1mm] |
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144 | % |
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145 | The angular distribution is mainly determined by the energy ratio |
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146 | of the secondary to the projectile\,(i.e. |
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147 | the emission angle and probability of the occurrence |
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148 | increase as the energy ratio decreases). |
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149 | The emission angle of the secondary particle with respect to the incident direction |
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150 | is given by |
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151 | \begin{equation}\label{eq:angle} |
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152 | \theta = - \log \bigl( 1 - \epsilon ( 1 - e^{-\pi\,\tau} ) \bigr)/ |
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153 | \tau~, |
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154 | \end{equation} |
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155 | where $\tau = E_0/5(E_0 + 1/2)$. \\ |
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156 | %% |
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157 | %% |
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158 | \subsection{Sampling statistical weight} |
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159 | The kinematics of the secondary particle are determined randomly |
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160 | using the above formulae\,(\ref{eq:energy},\ref{eq:angle}). |
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161 | A statistical weight is calculated and assigned to each generated particle |
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162 | to reproduce a true inclusive spectrum in the event. |
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163 | The weight is given by |
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164 | \begin{equation} |
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165 | D2N = V10({\rm JP}) \times DW\,(E) \times DA\,(\theta) \times |
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166 | V1\,(E,\theta,{\rm JP}), |
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167 | \end{equation} |
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168 | %% |
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169 | where \\ |
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170 | $\bullet~V10$ is the statistical weight for the production rate |
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171 | based on neutral pion production\,($V10 = 1$). |
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172 | \begin{equation} |
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173 | V10 = \left\{ |
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174 | \begin{array}{ll} |
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175 | 2.0\,(2.5) & {\rm nucleon~production\,(the~case~of~gamma~projectile)} \\ |
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176 | 2.1 & {\rm charged~pion/kaon~production} |
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177 | \end{array} \right. |
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178 | \end{equation}\vspace*{2mm}\\ |
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179 | %% |
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180 | %% |
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181 | $\bullet$~DW and DA are dominantly determined by the secondary energy and |
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182 | emission angle, individually. \\ [2mm] |
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183 | %% |
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184 | %% |
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185 | $\bullet$~V1 is a true double-differential production cross-section (divided by the total |
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186 | inelastic cross-section)~\cite{MARS98}, calculated in {\it G4Mars5GeV::D2N2} |
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187 | according to the projectile type and energy, target atomic mass, and simulated secondary |
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188 | energy, emission angle and particle type. \\ |
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189 | |
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190 | \section{Status of this document} |
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191 | 11.06.2002 created by N. Kanaya. \\ |
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192 | 20.06.2002 modified by N.V.Mokhov. \\ |
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193 | |
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194 | |
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195 | \begin{latexonly} |
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196 | |
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197 | \begin{thebibliography}{599} |
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198 | \bibitem{MARS98} N.V.~Mokhov, {\it The MARS Code System User's Guide, Version 13(98)}, |
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199 | Fermilab-FN-628. |
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200 | |
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201 | \bibitem{MARSWWW} {\it http://www-ap.fnal.gov/MARS/} |
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202 | \end{thebibliography} |
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203 | |
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204 | \end{latexonly} |
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205 | |
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206 | \begin{htmlonly} |
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207 | |
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208 | \section{Bibliography} |
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209 | |
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210 | \begin{enumerate} |
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211 | \item N.V.~Mokhov, {\it The MARS Code System User's Guide, Version 13(98)}, |
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212 | Fermilab-FN-628. |
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213 | |
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214 | \item {\it http://www-ap.fnal.gov/MARS/} |
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215 | \end{enumerate} |
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216 | |
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217 | \end{htmlonly} |
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218 | |
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