[1208] | 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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