1 | |
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2 | \section{Gflash Shower Parameterizations} |
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3 | |
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4 | The computing time needed for the simulation of high energy electromagnetic |
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5 | showers can become very large, since it increases approximately linearly with the |
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6 | energy absorbed in the detector. Using parameterizations instead of |
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7 | individual particle tracking for electromagnetic (sub)showers can speed up |
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8 | the simulations considerably without sacrificing much precision. The Gflash |
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9 | package allows the parameterization of electron and positron showers in |
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10 | homogeneous (for the time being) calorimeters and is based on the |
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11 | parameterization described in Ref. \cite{para.grind} . |
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12 | |
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13 | \subsection{Parameterization Ansatz} |
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14 | The spatial energy distribution of electromagnetic showers is given by |
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15 | three probability density functions (pdf), |
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16 | \begin{equation} |
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17 | dE (\vec{r}) \, = \, E\, f(t)dt\, f(r)dr\, f(\phi) d\phi , |
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18 | \end{equation} |
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19 | describing the longitudinal, radial, and azimuthal energy distributions. |
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20 | Here $t$ denotes the longitudinal shower depth in units of radiation length, |
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21 | $r$ measures the radial distance from the shower axis in Moli\`{e}re units, |
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22 | and $\phi$ is the azimuthal angle. The start of the shower is defined by the |
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23 | space point where the electron or positron enters the calorimeter, which |
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24 | is different from the original Gflash. |
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25 | A gamma distribution is used for the parameterization of the longitudinal |
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26 | shower profile, $f(t)$. The radial distribution $f(r)$, is described by a |
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27 | two-component ansatz. In $\phi$, it is assumed that the energy is distributed |
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28 | uniformly: $ f(\phi) = 1/2\pi $. |
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29 | |
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30 | \subsection{Longitudinal Shower Profiles } |
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31 | \label{sec_hom_long} |
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32 | The average longitudinal shower profiles can be described by a gamma |
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33 | dis\-tri\-bution \cite{para.longo}: |
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34 | \begin{equation} |
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35 | \left\langle \frac{1}{E} \frac{dE(t)}{dt} \right\rangle |
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36 | \, = \, f(t) \, = \, |
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37 | \frac{ (\beta t)^{\alpha -1} \beta \exp(-\beta t) } |
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38 | { \Gamma(\alpha) }. |
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39 | \end{equation} |
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40 | |
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41 | The center of gravity, $\langle t \rangle$, and the depth of the maximum, |
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42 | $T$, are calculated from the shape parameter $\alpha$ and the scaling |
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43 | parameter $\beta$ according~to: |
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44 | \begin{eqnarray} |
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45 | \langle t \rangle & = & \frac{\alpha}{\beta}\\ |
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46 | T & = & \frac{\alpha-1}{\beta}. |
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47 | \label{talp} |
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48 | \end{eqnarray} |
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49 | |
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50 | In the parameterization all lengths are measured in units of radiation length |
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51 | $(X_0)$, and energies in units of the critical energy |
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52 | ($E_c= 2.66 \left( X_0 \frac{Z}{A} \right)^{1.1}$~). |
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53 | This allows material independence, since the longitudinal shower moments are |
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54 | equal in different materials, according to Ref. \cite{para.rossi}. |
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55 | The following equations are used for the energy dependence of $T_{hom}$ and |
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56 | $(\alpha_{hom})$, |
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57 | with $y = E/E_c$ and $t=x/X_0$, x being the longitudinal shower depth: |
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58 | \begin{eqnarray} |
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59 | \label{e_thom} |
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60 | T_{hom} & = & \ln y + t_1 \\ |
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61 | \label{e_ahom} |
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62 | \alpha_{hom} & = & a_1 + (a_2 + a_3/Z) \ln y. |
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63 | \end{eqnarray} |
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64 | |
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65 | The $y$-dependence of the fluctuations can be described by: |
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66 | \begin{equation} |
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67 | \sigma \, = \, ( s_1 + s_2 \ln y )^{-1} . |
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68 | \label{lsighom} |
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69 | \end{equation} |
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70 | |
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71 | The correlation between $\ln T_{hom} $ and $\ln \alpha_{hom} $ |
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72 | is given by: |
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73 | \begin{equation} |
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74 | \rho(\ln T_{hom}, \ln \alpha_{hom}) \, \equiv \, \rho |
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75 | \, = \, r_1 + r_2 \ln y . |
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76 | \label{corrhom} |
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77 | \end{equation} |
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78 | From these formulae, correlated and varying parameters $\alpha_i$ and |
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79 | $\beta_i$ are generated according to |
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80 | \begin{equation} |
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81 | \left( |
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82 | \begin{array}{c} |
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83 | \ln T_i \\ |
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84 | \ln \alpha_i |
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85 | \end{array} \right) |
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86 | \, = \, |
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87 | \left( |
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88 | \begin{array}{c} |
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89 | \langle \ln T \rangle \\ |
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90 | \langle \ln \alpha \rangle |
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91 | \end{array} \right) |
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92 | + C |
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93 | \left( |
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94 | \begin{array}{c} |
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95 | z_1 \\ |
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96 | z_2 |
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97 | \end{array} \right) |
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98 | \end{equation} |
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99 | with |
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100 | $$ |
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101 | C \, = \, |
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102 | \left( |
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103 | \begin{array}{cc} |
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104 | \sigma (\ln T) & 0 \\ |
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105 | 0 & \sigma (\ln \alpha) |
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106 | \end{array} \right) |
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107 | \left( |
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108 | \begin{array}{cc} |
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109 | \sqrt{\frac{1+\rho}{2}} & \sqrt{\frac{1-\rho}{2}} \\ |
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110 | \sqrt{\frac{1+\rho}{2}} & - \sqrt{\frac{1-\rho}{2}} |
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111 | \end{array} \right) \, $$ |
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112 | $ \sigma (\ln \alpha)$ and $\sigma (\ln T)$ are the fluctuations of $T_{hom}$ |
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113 | and $(\alpha_{hom}$. |
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114 | The values of the coefficients can be found in Ref. \cite{para.grind}. |
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115 | |
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116 | |
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117 | \subsection{Radial Shower Profiles} |
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118 | \label{sec_hom_rad} |
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119 | |
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120 | For the description of average radial energy profiles, |
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121 | \begin{equation} |
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122 | f(r) \, = \, \frac{1}{dE(t)} \frac{dE(t,r)}{dr}, |
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123 | \end{equation} |
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124 | a variety of different functions can be found in the literature. |
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125 | In Gflash the following two-component ansatz, an extension of that in |
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126 | Ref.\cite{para.nim90}, was used: |
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127 | \begin{eqnarray} |
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128 | \label{frad} |
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129 | f(r) & = & p f_C(r) + (1-p) f_T(r) \\ |
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130 | & = & |
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131 | p \frac{2 r R_C^2}{(r^2 + R_C^2)^2} |
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132 | + (1-p) \frac{2 r R_T^2}{(r^2 + R_T^2)^2} |
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133 | \nonumber |
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134 | \end{eqnarray} |
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135 | |
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136 | with $$ 0 \leq p \leq 1 . $$ |
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137 | Here $R_C$ ($R_T$) is the median of the core (tail) component and $p$ |
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138 | is a probability giving the relative weight of the core component. |
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139 | The variable $\tau = t/T$, which measures the shower depth in |
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140 | units of the depth of the shower maximum, is used in order to generalize |
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141 | the radial profiles. This makes the parameterization more convenient and |
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142 | separates the energy and material dependence of various parameters. |
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143 | The median of the core distribution, $R_C$, increases linearly with $\tau$. |
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144 | The weight of the core, $p$, is maximal around the shower maximum, and the |
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145 | width of the tail, $R_T$, is minimal at $\tau \approx 1$. |
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146 | |
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147 | The following formulae are used to parameterize the radial energy density |
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148 | distribution for a given energy and material: |
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149 | \begin{eqnarray} |
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150 | \label{rz} |
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151 | R_{C,hom}(\tau) & = & |
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152 | z_1 + z_2 \tau \\ |
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153 | \label{rk} |
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154 | R_{T,hom}(\tau) & = & |
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155 | k_1 \{ \exp (k_3(\tau -k_2)) + \exp (k_4(\tau -k_2)) \} \\ |
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156 | \label{p} |
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157 | p_{hom}(\tau) & = & |
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158 | p_1 \exp \left\{ \frac{p_2-\tau}{p_3} - |
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159 | \exp \left( \frac{p_2-\tau}{p_3} \right) \right\} |
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160 | \end{eqnarray} |
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161 | The parameters $z_1 \cdots p_3$ are either constant or simple |
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162 | functions of $\ln E$ or $Z$. |
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163 | |
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164 | Radial shape fluctuations are also taken into account. A detailed |
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165 | explanation of this procedure, as well as a list of all the parameters used |
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166 | in Gflash, can be found in Ref. \cite{para.grind}. |
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167 | |
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168 | \subsection{Gflash Performance} |
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169 | |
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170 | The parameters used in this Gflash implementation were extracted from full |
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171 | simulation studies with Geant 3. They also give good results inside the |
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172 | Geant4 fast shower framework when compared with the full electromagnetic |
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173 | shower simulation. However, if more precision or higher particle energies |
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174 | are required, retuning may be necessary. |
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175 | For the longitudinal profiles the difference between full simulation |
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176 | and Gflash parameterization is at the level of a few percent. |
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177 | Because the radial profiles are slightly broader in Geant3 than in Geant4, |
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178 | the differences may reach $ > 10 \% $. |
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179 | The gain in speed, on the other hand, is impressive. The simulation of a |
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180 | 1~TeV electron in a $PbWO_4$ cube is 160 times faster with Gflash. |
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181 | Gflash can also be used to parameterize electromagnetic showers in sampling |
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182 | calorimeters. So far, however, only homogeneous materials are supported. |
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183 | |
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184 | |
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185 | \subsection{Status of this document} |
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186 | 02.12.04 created by J.Weng \\ |
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187 | 03.12.04 grammar check and minor re-wording by D.H. Wright \\ |
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188 | |
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189 | |
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190 | \begin{latexonly} |
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191 | |
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192 | \begin{thebibliography}{99} |
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193 | |
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194 | \bibitem{para.grind} G.~Grindhammer, S.~Peters, |
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195 | {\em The Parameterized Simulation of Electromagnetic Showers in |
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196 | Homogeneous and Sampling Calorimeters, hep-ex/0001020 } (1993). |
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197 | |
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198 | \bibitem {para.longo} E.~Longo and I.~Sestili,{\em |
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199 | Nucl.~Instrum.~Meth.~128, 283} (1975). |
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200 | |
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201 | \bibitem{para.rossi} ~Rossi {\em |
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202 | rentice Hall, New York} (1952). |
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203 | |
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204 | \bibitem{para.nim90} G.~Grindhammer, M.~Rudowicz, and |
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205 | S.~Peters, {\em Nucl.~Instrum.~Meth.~A290, 469} (1990). |
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206 | |
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207 | \end{thebibliography} |
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208 | |
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209 | \end{latexonly} |
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210 | |
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211 | \begin{htmlonly} |
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212 | |
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213 | \subsection{Bibliography} |
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214 | |
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215 | \begin{enumerate} |
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216 | \item G.~Grindhammer, S.~Peters, |
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217 | {\em The Parameterized Simulation of Electromagnetic Showers in |
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218 | Homogeneous and Sampling Calorimeters, hep-ex/0001020 } (1993). |
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219 | |
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220 | \item E.~Longo and I.~Sestili,{\em |
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221 | Nucl.~Instrum.~Meth.~128, 283} (1975). |
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222 | |
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223 | \item ~Rossi {\em Prentice Hall, New York} (1952). |
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224 | |
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225 | \item G.~Grindhammer, M.~Rudowicz, and |
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226 | S.~Peters, {\em Nucl.~Instrum.~Meth.~A290, 469} (1990). |
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227 | |
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228 | \end{enumerate} |
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229 | |
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230 | \end{htmlonly} |
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231 | |
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