[1208] | 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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