1 | \section{Computing the Mean Energy Loss} \label{en_loss} |
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2 | |
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3 | Energy loss processes are very similar for \(e+/e-\) , \(\mu+/\mu-\) and |
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4 | charged hadrons, so a common description for them was a natural choice in |
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5 | Geant4. Any energy loss process must calculate the continuous and discrete |
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6 | energy loss in a material. Below a given energy threshold the energy loss |
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7 | is continuous and above it the energy loss is simulated by the explicit |
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8 | production of secondary particles - gammas, electrons, and positrons. |
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9 | |
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10 | \subsection{Method} |
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11 | |
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12 | Let |
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13 | \[\frac{d\sigma(Z,E,T)}{dT}\] |
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14 | be the differential cross-section per atom (atomic number $Z$) for the ejection |
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15 | of a secondary particle with kinetic energy $T$ by an incident particle of |
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16 | total energy $E$ moving in a material of density $\rho$. The value of the |
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17 | {\em kinetic energy cut-off} or {\em production threshold} is denoted by |
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18 | $T_{cut}$. Below this threshold the soft secondaries ejected are simulated as |
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19 | continuous energy loss by the incident particle, and above it they are |
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20 | explicitly generated. The mean rate of energy loss is given by: |
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21 | \begin{equation} |
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22 | \label{comion.a} |
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23 | \frac{dE_{soft}(E,T_{cut})}{dx} = n_{at} \cdot |
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24 | \int_{0}^{T_{cut}} \frac{d \sigma (Z,E,T)}{dT} T \: dT |
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25 | \end{equation} |
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26 | where $n_{at}$ is the number of atoms per volume in the material. |
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27 | The total cross section per atom for the ejection of a secondary of |
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28 | energy \linebreak $T > T_{cut}$ is |
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29 | \begin{equation} |
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30 | \label{comion.b} |
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31 | \sigma (Z,E,T_{cut}) |
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32 | = \int_{T_{cut}}^{T_{max}}\frac{d \sigma (Z,E,T)} {dT} \: dT \, |
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33 | \end{equation} |
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34 | where $T_{max}$ is the maximum energy transferable to the secondary particle. |
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35 | |
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36 | \noindent |
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37 | If there are several processes providing energy loss for a given particle, then |
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38 | the total continuous part of the energy loss is the sum: |
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39 | \begin{equation} |
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40 | \label{comion.c} |
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41 | \frac{dE_{soft}^{tot}(E,T_{cut})}{dx} = \sum_i{\frac{dE_{soft,i}(E,T_{cut})}{dx}}. |
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42 | \end{equation} |
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43 | These values are pre-calculated during the initialization phase of {\sc Geant4} |
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44 | and stored in the $dE/dx$ table. Using this table the ranges of the particle in |
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45 | given materials are calculated and stored in the $Range$ table. The $Range$ |
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46 | table is then inverted to provide the $InverseRange$ table. At run time, |
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47 | values of the particle's continuous energy loss and range are obtained using |
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48 | these tables. Concrete processes contributing to the energy loss are not |
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49 | involved in the calculation at that moment. In contrast, the production of |
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50 | secondaries with kinetic energies above the production threshold is sampled |
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51 | by each concrete energy loss process. |
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52 | |
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53 | The default energy interval for these tables extends from 100 eV to 100 TeV and |
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54 | the default number of bins is 120. For muon energy loss processes models are |
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55 | valid for higher energies and this interval can be extended up to 1000 PeV. |
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56 | Note that this extention should be done for all three processes which |
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57 | contribute to muon energy loss. |
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58 | |
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59 | \subsection{Implementation Details} |
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60 | |
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61 | Common calculations are performed in the class $G4VEnergyLossProcess$ in which |
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62 | the following methods are implemented: |
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63 | \begin{itemize} |
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64 | \item |
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65 | BuildPhysicsTable; |
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66 | \item |
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67 | StorePhysicsTable; |
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68 | \item |
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69 | RetrievePhysicsTable; |
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70 | \item |
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71 | AlongStepDoIt; |
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72 | \item |
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73 | PostStepDoIt; |
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74 | \item |
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75 | GetMeanFreePath; |
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76 | \item |
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77 | GetContinuousStepLimit; |
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78 | \item |
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79 | MicroscopicCrossSection; |
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80 | \item |
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81 | GetDEDXDispersion; |
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82 | \item |
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83 | SetMinKinEnergy; |
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84 | \item |
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85 | SetMaxKinEnergy; |
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86 | \item |
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87 | SetDEDXBinning; |
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88 | \item |
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89 | SetLambdaBinning; |
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90 | \end{itemize} |
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91 | This interface is used by the following processes: |
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92 | \begin{itemize} |
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93 | \item |
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94 | G4eIonisation; |
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95 | \item |
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96 | G4eBremsstrahlung; |
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97 | \item |
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98 | G4hIonisation; |
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99 | \item |
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100 | G4hhIonisation; |
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101 | \item |
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102 | G4ionIonisation; |
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103 | \item |
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104 | G4MuIonisation; |
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105 | \item |
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106 | G4MuBremsstrahlung; |
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107 | \item |
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108 | G4MuPairProduction. |
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109 | \end{itemize} |
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110 | These processes mainly provide initialization. The physics models are |
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111 | implemented using the $G4VEmModel$ interface. Because a model is defined to |
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112 | be active over a given energy range and for a defined set of $G4Region$s, |
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113 | an energy loss process can have one or several models defined for a particle |
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114 | and $G4Region$. The following models are available: |
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115 | \begin{itemize} |
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116 | \item |
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117 | G4BetheBlochModel; |
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118 | \item |
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119 | G4BetheBlochNoDeltaModel; |
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120 | \item |
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121 | G4BraggModel; |
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122 | \item |
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123 | G4BraggIonModel; |
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124 | \item |
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125 | G4BraggNoDeltaModel; |
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126 | \item |
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127 | G4MollerBhabhaModel; |
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128 | \item |
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129 | G4PAIModel; |
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130 | \item |
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131 | G4PAIPhotonModel; |
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132 | \item |
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133 | G4eBremmstrahlungModel; |
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134 | \item |
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135 | G4MuBetheBlochModel; |
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136 | \item |
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137 | G4MuBremmstrahlungModel; |
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138 | \item |
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139 | G4MuPairProductionModel; |
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140 | \end{itemize} |
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141 | |
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142 | \subsubsection{Stepsize Limit Due to Continuous Energy Loss} |
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143 | |
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144 | Continuous energy loss imposes a limit on the stepsize because of the |
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145 | energy dependence of the cross sections. It is generally assumed in MC programs |
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146 | \cite{enloss.G3} |
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147 | that the particle cross sections are approximately constant along a step, i.e. |
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148 | the step size should be small enough that the change in cross section, from |
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149 | the beginning of the step to the end, is also small. In principle one must |
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150 | use very small steps in order to insure an accurate simulation, however |
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151 | the computing time increases as the stepsize decreases. A good compromise is |
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152 | to limit the stepsize by not allowing the stopping range of the particle to |
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153 | decrease by more than 20 \% during the step. This condition works well for |
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154 | particles with kinetic energies $> 1 MeV$, but for lower energies it gives |
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155 | very short stepsizes. |
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156 | |
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157 | To cure this problem a lower limit on the stepsize was introduced. There is a |
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158 | natural choice for this limit: the stepsize cannot be smaller than the |
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159 | {\em range cut} parameter of the program. The stepsize limit varies smoothly |
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160 | with decreasing energy from the value given by the |
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161 | condition \(\Delta range/range=0.20\) to the lowest possible value |
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162 | {\em range cut}. These are the default step limitation parameters; they can be |
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163 | overwritten using the UI command ``/process/eLoss/StepFunction 0.2 1 mm'', |
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164 | for example. |
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165 | |
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166 | |
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167 | \subsubsection{Energy Loss Computation} |
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168 | |
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169 | The computation of the {\em mean energy loss} after a given step is done |
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170 | by using the $dE/dx$, $Range$, and $InverseRange$ tables. The $dE/dx$ table |
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171 | is used if the energy deposition is less than 5 \% of kinetic energy of the |
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172 | particle. When a larger percentage of energy is lost, the mean loss |
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173 | \(\Delta T\) can be written as |
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174 | \begin{equation} |
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175 | \Delta T = T_0 - f_T(r_0-step) |
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176 | \end{equation} |
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177 | where \(T_0\) is the kinetic energy, |
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178 | \(r_0\) the range at the beginning of the step \(step\), |
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179 | the function \(f_T(r)\) is the inverse of the $Range$ table (i.e. it |
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180 | gives the kinetic energy of the particle for a range value of $r$) . |
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181 | \\ |
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182 | After the mean energy loss has been calculated, the process computes the |
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183 | {\em actual} energy loss, i.e. the loss with fluctuations. The fluctuation |
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184 | is computed from a model described in Section \ref{gen_fluctuations}. |
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185 | |
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186 | |
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187 | \subsection{Energy Loss by Heavy Charged Particles} |
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188 | |
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189 | To save memory |
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190 | in the case of hadron energy loss, $dE/dx$, $Range$ and $InverseRange$ tables |
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191 | are constructed only for {\em protons}. The energy loss for other heavy, |
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192 | charged particles is computed from these tables at the |
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193 | {\em scaled kinetic energy} $T_{scaled}$ : |
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194 | \begin{equation} |
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195 | T_{scaled} = q^2_{eff}\frac{ M_{proton} T}{ M_{particle}}, |
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196 | \end{equation} |
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197 | where $T$ is the kinetic energy of the particle, $q_{eff}$ its effective |
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198 | change in units of positron charge, $M_{proton}$ and |
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199 | $M_{particle}$ are the masses of the proton and particle. |
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200 | Note that in this approach the small differences between $T_{max}$ |
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201 | values calculated for different changed particles are neglected. |
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202 | This is acceptable for hadrons and ions, but is not very accurate |
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203 | for the simulation of muon energy loss, which is simulated by a |
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204 | separate process. For slow ions effective charge is different from the charge |
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205 | of ion nucleus, because of exchange between trasporting ion and the media. |
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206 | The effective charge approach is used \cite{enloss.Ziegler88}. |
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207 | |
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208 | \subsection{Status of this document} |
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209 | 09.10.98 created by L. Urb\'an.\\ |
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210 | 01.12.03 revised by V.Ivanchenko.\\ |
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211 | 02.12.03 spelling and grammar check by D.H. Wright \\ |
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212 | 09.12.05 minor update by V.Ivanchenko.\\ |
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213 | |
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214 | \begin{latexonly} |
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215 | |
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216 | \begin{thebibliography}{99} |
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217 | \bibitem{enloss.G3} {\sc geant3} manual |
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218 | {\em Cern Program Library Long Writeup W5013 (1994)} |
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219 | \bibitem{enloss.Ziegler88} J.F.~Ziegler and |
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220 | J.M.~Manoyan, Nucl. Instr. and Meth. |
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221 | B35 (1988) 215. |
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222 | \end{thebibliography} |
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223 | |
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224 | \end{latexonly} |
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225 | |
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226 | \begin{htmlonly} |
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227 | |
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228 | \subsection{Bibliography} |
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229 | |
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230 | \begin{enumerate} |
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231 | \item {\sc geant3} manual |
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232 | {\em Cern Program Library Long Writeup W5013} (1994). |
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233 | \item J.F.~Ziegler and |
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234 | J.M.~Manoyan, Nucl. Instr. and Meth. |
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235 | B35 (1988) 215. |
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236 | \end{enumerate} |
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237 | |
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238 | \end{htmlonly} |
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